Prof. Dr. Daniel Loss
ContactDepartment of PhysicsUniversity of Basel Klingelbergstrasse 82 CH4056 Basel, Switzerland

CV and Research Interests
Daniel Loss received a Ph.D. in Theoretical Physics at the University of Zurich in 1985 under the supervision of Prof. A. Thellung. He stayed there as postdoctoral researcher for four more years before moving to the US in 1989. From 1989 to 1991 he worked as postdoctoral researcher in the group of Prof. A. J. Leggett, Urbana, and from 1991 to 1993 at IBM Research Center, NY (USA). In 1993 he moved to Vancouver (Canada) to become Assistant and then Associate Professor of Physics at Simon Fraser University. In 1996 he returned to Switzerland to become full Professor of Theoretical Physics at the University of Basel. Loss is director of the Basel Center for Quantum Computing and Quantum Coherence (QC2), was codirector (20062013) of the Swiss National Center of Competence and Research (NCCR) in Nanoscale Science, since 2006 he is codirector of the Swiss Nanoscience Institute (SNI) at the University of Basel, and since 2020 he is codirector (and founding member) of the NCCR SPIN QUBITS IN SILICON. He received several prestigious fellowships, is a Fellow of the American Physical Society, a member of the European Academy of Sciences and of the German National Academy of Sciences Leopoldina. He has been awarded the Humboldt Research Prize in 2005, the Marcel Benoist Prize in 2010; the most prestigious science prize in Switzerland (see www.marcelbenoist.ch, and Uni news), the Blaise Pascal Medal in Physics 2014 from the European Academy of Sciences (Uni news), and the King Faisal International Prize in Science 2017 (see kingfaisalprize.org and Uni news). In 2021, he has been elected as External Scientific Member of the Max Planck Society (see here and Uni news). He is married and has two sons.Loss's research interests include many aspects of the theory of condensed matter systems with a particular focus on spindependent and phasecoherent phenomena ("mesoscopics") in semiconducting nanostructures and molecular magnets. A major portion of Loss's current research involves the theory of spin dynamics, spin coherence, spintronics in twodimensional electron gases, and spinrelated phenomena in semiconducting quantum dotsartificial atoms and molecules. Part of this work is related to quantum information processing (QIP)quantum computing and quantum communication in solid state systems with focus on spin qubits, where Loss and collaborators made seminal contributions. Their theoretical predictions and proposals have stimulated many further investigations, and in particular many experimental programs on spin qubits worldwide. Current research includes spin relaxation and decoherence in quantum dots due to spinorbit and hyperfine interaction; nonMarkovian spin dynamics in bosonic and nuclear spin environments; generation and characterization of nonlocal entanglement with quantum dots, superconductors, Luttinger liquids or Coulomb scattering in interacting 2DEGs; spin currents in magnetic insulators and in semiconductors; spin Hall effect in disordered systems; spin orbit effects in transport and noise; asymmetric quantum shot noise in quantum dots; entanglement transfer from electron spins to photons; QIP with spin qubits in quantum dots and molecular magnets; macroscopic quantum phenomena (spin tunneling and coherence) in molecular and nanoscale magnetism.
Extended CV
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Publications
Citation record.Show all abstracts.
Longrange crossed Andreev reflection in topological insulator nanowires proximitized by Nb 
Junya Feng, Henry F. Legg, Mahasweta Bagchi, Daniel Loss, Jelena Klinovaja, Yoichi Ando arXiv:2407.02383
Crossed Andreev reflection (CAR) is a nonlocal transport phenomenon that creates/detects Cooperpair correlations between distant places. It is also the basis of Cooperpair splitting to generate remote entanglement. Although CAR has been extensively studied in semiconductors proximitycoupled to a superconductor, it has been very difficult to observe it in a topological insulator (TI). Here we report the first observation of CAR in a proximitized TI nanowire (TINW). We performed local and nonlocal conductance spectroscopy on mesoscopic TINW devices in which superconducting (Nb) and metallic (Pt/Au) contacts are made on a bulkinsulating TINW. The local conductance detected a hard gap, accompanied by the appearance of Andreev bound states that can reach zerobias, while a negative nonlocal conductance was occasionally observed upon sweeping the chemical potential, giving evidence for CAR. Surprisingly, the CAR signal was detected even over 1.5 μm, which implies that pair correlations extend over a length scale much longer than the expected superconducting coherence length of either Nb or the proximitised TINW. Such a longrange CAR effect is possibly due to an intricate role of disorder in proximitized nanowires. Also, our 0.9μm device presented a decent Cooperpair splitting efficiency of up to 0.5.

Quantum computation with hybrid parafermionspin qubits 
Denis V. Kurlov, Melina Luethi, Anatoliy I. Lotkov, Katharina Laubscher, Jelena Klinovaja, Daniel Loss arXiv:2405.20950
We propose a universal set of single and twoqubit quantum gates acting on a hybrid qubit formed by coupling a quantum dot spin qubit to a Z2m parafermion qubit with arbitrary integer m. The special case m=1 reproduces the results previously derived for Majorana qubits. Our formalism utilizes Fock parafermions, facilitating a transparent treatment of hybrid parafermionspin systems. Furthermore, we highlight the previously overlooked importance of particlehole symmetry in these systems. We give concrete examples how the hybrid qubit system could be realized experimentally for Z4 and Z6 parafermions. In addition, we discuss a simple readout scheme for the fractional parafermion charge via the measurement of the spin qubit resonant frequency.

Heavyholelighthole mixing at the zone center 
Peter Stano, Daniel Loss arXiv:2405.16844
We investigate heavyholelighthole mixing in quasitwodimensional semiconducting structures. We propose and analyze several measures that characterize the mixing quantitatively in Ge, Si, and GaAs. We find that a mixing that is `weak' when considering the lighthole content can be `strong' when considering the induced spinorbit interactions (SOIs). We identify two different types of holeHamiltonian terms concerning mixing. The first type changes the direction of the pure spinors, without admixing lighthole components. It is efficient for Rabi driving the heavyhole spin. The second type induces mixing and changes the eigenvalues of the g tensor. We restrict ourselves to the zone center, what leads to a simple description, analytical results, and physical insights. We show how the model can be used to investigate heavyhole spin qubit g tensor, dephasing, relaxation, and Rabi frequencies.

A coherence sweet spot with enhanced dipolar coupling 
Jann H. Ungerer, Alessia Pally, Stefano Bosco, Artem Kononov, Deepankar Sarmah, Sebastian Lehmann, Claes Thelander, Ville F. Maisi, Pasquale Scarlino, Daniel Loss, Andreas Baumgartner, Christian Schönenberger arXiv:2405.10796
Qubits require a compromise between operation speed and coherence. Here, we demonstrate a compromisefree singlettriplet (ST) qubit, where the qubit couples maximally to the driving field while simultaneously coupling minimally to the dominant noise sources. The qubit is implemented in a crystalphase defined doublequantum dot in an InAs nanowire. Using a superconducting resonator, we measure the spinorbit interaction (SOI) gap, the spinphoton coupling strength and the qubit decoherence rate as a function of the inplane magneticfield orientation. We demonstrate a spin qubit sweet spot maximizing the dipolar coupling and simultaneously minimizing the decoherence. Our theoretical description postulates phonons as the most likely dominant noise source. The compromisefree sweet spot originates from the SOI suggesting that it is not restricted to this material platform, but might find applications in any material with SOI. These findings pave the way for enhanced engineering of these nanomaterials for nextgeneration qubit technologies.

Coherence limit due to hyperfine interaction with nuclei in the barrier material of Si spin qubits 
Lukas Cvitkovich, Peter Stano, Christoph Wilhelmer, Dominic Waldhör, Daniel Loss, YannMichel Niquet, Tibor Grasser arXiv:2405.10667
On the quest to understand and reduce environmental noise in Si spin qubits, hyperfine interactions between electron and nuclear spins impose a major challenge. Silicon is a promising host material because one can enhance the spin coherence time by removing spinful 29Si isotopes. As more experiments rely on isotopic purification of Si, the role of other spinful atoms in the device should be clarified. This is not a straightforward task, as the hyperfine interactions with atoms in the barrier layers are poorly understood. We utilize density functional theory to determine the hyperfine tensors of both Si and Ge in a crystalline epitaxial Si/SiGe quantum well as well as Si and O atoms in an amorphous Si/SiO2 (MOS) interface structure. Based on these results, we estimate the dephasing time T∗2 due to magnetic noise from the spin bath and show that the coherence is limited by interactions with nonSi barrier atoms to a few \textmu s in Si/SiGe (for nonpurified Ge) and about 100\,\textmu s in SiMOS. Expressing these numbers alternatively, in Si/SiGe the interactions with Ge dominate below 1000\,ppm of 29Si content, and, due to low natural concentration of the spinful oxygen isotopes, the interactions with oxygen in SiMOS become significant only below 1\,ppm of 29Si content.

Cavityenhanced superconductivity via band engineering 
Valerii K. Kozin, Even Thingstad, Daniel Loss, Jelena Klinovaja arXiv:2405.08642
We consider a twodimensional electron gas interacting with a quantized cavity mode. We find that the coupling between the electrons and the photons in the cavity enhances the superconducting gap. Crucially, all terms in the Peierls phase are kept, in contrast to more naive approaches, which may result in spurious superradiant phase transitions. We use a meanfield theory to show that the gap increases approximately linearly with the cavity coupling strength. The effect can be observed locally as an increase in the gap size via scanning tunneling microscopy (STM) measurements for a flake of a 2D material (or for a Moiré system where the enhancement is expected to be more pronounced due to a large lattice constant) interacting with a locallystructured electromagnetic field formed by splitring resonators. Our results are also relevant for quantum optics setups with cold atoms interacting with the cavity mode, where the lattice geometry and system parameters can be tuned in a vast range.

Topological Interlayer Superconductivity in a van der Waals Heterostructure 
Even Thingstad, Joel Hutchinson, Daniel Loss, Jelena Klinovaja arXiv:2405.07927
We show that when a honeycomb antiferromagnetic insulator (AFMI) is sandwiched between two transition metal dichalcogenide (TMD) monolayers in a commensurate way, magnons in the AFMI can mediate an interaction between electrons in the TMDs that gives rise to interlayer Cooper pairing. This interaction opens coexisting extended swave and chiral pwave superconducting gaps in the energy spectrum of the coupled system, and the latter give rise to topological Majorana edge modes.

Tunable Ultrafast Dynamics of Antiferromagnetic Vortices in Nanoscale Dots 
Ji Zou, Even Thingstad, Se Kwon Kim, Jelena Klinovaja, Daniel Loss arXiv:2404.18306
Topological vortex textures in magnetic disks have garnered great attention due to their interesting physics and diverse applications. However, up to now, the vortex state has mainly been studied in microsize ferromagnetic disks, which have oscillation frequencies confined to the GHz range. Here, we propose an experimentally feasible ultrasmall and ultrafast vortex state in an antiferromagnetic nanodot surrounded by a heavy metal, which is further harnessed to construct a highly tunable vortex network. We theoretically demonstrate that, interestingly, the interfacial DzyaloshinskiiMoriya interaction (iDMI) induced by the heavy metal at the boundary of the dot acts as an effective chemical potential for the vortices in the interior. Mimicking the creation of a superfluid vortex by rotation, we show that a magnetic vortex state can be stabilized by this iDMI. Subjecting the system to an electric current can trigger vortex oscillations via spintransfer torque, which reside in the THz regime and can be further modulated by external magnetic fields. Furthermore, we show that coherent coupling between vortices in different nanodisks can be achieved via an antiferromagnetic link. Remarkably, this interaction depends on the vortex polarity and topological charge and is also exceptionally tunable through the vortex resonance frequency. This opens up the possibility for controllable interconnected networks of antiferromagnetic vortices. Our proposal therefore introduces a new avenue for developing highdensity memory, ultrafast logic devices, and THz signal generators, which are ideal for compact integration into microchips.

Relations between normal state nonreciprocal transport and the superconducting diode effect in the trivial and topological phases 
Georg Angehrn, Henry F. Legg, Daniel Loss, Jelena Klinovaja Appl. Phys. Lett. 125, 023507 (2024) ; arXiv:2404.17501.
Nonreciprocal transport effects can occur in the normal state of conductors and in superconductors when both inversion and timereversal symmetry are broken. Here, we consider systems where magnetochiral anisotropy (MCA) of the energy spectrum due to an externally applied magnetic field results in a rectification effect in the normal state and a superconducting (SC) diode effect when the system is proximitised by a superconductor. Focussing on nanowire systems, we obtain analytic expressions for both normal state rectification and SC diode effects that reveal the commonalities  as well as differences  between these two phenomena. Furthermore, we consider the nanowire brought into an (almost) helical state in the normal phase or a topological superconducting phase when proximitised. In both cases this reveals that the topology of the system considerably modifies its nonreciprocal transport properties. Our results provide new insights into how to determine the origin of nonreciprocal effects and further evince the strong connection of nonreciprocal transport with the topological properties of a system

Spinresolved nonlocal transport in proximitized Rashba nanowires 
Paweł Szumniak, Daniel Loss, Jelena Klinovaja arXiv:2404.08527
Nonequilibrium transport in hybrid semiconductorsuperconductor nanowires is crucial for many quantum phenomena such as generating entangled states via cross Andreev reflection (CAR) processes, detecting topological superconductivity, reading out Andreev spin qubits, coupling spin qubits over long distances and so on. Here, we investigate numerically transport properties of a proximitized Rashba nanowire that hosts spinpolarized lowenergy quasiparticle states. We show that the spin polarization in such onedimensional Andreev bands, extended over the entire nanowire length, can be detected in nonlocal transport measurements with tunnelcoupled side leads that are spin polarized. Remarkably, we find an exact correspondence between the sign of the nonlocal conductance and the spin density of the superconducting quasiparticles at the side lead position. We demonstrate that this feature is robust to moderate static disorder. As an example, we show that such a method can be used to detect spin inversion of the bands, accompanying the topological phase transition (TPT) for realistic system parameters. Furthermore, we show that such effects can be used to switch between CAR and elastic cotunneling (ECT) processes by tuning the strength of either the electric or the magnetic field. These findings hold significant practical implications for stateoftheart transport experiments in such hybrid systems.

Magnonic φ Josephson Junctions and Synchronized Precession 
Kouki Nakata, Ji Zou, Jelena Klinovaja, Daniel Loss arXiv:2403.01625
There has been a growing interest in nonHermitian physics. One of its main goals is to engineer dissipation and to explore ensuing functionality. In magnonics, the effect of dissipation due to local damping on magnon transport has been explored. However, the effects of nonlocal damping on the magnonic analog of the Josephson effect remain missing, despite that nonlocal damping is inevitable and has been playing a central role in magnonics. Here, we uncover theoretically that a surprisingly rich dynamics can emerge in magnetic junctions due to intrinsic nonlocal damping, using analytical and numerical methods. In particular, under microwave pumping, we show that coherent spin precession in the right and left insulating ferromagnet (FM) of the junction becomes synchronized by nonlocal damping and thereby a magnonic analog of the φ Josephson junction emerges, where φ stands here for the relative precession phase of right and left FM in the stationary limit. Remarkably, φ decreases monotonically from π to π/2 as the magnonmagnon interaction, arising from spin anisotropies, increases. Moreover, we also find a magnonic diode effect giving rise to rectification of magnon currents. Our predictions are readily testable with current device and measurement technologies at room temperatures.

Fractional Spin Quantum Hall Effect in Weakly Coupled Spin Chain Arrays 
Even Thingstad, Pierre Fromholz, Flavio Ronetti, Daniel Loss, Jelena Klinovaja arXiv:2402.10849
Topological magnetic insulators host chiral gapless edge modes. In the presence of strong interaction effects, the spin of these modes may fractionalize. Studying a 2D array of coupled insulating spin1/2 chains, we show how spatially modulated magnetic fields and DzyaloshinskiiMoriya interactions can be exploited to realize chiral spin liquids or integer and fractional spin quantum Hall effect phases. These are characterized by a gapped bulk spectrum and gapless chiral edge modes with fractional spin. The spin fractionalization is manifested in the quantized spin conductance, which can be used to probe the fractional spin quantum Hall effect. We analyze the system via bosonization and perturbative renormalization group techniques that allow us to identify the most relevant terms induced by the spinspin interactions that open gaps and render the system topological under wellspecified resonance conditions. We show explicitly that the emerging phase is a genuine chiral spin liquid. We suggest that the phases can be realized experimentally in synthetic spin chains and ultracold atom systems.

A Qubit with Simultaneously Maximized Speed and Coherence 
Miguel J. Carballido, Simon Svab, Rafael S. Eggli, Taras Patlatiuk, Pierre Chevalier Kwon, Jonas Schuff, Rahel M. Kaiser, Leon C. Camenzind, Ang Li, Natalia Ares, Erik P.A.M Bakkers, Stefano Bosco, J. Carlos Egues, Daniel Loss, Dominik M. Zumbühl arXiv:2402.07313
To overcome the threshold for faulttolerant quantum computation, qubits have to be protected from their noisy environment to attain the necessary high fidelities. Recent experiments discovered sweet spots with strongly enhanced coherence. However, decoupling a qubit from its surroundings also limits the control over the qubit's state, typically leading to either coherent but slow or fast but shortlived qubits. This tradeoff appears to be a severe fundamental limitation hampering the performance of qubits. Here, we show how this can be circumvented by demonstrating a simultaneously fast and coherent tunable regime in a hole spin qubit. In this regime, we can triple the operation speed, while simultaneously quadrupling the coherence time when tuning a local electric field, demonstrating that the qubit speed and coherence scale together without compromise. This relies on strong, quasi 1D confinement providing a local maximum in drive strength, where charge fluctuations are decoupled and thus the coherence is enhanced, yet the drive speed is maximal. A Ge/Si core/shell nanowire, operated at 1.5 K, provides the strong confinement. The driving mechanism here is the strong and tunable direct Rashba spinorbit interaction, achieving a maximal strength at finite electrical field due to gatedependent heavyhole lighthole mixing. Breaking the speedcoherence tradeoff makes it possible to boost fidelity and speed of one and twoqubit gates. This concept can be expanded to planar arrays of hole or electron spin qubits as well. In this regime, the coupling to a microwave resonator is also predicted to be both strong and coherent. Altogether, this is opening a new path towards faulttolerant quantum computation.

Microscopic Mechanism of Pair, Charge and SpinDensityWave Instabilities in Interacting DDimensional Fermi Liquids 
Dmitry Miserev, Herbert Schoeller, Jelena Klinovaja, Daniel Loss arXiv:2312.17208
We present an analytic theory unraveling the microscopic mechanism of instabilities within interacting Ddimensional Fermi liquid. Our model consists of a Ddimensional electron gas subject to an instantaneous electronelectron interaction of a finite range exceeding the average interparticle distance. Pair, charge and spin susceptibilities are evaluated via the oneloop renormalization group theory and via the bosonization approach, giving identical results. In case of a repulsive interaction, we identify an intrinsic Fermi liquid instability towards insulating spin/charge density wave order when the interaction coupling strength reaches a universal critical value. If both electron and hole pockets of the same size are present, the ground state is an excitonic insulator at arbitrarily small repulsive interaction. If the interaction is attractive, the ground state is a singlet nonBCS superconductor with a uniform condensate. In case if both electron and hole Fermi surfaces are present, we predict an instability towards the interpocket pairdensitywave ordering at the critical coupling. This prediction lends strong theoretical support to the pairdensitywave scenario of superconductivity in cuprate materials. Due to its simple and universal nature, presented microscopic mechanism of intrinsic instabilities of interacting Ddimensional Fermi liquids constitutes a solid theoretical ground for understanding quantum phase transitions in a variety of quantum materials, from ultraclean semiconductor quantum wells to hightemperature superconductors.

Highfidelity spin qubit shuttling via large spinorbit interaction 
Stefano Bosco, Ji Zou, Daniel Loss PRX Quantum 5, 020353 (2024); arXiv:2311.15970.
Shuttling spins with high fidelity is a key requirement to scale up semiconducting quantum computers, enabling qubit entanglement over large distances and favoring the integration of control electronics onchip. To decouple the spin from the unavoidable charge noise, stateoftheart spin shuttlers try to minimize the inhomogeneity of the Zeeman field. However, this decoupling is challenging in otherwise promising quantum computing platforms such as hole spin qubits in silicon and germanium, characterized by a large spinorbit interaction and electricallytunable qubit frequency. In this work, we show that, surprisingly, the large inhomogeneity of the Zeeman field stabilizes the coherence of a moving spin state, thus enabling highfidelity shuttling also in these systems. We relate this enhancement in fidelity to the deterministic dynamics of the spin which filters out the dominant lowfrequency contributions of the charge noise. By simulating several different scenarios and noise sources, we show that this is a robust phenomenon generally occurring at large field inhomogeneity. By appropriately adjusting the motion of the quantum dot, we also design realistic protocols enabling faster and more coherent spin shuttling. Our findings are generally applicable to a wide range of setups and could pave the way toward largescale quantum processors.

Strong holephoton coupling in planar Ge: probing the charge degree and Wigner molecule states 
Franco De Palma, Fabian Oppliger, Wonjin Jang, Stefano Bosco, Marián Janík, Stefano Calcaterra, Georgios Katsaros, Giovanni Isella, Daniel Loss, Pasquale Scarlino arXiv:2310.20661
Semiconductor quantum dots (QDs) in planar germanium (Ge) heterostructures have emerged as frontrunners for future holebased quantum processors. Notably, the large spinorbit interaction of holes offers rapid, coherent electrical control of spin states, which can be further beneficial for interfacing hole spins to microwave photons in superconducting circuits via coherent chargephoton coupling. Here, we present strong coupling between a hole charge qubit, defined in a double quantum dot (DQD) in a planar Ge, and microwave photons in a highimpedance (Z_r=1.3 kΩ) superconducting quantum interference device (SQUID) array resonator. Our investigation reveals vacuumRabi splittings with coupling strengths up to g_0/2\pi=260 MHz, and a cooperativity of C∼100, dependent on DQD tuning, confirming the strong chargephoton coupling regime within planar Ge. Furthermore, utilizing the frequency tunability of our resonator, we explore the quenched energy splitting associated with stronglycorrelated Wigner molecule (WM) states that emerge in Ge QDs. The observed enhanced coherence of the WM excited state signals the presence of distinct symmetries within related spin functions, serving as a precursor to the strong coupling between photons and spincharge hybrid qubits in planar Ge. This work paves the way towards coherent quantum connections between remote hole qubits in planar Ge, required to scale up holebased quantum processors.

Gatetunable topological superconductivity in a supramolecular electron spin lattice 
JungChing Liu, Chao Li, Richard Hess, Hongyan Chen, Carl Drechsel, Ping Zhou, Robert Häner, Ulrich Aschauer, Thilo Glatzel, Silvio Decurtins, Daniel Loss, Jelena Klinovaja, ShiXia Liu, Wulf Wulfhekel, Ernst Meyer, Rémy Pawlak arXiv:2310.18134
Topological superconductivity emerges in chains or arrays of magnetic atoms coupled to a superconductor. However, the external controllability of such systems with gate voltages is detrimental for their future implementation in a topological quantum computer. Here we showcase the supramolecular assembly of radical molecules on Pb(111), whose discharge is controlled by the tip of a scanning tunneling microscope. Charged molecules carry a spin1/2 state, as confirmed by observing YuShibaRusinov ingap states by tunneling spectroscopy at millikelvin temperature. Low energy modes are localized at island boundaries with a long decay towards the interior, whose spectral signature is consistent with Majorana zero modes protected by mirror symmetry. Our results open up a vast playground for the synthesis of gatetunable organic topological superconductors.

Quantum phase transitions and cat states in cavitycoupled quantum dots 
Valerii K. Kozin, Dmitry Miserev, Daniel Loss, Jelena Klinovaja arXiv:2310.15167
We study double quantum dots coupled to a quasistatic cavity mode with high modevolume compression allowing for strong lightmatter coupling. Besides the cavitymediated interaction, electrons in different double quantum dots interact with each other via dipoledipole (Coulomb) interaction. For attractive dipolar interaction, a cavityinduced ferroelectric quantum phase transition emerges leading to ordered dipole moments. Surprisingly, we find that the phase transition can be either continuous or discontinuous, depending on the ratio between the strengths of cavitymediated and Coulomb interactions. We show that, in the strong coupling regime, both the ground and the first excited states of an array of double quantum dots are squeezed Schrödinger cat states. Such states are actively discussed as highfidelity qubits for quantum computing, and thus our proposal provides a platform for semiconductor implementation of such qubits. We also calculate gaugeinvariant observables such as the net dipole moment, the optical conductivity, and the absorption spectrum beyond the semiclassical approximation.

Spin susceptibility in interacting twodimensional semiconductors and bilayer systems at first order: Kohn anomalies and spin density wave ordering 
Joel Hutchinson, Dmitry Miserev, Jelena Klinovaja, Daniel Loss Phys. Rev. B 109, 075139 (2024); arXiv:2310.05555.
This work is an analytic theoretical study of a 2D semiconductor with a Fermi surface that is split by the Zeeman coupling of electron spins to an external magnetic field in the presence of electronelectron interactions. For the first time, we calculate the spin susceptibility for longrange and finiterange interactions diagrammatically, and find a resonant peak structure at the Kohn anomaly already in firstorder perturbation theory. In contrast to the densitydensity correlator that is suppressed due to the large electrostatic energy required to stabilize charge density order, the spin susceptibility does not suffer from electrostatic screening effects, thus favouring spindensitywave order in 2D semiconductors. Our results impose significant consequences for determining magnetic phases in 2D semiconductors. For example, a strongly enhanced Kohn anomaly may result in helical ordering of magnetic impurities due to the RKKY interaction. Furthermore, the spin degree of freedom can equally represent a layer pseudospin in the case of bilayer materials. In this case, the external "magnetic field" is a combination of layer bias and interlayer hopping. The sharp peak of the 2D static spin susceptibility may then be responsible for dipoledensitywave order in bilayer materials at large enough electronphonon coupling.

Valleyfree silicon fins by shear strain 
Christoph Adelsberger, Stefano Bosco, Jelena Klinovaja, Daniel Loss Phys. Rev. Lett. 133, 037001 (2024); arXiv:2308.13448.
Electron spins confined in silicon quantum dots are promising candidates for largescale quantum computers. However, the degeneracy of the conduction band of bulk silicon introduces additional levels dangerously close to the window of computational energies, where the quantum information can leak. The energy of the valley states  typically 0.1 meV  depends on hardly controllable atomistic disorder and still constitutes a fundamental limit to the scalability of these architectures. In this work, we introduce designs of CMOScompatible silicon fin fieldeffect transistors that enhance the energy gap to noncomputational states by more than one order of magnitude. Our devices comprise realistic silicongermanium nanostructures with a large shear strain, where troublesome valley degrees of freedom are completely removed. The energy of noncomputational states is therefore not affected by unavoidable atomistic disorder and can further be tuned insitu by applied electric fields. Our design ideas are directly applicable to a variety of setups and will offer a blueprint towards siliconbased largescale quantum processors.

Josephson transistor from the superconducting diode effect in domain wall and skyrmion magnetic racetracks 
Richard Hess, Henry F. Legg, Daniel Loss, Jelena Klinovaja Phys. Rev. B 108, 174516 (2023); arXiv:2308.04817.
In superconductors, the combination of broken timereversal and broken inversion symmetries can result in a critical current being dependent on the direction of current flow. This phenomenon is known as superconducting diode effect (SDE) and has great potential for applications in future lowtemperature electronics. Here, we investigate how magnetic textures such as domain walls or skyrmions on a racetrack can be used to control the SDE in a Josephson junction and how the SDE can be used as a lowtemperature readout of the data in racetrack memory devices. First, we consider a twodimensional electron gas (2DEG) with strong spinorbitinteraction (SOI) coupled to a magnetic racetrack, which forms the weaklink in a Josephson junction. In this setup, the exchange coupling between the magnetic texture and the itinerant electrons in the 2DEG breaks timereversal symmetry and enables the SDE. When a magnetic texture, such as a domain wall or skyrmion enters the Josephson junction, the local exchange field within the junction is changed and, consequently, the strength of the SDE is altered. In particular, depending on the position and form of the magnetic texture, moving the magnetic texture can cause the SDE coefficient to change its sign, enabling a Josephson transistor effect with potentially fast switching frequencies. Further, we find that the SDE is enhanced if the junction lengthscales are comparable with the lengthscale of the magnetic texture. Furthermore, we show that, under certain circumstances, the symmetry breaking provided by particular magnetic textures, such as skyrmions, can lead to an SDE even in the absence of Rashba SOI in the 2DEG. Our results provide a basis for new forms of readout in lowtemperature memory devices as well as demonstrating how a Josephson transistor effect can be achieved even in the absence of an external magnetic field and intrinsic Rashba SOI.

Spatially correlated classical and quantum noise in driven qubits 
Ji Zou, Stefano Bosco, Daniel Loss npj Quantum Inf 10, 46 (2024); arXiv:2308.03054.
Correlated noise across multiple qubits poses a significant challenge for achieving scalable and faulttolerant quantum processors. Despite recent experimental efforts to quantify this noise in various qubit architectures, a comprehensive understanding of its role in qubit dynamics remains elusive. Here, we present an analytical study of the dynamics of driven qubits under spatially correlated noise, including both Markovian and nonMarkovian noise. Surprisingly, we find that, while correlated classical noise only leads to correlated decoherence without increasing the quantum coherence in the system, the correlated quantum noise can be exploited to generate entanglement. In particular, we reveal that, in the quantum limit, pure dephasing noise induces a coherent longrange twoqubit Ising interaction that correlates distant qubits. In contrast, for purely transverse noise when qubits are subjected to coherent drives, the correlated quantum noise induces both coherent symmetric exchange and DzyaloshinskiiMoriya interaction between the qubits, as well as correlated relaxation, both of which give rise to significant entanglement. Remarkably, in this case, we uncover that the system exhibits distinct dynamical phases in different parameter regimes. Finally, we reveal the impact of spatiotemporally correlated 1/f noise on the decoherence rate, and how its temporal correlations restore lost entanglement. Our analysis not only offers critical insights into designing effective error mitigation strategies to reduce harmful effects of correlated noise, but also enables tailored protocols to leverage and harness noiseinduced correlations for quantum information processing.

Reply to "Comment on 'Trivial Andreev Band Mimicking Topological Bulk Gap Reopening in the Nonlocal Conductance of Long Rashba Nanowires'" 
Richard Hess, Henry F. Legg, Daniel Loss, Jelena Klinovaja arXiv:2306.16853 (reply 1a); Phys. Rev. Lett. 132, 099602 (2024) (reply 1b); arXiv:2308.10669 (reply 2).
Reply 1a:In this Reply we respond to the comment by Das Sarma and Pan [1] on Hess et al., Phys. Rev. Lett. 130, 207001, "Trivial Andreev Band Mimicking Topological Bulk Gap Reopening in the Nonlocal Conductance of Long Rashba Nanowires" [2]. First, we note that Das Sarma and Pan reproduce the key results of Hess et al., substantiating that our findings are entirely valid. Next, we clarify the incorrect statement by Das Sarma and Pan that the main result of Hess et al. requires a "contrived periodic pristine system", pointing out the extensive discussion of positional disorder in the Hess et al. We also demonstrate that nonlocal conductance features are generically reduced by disorder. This applies to both an Andreev band and to a genuine topological bulk gap reopening signature (BRS). In fact, the suppression of nonlocal conductance of a genuine BRS by disorder was discussed in, e.g., Pan, Sau, Das Sarma, PRB 103, 014513 (2021) [3]. We conclude that, contrary to the claims of Das Sarma and Pan, the minimal model of Hess et al. is relevant to current realistic nanowire devices where only a few overlapping ABSs would be required to mimic a BRS.
Reply 2: In this Reply we respond to the comment by Antipov et al. from Microsoft Quantum on Hess et al., PRL 130, 207001 (2023). Antipov et al. reported only a single simulation and claimed it did not pass the Microsoft Quantum topological gap protocol (TGP). They have provided no parameters or data for this simulation (despite request). Regardless, in this reply we demonstrate that the trivial bulk gap reopening mechanism outlined in Hess et al., in combination with trivial ZBPs, passes the TGP and therefore can result in TGP false positives.

Dissipative Spinwave Diode and Nonreciprocal Magnonic Amplifier 
Ji Zou, Stefano Bosco, Even Thingstad, Jelena Klinovaja, Daniel Loss Phys. Rev. Lett. 132, 036701 (2024); arXiv:2306.15916.
We propose an experimentally feasible dissipative spinwave diode comprising two magnetic layers coupled via a nonmagnetic spacer. We theoretically demonstrate that the spacer mediates not only coherent interactions but also dissipative coupling. Interestingly, an appropriately engineered dissipation engenders a nonreciprocal device response, facilitating the realization of a spinwave diode. This diode permits wave propagation in one direction alone, given that the coherent Dzyaloshinskii Moriya (DM) interaction is balanced with the dissipative coupling. The polarity of the diode is determined by the sign of the DM interaction. Furthermore, we show that when the magnetic layers undergo incoherent pumping, the device operates as a unidirectional spinwave amplifier. The amplifier gain is augmented by cascading multiple magnetic bilayers. By extending our model to a onedimensional ring structure, we establish a connection between the physics of spinwave amplification and nonHermitian topology. Our proposal opens up a new avenue for harnessing inherent dissipation in spintronic applications.

Dynamical nuclear spin polarization in a quantum dot with an electron spin driven by electric dipole spin resonance 
Peter Stano, Takashi Nakajima, Akito Noiri, Seigo Tarucha, Daniel Loss Phys. Rev. B 108, 155306 (2023); Editor's suggestion; arXiv:2306.11253.
We analyze the polarization of nuclear spins in a quantum dot induced by a singleelectron spin that is
electrically driven to perform coherent Rabi oscillations. We derive the associated nuclearspin polarization rate
and analyze its dependence on the accessible control parameters, especially the detuning of the driving frequency
from the electron Larmor frequency. The arising nuclearspin polarization is related to the HartmannHahn effect
known from the NMR literature with two important differences. First, in quantum dots, one typically uses a
micromagnet, leading to a small deflection of the quantization axes of the electron and nuclear spins. Second,
the electric driving wiggles the electron with respect to the atomic lattice. The two effects, absent in the traditional
HartmannHahn scenario, give rise to two mechanisms of nuclearspin polarization in gated quantum dots. The
arising nuclearspin polarization is a resonance phenomenon, achieving maximal efficiency at the resonance of
the electron Rabi and nuclear Larmor frequency (typically a few or a few tens of MHz). As a function of the
driving frequency, the polarization rate can develop sharp peaks and reach large values at them. Since the nuclear
polarization is experimentally detected as changes of the electron Larmor frequency, we often convert the former
to the latter in our formulas and figures. In these units, the polarization can reach hundreds of MHz/s in GaAs
quantum dots and at least tens of kHz/s in Si quantum dots. We analyze possibilities to exploit the resonant
polarization effects for achieving large nuclear polarization and for stabilizing the Overhauser field through
feedback.

Microscopic analysis of proximityinduced superconductivity and metallization effects in superconductorgermanium hole nanowires 
Christoph Adelsberger, Henry F. Legg, Daniel Loss, Jelena Klinovaja Phys. Rev. B 108, 155433 (2023); arXiv:2306.06944.
Lowdimensional Ge hole devices are promising systems with many potential applications such as hole spin qubits, Andreev spin qubits, Josephson junctions, and can serve as a basis for the realization of topological superconductivity. This vast array of potential uses for Ge largely stems from the exceptionally strong and controllable spinorbit interaction (SOI), ultralong mean free paths, long coherence times, and CMOS compatibility. However, when brought into proximity with a superconductor (SC), metallization normally diminishes many useful properties of a semiconductor, for instance, typically reducing the g factor and SOI energy, as well as renormalizing the effective mass. In this paper we consider metallization of a Ge nanowire (NW) in proximity to a SC, explicitly taking into account the 3D geometry of the NW. We find that proximitized Ge exhibits a unique phenomenology of metallization effects, where the 3D cross section plays a crucial role. For instance, in contrast to expectations, we find that SOI can be enhanced by strong coupling to the superconductor. We also show that the thickness of the NW plays a critical role in determining both the size of the proximity induced pairing potential and metallization effects, since the coupling between NW and SC strongly depends on the distance of the NW wave function from the interface with the SC. In the absence of electrostatic effects, we find that a sizable gap opens only in thin NWs (d≲3 nm). In thicker NWs, the wave function must be pushed closer to the SC by electrostatic effects in order to achieve a sizable proximity gap such that the required electrostatic field strength can simultaneously induce a strong SOI. The unique and sometimes beneficial nature of metallization effects in SCGe NW devices evinces them as ideal platforms for future applications in quantum information processing.

Majorana Bound States in Germanium Josephson Junctions via Phase Control 
Melina Luethi, Henry F. Legg, Katharina Laubscher, Daniel Loss, Jelena Klinovaja Phys. Rev. B 108, 195406 (2023); arXiv:2304.12689.
We consider superconductornormalsuperconductornormalsuperconductor (SNSNS) planar Josephson junctions in hole systems with spinorbit interaction that is cubic in momentum (CSOI). Utilizing only the superconducting phase difference, we find parameter `sweet spots' for reasonable junction transparencies that result in a topological region of phase space, within which Majorana bound states (MBSs) appear at the ends of the junction. In planar germanium hetereostructures CSOI can be the dominant form of SOI and extremely strong. We show analytically and numerically that, within experimental regimes, our results provide an achievable roadmap for a new MBS platform with low disorder, minimal magnetic fields, and very strong spinorbit interaction, overcoming many of the key deficiencies that have so far prevented the conclusive observation of MBSs.

Cavityinduced charge transfer in periodic systems: lengthgauge formalism 
Ekaterina Vlasiuk, Valerii K. Kozin, Jelena Klinovaja, Daniel Loss, Ivan V. Iorsh, Ilya V. Tokatly Phys. Rev. B 108, 085410 (2023); arXiv:2304.11364.
We develop a lengthgauge formalism for treating onedimensional periodic lattice systems in the presence of a photon cavity inducing lightmatter interaction. The purpose of the formalism is to remove mathematical ambiguities that occur when defining the position operator in the context of the PowerZienauWoolley Hamiltonian. We then use a diagrammatic approach to analyze perturbatively the interaction between an electronic quantum system and a photonic cavity mode of long wavelength. We illustrate the versatility of the formalism by studying the cavityinduced electric charge imbalance and polarization in the RiceMele model with broken inversion symmetry.

Dimensional reduction of the LuttingerWard functional for spindegenerate Ddimensional electron gases 
D. Miserev, J. Klinovaja, D. Loss Phys. Rev. B 108, 235116 (2023); arXiv:2303.16732.
We consider an isotropic spindegenerate interacting uniform Ddimensional electron gas (DDEG) with D>1 within the LuttingerWard (LW) formalism. We derive the asymptotically exact semiclassical/infrared limit of the LW functional at large distances, r≫λF, and large times, τ≫1/EF, where λF and EF are the Fermi wavelength and the Fermi energy, respectively. The LW functional is represented by skeleton diagrams, each skeleton diagram consists of appropriately connected dressed fermion loops. First, we prove that every Ddimensional skeleton diagram consisting of a single fermion loop is reduced to a onedimensional (1D) fermion loop with the same diagrammatic structure, which justifies the name dimensional reduction. This statement, combined with the fermion loop cancellation theorem (FLCT), agrees with results of multidimensional bosonization. Here we show that the backscattering and the spectral curvature, both explicitly violate the FLCT and both are irrelevant for a 1DEG, become relevant at D>1 and D>2, respectively. The reason for this is a strong infrared divergence of the skeleton diagrams containing multiple fermion loops at D>1. These diagrams, which are omitted within the multidimensional bosonization approaches, account for the noncollinear scattering processes. Thus, the dimensional reduction provides the framework to go beyond predictions of the multidimensional bosonization. A simple diagrammatic structure of the reduced LW functional is another advantage of our approach. The dimensional reduction technique is also applicable to the thermodynamic potential and various approximations, from perturbation theory to selfconsistent approaches.

Highfidelity twoqubit gates of hybrid superconductingsemiconducting singlettriplet qubits 
Maria Spethmann, Stefano Bosco, Andrea Hofmann, Jelena Klinovaja, Daniel Loss Phys. Rev. B 109, 085303 (2024); arXiv:2304.05086.
Hybrid systems comprising superconducting and semiconducting materials are promising architectures for quantum computing. Superconductors induce longrange interactions between the spin degrees of freedom of semiconducting quantum dots. These interactions are widely anisotropic when the semiconductor material has strong spinorbit interactions. We show that this anisotropy is tunable and enables fast and highfidelity twoqubit gates between singlettriplet (ST) spin qubits. Our design is immune to leakage of the quantum information into noncomputational states and removes alwayson interactions between the qubits, thus resolving key open challenges for these architectures. Our ST qubits do not require additional technologicallydemanding components nor finetuning of parameters. They operate at low magnetic fields of a few milli Tesla and are fully compatible with superconductors. In realistic devices, we estimate infidelities below 10−3, that could pave the way toward largescale hybrid superconductingsemiconducting quantum processors.

Phase driving hole spin qubits 
Stefano Bosco, Simon Geyer, Leon C. Camenzind, Rafael S. Eggli, Andreas Fuhrer, Richard J. Warburton, Dominik M. Zumbühl, J. Carlos Egues, Andreas V. Kuhlmann, Daniel Loss Phys. Rev. Lett. 131, 197001 (2023); Editor's suggestion; arXiv:2303.03350.
The spinorbit interaction in spin qubits enables spinflip transitions, resulting in Rabi oscillations when an external microwave field is resonant with the qubit frequency. Here, we introduce an alternative driving mechanism of hole spin qubits, where a fardetuned oscillating field couples to the qubit phase. Phase driving at radio frequencies, orders of magnitude slower than the microwave qubit frequency, induces highly nontrivial spin dynamics, violating the Rabi resonance condition. By using a qubit integrated in a silicon fin fieldeffect transistor (Si FinFET), we demonstrate a controllable suppression of resonant Rabi oscillations, and their revivals at tunable sidebands. These sidebands enable alternative qubit control schemes using global fields and local fardetuned pulses, facilitating the design of dense largescale qubit architectures with local qubit addressability. Phase driving also decouples Rabi oscillations from noise, an effect due to a gapped Floquet spectrum and can enable Floquet engineering highfidelity gates in future quantum processors.

General scatterings and electronic states in the quantumwire network of moiré systems 
ChenHsuan Hsu, Daniel Loss, Jelena Klinovaja Phys. Rev. B 108, L121409 (2023); arXiv:2303.00759.
We investigate electronic states in a twodimensional network consisting of interacting quantum wires, a model adopted for twisted bilayer systems. We construct general operators which describe various scattering processes in the system. In a twisted bilayer structure, the moiré periodicity allows for generalized umklapp scatterings, leading to a class of correlated states at fractional fillings. We identify scattering processes which can lead to an insulating bulk with gapless chiral edge modes at certain fractional fillings, resembling the quantum anomalous Hall effect recently observed in twisted bilayer graphene. Finally, we propose experimental setups to detect and characterize the edge modes through spectroscopic and transport measurements.

Spatial noise correlations beyond nearestneighbor in 28Si/SiGe spin qubits 
Juan S. RojasArias, Akito Noiri, Peter Stano, Takashi Nakajima, Jun Yoneda, Kenta Takeda, Takashi Kobayashi, Amir Sammak, Giordano Scappucci, Daniel Loss, Seigo Tarucha Phys. Rev. Applied 20, 054024 (2023); arXiv:2302.11717.
We detect correlations in qubitenergy fluctuations of nonneighboring qubits defined in isotopically purified Si/SiGe quantum dots. At low frequencies (where the noise is strongest), the correlation coefficient reaches 10% for a nextnearestneighbor qubitpair separated by 200 nm. Assigning the observed noise to be of electrical origin, a simple theoretical model quantitatively reproduces the measurements and predicts a polynomial decay of correlations with interqubit distance. Our results quantify longrange correlations of noise dephasing quantumdot spin qubits arranged in arrays, essential for scalability and faulttolerance of such systems.

Parity protected superconducting diode effect in topological Josephson junctions 
Henry F. Legg, Katharina Laubscher, Daniel Loss, Jelena Klinovaja Phys. Rev. B 108, 214520 (2023); arXiv:2301.13740.
In bulk superconductors or Josephson junctions formed in materials with spinorbit interaction, the critical current can depend on the direction of current flow and applied magnetic field, an effect known as the superconducting (SC) diode effect. Here, we consider the SC diode effect in Josephson junctions in nanowire devices. We find that the 4πperiodic contribution of Majorana bound states (MBSs) to the current phase relation (CPR) of single junctions results in a significant enhancement of the SC diode effect when the device enters the topological phase. Crucially, this enhancement of the SC diode effect is independent of the parity of the junction and therefore protected from parity altering events, such as quasiparticle poisoning, which have hampered efforts to directly observe the 4πperiodic CPR of MBSs. We show that this effect can be generalized to SQUIDs and that, in such devices, the parityprotected SC diode effect can provide a highly controllable probe of the topology in a Josephson junction.

Enhancement of the Kondo effect in a quantum dot formed in a fullshell nanowire 
Aleksandr E. Svetogorov, Daniel Loss, Jelena Klinovaja Phys. Rev. B 107, 134505 (2023); arXiv:2301.12442.
We analyze results of a recent experiment [D. Razmadze et al., Phys. Rev. Lett., 125, 116803 (2020)] on transport through a quantum dot between two fullshell nanowires and show that the observed effects are caused by the Kondo effect enhancement due to a nontrivial geometry (magnetic flux in a fullshell nanowire) rather than the presence of Majorana bound states. Moreover, we propose that such a setup presents a unique and convenient system to study the competition between superconductivity and the Kondo effect and has significant advantages in comparison to other known approaches, as the important parameter is controlled by the magnetic flux through the fullshell nanowire, which can be significantly varied with small changes of magnetic field, and does not require additional gates. This competition is of fundamental interest as it results in a quantum phase transition between an unscreened doublet and a manybody Kondo singlet ground states of the system.

Domain wall qubits on magnetic racetracks 
Ji Zou, Stefano Bosco, Banabir Pal, Stuart S. P. Parkin, Jelena Klinovaja, Daniel Loss Phys. Rev. Research 5, 033166 (2023); arXiv:2212.12019.
We propose a scalable implementation of a quantum computer based on hardwareefficient mobile domain walls on magnetic racetracks. In our proposal, the quantum information is encoded in the chirality of the spin structure of nanoscale domain walls. We estimate that these qubits are longlived and could be operated at sweet spots reducing possible noise sources. Singlequbit gates are implemented by controlling the movement of the walls in magnetic nanowires, and twoqubit entangling gates take advantage of naturally emerging interactions between different walls. These gates are sufficient for universal quantum computing and are fully compatible with current stateoftheart experiments on racetrack memories. Possible schemes for qubit readout and initialization are also discussed.

Realization of a threedimensional quantum Hall effect in a Zeemaninduced second order topological insulator on a torus 
Zhe Hou, Clara S. Weber, Dante M. Kennes, Daniel Loss, Herbert Schoeller, Jelena Klinovaja, Mikhail Pletyukhov Phys. Rev. B 107, 075437 (2023); arXiv:2212.09053.
We propose a realization of a quantum Hall effect (QHE) in a secondorder topological insulator (SOTI) in three dimensions (3D), which is mediated by hinge states on a torus surface. It results from the nontrivial interplay of the material structure, Zeeman effect, and the surface curvature. In contrast to the conventional 2D and 3DQHE, we show that the 3DSOTI QHE is not affected by orbital effects of the applied magnetic field and exists in the presence of a Zeeman term only, induced e.g. by magnetic doping. To explain the 3DSOTI QHE, we analyze the boundary charge for a 3DSOTI and establish its universal dependence on the AharonovBohm flux threading through the torus hole. Exploiting the fundamental relation between the boundary charge and the Hall conductance, we demonstrate the universal quantization of the latter, as well as its stability against random disorder potentials and continuous deformations of the torus surface.

Anisotropic exchange interaction of two holespin qubits 
Simon Geyer, Bence Hetényi, Stefano Bosco, Leon C. Camenzind, Rafael S. Eggli, Andreas Fuhrer, Daniel Loss, Richard J. Warburton, Dominik M. Zumbühl, and Andreas V. Kuhlmann Nature Physics (2024); arXiv:2212.02308.
Semiconductor spin qubits offer the potential to employ industrial transistor technology to produce largescale quantum computers. Silicon hole spin qubits benefit from fast allelectrical qubit control and sweet spots to counteract charge and nuclear spin noise. However, the demonstration of a twoqubit interaction has remained an open challenge. One missing factor is an understanding of the exchange coupling in the presence of a strong spin–orbit interaction. Here we study two holespin qubits in a silicon fin fieldeffect transistor, the workhorse device of today’s semiconductor industry. We demonstrate electrical tunability of the exchange splitting from above 500 MHz to closetooff and perform a conditional spinflip in 24 ns. The exchange is anisotropic because of the spin–orbit interaction. Upon tunnelling from one quantum dot to the other, the spin is rotated by almost 180 degrees. The exchange Hamiltonian no longer has the Heisenberg form and can be engineered such that it enables twoqubit controlled rotation gates without a tradeoff between speed and fidelity. This ideal behaviour applies over a wide range of magnetic field orientations, rendering the concept robust with respect to variations from qubit to qubit, indicating that it is a suitable approach for realizing a largescale quantum computer.

Secondorder topology and supersymmetry in twodimensional topological insulators 
Clara S. Weber, Mikhail Pletyukhov, Zhe Hou, Dante M. Kennes, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller Phys. Rev. B 107, 235402 (2023); arXiv:2212.01307.
We unravel a fundamental connection between a supersymmetry and a wide class of two dimensional secondorder topological insulators (SOTI). This particular supersymmetry is induced by applying a halfinteger AharonovBohm flux f=Φ/Φ0=1/2 through a hole in the system. Here, three symmetries are essential to establish this fundamental link: chiral symmetry, inversion symmetry, and mirror symmetry. At such a flux of halfinteger value the mirror symmetry anticommutes with the inversion symmetry leading to a nontrivial n=1SUSY representation for the absolute value of the Hamiltonian in each chiral sector, separately. This implies that a unique zeroenergy state and an exact 2fold degeneracy of all eigenstates with nonzero energy is found even at finite system size. For arbitrary smooth surfaces the link between 2DSOTI and SUSY can be described within a universal lowenergy theory in terms of an effective surface Hamiltonian which encompasses the whole class of supersymmetric periodic Witten models. Applying this general link to the prototypical example of a BernevigHughesZhangmodel with an inplane Zeeman field, we analyse the entire phase diagram and identify a gapless Weyl phase separating the topological from the nontopological gapped phase. Surprisingly, we find that topological states localized at the outer surface remain in the Weyl phase, whereas topological hole states move to the outer surface and change their spatial symmetry upon approaching the Weyl phase. Therefore, the topological hole states can be tuned in a versatile manner opening up a route towards magneticfieldinduced topological engineering in multihole systems. Finally, we demonstrate the stability of localized states against deviation from halfinteger flux, flux penetration into the sample, surface distortions, and random impurities for impurity strengths up to the order of the surface gap.

RKKY interaction in onedimensional flat band lattices 
Katharina Laubscher, Clara S. Weber, Maximilian Hünenberger, Herbert Schoeller, Dante M. Kennes, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 108, 155429 (2023); arXiv:2210.10025.
We study the RudermanKittelKasuyaYosida (RKKY) interaction between two classical magnetic impurities in onedimensional lattice models with flat bands. As two representative examples, we pick the stub lattice and the diamond lattice at half filling. We first calculate the exact RKKY interaction numerically and then compare our data to results obtained via different analytical techniques. In both our examples, we find that the RKKY interaction exhibits peculiar features that can directly be traced back to the presence of a flat band. Importantly, these features are not captured by the conventional RKKY approximation based on nondegenerate perturbation theory. Instead, we find that degenerate perturbation theory correctly reproduces our exact results if there is an energy gap between the flat and the dispersive bands, while a nonperturbative approach becomes necessary in the absence of a gap.

Determination of spinorbit interaction in semiconductor nanostructures via nonlinear transport 
Renato M. A. Dantas, Henry F. Legg, Stefano Bosco, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 107, L241202 (2023); arXiv:2210.05429.
We investigate nonlinear transport signatures stemming from linear and cubic spinorbit interactions in one and twodimensional systems. The analytical zerotemperature response to external fields is complemented by finite temperature numerical analysis, establishing a way to distinguish between linear and cubic spinorbit interactions. We also propose a protocol to determine the relevant material parameters from transport measurements attainable in realistic conditions, illustrated by values for Ge heterostructures. Our results establish a method for the fast benchmarking of spinorbit properties in semiconductor nanostructures.

Trivial Andreev band mimicking topological bulk gap reopening in the nonlocal conductance of long Rashba nanowires 
Richard Hess, Henry F. Legg, Daniel Loss, and Jelena Klinovaja Phys. Rev. Lett. 130, 207001 (2023); Editor's Suggestion; arXiv:2210.03507.
We consider a onedimensional Rashba nanowire in which multiple Andreev bound states in the bulk of the nanowire form an Andreev band. We show that, under certain circumstances, this trivial Andreev band can produce an apparent closing and reopening signature of the bulk band gap in the nonlocal conductance of the nanowire. Furthermore, we show that the existence of the trivial bulk reopening signature (BRS) in nonlocal conductance is essentially unaffected by the additional presence of trivial zerobias peaks (ZBPs) in the local conductance at either end of the nanowire. The simultaneous occurrence of a trivial BRS and ZBPs mimics the basic features required to pass the socalled "topological gap protocol". Our results therefore provide a topologically trivial minimal model by which the applicability of this protocol can be benchmarked.

Planar Josephson junctions in germanium: Effect of cubic spinorbit interaction 
Melina Luethi, Katharina Laubscher, Stefano Bosco, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 107, 035435 (2023); arXiv:2209.12745.
Planar Josephson junctions comprising semiconductors with strong spinorbit interaction (SOI) are promising platforms to host Majorana bound states (MBSs). In this context, twodimensional hole gases in germanium (Ge) are favorable candidates due to their particularly large SOI. In contrast to electron gases, where the SOI is a linear function of momentum, the SOI is cubic in momentum for a hole gas in planar Ge. Using a discretized model, we numerically simulate a Ge planar Josephson junction and show that it can host MBSs. Interestingly, we find that the cubic SOI yields an asymmetric phase diagram as a function of the superconducting phase difference across the junction. We also find that trivial Andreev bound states can imitate the signatures of MBSs in a Ge planar Josephson junction, therefore making the experimental detection of MBSs difficult. We use experimentally realistic parameters to assess if the topological phase is accessible within experimental limitations. Our analysis shows that twodimensional Ge is an auspicious candidate for topological phases.

Noisecorrelation spectrum for a pair of spin qubits in silicon 
J. Yoneda, J. S. RojasArias, P. Stano, K. Takeda, A. Noiri, T. Nakajima, D. Loss, and S. Tarucha Nat. Phys. 19, 1793 (2023); arXiv:2208.14150.
Semiconductor qubits are appealing for building quantum processors as they may be densely integrated due to small footprint. However, a high density raises the issue of noise correlated across different qubits, which is of practical concern for scalability and fault tolerance. Here, we analyse and quantify in detail the degree of noise correlation in a pair of neighbouring silicon spin qubits ~100 nm apart. We evaluate all apriori independent auto and cross power spectral densities of noise as a function of frequency. We reveal strong interqubit noise correlation with a correlation strength as large as ~0.7 at ~1 Hz (70% of the maximum inphase correlation), even in the regime where the spinspin exchange interaction contributes negligibly. We furthermore find that fluctuations of singlespin precession rates are strongly correlated with exchange noise, giving away their electrical origin. Noise crosscorrelations have thus enabled us to pinpoint the most influential noise in the present device among compelling mechanisms including nuclear spins. Our work presents a powerful tool set to assess and identify the noise acting on multiple qubits and highlights the importance of longrange electric noise in densely packed silicon spin qubits.

Enhanced orbital magnetic field effects in Ge hole nanowires 
Christoph Adelsberger, Stefano Bosco, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 106, 235408 (2022); arXiv:2207.12050.
Hole semiconductor nanowires (NW) are promising platforms to host spin qubits and Majorana bound states for topological qubits because of their strong spinorbit interactions (SOI). The properties of these systems depend strongly on the design of the cross section and on strain, as well as on external electric and magnetic fields. In this work, we analyze in detail the dependence of the SOI and g factors on the orbital magnetic field. We focus on magnetic fields aligned along the axis of the NW, where orbital effects are enhanced and result in a renormalization of the effective g factor up to 400%, even at small values of magnetic field. We provide an exact analytical solution for holes in Ge NWs and we derive an effective lowenergy model that enables us to investigate the effect of electric fields applied perpendicular to the NW. We also discuss in detail the role of strain, growth direction, and high energy valence bands in different architectures, including Ge/Si core/shell NWs, gatedefined onedimensional channels in planar Ge, and curved Ge quantum wells. In core/shell NWs grown along the [110] direction the g factor can be twice larger than for other growth directions which makes this growth direction advantageous for Majorana bound states. Also curved Ge quantum wells feature large effective g factors and SOI, again ideal for hosting Majorana bound states. Strikingly, because these quantities are independent of the electric field, hole spin qubits encoded in curved quantum wells are to good approximation not susceptible to charge noise, significantly boosting their coherence time.

Sector length distributions of graph states 
Daniel Miller, Daniel Loss, Ivano Tavernelli, Hermann Kampermann, Dagmar Bruß, and Nikolai Wyderka J. Phys. A: Math. Theor. 56, 335303 (2023); arXiv:2207.07665.
The sector length distribution (SLD) of a quantum state is a collection of local unitary invariants that quantify kbody correlations. We show that the SLD of graph states can be derived by solving a graphtheoretical problem. In this way, the mean and variance of the SLD are obtained as simple functions of efficiently computable graph properties. Furthermore, this formulation enables us to derive closed expressions of SLDs for some graph state families. For cluster states, we observe that the SLD is very similar to a binomial distribution, and we argue that this property is typical for graph states in general. Finally, we derive an SLDbased entanglement criterion from the majorization criterion and apply it to derive meaningful noise thresholds for entanglement.

Longdistance coupling of spin qubits via topological magnons 
Bence Hetényi, Alexander Mook, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 106, 235409 (2022); arXiv:2207.01264.
We consider two distant spin qubits in quantum dots, both coupled to a twodimensional topological ferromagnet hosting chiral magnon edge states at the boundary. The chiral magnon is used to mediate entanglement between the spin qubits, realizing a fundamental building block of scalable quantum computing architectures: a longdistance twoqubit gate. Previous proposals for longdistance coupling with magnons involved offresonant coupling, where the detuning of the spinqubit frequency from the magnonic band edge provides protection against spontaneous relaxation. The topological magnon mode, on the other hand, lies inbetween two magnonic bands far away from any bulk magnon resonances, facilitating strong and highly tuneable coupling between the two spin qubits. Even though the coupling between the qubit and the chiral magnon is resonant for a wide range of qubit splittings, we find that the magnoninduced qubit relaxation is vastly suppressed if the coupling between the qubit and the ferromagnet is antiferromagnetic. A fast and highfidelity longdistance coupling protocol is presented capable of achieving spinqubit entanglement over micrometer distances with 1MHz gate speed and up to 99.9% fidelities. The resulting spinqubit entanglement may be used as a probe for the longsought detection of topological edge magnons.

Superconducting diode effect due to magnetochiral anisotropy in topological insulator and Rashba nanowires 
Henry F. Legg, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 106, 104501 (2022); arXiv:2205.12939.
The critical current of a superconductor can depend on the direction of current flow due to magnetochiral anisotropy when both inversion and timereversal symmetry are broken, an effect known as the superconducting (SC) diode effect. Here, we consider onedimensional (1D) systems in which superconductivity is induced via the proximity effect. In both topological insulator and Rashba nanowires, the SC diode effect due to a magnetic field applied along the spinpolarization axis and perpendicular to the nanowire provides a measure of inversion symmetry breaking in the presence of a superconductor. Furthermore, a strong dependence of the SC diode effect on an additional component of magnetic field applied parallel to the nanowire as well as on the position of the chemical potential can be used to detect that a device is in the region of parameter space where the phase transition to topological superconductivity is expected to arise.

Coupled superconducting spin qubits with spinorbit interaction 
Maria Spethmann, XianPeng Zhang, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 106, 115411 (2022); arXiv:2205.03843.
Superconducting spin qubits, also known as Andreev spin qubits, promise to combine the benefits of superconducting qubits and spin qubits defined in quantum dots. While most approaches to control these qubits rely on controlling the spin degreeoffreedom via the supercurrent, superconducting spin qubits can also be coupled to each other via the superconductor to implement twoqubit quantum gates. We theoretically investigate the interaction between superconducting spin qubits in the weak tunneling regime and concentrate on the effect of spinorbit interaction (SOI), which can be large in semiconductorbased quantum dots and thereby offers an additional tuning parameter for quantum gates. We find analytically that the effective interaction between two superconducting spin qubits consists of Ising, Heisenberg, and DzyaloshinskiiMoriya interactions and can be tuned by the superconducting phase difference, the tunnel barrier strength, or the SOI parameters. The Josephson current becomes dependent on SOI and spin orientations. We demonstrate that this interaction can be used for fast controlled phaseflip gates with a fidelity >99.99%. We propose a scalable network of superconducting spin qubits which is suitable for implementing the surface code.

Bayesian estimation of correlation functions 
Angel GutierrezRubio, Daniel Loss, and Peter Stano Phys. Rev. Research 4, 043166 (2022); arXiv:2205.03611.
We apply Bayesian statistics to the estimation of correlation functions. We give the probability distributions of auto and cross correlations as functions of the data. Our procedure uses the measured data optimally and informs about the certainty level of the estimation. Our results apply to general stationary processes and their essence is a nonparametric estimation of spectra. It allows one to understand better the noise statistical fluctuations, assess the correlation between two variables, and postulate parametric models of spectra that can be further tested. We also propose a method to numerically generate correlated noise with a given spectrum.

Anomalous zerofield splitting for hole spin qubits in Si and Ge quantum dots 
Bence Hetényi, Stefano Bosco, and Daniel Loss Phys. Rev. Lett. 129, 116805 (2022); arXiv:2205.02582.
An anomalous energy splitting of spin triplet states at zero magnetic field has recently been measured in germanium quantum dots. This zerofield splitting could crucially alter the coupling between tunnelcoupled quantum dots, the basic building blocks of stateoftheart spinbased quantum processors, with profound implications for semiconducting quantum computers. We develop an analytical model linking the zerofield splitting to spinorbit interactions that are cubic in momentum. Such interactions naturally emerge in hole nanostructures, where they can also be tuned by external electric fields, and we find them to be particularly large in silicon and germanium, resulting in a significant zerofield splitting in the μeV range. We confirm our analytical theory by numerical simulations of different quantum dots, also including other possible sources of zerofield splitting. Our findings are applicable to a broad range of current architectures encoding spin qubits and provide a deeper understanding of these materials, paving the way towards the next generation of semiconducting quantum processors.

Hole spin qubits in thin curved quantum wells 
Stefano Bosco and Daniel Loss Phys. Rev. Applied 18, 044038 (2022); arXiv:2204.08212.
Hole spin qubits are frontrunner platforms for scalable quantum computers because of their large spinorbit interaction which enables ultrafast allelectric qubit control at low power. The fastest spin qubits to date are defined in long quantum dots with two tight confinement directions, when the driving field is aligned to the smooth direction. However, in these systems the lifetime of the qubit is strongly limited by charge noise, a major issue in hole qubits. We propose here a different, scalable qubit design, compatible with planar CMOS technology, where the hole is confined in a curved germanium quantum well surrounded by silicon. This design takes full advantage of the strong spinorbit interaction of holes, and at the same time suppresses charge noise in a wide range of configurations, enabling highly coherent, ultrafast qubit gates. While here we focus on a Si/Ge/Si curved quantum well, our design is also applicable to different semiconductors. Strikingly, these devices allow for ultrafast operations even in short quantum dots by a transversal electric field. This additional driving mechanism relaxes the demanding design constraints, and opens up a new way to reliably interface spin qubits in a single quantum dot to microwave photons. By considering stateoftheart highimpedance resonators and realistic qubit designs, we estimate interaction strengths of a few hundreds of MHz, largely exceeding the decay rate of spins and photons. Reaching such a strong coupling regime in hole spin qubits will be a significant step towards highfidelity entangling operations between distant qubits, as well as fast singleshot readout, and will pave the way towards the implementation of a largescale semiconducting quantum processor.

Observation of fractional spin textures in a Heusler material 
Jagannath Jena, Börge Göbel, Tomoki Hirosawa, Sebastian A. Diaz, Daniel Wolf, Taichi Hinokihara, Vivek Kumar, Ingrid Mertig, Claudia Felser, Axel Lubk, Daniel Loss, and Stuart S. P. Parkin Nat Commun 13, 2348 (2022)
Recently a zoology of noncollinear chiral spin textures has been discovered, most of which, such as skyrmions and antiskyrmions, have integer topological charges. Here we report the
experimental realspace observation of the formation and stability of fractional antiskyrmions
and fractional elliptical skyrmions in a Heusler material. These fractional objects appear, over
a wide range of temperature and magnetic field, at the edges of a sample, whose interior is
occupied by an array of nanoobjects with integer topological charges, in agreement with our
simulations. We explore the evolution of these objects in the presence of magnetic fields and
show their interconversion to objects with integer topological charges. This means the
topological charge can be varied continuously. These fractional spin textures are not just
another type of skyrmion, but are essentially a new state of matter that emerges and lives
only at the boundary of a magnetic system. The coexistence of both integer and fractionally
charged spin textures in the same material makes the Heusler family of compounds unique
for the manipulation of the realspace topology of spin textures and thus an exciting platform
for spintronic and magnonic applications.

Prevalence of trivial zeroenergy subgap states in nonuniform helical spin chains on the surface of superconductors 
Richard Hess, Henry F. Legg, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 106, 104503 (2022); arXiv:2204.02324.
Helical spin chains, consisting of magnetic (ad)atoms, on the surface of bulk superconductors are predicted to host Majorana bound states (MBSs) at the ends of the chain. Here, we investigate the prevalence of trivial zeroenergy bound states in these helical spin chain systems. First, we show that the Hamiltonian of a helical spin chain on a superconductor can be mapped to an effective Hamiltonian reminiscent of a semiconductor nanowire with strong Rashba spinorbit coupling. In particular, we show that a varying rotation rate between neighbouring magnetic moments maps to smooth nonuniform potentials in the effective nanowire Hamiltonian. Previously it has been found that trivial zeroenergy states are abundant in nanowire systems with smooth potentials. Therefore, we perform an extensive search for zeroenergy bound states in helical spin chain systems with varying rotation rates. Although bound states with near zeroenergy do exist for certain dimensionalities and rotation profiles, we find that zeroenergy bound states are far less prevalent than in semiconductor nanowire systems with equivalent nonuniformities. In particular, utilising varying rotation rates, we do not find zeroenergy bound states in the most experimentally relevant setup consisting of a onedimensional helical spin chain on the surface of a threedimensional superconductor, even for profiles that produce near zeroenergy states in equivalent one and two dimensional systems. Although our findings do not rule them out, the much reduced prevalence of zeroenergy bound states in long nonuniform helical spin chains compared with equivalent semiconductor nanowires, as well as the ability to measure states locally via STM, should reduce the experimental barrier to identifying MBSs in such systems.

Fully tunable longitudinal spinphoton interactions in Si and Ge quantum dots 
Stefano Bosco, Pasquale Scarlino, Jelena Klinovaja, and Daniel Loss Phys. Rev. Lett. 129, 066801 (2022); arXiv:2203.17163.
Spin qubits in silicon and germanium quantum dots are promising platforms for quantum computing, but entangling spin qubits over micrometer distances remains a critical challenge. Current prototypical architectures maximize transversal interactions between qubits and microwave resonators, where the spin state is flipped by nearly resonant photons. However, these interactions cause backaction on the qubit, that yield unavoidable residual qubitqubit couplings and significantly affect the gate fidelity. Strikingly, residual couplings vanish when spinphoton interactions are longitudinal and photons couple to the phase of the qubit. We show that large longitudinal interactions emerge naturally in stateoftheart hole spin qubits. These interactions are fully tunable and can be parametrically modulated by external oscillating electric fields. We propose realistic protocols to measure these interactions and to implement fast and highfidelity twoqubit entangling gates. These protocols work also at high temperatures, paving the way towards the implementation of largescale quantum processors.

Magnons, magnon bound pairs, and their hybrid spinmultipolar topology 
Alexander Mook, Rhea Hoyer, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 107, 064429 (2023); arXiv:2203.12374.
We consider quantum condensed matter systems without particlenumber conservation. Since the particle number is not a good quantum number, states belonging to different particlenumber sectors can hybridize, which causes topological anticrossings in the spectrum. The resulting spectral gaps support chiral edge excitations whose wavefunction is a superposition of states in the two hybridized sectors. This situation is realized in fully saturated spinanisotropic quantum magnets without spin conservation, in which single magnons hybridize with magnon bound pairs, i.e., twomagnon bound states. The resulting chiral edge excitations are exotic composites that carry mixed spinmultipolar character, inheriting spindipolar and spinquadrupolar character from their singleparticleness and twoparticleness, respectively. In contrast to established topological magnons, the topological effects discussed here are of genuine quantum mechanical origin and vanish in the classical limit. We discuss implications for intrinsic anomalous Halltype transport and estimate that the thermal Hall conductivity brought about by the hybridization of magnons and magnon bound pairs can be as large as that of magnons with other magnons. We conclude that fully polarized quantum magnets are a promising platform for topology caused by hybridizations between particlenumber sectors, complementing the field of ultracold atoms working with a conserved number of particles.

RKKY interaction at helical edges of topological superconductors 
Katharina Laubscher, Dmitry Miserev, Vardan Kaladzhyan, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 107, 115421 (2023); arXiv:2203.08137.
We study spin configurations of magnetic impurities placed close to the edge of a twodimensional topological superconductor both analytically and numerically. First, we demonstrate that the spin of a single magnetic impurity close to the edge of a topological superconductor tends to align along the edge. The strong easyaxis spin anisotropy behind this effect originates from the interaction between the impurity and the gapless helical Majorana edge states. We then compute the RudermanKittelKasuyaYosida (RKKY) interaction between two magnetic impurities placed close to the edge. We show that, in the limit of large interimpurity distances, the RKKY interaction between the two impurities is mainly mediated by the Majorana edge states and leads to a ferromagnetic alignment of both spins along the edge. This effect can be used to detect helical Majorana edge states.

Crossed Andreev reflection in spinpolarized chiral edge states due to Meissner effect 
Tamás Haidekker Galambos, Flavio Ronetti, Bence Hetényi, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 106, 075410 (2022); arXiv:2203.05894.
We consider a hybrid quantum Hallsuperconductor system, where a superconducting finger with oblique profile is wedged into a twodimensional electron gas in the presence of a perpendicular magnetic field, as considered by Lee et al., Nat. Phys. 13, 693 (2017). The electron gas is in the quantum Hall regime at filling factor ν=1. Due to the Meissner effect, the perpendicular magnetic field close to the quantum Hallsuperconductor boundary is distorted and gives rise to an inplane component of the magnetic field. This component enables nonlocal crossed Andreev reflection between the spinpolarized chiral edge states running on opposite sides of the superconducting finger, thus opening a gap in the spectrum of the edge states without the need of spinorbit interaction or nontrivial magnetic textures. We compute numerically the transport properties of this setup and show that a negative resistance exists as consequence of nonlocal Andreev processes. We also obtain numerically the zeroenergy local density of states, which systematically shows peaks stable to disorder. The latter result is compatible with the emergence of Majorana bound states.

Magnetoelectric Cavity Magnonics in Skyrmion Crystals 
Tomoki Hirosawa, Alexander Mook, Jelena Klinovaja, and Daniel Loss PRX Quantum 3, 040321 (2022); arXiv:2203.03241.
We present a theory of magnetoelectric magnonphoton coupling in cavities hosting noncentrosymmetric magnets. Analogously to nonreciprocal phenomena in multiferroics, the magnetoelectric coupling is timereversal and inversion asymmetric. This asymmetry establishes a means for exceptional tunability of magnonphoton coupling, which can be switched on and off by reversing the magnetization direction. Taking the multiferroic skyrmionhost Cu2OSeO3 as an example, we reveal the electrical activity of skyrmion eigenmodes and propose it for magnonphoton splitting of "magnetically dark" elliptic modes. Furthermore, we predict a cavityinduced magnonmagnon coupling between magnetoelectrically active skyrmion excitations. Our study highlights magnetoelectric cavity magnonics as a novel platform for realizing quantumhybrid systems and the quantum control of topological magnetic textures.

QuantumCoherent Nanoscience 
Andreas J. Heinrich, William D. Oliver, Lieven Vandersypen, Arzhang Ardavan, Roberta Sessoli, Daniel Loss, Ania Bleszynski Jayich, Joaquin FernandezRossier, Arne Laucht, and Andrea Morello Nature Nanotechnology 16, 1318 (2021); arXiv:2202.01431.
For the past three decades, nanoscience has widely affected many areas in physics, chemistry, and engineering, and has led to numerous fundamental discoveries as well as applications and products. Concurrently, quantum science and technology has developed into a crossdisciplinary research endeavour connecting these same areas and holds a burgeoning commercial promise. Although quantum physics dictates the behaviour of nanoscale objects, quantum coherence, which is central to quantum information, communication and sensing has not played an explicit role in much of nanoscience. This Review describes fundamental principles and practical applications of quantum coherence in nanoscale systems, a research area we call quantumcoherent nanoscience. We structure this manuscript according to specific degrees of freedom that can be quantumcoherently controlled in a given nanoscale system such as charge, spin, mechanical motion, and photons. We review the current state of the art and focus on outstanding challenges and opportunities unlocked by the merging of nanoscience and coherent quantum operations.

Instability of the ferromagnetic quantum critical point in strongly interacting 2D and 3D electron gases with arbitrary spinorbit splitting 
Dmitry Miserev, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 106, 134417 (2022); arXiv:2201.10995.
In this work we revisit itinerant ferromagnetism in 2D and 3D electron gases with arbitrary spinorbit splitting and strong electronelectron interaction. We identify the resonant scattering processes close to the Fermi surface that are responsible for the instability of the ferromagnetic quantum critical point at low temperatures. In contrast to previous theoretical studies, we show that such processes cannot be fully suppressed even in presence of arbitrary spinorbit splitting. A fully selfconsistent nonperturbative treatment of the electronelectron interaction close to the phase transition shows that these resonant processes always destabilize the ferromagnetic quantum critical point and lead to a firstorder phase transition. Characteristic signatures of these processes can be measured via the nonanalytic dependence of the spin susceptibility on magnetic field both far away or close to the phase transition

Metallization and proximity superconductivity in topological insulator nanowires 
Henry F. Legg, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 105, 155413 (2022); arXiv:2201.02918.
A heterostructure consisting of a topological insulator (TI) nanowire brought into proximity with a superconducting layer provides a promising route to achieve topological superconductivity and associated Majorana bound states (MBSs). Here, we study effects caused by such a coupling between a thin layer of an swave superconductor and a TI nanowire. We show that there is a distinct phenomenology arising from the metallization of states in the TI nanowire by the superconductor. In the strong coupling limit, required to induce a large superconducting pairing potential, we find that metallization results in a shift of the TI nanowire subbands (∼20 meV) as well as it leads to a small reduction in the size of the subband gap opened by a magnetic field applied parallel to the nanowire axis. Surprisingly, we find that metallization effects in TI nanowires can also be beneficial. Most notably, coupling to the superconductor induces a potential in the portion of the TI nanowire close to the interface with the superconductor, this breaks inversion symmetry and at finite momentum lifts the spin degeneracy of states within a subband. As such coupling to a superconductor can create or enhance the subband splitting that is key to achieving topological superconductivity. This is in stark contrast to semiconductors, where it has been shown that metallization effects always reduce the equivalent subbandsplitting caused by spinorbit coupling. We also find that in certain geometries metallization effects can reduce the critical magnetic required to enter the topological phase. We conclude that, unlike in semiconductors, the metallization effects that occur in TI nanowires can be relatively easily mitigated, for instance by modifying the geometry of the attached superconductor or by compensation of the TI material.

Chargenoise induced dephasing in silicon holespin qubits 
Ognjen Malkoc, Peter Stano, and Daniel Loss Phys. Rev. Lett. 129, 247701 (2022); arXiv:2201.06181.
We investigate theoretically chargenoise induced spin dephasing of a hole confined in a quasitwodimensional silicon quantum dot. Central to our treatment is accounting for higherorder corrections to the Luttinger Hamiltonian. Using experimentally reported parameters, we find that the new terms give rise to sweetspots for the holespin dephasing, which are sensitive to device details: dot size and asymmetry, growth direction, and applied magnetic and electric fields. Furthermore, we estimate that the dephasing time at the sweetspots is boosted by several orders of magnitude, up to order of milliseconds.

NonMajorana zero energy modes in diluted spin chains proximitized to a superconductor 
Felix Küster, Sascha Brinker, Richard Hess, Daniel Loss, Stuart Parkin, Jelena Klinovaja, Samir Lounis, and Paolo Sessi PNAS 2022 Vol. 119 No. 42 e2210589119; arXiv:2112.05708.
Spin chains proximitized with superconducting condensates have emerged as one of the most promising platforms for the realization of Majorana modes. The recent use of atomic manipulation techniques raised great expectations for successfully creating and controlling such chains. Here, we craft diluted spin chains atombyatom following seminal theoretical proposal suggesting indirect coupling mechanisms as a viable route to trigger topological superconductivity. We demonstrate that, starting from deep Shiba states, it is possible to cross the quantum phase transition, a necessary condition for the emergence of topological superconductivity, for very short chains. This transition is associated with the emergence of highly localized zero energy end modes. The use of a substrate with highly anisotropic Fermi surface enables to create spin chains characterized by distinct magnetic configurations along various crystallographic directions. By scrutinizing a large set of parameters we reveal the ubiquitous existence of zero energy boundary modes. Although mimicking signatures generally assigned to Majorana modes, the end modes are identified as topologically trivial Shiba states. These results highlight the important role of the local environment, showing that it cannot be completely eliminated also in diluted systems where the effect is expected to be minimized. Our work demonstrates that zero energy modes in spin chains proximitized to supercondcutors are not necessarily a link to Majorana modes while simultaneously identifying new experimental platforms, driving mechanisms, and test protocols for the determination of topologically non trivial superconducting phases.

Quasiparticle poisoning in trivial and topological Josephson junctions 
Aleksandr E. Svetogorov, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 105, 174519 (2022); arXiv:2112.05650.
We study theoretically a short singlechannel Josephson junction between superconductors in the trivial and topological phases. The junction is assumed to be biased by a small current and subjected to quasiparticle poisoning. We find that the presence of quasiparticles leads to a voltage signal from the Josephson junction that can be observed both in the trivial and in the topological phase. Quite remarkably, these voltage signatures are sufficiently different in the two phases such that they can serve as means to clearly distinguish between trivial Andreev and topological Majorana bound states in the system. Moreover, these voltage signatures, in the trivial and topological phase, would allow one to measure directly the quasiparticle poisoning rates and to test various approaches for protection against quasiparticle poisoning.

Hole Spin Qubits in Ge Nanowire Quantum Dots: Interplay of Orbital Magnetic Field, Strain, and Growth Direction 
Christoph Adelsberger, Mónica Benito, Stefano Bosco, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 105, 075308 (2022); arXiv:2110.15039.
Hole spin qubits in quasi onedimensional structures are a promising platform for quantum information processing because of the strong spinorbit interaction (SOI). We present analytical results and discuss device designs that optimize the SOI in Ge semiconductors. We show that at the magnetic field values at which qubits are operated, orbital effects of magnetic fields can strongly affect the response of the spin qubit. We study onedimensional hole systems in Ge under the influence of electric and magnetic fields applied perpendicularly to the device. In our theoretical description, we include these effects exactly. The orbital effects lead to a strong renormalization of the gfactor. We find a sweetspot of the nanowire (NW) gfactor where charge noise is strongly suppressed and present an effective lowenergy model that captures the dependence of the SOI on the electromagnetic fields. Moreover, we compare properties of NWs with square and circular crosssections with ones of gatedefined onedimensional channels in twodimensional Ge heterostructures. Interestingly, the effective model predicts a flat band ground state for finetuned electric and magnetic fields. By considering a quantum dot (QD) harmonically confined by gates, we demonstrate that the NW gfactor sweet spot is retained in the QD. Our calculations reveal that this sweet spot can be designed to coincide with the maximum of the SOI, yielding highly coherent qubits with large Rabi frequencies. We also study the effective gfactor of NWs grown along different high symmetry axes and find that our model derived for isotropic semiconductors is valid for the most relevant growth directions of nonisotropic Ge NWs. Moreover, a NW grown along one of the three main crystallographic axes shows the largest SOI. Our results show that the isotropic approximation is not justified in Ge in all cases.

Fractional spin excitations and conductance in the spiral staircase Heisenberg ladder 
Flavio Ronetti, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 105, 134413 (2022); arXiv:2110.02130.
We investigate theoretically the spiral staircase Heisenberg spin1/2 ladder in the presence of antiferromagnetic longrange spin interactions and a uniform magnetic field. As a special case we also consider the Kondo necklace model. If the magnetizations of the two chains forming the ladder satisfy a certain resonance condition, involving interchain couplings as perturbations, the system is in a partially gapped magnetic phase hosting excitations characterized by fractional spins, whose values can be changed by the magnetic field. We show that these fractional spin excitations can be probed by spin currents in a transport setup with a spin conductance that reveals the fractionalized spin. In some special cases, the spin conductance reaches universal values in units of (gμ_B)^2/h, where g is the gfactor, μ_B the Bohr magneton, and h the Planck constant. We obtain our results with the help of bosonization and numerical density matrix renormalization group methods.

Giant magnetochiral anisotropy from quantum confined surface states of topological insulator nanowires 
Henry F. Legg, Matthias Rößler, Felix Münning, Dingxun Fan, Oliver Breunig, Andrea Bliesener, Gertjan Lippertz, Anjana Uday, A. A. Taskin, Daniel Loss, Jelena Klinovaja, and Yoichi Ando Nature Nanotechnology 17, 696 (2022); arXiv:2109.05188.
Wireless technology relies on the conversion of alternating electromagnetic fields to direct currents, a process known as rectification. While rectifiers are normally based on semiconductor diodes, quantum mechanical nonreciprocal transport effects that enable highly controllable rectification have recently been discovered. One such effect is magnetochiral anisotropy (MCA), where the resistance of a material or a device depends on both the direction of current flow and an applied magnetic field. However, the size of rectification possible due to MCA is usually extremely small, because MCA relies on electronic inversion symmetry breaking which typically stems from intrinsic spinorbit coupling  a relativistic effect  in a noncentrosymmetric environment. Here, to overcome this limitation, we artificially break inversion symmetry via an applied gate voltage in thin topological insulator (TI) nanowire heterostructures and theoretically predict that such a symmetry breaking can lead to a giant MCA effect. Our prediction is confirmed via experiments on thin bulkinsulating (Bi1−xSbx)2Te3 TI nanowires, in which we observe the largest ever reported size of MCA rectification effect in a normal conductor  over 10000 times greater than in a typical material with a large MCA  and its behaviour is consistent with theory. Our findings present new opportunities for future technological applications of topological devices.

Lasercontrolled real and reciprocalspace topology in multiferroic insulators 
Tomoki Hirosawa, Jelena Klinovaja, Daniel Loss, and Sebastian A. Diaz Phys. Rev. Lett. 128, 037201 (2022); arXiv:2108.06535.
Magnetic materials in which it is possible to control the topology of their magnetic order in real space or the topology of their magnetic excitations in reciprocal space are highly soughtafter as platforms for alternative data storage and computing architectures. Here we show that multiferroic insulators, owing to their magnetoelectric coupling, offer a natural and advantageous way to address these two different topologies using laser fields. We demonstrate that via a delicate balance between the energy injection from a highfrequency laser and dissipation, single skyrmionsarchetypical topological magnetic texturescan be set into motion with a velocity and propagation direction that can be tuned by the laser field amplitude and polarization, respectively. Moreover, we uncover an ultrafast Floquet magnonic topological phase transition in a laserdriven skyrmion crystal and we propose a new diagnostic tool to reveal it using the magnonic thermal Hall conductivity.

Helical Liquids in Semiconductors 
ChenHsuan Hsu, Peter Stano, Jelena Klinovaja, and Daniel Loss Semicond. Sci. Technol. 36, 123003 (2021); arXiv:2107.13553.
Onedimensional helical liquids can appear at boundaries of certain condensed matter systems. Two prime examples are the edge of a quantum spin Hall insulator, also known as a twodimensional topological insulator, and the hinge of a threedimensional secondorder topological insulator. For these materials, the presence of a helical state at the boundary serves as a signature of their nontrivial bulk topology. Additionally, these boundary states are of interest themselves, as a novel class of strongly correlated lowdimensional systems with interesting potential applications. Here, we review existing results on such helical liquids in semiconductors. Our focus is on the theory, though we confront it with existing experiments. We discuss various aspects of the helical states, such as their realization, topological protection and stability, or possible experimental characterization. We lay emphasis on the hallmark of these states, being the prediction of a quantized electrical conductance. Since so far reaching a wellquantized conductance remained challenging experimentally, a large part of the review is a discussion of various backscattering mechanisms which have been invoked to explain this discrepancy. Finally, we include topics related to proximityinduced topological superconductivity in helical states, as an exciting application towards topological quantum computation with the resulting Majorana bound states.

Review of performance metrics of spin qubits in gated semiconducting nanostructures 
Peter Stano and Daniel Loss Nat Rev Phys 4, 672 (2022); arXiv:2107.06485.
This Technical Review collects values of selected performance characteristics of
semiconductor spin qubits defined in electrically controlled nanostructures. The characteristics
are envisaged to serve as a community source for the values of figures of merit with agreed
definitions allowing the comparison of different spin qubit platforms. We include characteristics
on the qubit coherence, speed, fidelity and qubit size of multi qubit devices. The focus is on
collecting and curating the values of these characteristics as reported in the literature, rather
than on their motivation or significance.

Fully tunable hyperfine interactions of hole spin qubits in Si and Ge quantum dots 
Stefano Bosco and Daniel Loss Phys. Rev. Lett. 127, 190501 (2021); arXiv:2106.13744.
Hole spin qubits are frontrunner platforms for scalable quantum computers, but stateoftheart devices suffer from noise originating from the hyperfine interactions with nuclear defects. We show that these interactions have a highly tunable anisotropy that is controlled by device design and external electric fields. This tunability enables sweet spots where the hyperfine noise is suppressed by an order of magnitude and is comparable to isotopically purified materials. We identify surprisingly simple designs where the qubits are highly coherent and are largely unaffected by both charge and hyperfine noise. We find that the large spinorbit interaction typical of elongated quantum dots not only speeds up qubit operations, but also dramatically renormalizes the hyperfine noise, altering qualitatively the dynamics of driven qubits and enhancing the fidelity of qubit gates. Our findings serve as guidelines to design high performance qubits for scaling up quantum computers.

Local and nonlocal quantum transport due to Andreev bound states in finite Rashba nanowires with superconducting and normal sections 
Richard Hess, Henry F. Legg, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 104, 075405 (2021) (Editors' Suggestion); arXiv:2105.02791.
We analyze Andreev bound states (ABSs) that form in normal sections of a Rashba nanowire that is only partially covered by a superconducting layer. These ABSs are localized close to the ends of the superconducting section and can be pinned to zero energy over a wide range of magnetic field strengths even if the nanowire is in the nontopological regime. For finitesize nanowires (typically ≲1 μm in current experiments), the ABS localization length is comparable to the length of the nanowire. The probability density of an ABS is therefore nonzero throughout the nanowire and differentialconductance calculations reveal a correlated zerobias peak (ZBP) at both ends of the nanowire. When a second normal section hosts an additional ABS at the opposite end of the superconducting section, the combination of the two ABSs can mimic the closing and reopening of the bulk gap in local and nonlocal conductances accompanied by the appearance of the ZBP. These signatures are reminiscent of those expected for Majorana bound states (MBSs) but occur here in the nontopological regime. Our results demonstrate that conductance measurements of correlated ZBPs at the ends of a typical superconducting nanowire or an apparent closing and reopening of the bulk gap in the local and nonlocal conductance are not conclusive indicators for the presence of MBSs.

Squeezed hole spin qubits in Ge quantum dots with ultrafast gates at low power 
Stefano Bosco, Mónica Benito, Christoph Adelsberger, and Daniel Loss Phys. Rev. B 104, 115425 (2021); arXiv:2103.16724.
Hole spin qubits in planar Ge heterostructures are one of the frontrunner platforms for scalable quantum computers. In these systems, the spinorbit interactions permit efficient allelectric qubit control. We propose a minimal design modification of planar devices that enhances these interactions by orders of magnitude and enables low power ultrafast qubit operations in the GHz range. Our approach is based on an asymmetric potential that strongly squeezes the quantum dot in one direction. This confinementinduced spinorbit interaction does not rely on microscopic details of the device such as growth direction or strain, and could be turned on and off on demand in stateoftheart qubits.

Tuning interactions between spins in a superconductor 
Hao Ding, Yuwen Hu, Mallika T. Randeria, Silas Hoffman, Oindrila Deb, Jelena Klinovaja, Daniel Loss, and Ali Yazdani Proc. Natl. Acad. Sci. (PNAS) 118, e2024837118 (2021); arXiv:2103.14656.
Novel manybody and topological electronic phases can be created in assemblies of interacting spins coupled to a superconductor, such as onedimensional topological superconductors with Majorana zero modes (MZMs) at their ends. Understanding and controlling interactions between spins and the emergent band structure of the ingap YuShibaRusinov (YSR) states they induce in a superconductor are fundamental for engineering such phases. Here, by precisely positioning magnetic adatoms with a scanning tunneling microscope (STM), we demonstrate both the tunability of exchange interaction between spins and precise control of the hybridization of YSR states they induce on the surface of a bismuth (Bi) thin film that is made superconducting with the proximity effect. In this platform, depending on the separation of spins, the interplay between RudermanKittelKasuyaYosida (RKKY) interaction, spinorbit coupling, and surface magnetic anisotropy stabilizes different types of spin alignments. Using highresolution STM spectroscopy at millikelvin temperatures, we probe these spin alignments through monitoring the spininduced YSR states and their energy splitting. Such measurements also reveal a quantum phase transition between the ground states with different electron number parity for a pair of spins in a superconductor tuned by their separation. Experiments on larger assemblies show that spinspin interactions can be mediated in a superconductor over long distances. Our results show that controlling hybridization of the YSR states in this platform provides the possibility of engineering the band structure of such states for creating topological phases.

Majorana bound states in topological insulators without a vortex 
Henry F. Legg, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 104, 165405 (2021); arXiv:2103.13412.
We consider a threedimensional topological insulator (TI) wire with a nonuniform chemical potential induced by gating across the crosssection. This inhomogeneity in chemical potential lifts the degeneracy between two onedimensional surface state subbands. A magnetic field applied along the wire, due to orbital effects, breaks timereversal symmetry and lifts the Kramers degeneracy at zeromomentum. If placed in proximity to an swave superconductor, the system can be brought into a topological phase at relatively weak magnetic fields. Majorana bound states (MBSs), localized at the ends of the TI wire, emerge and are present for an exceptionally large region of parameter space in realistic systems. Unlike in previous proposals, these MBSs occur without the requirement of a vortex in the superconducting pairing potential, which represents a significant simplification for experiments. Our results open a pathway to the realisation of MBSs in present day TI wire devices.

Majorana Bound States Induced by Antiferromagnetic Skyrmion Textures 
Sebastián A. Díaz, Jelena Klinovaja, Daniel Loss, and Silas Hoffman Phys. Rev. B 104, 214501 (2021); arXiv:2102.03423.
Majorana bound states are zeroenergy states predicted to emerge in topological superconductors and intense efforts seeking a definitive proof of their observation are still ongoing. A standard route to realize them involves antagonistic orders: a superconductor in proximity to a ferromagnet. Here we show this issue can be resolved using antiferromagnetic rather than ferromagnetic order. We propose to use a chain of antiferromagnetic skyrmions, in an otherwise collinear antiferromagnet, coupled to a bulk conventional superconductor as a novel platform capable of supporting Majorana bound states that are robust against disorder. Crucially, the collinear antiferromagnetic region neither suppresses superconductivity nor induces topological superconductivity, thus allowing for Majorana bound states localized at the ends of the chain. Our model introduces a new class of systems where topological superconductivity can be induced by editing antiferromagnetic textures rather than locally tuning material parameters, opening avenues for the conclusive observation of Majorana bound states.

Fractional boundary charges with quantized slopes in interacting one and twodimensional systems 
Katharina Laubscher, Clara S. Weber, Dante M. Kennes, Mikhail Pletyukhov, Herbert Schoeller, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 104, 035432 (2021); arXiv:2101.10301.
We study fractional boundary charges (FBCs) for two classes of strongly interacting systems. First, we study strongly interacting nanowires subjected to a periodic potential with a period that is a rational fraction of the Fermi wavelength. For sufficiently strong interactions, the periodic potential leads to the opening of a charge density wave gap at the Fermi level. The FBC then depends linearly on the phase offset of the potential with a quantized slope determined by the period. Furthermore, different possible values for the FBC at a fixed phase offset label different degenerate ground states of the system that cannot be connected adiabatically. Next, we turn to the fractional quantum Hall effect (FQHE) at odd filling factors ν=1/(2l+1), where l is an integer. For a Corbino disk threaded by an external flux, we find that the FBC depends linearly on the flux with a quantized slope that is determined by the filling factor. Again, the FBC has 2l+1 different branches that cannot be connected adiabatically, reflecting the (2l+1)fold degeneracy of the ground state. These results allow for several promising and strikingly simple ways to probe strongly interacting phases via boundary charge measurements.

YuShibaRusinov States and Ordering of Magnetic Impurities Near the Boundary of a Superconducting Nanowire 
Oindrila Deb, Silas Hoffman, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 103, 165403 (2021); arXiv:2012.15205.
We theoretically study the spectrum induced by one and two magnetic impurities near the boundary of a onedimensional nanowire in proximity to a conventional swave superconductor and extract the ground state magnetic configuration. We show that the energies of the subgap states, supported by the magnetic impurities, are strongly affected by the boundary for distances less than the superconducting coherence length. In particular, when the impurity is moved towards the boundary, multiple quantum phase transitions periodically occur in which the parity of the superconducting condensate oscillates between even and odd. We find that the magnetic ground state configuration of two magnetic impurities depends not only on the distance between them but also explicitly on their distance away from the boundary of the nanowire. As a consequence, the magnetic ground state can switch from ferromagnetic to antiferromagnetic while keeping the interimpurity distance unaltered by simultaneously moving both impurities away from the boundary. The ground state magnetic configuration of two impurities is found analytically in the weak coupling regime and exactly for an arbitrary impurity coupling strength using numerical tightbinding simulations.

Insulating regime of an underdamped currentbiased Josephson junction supporting Z3 and Z4 parafermions 
Aleksandr E. Svetogorov, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 103, L180505 (2021); arXiv:2012.09062.
We study analytically a currentbiased topological Josephson junction supporting Zn parafermions. First, we show that in an infinitesize system a pair of parafermions on the junction can be in n different states; the (2pi)n periodicity of the phase potential of the junction results in a significant suppression of the maximal current Im for an insulating regime of the underdamped junction. Second, we study the behaviour of a realistic finitesize system with avoided level crossings characterized by splitting delta. We consider two limiting cases: when the phase evolution may be considered adiabatic, which results in decreased periodicity of the effective potential, and the opposite case, when LandauZener transitions restore the (2pi)n periodicity of the phase potential. The resulting current Im is exponentially different in the opposite limits, which allows us to propose a new detection method to establish the appearance of parafermions in the system experimentally, based on measuring Im at different values of the splitting delta.

Hole spin qubits in Si FinFETs with fully tunable spinorbit coupling and sweet spots for charge noise 
Stefano Bosco, Bence Hetényi, and Daniel Loss PRX Quantum 2, 010348 (2021); arXiv:2011.09417.
The strong spinorbit coupling in hole spin qubits enables fast and electrically tunable gates, but at the same time enhances the susceptibility of the qubit to charge noise. Suppressing this noise is a significant challenge in semiconductor quantum computing. Here, we show theoretically that hole Si FinFETs are not only very compatible with modern CMOS technology, but they present operational sweet spots where the charge noise is completely removed. The presence of these sweet spots is a result of the interplay between the anisotropy of the material and the triangular shape of the FinFET crosssection, and it does not require an extreme finetuning of the electrostatics of the device. We present how the sweet spots appear in FinFETs grown along different crystallographic axes and we study in detail how the behaviour of these devices change when the crosssection area and aspect ratio are varied. We identify designs that maximize the qubit performance and could pave the way towards a scalable spinbased quantum computer.

Interactionstabilized topological magnon insulator in ferromagnets 
Alexander Mook, Kirill Plekhanov, Jelena Klinovaja, and Daniel Loss Phys. Rev. X 11, 021061 (2021); arXiv:2011.06543.
Condensed matter systems admit topological collective excitations above a trivial ground state, an example being Chern insulators formed by Dirac bosons with a gap at finite energies. However, in contrast to electrons, there is no particlenumber conservation law for collective excitations. This gives rise to particle numbernonconserving manybody interactions whose influence on singleparticle topology is an open issue of fundamental interest in the field of topological quantum materials. Taking magnons in ferromagnets as an example, we uncover topological magnon insulators that are stabilized by interactions through opening Cherninsulating gaps in the magnon spectrum. This can be traced back to the fact that the particlenumber nonconserving interactions break the effective timereversal symmetry of the harmonic theory. Hence, magnonmagnon interactions are a source of topology that can introduce chiral edge states, whose chirality depends on the magnetization direction. Importantly, interactions do not necessarily cause detrimental damping but can give rise to topological magnons with exceptionally long lifetimes. We identify two mechanisms of interactioninduced topological phase transitionsone driven by an external field, the other by temperatureand show that they cause unconventional sign reversals of transverse transport signals, in particular of the thermal Hall conductivity. We identify candidate materials where this manybody mechanism is expected to occur, such as the metalorganic kagomelattice magnet Cu(1,3benzenedicarboxylate), the van der Waals honeycomblattice magnet CrI3, and the multiferroic kamiokite (Fe2Mo3O8). Our results demonstrate that interactions can play an important role in generating nontrivial topology.

Clock model and parafermions in Rashba nanowires 
Flavio Ronetti, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 103, 235410 (2021); arXiv:2011.06367.
We consider a semiconducting nanowire with Rashba spinorbit interaction subjected to a magnetic field and in the presence of strong electronelectron interactions. When the ratio between Fermi and Rashba momenta is tuned to 1/2, two competing resonant multiparticle scattering processes are present simultaneously and the interplay between them brings the system into a gapless critical parafermion phase. This critical phase is described by a selfdual sineGordon model, which we are able to map explicitly onto the lowenergy sector of the ℤ4 parafermion clock chain model. Finally, we show that by alternating regions in which only one of these two processes is present one can generate localized zeroenergy parafermion bound states.

Isotropic and Anisotropic gfactor Corrections in GaAs Quantum Dots 
Leon C. Camenzind, Simon Svab, Peter Stano, Liuqi Yu, Jeramy D. Zimmerman, Arthur C. Gossard, Daniel Loss, and Dominik M. Zumbühl Phys. Rev. Lett. 127, 057701 (2021); arXiv:2010.11185.
We experimentally determine isotropic and anisotropic gfactor corrections in lateral GaAs singleelectron quantum dots. We extract the Zeeman splitting by measuring the tunnel rates into the individual spin states of an empty quantum dot for an inplane magnetic field with various strengths and directions. We quantify the Zeeman energy and find a linear dependence on the magnetic field strength which allows us to extract the gfactor. The measured gfactor is understood in terms of spinorbit interaction induced isotropic and anisotropic corrections to the GaAs bulk gfactor. Because this implies a dependence of the spin splitting on the magnetic field direction, these findings are of significance for spin qubits in GaAs quantum dots.

Chiral Hinge Magnons in SecondOrder Topological Magnon Insulators 
Alexander Mook, Sebastián A. Díaz, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 104, 024406 (2021); arXiv:2010.04142.
When interacting spins in condensed matter order ferromagnetically, their ground state wave function is topologically trivial. Nonetheless, in two dimensions, the ferromagnetic state can support spin excitations with nontrivial topology, an exotic state known as topological magnon insulator (TMI). Here, we theoretically unveil and numerically confirm a novel ferromagnetic state in three dimensions dubbed secondorder TMI, whose hallmarks are excitations at its hinges, where facets intersect. Since ferromagnetism naturally comes with broken timereversal symmetry, the hinge magnons are chiral, rendering backscattering impossible. Hence, they trace out a threedimensional path about the sample unimpeded by defects and are topologically protected by the spectral gap. They are remarkably robust against disorder and simultaneously highly tunable by atomiclevel engineering of the sample termination. Our findings empower magnonics with the tools of higherorder topology, a promising route to combine lowenergy information transfer free of Joule heating with threedimensional vertical integration.

Universality of Boundary Charge Fluctuations 
Clara S. Weber, Kiryl Piasotski, Mikhail Pletyukhov, Jelena Klinovaja, Daniel Loss, Herbert Schoeller, and Dante M. Kennes Phys. Rev. Lett. 126, 016803 (2021); Editor's suggestion; arXiv:2008.08431.
We establish the quantum fluctuations ΔQ2B of the charge QB accumulated at the boundary of an insulator as an integral tool to characterize phase transitions where a direct gap closes (and reopens), typically occurring for insulators with topological properties. The power of this characterization lies in its capability to treat different kinds of insulators on equal footing; being applicable to transitions between topological and nontopological band, Anderson, and Mott insulators alike. In the vicinity of the phase transition we find a universal scaling ΔQ2B(Eg) as function of the gap size Eg and determine its generic form in various dimensions. For prototypical phase transitions with a massive Diraclike bulk spectrum we demonstrate a scaling with the inverse gap in one dimension and a logarithmic one in two dimensions.

Magnetic phase transitions in twodimensional twovalley semiconductors with inplane magnetic field 
Dmitry Miserev, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 103, 024401 (2021); arXiv:2008.07518.
A twodimensional electron gas (2DEG) in twovalley semiconductors has two discrete degrees of freedom given by the spin and valley quantum numbers. We analyze the zerotemperature magnetic instabilities of twovalley semiconductors with SOI, inplane magnetic field, and electronelectron interaction. The interplay of an applied inplane magnetic field and the SOI results in noncollinear spin quantization in different valleys. Together with the exchange intervalley interaction this results in a rich phase diagram containing four nontrivial magnetic phases. The negative nonanalytic cubic correction to the free energy, which is always present in an interacting 2DEG, is responsible for first order phase transitions. Here, we show that nonzero ground state values of the order parameters can cut this cubic nonanalyticity and drive certain magnetic phase transitions second order. We also find two tricritical points at zero temperature which together with the line of second order phase transitions constitute the quantum critical sector of the phase diagram. The phase transitions can be tuned externally by electrostatic gates or by the inplane magnetic field.

Quadrupole spin polarization as signature of secondorder topological superconductors 
Kirill Plekhanov, Niclas Müller, Yanick Volpez, Dante M. Kennes, Herbert Schoeller, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 103, L041401 (2021); arXiv:2008.03611.
We study theoretically secondorder topological superconductors characterized by the presence of pairs of zeroenergy Majorana corner states. We uncover a quadrupole spin polarization at the system edges that provides a striking signature to identify topological phases, thereby complementing standard approaches based on zerobias conductance peaks due to Majorana corner states. We consider two different classes of secondorder topological superconductors with broken timereversal symmetry and show that both classes are characterized by a quadrupolar structure of the spin polarization that disappears as the system passes through the topological phase transition. This feature can be accessed experimentally using spinpolarized scanning tunneling microscopes. We study different models hosting secondorder topological phases, both analytically and numerically, and using Keldysh techniques we provide numerical simulations of the spinpolarized currents probed by scanning tips.

Pinning of Andreev bound states to zero energy in twodimensional superconductorsemiconductor Rashba heterostructures 
Olesia Dmytruk, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 102, 245431 (2020); arXiv:2007.14369.
We consider a twodimensional electron gas (2DEG) with Rashba spinorbit interaction (SOI) partially covered by an swave superconductor, where the uncovered region remains normal but is exposed to a perpendicular
Zeeman field. We find analytically and numerically Andreev bound states (ABSs) formed in the normal region
and show that, by tuning the SOI to certain values, one can reach a regime where the energy of the lowest ABS
becomes pinned close to zero as a function of Zeeman field. In addition, we also consider a configuration with a
superconducting vortex and find again ABSs pinned close to zero energy in the topologically trivial phase for a
wide range of parameters. Thus, ABSs and Majorana bound states show similar behavior in such structures.

Kramers pairs of Majorana corner states in a topological insulator bilayer 
Katharina Laubscher, Danial Chughtai, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 102, 195401 (2020); arXiv:2007.13579.
We consider a system consisting of two tunnelcoupled twodimensional topological insulators proximitized by a top and bottom superconductor with a phase difference of π between them. We show that this system exhibits a timereversal invariant secondorder topological superconducting phase characterized by the presence of a Kramers pair of Majorana corner states at all four corners of a rectangular sample. We furthermore investigate the effect of a weak timereversal symmetry breaking perturbation and show that an inplane Zeeman field leads to an even richer phase diagram exhibiting two nonequivalent phases with two Majorana corner states per corner as well as an intermediate phase with only one Majorana corner state per corner. We derive our results analytically from continuum models describing our system. In addition, we also provide independent numerical confirmation of the resulting phases using discretized lattice representations of the models, which allows us to demonstrate the robustness of the topological phases and the Majorana corner states against parameter variations and potential disorder.

Fermi Surface Resonance and Quantum Criticality in Strongly Interacting Fermi Gases 
Dmitry Miserev, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 103, 075104 (2021); arXiv:2007.04913.
Fermions in the Fermi gas obey the Pauli exclusion principle restricting any two fermions from filling the same quantum state. Strong interaction between fermions can completely change the properties of the Fermi gas. In our theoretical study we find a new exotic quantum phase in strongly interacting Fermi gases constrained to a certain condition imposed on the Fermi surfaces which we call the Fermi surface resonance. The new phase is quantum critical which can be identified by the powerlaw frequency tail of the spectral density and divergent static susceptibilities. An especially striking feature of the new phase is the anomalous powerlaw temperature dependence of the dc resistivity that is similar to strange metals. The new quantum critical phase can be experimentally found in ordinary semiconductor heterostructures.

Strong spinorbit interaction and gfactor renormalization of hole spins in Ge/Si nanowire quantum dots 
F. N. M. Froning, M. J. Rančić, B. Hetényi, S. Bosco, M. K. Rehmann, A. Li, E. P. A. M. Bakkers, F. A. Zwanenburg, D. Loss, D. M. Zumbühl, and F. R. Braakman Phys. Rev. Res. 3, 013081 (2021); arXiv:2007.04308.
The spinorbit interaction lies at the heart of quantum computation with spin qubits, research on topologically nontrivial states, and various applications in spintronics. Hole spins in Ge/Si core/shell nanowires experience a spinorbit interaction that has been predicted to be both strong and electrically tunable, making them a particularly promising platform for research in these fields. We experimentally determine the strength of spinorbit interaction of hole spins confined to a double quantum dot in a Ge/Si nanowire by measuring spinmixing transitions inside a regime of spinblockaded transport. We find a remarkably short spinorbit length of ∼65 nm, comparable to the quantum dot length and the interdot distance. We additionally observe a large orbital effect of the applied magnetic field on the hole states, resulting in a large magnetic field dependence of the spinmixing transition energies. Strikingly, together with these orbital effects, the strong spinorbit interaction causes a significant enhancement of the gfactor with magnetic field.The large spinorbit interaction strength demonstrated is consistent with the predicted direct Rashba spinorbit interaction in this material system and is expected to enable ultrafast Rabi oscillations of spin qubits and efficient qubitqubit interactions, as well as provide a platform suitable for studying Majorana zero modes

Critical current for an insulating regime of an underdamped currentbiased topological Josephson junction 
Aleksandr E. Svetogorov, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 2, 033448 (2020); arXiv:2006.16643.
We study analytically an underdamped currentbiased topological Josephson junction. First, we consider a simplified model at zero temperature, where the parity of the nonlocal fermionic state formed by Majorana bound states (MBSs) localized on the junction is fixed, and show that a transition from insulating to conducting state in this case is governed by singlequasiparticle tunneling rather than by Cooper pair tunneling in contrast to a nontopological Josephson junction. This results in a significantly lower critical current for the transition from insulating to conducting state. We propose that, if the length of the system is finite, the transition from insulating to conducting state occurs at exponentially higher bias current due to hybridization of the states with different parities as a result of the overlap of MBSs localized on the junction and at the edges of the topological nanowire forming the junction. Finally, we discuss how the appearance of MBSs can be established experimentally by measuring the critical current for an insulating regime at different values of the applied magnetic field.

Majorana zero modes and their bosonization 
Victor Chua, Katharina Laubscher, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 102, 155416 (2020); arXiv:2006.03344.
The simplest continuum model of a onedimensional noninteracting superconducting fermionic symmetryprotected topological (SPT) phase is analyzed in great detail using analytic methods. A full exact diagonalization of the meanfield Bogoliubovde Gennes Hamiltonian is carried out with open boundaries and finite lengths. Majorana zero modes are derived and studied in great detail. Thereafter exact operator bosonization in both open and closed geometries is carried out. The complementary viewpoints provided by fermionic and bosonic formulations of the superconducting SPT phase are then reconciled. In particular, we provide a complete and exact account of how the topological Majorana zero modes manifest in a bosonized description of an SPT phase.

Magnonic Quadrupole Topological Insulator in Antiskyrmion Crystals 
Tomoki Hirosawa, Sebastian A. Diaz, Jelena Klinovaja, and Daniel Loss Phys. Rev. Lett. 125, 207204 (2020); arXiv:2005.05884.
We uncover that antiskyrmion crystals provide an experimentally accessible platform to realize a magnonic quadrupole topological insulator, whose hallmark signatures are robust magnonic corner states. Furthermore, we show that tuning an applied magnetic field can trigger the selfassembly of antiskyrmions carrying a fractional topological charge along the sample edges. Crucially, these fractional antiskyrmions restore the symmetries needed to enforce the emergence of the magnonic corner states. Using the machinery of nested Wilson loops, adapted to magnonic systems supported by noncollinear magnetic textures, we demonstrate the quantization of the bulk quadrupole moment, edge dipole moments, and corner charges.

Optimal frequency estimation and its application to quantum dots 
Angel GutierrezRubio, Peter Stano, and Daniel Loss arXiv:2004.12049
We address the interactiontime optimization for frequency estimation in a general twolevel system. The goal is to track with maximum precision a stochastic perturbation with arbitrary dynamics. Our approach is valid for any figure of merit used to define optimality, and is illustrated for the variance and entropy. For the entropy, we clarify the connection to maximumlikelihood estimation. We devise novel estimation protocols with and without feedback. They outperform common protocols given in the literature. We design a probabilistic selfconsistent protocol as a generically optimal estimation without feedback. It can improve current experimental techniques and boost coherence times in quantum computing.

The germanium quantum information route 
Giordano Scappucci, Christoph Kloeffel, Floris A. Zwanenburg, Daniel Loss, Maksym Myronov, JianJun Zhang, Silvano De Franceschi, Georgios Katsaros, and Menno Veldhorst Nat Rev Mater (2020); arXiv:2004.08133.
In the worldwide endeavor for disruptive quantum technologies, germanium is emerging as a versatile material to realize devices capable of encoding, processing, or transmitting quantum information. These devices leverage special properties of the germanium valenceband states, commonly known as holes, such as their inherently strong spinorbit coupling and the ability to host superconducting pairing correlations. In this Review, we initially introduce the physics of holes in lowdimensional germanium structures with key insights from a theoretical perspective. We then examine the material science progress underpinning germaniumbased planar heterostructures and nanowires. We review the most significant experimental results demonstrating key building blocks for quantum technology, such as an electrically driven universal quantum gate set with spin qubits in quantum dots and superconductorsemiconductor devices for hybrid quantum systems. We conclude by identifying the most promising prospects toward scalable quantum information processing.

Majorana bound states in topological insulators with hidden Dirac points 
Ferdinand Schulz, Kirill Plekhanov, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 2, 033215 (2020); arXiv:2004.10623.
We address the issue whether it is possible to generate Majorana bound states at the magneticsuperconducting interface in twodimensional topological insulators with hidden Dirac points in the spectrum. In this case, the Dirac point of edge states is located at the energies of the bulk states such that two types of states are strongly hybridized. Here, we show that welldefined Majorana bound states can be obtained even in materials with hidden Dirac point provided that the width of the magnetic strip is chosen to be comparable with the localization length of the edge states. The obtained topological phase diagram allows one to extract precisely the position of the Dirac point in the spectrum. In addition to standard zerobias peak features caused by Majorana bound states in transport experiments, we propose to supplement future experiments with measurements of charge and spin polarization. In particular, we demonstrate that both observables flip their signs at the topological phase transition, thus, providing an independent signature of the presence of topological superconductivity. All features remain stable against substantially strong disorder.

Exchange interaction of holespin qubits in double quantum dots in highly anisotropic semiconductors 
Bence Hetényi, Christoph Kloeffel, and Daniel Loss Phys. Rev. Research 2, 033036 (2020); arXiv:2004.07658.
We study the exchange interaction between two holespin qubits in a double quantum dot setup in a silicon nanowire in the presence of magnetic and electric fields. Based on symmetry arguments we show that there exists an effective spin that is conserved even in highly anisotropic semiconductors, provided that the system has a twofold symmetry with respect to the direction of the applied magnetic field. This finding facilitates the definition of qubit basis states and simplifies the form of exchange interaction for twoqubit gates in coupled quantum dots. If the magnetic field is applied along a generic direction, cubic anisotropy terms act as an effective spinorbit interaction introducing novel exchange couplings even for an inversion symmetric setup. Considering the example of a silicon nanowire double dot, we present the relative strength of these anisotropic exchange interaction terms and calculate the fidelity of the SQRTofSWAP gate. Furthermore, we show that the anisotropyinduced spinorbit effects can be comparable to that of the direct Rashba spinorbit interaction for experimentally feasible electric field strengths.

Superconducting Quantum Interference in Edge State Josephson Junctions 
Tamás Haidekker Galambos, Silas Hoffman, Patrik Recher, Jelena Klinovaja, and Daniel Loss Phys. Rev. Lett. 125, 157701 (2020); arXiv:2004.01733.
We study superconducting quantum interference in a Josephson junction linked via edge states in twodimensional (2D) insulators. We consider two scenarios in which the 2D insulator is either a topological or a trivial insulator supporting onedimensional (1D) helical or nonhelical edge states, respectively. In equilibrium, we find that the qualitative dependence of critical supercurrent on the flux through the junction is insensitive to the helical nature of the mediating states and can, therefore, not be used to verify the topological features of the underlying insulator. However, upon applying a finite voltage bias smaller than the superconducting gap to a relatively long junction, the finitefrequency interference pattern in the nonequilibrium transport current is qualitatively different for helical edge states as compared to nonhelical ones.

Rational boundary charge in onedimensional systems with interaction and disorder 
Mikhail Pletyukhov, Dante M. Kennes, Kiryl Piasotski, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller Phys. Rev. Res. 2, 033345 (2020); arXiv:2004.00463.
We study the boundary charge QB of generic semiinfinite onedimensional insulators with translational invariance and show that nonlocal symmetries (i.e., including translations) lead to rational quantizations p/q of QB. In particular, we find that (up to an unknown integer) the quantization of QB is given in integer units of 12ρ¯ and 12(ρ¯−1), where ρ¯ is the average charge per site (which is a rational number for an insulator). This is a direct generalization of the known halfinteger quantization of QB for systems with local inversion or local chiral symmetries to any rational value. Quite remarkably, this rational quantization remains valid even in the presence of shortranged electronelectron interactions as well as static random disorder (breaking translational invariance). This striking stability can be traced back to the fact that local perturbations in insulators induce only local charge redistributions. We establish this result with complementary methods including density matrix renormalization group calculations, bosonization methods, and exact solutions for particular lattice models. Furthermore, for the special case of halffilling ρ¯=12, we present explicit results in singlechannel and nearestneighbor hopping models and identify Weyl semimetal physics at gap closing points. Our general framework also allows us to shed new light on the wellknown rational quantization of soliton charges at domain walls.

Spin orbit field in a physically defined p type MOS silicon double quantum dot 
Marian Marx, Jun Yoneda, Ángel Gutiérrez Rubio, Peter Stano, Tomohiro Otsuka, Kenta Takeda, Sen Li, Yu Yamaoka, Takashi Nakajima, Akito Noiri, Daniel Loss, Tetsuo Kodera, and Seigo Tarucha arXiv:2003.07079
We experimentally and theoretically investigate the spin orbit (SO) field in a physically defined, p type metal oxide semiconductor double quantum dot in silicon. We measure the magnetic field dependence of the leakage current through the double dot in the Pauli spin blockade. A finite magnetic field lifts the blockade, with the lifting least effective when the external and SO fields are parallel. In this way, we find that the spin flip of a tunneling hole is due to a SO field pointing perpendicular to the double dot axis and almost fully out of the quantum well plane. We augment the measurements by a derivation of SO terms using group symmetric representations theory. It predicts that without in plane electric fields (a quantum well case), the SO field would be mostly within the plane, dominated by a sum of a Rashba and a Dresselhaus like term. We, therefore, interpret the observed SO field as originated in the electric fields with substantial in plane components.

Quantum Damping of Skyrmion Crystal Eigenmodes due to Spontaneous Quasiparticle Decay 
Alexander Mook, Jelena Klinovaja, and Daniel Loss Phys. Rev. Res. 2, 033491 (2020); arXiv:2002.12676.
The elementary excitations of skyrmion crystals experience both emergent magnetic fields and anharmonic interactions brought about by the topologically nontrivial noncollinear texture. The resulting flat bands cause strong spontaneous quasiparticle decay, dressing the eigenmodes of skyrmion crystals with a finite zerotemperature quantum lifetime. Sweeping the flat bands through the spectrum by changing the magnetic field leads to an externally controllable energyselective magnon breakdown. In particular, we uncover that the three fundamental modes, i.e., the anticlockwise, breathing, and clockwise mode, exhibit distinct decay behavior, with the clockwise (anticlockwise) mode being the least (most) stable mode out of the three.

Spin Wave Radiation by a Topological Charge Dipole 
Sebastian A. Diaz, Tomoki Hirosawa, Daniel Loss, and Christina Psaroudaki Nano Lett. 20, 6556 (2020); arXiv:2002.12282.
The use of spin waves (SWs) as data carriers in spintronic and magnonic logic devices offers operation at low power consumption, free of Joule heating. Nevertheless, the controlled emission and propagation of SWs in magnetic materials remains a significant challenge. Here, we propose that skyrmionantiskyrmion bilayers form topological charge dipoles and act as efficient sub100 nm SW emitters when excited by inplane ac magnetic fields. The propagating SWs have a preferred radiation direction, with clear dipole signatures in their radiation pattern, suggesting that the bilayer forms a SW antenna. Bilayers with the same topological charge radiate SWs with spiral and antispiral spatial profiles, enlarging the class of SW patterns. We demonstrate that the characteristics of the emitted SWs are linked to the topology of the source, allowing for full control of the SW features, including their amplitude, preferred direction of propagation, and wavelength.

Site‐Controlled Uniform Ge/Si Hut Wires with Electrically Tunable Spin–Orbit Coupling 
Fei Gao, JianHuan Wang, Hannes Watzinger, Hao Hu, Marko J. Rancic, JieYin Zhang, Ting Wang, Yuan Yao, GuiLei Wang, Josip Kukucka, Lada Vukusic, Christoph Kloeffel, Daniel Loss, Feng Liu, Georgios Katsaros, and JianJun Zhang Adv. Mater. 2020, 1906523; arXiv:2001.11305.
Semiconductor nanowires have been playing a crucial role in the development of nanoscale devices for the realization of spin qubits, Majorana fermions, single photon emitters, nanoprocessors, etc. The monolithic growth of site‐controlled nanowires is a prerequisite toward the next generation of devices that will require addressability and scalability. Here, combining top‐down nanofabrication and bottom‐up self‐assembly, the growth of Ge wires on prepatterned Si (001) substrates with controllable position, distance, length, and structure is reported. This is achieved by a novel growth process that uses a SiGe strain‐relaxation template and can be potentially generalized to other material combinations. Transport measurements show an electrically tunable spin–orbit coupling, with a spin–orbit length similar to that of III–V materials. Also, charge sensing between quantum dots in closely spaced wires is observed, which underlines their potential for the realization of advanced quantum devices. The reported results open a path toward scalable qubit devices using nanowires on silicon.

Magnetic field independent subgap states in hybrid Rashba nanowires 
Christian Juenger, Raphaelle Delagrange, Denis Chevallier, Sebastian Lehmann, Kimberly A. Dick, Claes Thelander, Jelena Klinovaja, Daniel Loss, Andreas Baumgartner, and Christian Schoenenberger Phys. Rev. Lett. 125, 017701 (2020); arXiv:2001.07666.
Subgap states in semiconductingsuperconducting nanowire hybrid devices are controversially discussed as potential topologically nontrivial quantum states. One source of ambiguity is the lack of an energetically and spatially well defined tunnel spectrometer. Here, we use quantum dots directly integrated into the nanowire during the growth process to perform tunnel spectroscopy of discrete subgap states in a long nanowire segment. In addition to subgap states with a standard magnetic field dependence, we find topologically trivial subgap states that are independent of the external magnetic field, i.e. that are pinned to a constant energy as a function of field. We explain this effect qualitatively and quantitatively by taking into account the strong spinorbit interaction in the nanowire, which can lead to a decoupling of Andreev bound states from the field due to a spatial spin texture of the confined eigenstates.

Transport signatures of topological phases in double nanowires probed by spinpolarized STM 
Manisha Thakurathi, Denis Chevallier, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 2, 023197 (2020); arXiv:2001.05470.
We study a doublenanowire setup proximity coupled to an swave superconductor and search for the bulk signatures of the topological phase transition that can be observed experimentally, for example, with an STM tip. Three bulk quantities, namely, the charge, the spin polarization, and the pairing amplitude of intrawire superconductivity are studied in this work. The spin polarization and the pairing amplitude flip sign as the system undergoes a phase transition from the trivial to the topological phase. In order to identify promising ways to observe bulk signatures of the phase transition in transport experiments, we compute the spin current flowing between a local spinpolarized probe, such as an STM tip, and the doublenanowire system in the Keldysh formalism. We find that the spin current contains information about the sign flip of the bulk spin polarization and can be used to determine the topological phase transition point.

Coherence of a driven electron spin qubit actively decoupled from quasistatic noise 
Takashi Nakajima, Akito Noiri, Kento Kawasaki, Jun Yoneda, Peter Stano, Shinichi Amaha, Tomohiro Otsuka, Kenta Takeda, Matthieu R. Delbecq, Giles Allison, Arne Ludwig, Andreas D. Wieck, Daniel Loss, and Seigo Tarucha Phys. Rev. X 10, 011060 (2020); arXiv:2001.02884.
The coherence of electron spin qubits in semiconductor quantum dots suffers mostly from lowfrequency noise. During the last decade, efforts have been devoted to mitigate such noise by material engineering, leading to substantial enhancement of the spin dephasing time for an idling qubit. However, the role of the environmental noise during spin manipulation, which determines the control fidelity, is less understood. We demonstrate an electron spin qubit whose coherence in the driven evolution is limited by highfrequency charge noise rather than the quasistatic noise inherent to any semiconductor device. We employed a feedback control technique to actively suppress the latter, demonstrating a πflip gate fidelity as high as 99.04±0.23% in a gallium arsenide quantum dot. We show that the drivenevolution coherence is limited by the longitudinal noise at the Rabi frequency, whose spectrum resembles the 1/f noise observed in isotopically purified silicon qubits.

Universal conductance dips and fractional excitations in a twosubband quantum wire 
ChenHsuan Hsu, Flavio Ronetti, Peter Stano, Jelena Klinovaja, and Daniel Loss Phys. Rev. Research 2, 043208 (2020); arXiv:1912.11592.
We theoretically investigate a quantum wire based on a quasionedimensional Kondo lattice formed by localized spins and itinerant electrons, where the lowest two subbands of the quantum wire are populated. We uncover a backscattering mechanism involving helically ordered spins and Coulomb interaction between the electrons. The combination of these ingredients results in scattering resonances and partial gaps which give rise to nonstandard plateaus and conductance dips at certain electron densities. The positions and values of these dips are independent of material parameters, serving as direct transport signatures of this mechanism. While our theory describes a generic Kondo lattice, an experimentally relevant realization is provided by quantum wires made out of IIIV semiconductors hosting nuclear spins such as InAs. Observation of the universal conductance dips would not only confirm the presence of a nuclear spin helix but also identify a strongly correlated fermion system hosting fractional excitations, resembling the fractional quantum Hall states even without external magnetic fields.

Majorana and parafermion corner states from two coupled sheets of bilayer graphene 
Katharina Laubscher, Daniel Loss, and Jelena Klinovaja Phys. Rev. Res. 2, 013330 (2020); arXiv:1912.10931.
We consider a setup consisting of two coupled sheets of bilayer graphene in the regime of strong spinorbit interaction, where electrostatic confinement is used to create an array of effective quantum wires. We show that for suitable interwire couplings the system supports a topological insulator phase exhibiting Kramers partners of gapless helical edge states, while the additional presence of a small inplane magnetic field and weak proximityinduced superconductivity leads to the emergence of zeroenergy Majorana corner states at all four corners of a rectangular sample, indicating the transition to a secondorder topological superconducting phase. The presence of strong electronelectron interactions is shown to promote the above phases to their exotic fractional counterparts. In particular, we find that the system supports a fractional topological insulator phase exhibiting fractionally charged gapless edge states and a fractional secondorder topological superconducting phase exhibiting zeroenergy Z2m parafermion corner states, where m is an odd integer determined by the position of the chemical potential.

Firstorder magnetic phasetransition of mobile electrons in monolayer MoS2 
Jonas Gaël Roch, Dmitry Miserev, Guillaume Froehlicher, Nadine Leisgang, Lukas Sponfeldner, Kenji Watanabe, Takashi Taniguchi, Jelena Klinovaja, Daniel Loss, and Richard John Warburton Phys. Rev. Lett. 124, 187602 (2020); arXiv:1911.10238.
Evidence is presented for a firstorder magnetic phase transition in a gated twodimensional semiconductor, monolayerMoS2. The phase boundary separates a spinpolarised (ferromagnetic) phase at low electron density and a paramagnetic phase at high electron density. Abrupt changes in the optical response signal an abrupt change in the magnetism. The magnetic order is thereby controlled via the voltage applied to the gate electrode of the device. Accompanying the change in magnetism is a large change in the electron effective mass.

Topological invariants to characterize universality of boundary charge in onedimensional insulators beyond symmetry constraints 
Mikhail Pletyukhov, Dante M. Kennes, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller Phys. Rev. B 101, 161106 (2020); arXiv:1911.06890.
In the absence of any symmetry constraints we address universal properties of the boundary charge QB for a wide class of tightbinding models with nondegenerate bands in one dimension. We provide a precise formulation of the bulkboundary correspondence by splitting QB via a gauge invariant decomposition in a Friedel, polarisation, and edge part. We reveal the topological nature of QB by proving the quantization of a topological index I=ΔQB−ρ, where ΔQB is the change of QB when shifting the lattice by one site towards a boundary and ρ is the average charge per site. For a single band we find this index to be given by the winding number of the fundamental phase difference of the Bloch wave function between two adjacent sites. For a given chemical potential we establish a central topological constraint I in {−1,0} related to charge conservation and particlehole duality. Our results are shown to be stable against disorder and we propose generalizations to multichannel and interacting systems.

Surface charge theorem and topological constraints for edge states: Analytical study of onedimensional nearestneighbor tightbinding models 
Mikhail Pletyukhov, Dante M. Kennes, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller Phys. Rev. B 101, 165304 (2020); arXiv:1911.06886.
For a wide class of noninteracting tightbinding models in one dimension, we present an analytical solution
for all scattering and edge states on a halfinfinite system. Without assuming any symmetry constraints, we
consider models with nearestneighbor hoppings and one orbital per site but the arbitrary size of the unit cell and
generic modulations of onsite potentials and hoppings. The solutions are parametrized by determinants that can
be straightforwardly calculated from recursion relations. We show that this representation allows for an elegant
analytic continuation to complex quasimomentum consistent with previous treatments for continuum models.
Two important analytical results are obtained based on the explicit knowledge of all eigenstates. (1) An explicit
proof of the surface charge theorem is presented including a unique relationship between the boundary charge
Q_B(a) of a single band a and the bulk polarization in terms of the ZakBerry phase. In particular, the ZakBerry
phase is determined within a special gauge of the Bloch states such that no unknown integer is left. This
establishes a precise form of a bulkboundary correspondence relating the boundary charge of a single band to
bulk properties. (2) We derive a topological constraint for the phase dependence of the edge state energies, where
the phase variable describes a continuous shift of the lattice towards the boundary. The topological constraint
is shown to be equivalent to the quantization of a topological index I = Q_B − ρ= −1, 0, introduced in an
accompanying paper [M. Pletyukhov et al., Phys. Rev. B 101, 161106 (2020)]. Here, Q_B is the change of the
boundary charge Q_B for a given chemical potential in the insulating regime when the lattice is shifted by one
site towards the boundary, and ρ is the average charge per site (both in units of the elementary charge e = 1).
This establishes an interesting link between universal properties of the boundary charge and edge state physics
discussed within the field of topological insulators. In accordance with previous results for continuum systems,
we also establish the localization of the boundary charge and determine the explicit form of the density given by
an exponential decay and a preexponential function following a power law with generic exponent −1/2 at large
distances

From Andreev to Majorana bound states in hybrid superconductorsemiconductor nanowires 
Elsa Prada, Pablo SanJose, Michiel W. A. de Moor, Attila Geresdi, Eduardo J. H. Lee, Jelena Klinovaja, Daniel Loss, Jesper Nygard, Ramon Aguado, and Leo P. Kouwenhoven Nat. Rev. Phys. 2, 575 (2020); arXiv:1911.04512.
Electronic excitations above the ground state must overcome an energy gap in superconductors with spatiallyhomogeneous pairing. In contrast, inhomogeneous superconductors such as those with magnetic impurities, weak links or heterojunctions containing normal metals can host subgap electronic excitations that are generically known as Andreev bound states (ABSs). With the advent of topological superconductivity, a new kind of ABS with exotic qualities, known as Majorana bound state (MBS), has been discovered. We review the main properties of all such subgap states and the stateoftheart techniques for their detection. We focus on hybrid superconductorsemiconductor nanowires, possibly coupled to quantum dots, as one of the most flexible and promising experimental platforms. We discuss how the combined effect of spinorbit coupling and Zeeman energy in these wires triggers the transition from ABSs into MBSs and show theoretical progress beyond minimal models in understanding experiments, including the possibility of a new type of robust Majorana zero mode without the need of a band topological transition. We examine the role of spatial nonlocality, a special property of MBS wavefunctions that, together with nonAbelian braiding, is the key ingredient for realizing topological quantum computing.

MagneticallyConfined Bound States in Rashba Systems 
Flavio Ronetti, Kirill Plekhanov, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 2, 022052(R) (2020); arXiv:1911.03133.
A Rashba nanowire is subjected to a magnetic field that assumes opposite signs in two sections of the nanowire,
and, thus, creates a magnetic domain wall. The direction of magnetic field is chosen to be perpendicular to the
Rashba spinorbit vector such that there is only a partial gap in the spectrum. Nevertheless, we prove analytically
and numerically that such a domain wall hosts a bound state whose energy is at bottom of the spectrum
below the energy of all bulk states. Thus, this magnetically confined bound state is wellisolated and can be
accessed experimentally. We further show that the same type of magnetic confinement can be implemented in
twodimensional systems with strong spinorbit interaction. A quantum channel along the magnetic domain wall
emerges with a nondegenerate dispersive band that lies energetically below the bulk states. We show that this
magnetic confinement is robust against disorder and various parameter variations.

Electronic transport in onedimensional Floquet topological insulators via topological and nontopological edge states 
Niclas Müller, Dante M. Kennes, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller Phys. Rev. B 101, 155417 (2020); arXiv:1911.02295.
Based on probing electronic transport properties, we propose an experimental test for the recently discovered rich topological phase diagram of onedimensional Floquet topological insulators with Rashba spinorbit interaction [Kennes et al., Phys. Rev. B 100, 041104(R) (2019)]. Using the KeldyshFloquet formalism, we compute electronic transport properties of these nanowires, where we propose to couple the leads in such a way, as to primarily address electronic states with a large relative weight at one edge of the system. By tuning the Fermi energy of the leads to the center of the topological gap, we are able to directly address the topological edge states, granting experimental access to the topological phase diagram. Surprisingly, we find conductance values similar or even larger in magnitude to those corresponding to topological edge states, when tuning the lead Fermi energy to special values in the bulk, which coincide with bifurcation points of the dispersion relation in complex quasimomentum space. These peaks reveal the presence of narrow bands of states whose wave functions are linear combinations of delocalized bulk states and exponentially localized edge states, where the amplitude of the edgestate component is sharply peaked at the aforementioned bifurcation point, resulting in an unusually large relative edgeweight. We discuss the transport properties of these \emph{nontopological edge states} and explain their emergence in terms of an intuitive yet quantitative physical picture. The mechanism giving rise to these states is not specific to the model we consider here, suggesting that they may be present in a wide class of systems.

Quantum Depinning of a Magnetic Skyrmion 
Christina Psaroudaki and Daniel Loss Phys. Rev. Lett. 124, 097202 (2020); Editor's suggestion; arXiv:1910.09585.
We investigate the quantum depinning of a weakly driven skyrmion out of an impurity potential in a mesoscopic magnetic insulator. For small barrier height, the Magnus force dynamics dominates over the inertial one, and the problem is reduced to a massless charged particle in a strong magnetic field. The universal form of the WKB exponent, the rate of tunneling, and the crossover temperature between thermal and quantum tunneling is provided, independently of the detailed form of the pinning potential. The results are discussed in terms of macroscopic parameters of the insulator Cu2OSeO3 and various skyrmion radii. We demonstrate that small enough magnetic skyrmions, with a radius of ~ 10 lattice sites, consisting of some thousands of spins, can behave as quantum objects at low temperatures in the mK regime.

TimeReversal Invariant Topological Superconductivity in Planar Josephson Bijunction 
Yanick Volpez, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 2, 023415 (2020); arXiv:1910.06921.
We consider a Josephson bijunction consisting of a thin SIS πJosephson junction sandwiched between twodimensional semiconducting layers with strong Rashba spinorbit interaction. Each of these layers forms an SNS junction due to proximityinduced superconductivity. The SIS junction is assumed to be thin enough such that the two Rashba layers are tunnelcoupled. We show that, by tuning external gates, this system can be controllably brought into a timereversal invariant topological superconducting phase with a Kramers pair of Majorana bound states being localized at the end of the normal region for a large parameter phase space. In particular, in the strong spinorbit interaction limit, the topological phase can be accessed already in the regime of small tunneling amplitudes.

Chiral Magnonic Edge States in Ferromagnetic Skyrmion Crystals Controlled by Magnetic Fields 
Sebastian A. Diaz, Tomoki Hirosawa, Jelena Klinovaja, and Daniel Loss Phys. Rev. Res. 2, 013231 (2020); arXiv:1910.05214.
Achieving control over magnon spin currents in insulating magnets  where dissipation due to Joule heating is highly suppressed  is an active area of research that could lead to energyefficient spintronics applications. However, magnon spin currents supported by conventional systems with uniform magnetic order have proven hard to control. An alternative approach that relies on topologically protected magnonic edge states of spatially periodic magnetic textures has recently emerged. A prime example of such textures is the ferromagnetic skyrmion crystal which hosts chiral edge states providing a platform for magnon spin currents. Here, we show, for the first time, an external magnetic field can drive a topological phase transition in the spin wave spectrum of a ferromagnetic skyrmion crystal. The topological phase transition is signaled by the closing of a lowenergy bulk magnon gap at a critical field. In the topological phase, below the critical field, two topologically protected chiral magnonic edge states lie within this gap, but they unravel in the trivial phase, above the critical field. Remarkably, the topological phase transition involves an inversion of two magnon bands that at the Γ point correspond to the breathing and anticlockwise modes of the skyrmions in the crystal. Our findings suggest that an external magnetic field could be used as a knob to switch on and off magnon spin currents carried by topologically protected chiral magnonic edge states.

Hinge Modes and Surface States in SecondOrder Topological ThreeDimensional Quantum Hall Systems induced by Charge Density 
Pawel Szumniak, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 102, 125126 (2020); arXiv:1910.05090.
We consider a system of weakly coupled onedimensional wires forming a threedimensional stack in the presence of a spatially periodic modulation of the chemical potential along the wires, equivalent to a charge density wave (CDW). An external static magnetic field is applied parallel to the wire axes. We show that, for a certain parameter regime, due to interplay between the CDW and magnetic field, the system can support a secondorder topological phase characterized by the presence of chiral quasi1D Quantum Hall Effect (QHE) hinge modes. Interestingly, we demonstrate that direction of propagation of the hinge modes depends on the phase of the CDW and can be reversed only by electrical means without the need of changing the orientation of the magnetic field. Furthermore, we show that the system can also support 2D chiral surface QHE states, which can coexist with onedimensional hinge modes, realizing a scenario of a hybrid highorder topology. We show that the hinge modes are robust against static disorder.

Interaction Driven Floquet Engineering of Topological Superconductivity in Rashba Nanowires 
Manisha Thakurathi, Pavel P. Aseev, Daniel Loss, and Jelena Klinovaja Phys. Rev. Res. 2, 013292 (2020); arXiv:1910.03730.
We analyze, analytically and numerically, a periodically driven Rashba nanowire proximity coupled to an swave superconductor using bosonization and renormalization group analysis in the regime of strong electronelectron interactions. Due to the repulsive interactions, the superconducting gap is suppressed, whereas the Floquet Zeeman gap is enhanced, resulting in a higher effective value of gfactor compared to the noninteracting case. The flow equations for different coupling constants, velocities, and Luttingerliquid parameters explicitly establish that even for small initial values of the Floquet Zeeman gap compared to the superconducting proximity gap, the interactions drive the system into the topological phase and the interband interaction term helps to achieve larger regions of the topological phase in parameter space.

Hinge states in a system of coupled Rashba layers 
Kirill Plekhanov, Flavio Ronetti, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 2, 013083 (2020); arXiv:1910.01655.
We consider a system of stacked tunnelcoupled twodimensional electron and holegas layers with Rashba spinorbit interactions subjected to a staggered Zeeman field. The interplay of different intralayer tunnel couplings results in a phase transition to a topological insulator phase in three dimensions hosting gapless surface states. The staggered Zeeman field further enriches the topological phase diagram by generating a secondorder topological insulator phase hosting gapless hinge states. The emergence of the topological phases is proven analytically in the regime of small Zeeman field and confirmed by numerical simulations in the nonperturbative region of the phase diagram. The topological phases are stable against external perturbations and disorder.

Lowsymmetry nanowire crosssections for enhanced Dresselhaus spinorbit interaction 
Miguel J. Carballido, Christoph Kloeffel, Dominik M. Zumbuehl, and Daniel Loss Phys. Rev. B 103, 195444 (2021); arXiv:1910.00562.
We study theoretically the spinorbit interaction of lowenergy electrons in semiconducting nanowires with a zincblende lattice. The effective Dresselhaus term is derived for various growth directions, including <11(2)>oriented nanowires. While a specific configuration exists where the Dresselhaus spinorbit coupling is suppressed even at confinement potentials of low symmetry, many configurations allow for a strong Dresselhaus coupling. In particular, we discuss qualitative and quantitative results for nanowire crosssections modeled after sectors of rings or circles. The parameter dependence is analyzed in detail, enabling predictions for a large variety of setups. For example, we gain insight into the spinorbit coupling in recently fabricated GaAsInAs nanomembranenanowire structures. By combining the effective Dresselhaus and Rashba terms, we find that such structures are promising platforms for applications where an electrically controllable spinorbit interaction is needed. If the nanowire crosssection is scaled down and InAs replaced by InSb, remarkably high Dresselhausbased spinorbit energies of the order of millielectronvolt are expected. A Rashba term that is similar to the effective Dresselhaus term can be induced via electric gates, providing means to switch the spinorbit interaction on and off. By varying the central angle of the circular sector, we find, among other things, that particularly strong Dresselhaus couplings are possible when nanowire crosssections resemble halfdisks.

Majorana Fermions in Magnetic Chains 
Rémy Pawlak, Silas Hoffman, Jelena Klinovaja, Daniel Loss, and Ernst Meyer Progress in Particle and Nuclear Physics 107, 119 (2019); arXiv:1909.10778.
Majorana fermions have recently garnered a great attention outside the field of particle physics, in condensed matter physics. In contrast to their particle physics counterparts, Majorana fermions are zero energy, chargeless, spinless, composite quasiparticles, residing at the boundaries of socalled topological superconductors. Furthermore, in opposition to any particles in the standard model, Majorana fermions in solidstate systems obey nonAbelian exchange statistics that make them attractive candidates for decoherencefree implementations of quantum computers. In this review, we report on the recent advances to realize synthetic topological superconductors supporting Majorana fermions with an emphasis on chains of magnetic impurities on the surface of superconductors. After outlining the theoretical underpinning responsible for the formation of Majorana fermions, we report on the subsequent experimental efforts to build topological superconductors and the resulting evidence in favor of Majorana fermions, focusing on scanning tunneling microscopy and the hunt for zerobias peaks in the measured current. We conclude by summarizing the open questions in the field and propose possible experimental measurements to answer them.

Coherent backaction between spins and an electronic bath: NonMarkovian dynamics and low temperature quantum thermodynamic electron cooling 
Stephanie Matern, Daniel Loss, Jelena Klinovaja, and Bernd Braunecker Phys. Rev. B 100, 134308 (2019); arXiv:1905.11422.
We provide a general analytical framework for calculating the dynamics of a spin system in contact with a bath beyond the Markov approximation. The approach is based on a systematic expansion of the NakashimaZwanzig master equation in the weakcoupling limit but makes no assumption on the time dynamics and includes all quantum coherent memory effects leading to nonMarkovian dynamics. Our results describe, for the free induction decay, the full time range from the nonMarkovian dynamics at short times, to the wellknown exponential thermal decay at long times. We provide full analytic results for the entire time range using a bath of itinerant electrons as an archetype for universal quantum fluctuations. Furthermore, we propose a quantum thermodynamic scheme to employ the temperature insensitivity of the nonMarkovian decay to transport heat out of the electron system and thus, by repeated reinitialisation of a cluster of spins, to efficiently cool the electrons at very low temperatures.

Floquet SecondOrder Topological Superconductor Driven via Ferromagnetic Resonance 
Kirill Plekhanov, Manisha Thakurathi, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 1, 032013 (2019); arXiv:1905.09241.
We consider a Floquet triplelayer setup composed of a twodimensional electron gas with spinorbit interactions, proximity coupled to an swave superconductor and to a ferromagnet driven at resonance. The ferromagnetic layer generates a timeoscillating Zeeman field which competes with the induced superconducting gap and leads to a topological phase transition. The resulting Floquet states support a secondorder topological superconducting phase with a pair of localized zeroenergy Floquet Majorana corner states. Moreover, the phase diagram comprises a Floquet helical topological superconductor, hosting a Kramers pair of Majorana edge modes protected by an effective timereversal symmetry, as well as a gapless Floquet Weyl phase. The topological phases are stable against disorder and parameter variations and are within experimental reach.

Fractional Topological Superconductivity and Parafermion Corner States 
Katharina Laubscher, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 1, 032017(R) (2019); arXiv:1905.00885.
We consider a system of weakly coupled Rashba nanowires in the strong spinorbit interaction (SOI) regime. The nanowires are arranged into two tunnelcoupled layers proximitized by a top and bottom superconductor such that the superconducting phase difference between them is pi. We show that in such a system strong electronelectron interactions can stabilize a helical topological superconducting phase hosting Kramers partners of Z_2m parafermion edge modes, where m is an odd integer determined by the position of the chemical potential. Furthermore, upon turning on a weak inplane magnetic field, the system is driven into a secondorder topological superconducting phase hosting zeroenergy Z_2m parafermion bound states localized at two opposite corners of a rectangular sample. As a special case, zeroenergy Majorana corner states emerge in the noninteracting limit m=1, where the chemical potential is tuned to the SOI energy of the single nanowires.

Quantum nondemolition measurement of an electron spin qubit 
Takashi Nakajima, Akito Noiri, Jun Yoneda, Matthieu R. Delbecq, Peter Stano, Tomohiro Otsuka, Kenta Takeda, Shinichi Amaha, Giles Allison, Kento Kawasaki, Arne Ludwig, Andreas D. Wieck, Daniel Loss, and Seigo Tarucha Nature Nanotechnology, 14, 555 (2019); arXiv:1904.11220.
Measurement of quantum systems inevitably involves disturbance in various forms. Within the limits imposed by quantum mechanics, however, one can design an "ideal" projective measurement that does not introduce a back action on the measured observable, known as a quantum nondemolition (QND) measurement. Here we demonstrate an allelectrical QND measurement of a single electron spin in a gatedefined quantum dot via an exchangecoupled ancilla qubit. The ancilla qubit, encoded in the singlettriplet twoelectron subspace, is entangled with the single spin and subsequently read out in a single shot projective measurement at a rate two orders of magnitude faster than the spin relaxation. The QND nature of the measurement protocol is evidenced by observing a monotonic increase of the readout fidelity over one hundred repetitive measurements against arbitrary input states. We extract information from the measurement record using the method of optimal inference, which is tolerant to the presence of the relaxation and dephasing. The QND measurement allows us to observe spontaneous spin flips (quantum jumps) in an isolated system with small disturbance. Combined with the highfidelity control of spin qubits, these results pave the way for various measurementbased quantum state manipulations including quantum error correction protocols.

Quantum Brownian Motion of a Magnetic Skyrmion 
Christina Psaroudaki, Pavel Aseev, and Daniel Loss Phys. Rev. B 100, 134404 (2019); arXiv:1904.09215.
Within a microscopic theory, we study the quantum Brownian motion of a skyrmion in a magnetic insulator coupled to a bath of magnonlike quantum excitations. The intrinsic skyrmionbath coupling gives rise to damping terms for the skyrmion centerofmass, which remain finite down to zero temperature due to the quantum nature of the magnon bath. We show that the quantum version of the fluctuationdissipation theorem acquires a nontrivial temperature dependence. As a consequence, the skyrmion mean square displacement is finite at zero temperature and has a fast thermal activation that scales quadratically with temperature, contrary to the linear increase predicted by the classical phenomenological theory. The effects of an external oscillating drive which couples directly on the magnon bath are investigated. We generalize the standard quantum theory of dissipation and we show explicitly that additional timedependent dissipation terms are generated by the external drive. From these we emphasize a friction and a topological charge renormalization term, which are absent in the static limit. The skyrmion response function inherits the time periodicity of the driving field and it is thus enhanced and lowered over a driving cycle. Finally, we provide a generalized version of the nonequilibrium fluctuationdissipation theorem valid for weakly driven baths.

Charge transport of a spinorbitcoupled TomonagaLuttinger liquid 
ChenHsuan Hsu, Peter Stano, Yosuke Sato, Sadashige Matsuo, Seigo Tarucha, and Daniel Loss Phys. Rev. B 100, 195423 (2019); arXiv:1904.06869.
The charge transport of a (Tomonaga) Luttinger liquid with tunnel barriers exhibits universal scaling: the currentvoltage curves measured at various temperatures collapse into a single curve upon rescaling. The exponent characterizing this single curve can be used to extract the strength of electronelectron interaction. Motivated by a recent experiment on InAs nanowires [Sato et al., Phys. Rev. B 99, 155304 (2019)], we theoretically investigate the analogous behavior of a spinorbitcoupled Luttinger liquid. We find that the scaling exponent differs for different impurity strengths, being weak (disorder potential) or strong (tunnel barriers), and their positions, either in the bulk or near the edge of the wire. For each case we quantify the exponent of the universal scaling and its modification due to the spinorbit coupling. Our findings serve as a guide in the determination of the interaction strength of quasionedimensional spinorbitcoupled quantum wires from transport measurements.

Degeneracy lifting of Majorana bound states due to electronphonon interactions 
Pavel P. Aseev, Pasquale Marra, Peter Stano, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 99, 205435 (2019); arXiv:1903.12066.
We study theoretically how electronphonon interaction affects the energies and level broadening (inverse lifetime) of Majorana bound states (MBSs) in a clean topological nanowire at low temperatures. At zero temperature, the energy splitting between the right and left MBSs remains exponentially small with increasing nanowire length L. At finite temperatures, however, the absorption of thermal phonons leads to the broadening of energy levels of the MBSs that does not decay with system length, and the coherent absorption/emission of phonons at opposite ends of the nanowire results in MBSs energy splitting that decays only as an inverse powerlaw in L. Both effects remain exponential in temperature. In the case of quantized transverse motion of phonons, the presence of Van Hove singularities in the phonon density of states causes additional resonant enhancement of both the energy splitting and the level broadening of the MBSs. This is the most favorable case to observe the phononinduced energy splitting of MBSs as it becomes much larger than the broadening even if the topological nanowire is much longer than the coherence length. We also calculate the charge and spin associated with the energy splitting of the MBSs induced by phonons. We consider both a spinless lowenergy continuum model, which we evaluate analytically, as well as a spinful lattice model for a Rashba nanowire, which we evaluate numerically.

Majorana Bound States in Double Nanowires with Reduced Zeeman Thresholds due to Supercurrents 
Olesia Dmytruk, Manisha Thakurathi, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 99, 245416 (2019); arXiv:1902.11232.
We study the topological phase diagram of a setup composed of two nanowires with strong Rashba spinorbit interaction subjected to an external magnetic field and brought into the proximity to a bulk swave superconductor in the presence of a supercurrent flowing through it. The supercurrent reduces the critical values of the Zeeman energy and crossed Andreev superconducting pairing required to reach the topological phase characterized by the presence of one Majorana bound state localized at each system end. We demonstrate that, even in the regime of the crossed Andreev pairing being smaller than the direct proximity pairing, a relatively weak magnetic field drives the system into the topological phase due to the presence of the supercurrent.

Entangling Spins in Double Quantum Dots and Majorana Bound States 
Marko J. Rancic, Silas Hoffman, Constantin Schrade, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 99, 165306 (2019); arXiv:1902.10251.
We study the coupling between a singlettriplet qubit realized in a double quantum dot to a topological qubit realized by spatially wellseparated Majorana bound states. We demonstrate that the singlettriplet qubit can be leveraged for readout of the topological qubit and for supplementing the gate operations that cannot be performed by braiding of Majorana bound states. Furthermore, we extend our setup to a network of singlettriplet and topological hybrid qubits that paves the way to scalable faulttolerant quantum computing.

Spontaneous Symmetry Breaking in Monolayers of Transition Metal Dichalcogenides 
Dmitry Miserev, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 100, 014428 (2019); arXiv:1902.07961.
We analyze magnetic phases of monolayers of transition metal dichalcogenides that are twovalley materials
with electronelectron interactions. The exchange intervalley scattering makes twovalley systems less stable
to the spin fluctuations but more stable to the valley fluctuations. We predict a firstorder ferromagnetic
phase transition governed by the nonanalytic and negative cubic term in the free energy that results in a
large spontaneous spin magnetization. Finite spinorbit interaction leads to the outofplane Ising order of the
ferromagnetic phase. Our theoretical prediction is consistent with the recent experiment on the electrondoped
monolayers of MoS2 reported by Roch et al. [Nat. Nanotechnol. 14, 432 (2019)]. The proposed firstorder phase
transition can also be tested by measuring the linear magnetic field dependence of the spin susceptibility in the
paramagnetic phase which is a direct consequence of the nonanalyticity of the free energy.

Topological Magnons and Edge States in Antiferromagnetic Skyrmion Crystals 
Sebastian A. Diaz, Jelena Klinovaja, and Daniel Loss Phys. Rev. Lett. 122, 187203 (2019); arXiv:1812.11125.
Antiferromagnetic skyrmion crystals are magnetic phases predicted to exist in antiferromagnets with DzyaloshinskiiMoriya interactions. Their spatially periodic noncollinear magnetic texture gives rise to topological bulk magnon bands characterized by nonzero Chern numbers. We find topologicallyprotected chiral magnonic edge states over a wide range of magnetic fields and DzyaloshinskiiMoriya interaction values. Moreover, and of particular importance for experimental realizations, edge states appear at the lowest possible energies, namely, within the first bulk magnon gap. Thus, antiferromagnetic skyrmion crystals show great promise as novel platforms for topological magnonics.

Chiral 1D Floquet topological insulators beyond rotating wave approximation 
Dante M. Kennes, Niclas Mueller, Mikhail Pletyukhov, Clara Weber, Christoph Bruder, Fabian Hassler, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller Phys. Rev. B 100, 041103(R) (2019); arXiv:1811.12062.
We study onedimensional (1D) Floquet topological insulators with chiral symmetry going beyond the standard rotating wave approximation. The occurrence of many anticrossings between Floquet replicas leads to a dramatic extension of phase diagram regions with stable topological edge states (TESs). We present an explicit construction of all TESs in terms of a truncated Floquet Hamiltonian in frequency space, prove the bulkboundary correspondence, and analyze the stability of the TESs in terms of their localization lengths. We propose experimental tests of our predictions in curved bilayer graphene.

Universal quantum computation in the surface code using nonAbelian islands 
Katharina Laubscher, Daniel Loss, and James R. Wootton Phys. Rev. A 100, 012338 (2019); arXiv:1811.06738.
The surface code is currently the primary proposed method for performing quantum error correction. However, despite its many advantages, it has no native method to faulttolerantly apply nonClifford gates. Additional techniques are therefore required to achieve universal quantum computation. Here we propose a new method, using small islands of a qudit variant of the surface code. This allows the nontrivial action of the nonAbelian anyons in the latter to process information stored in the former. Specifically, we show that a nonstabilizer state can be prepared, which allows universality to be achieved.

Second Order Topological Superconductivity in piJunction Rashba Layers 
Yanick Volpez, Daniel Loss, and Jelena Klinovaja Phys. Rev. Lett. 122, 126402 (2019); arXiv:1811.01827.
We consider a Josephson junction bilayer consisting of two tunnelcoupled twodimensional electron gas layers with Rashba spinorbit interaction, proximitized by a top and bottom swave superconductor with phase difference phi close to pi. We show that, in the presence of a finite weak inplane Zeeman field, the bilayer can be driven into a second order topological superconducting phase, hosting two Majorana corner states (MCSs). If phi=pi, in a rectangular geometry, these zeroenergy bound states are located at two opposite corners determined by the direction of the Zeeman field. If the phase difference phi deviates from pi by a critical value, one of the two MCSs gets relocated to an adjacent corner. As the phase difference phi increases further, the system becomes trivially gapped. The obtained MCSs are robust against static and magnetic disorder. We propose two setups that could realize such a model: one is based on controlling phi by magnetic flux, the other involves an additional layer of randomlyoriented magnetic impurities responsible for the phase shift of pi in the proximityinduced superconducting pairing.

Superfluid Transport in Quantum Spin Chains 
Silas Hoffman, Daniel Loss, and Yaroslav Tserkovnyak Phys. Rev. B 107, 085403 (2023); arXiv:1810.11470.
Spin superfluids enable longdistance spin transport through classical ferromagnets by developing topologically stable magnetic textures. For small spins at low dimensions, however, the topological protection suffers from strong quantum fluctuations. We study the remanence of spin superfluidity inherited from the classical magnet by considering the twoterminal spin transport through a finite spin1/2 magnetic chain with planar exchange. By fermionizing the system, we recast the spintransport problem in terms of quasiparticle transmission through a superconducting region. We show that the topological underpinnings of a semiclassical spin superfluid relate to the topological superconductivity in the fermionic representation. In particular, we find an efficient spin transmission through the magnetic region of a characteristic resonant length, which can be related to the properties of the boundary Majorana zero modes.

Zeroenergy Andreev bound states from quantum dots in proximitized Rashba nanowires 
Christopher Reeg, Olesia Dmytruk, Denis Chevallier, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 98, 245407 (2018); arXiv:1810.09840.
We study an analytical model of a Rashba nanowire that is partially covered by and coupled to a thin superconducting layer, where the uncovered region of the nanowire forms a quantum dot. We find that, even if there is no topological superconducting phase possible, there is a trivial Andreev bound state that becomes pinned exponentially close to zero energy as a function of magnetic field strength when the length of the quantum dot is tuned with respect to its spinorbit length such that a resonance condition of FabryPerot type is satisfied. In this case, we find that the Andreev bound state remains pinned near zero energy for Zeeman energies that exceed the characteristic spacing between Andreev bound state levels but that are smaller than the spinorbit energy of the quantum dot. Importantly, as the pinning of the Andreev bound state depends only on properties of the quantum dot, we conclude that this behavior is unrelated to topological superconductivity. To support our analytical model, we also perform a numerical simulation of a hybrid system while explicitly incorporating a thin superconducting layer, showing that all qualitative features of our analytical model are also present in the numerical results.

Strong ElectronElectron Interactions of a TomonagaLuttinger Liquid Observed in InAs Quantum Wires 
Yosuke Sato, Sadashige Matsuo, ChenHsuan Hsu, Peter Stano, Kento Ueda, Yuusuke Takeshige, Hiroshi Kamata, Joon Sue Lee, Borzoyeh Shojaei, Kaushini Wickramasinghe, Javad Shabani, Chris Palmstroem, Yasuhiro Tokura, Daniel Loss, and Seigo Tarucha Phys. Rev. B 99, 155304 (2019); arXiv:1810.06259.
We report strong electronelectron interactions in quantum wires etched from an InAs quantum well, a material known to have strong spinorbit interactions. We find that the current through the wires as a function of the bias voltage and temperature follows the universal scaling behavior of a TomonagaLuttinger liquid. Using a universal scaling formula, we extract the interaction parameter and find strong electronelectron interactions, increasing as the wires become more depleted. We establish theoretically that spinorbit interactions cause only minor modifications of the interaction parameter in this regime, indicating that genuinely strong electronelectron interactions are indeed achieved in the device. Our results suggest that etched InAs wires provide a platform with both strong electronelectron and strong spinorbit interactions.

From fractional boundary charges to quantized Hall conductance 
Manisha Thakurathi, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 98, 245404 (2018); arXiv:1809.00538.
We study the fractional boundary charges (FBCs) occurring in nanowires in the presence of periodically modulated chemical potentials and connect them to the FBCs occurring in a twodimensional electron gas in the presence of a perpendicular magnetic field in the integer quantum Hall effect (QHE) regime. First, we show that in nanowires the FBCs take fractional values and change linearly as a function of phase offset of the modulated chemical potential. This linear slope takes quantized values determined by the period of the modulation and depends only on the number of the filled bands. Next, we establish a mapping from the onedimensional system to the QHE setup, where we again focus on the properties of the FBCs. By considering a cylinder topology with an external flux similar to the Laughlin construction, we find that the slope of the FBCs as function of flux is linear and assumes universal quantized values, also in the presence of arbitrary disorder. We establish that the quantized slopes give rise to the quantization of the Hall conductance. Importantly, the approach via FBCs is valid for arbitrary flux values and disorder. The slope of the FBCs plays the role of a topological invariant for clean and disordered QHE systems. Our predictions for the FBCs can be tested experimentally in nanowires and in Corbino disk geometries in the integer QHE regime.

Tunable Magnonic Thermal Hall Effect in Skyrmion Crystal Phases of Ferrimagnets 
Se Kwon Kim, Kouki Nakata, Daniel Loss, and Yaroslav Tserkovnyak Phys. Rev. Lett. 122, 057204 (2019); arXiv:1808.06690.
We theoretically study the thermal Hall effect by magnons in skyrmion crystal phases of ferrimagnets in the vicinity of the angular momentum compensation point (CP). To this end, we start by deriving the equation of motion for magnons in the background of an arbitrary equilibrium spin texture, which gives rise to the fictitious electromagnetic field for magnons. As the net spin density varies, the resultant equation of motion interpolates between the relativistic KleinGordon equation at CP and the nonrelativistic Schrodingerlike equation away from it. In skyrmion crystal phases, the right and the leftcircularly polarized magnons with respect to the order parameter are shown to form the Landau levels separately within the uniform skyrmiondensity approximation. For an experimental proposal, we predict that the magnonic thermal Hall conductivity changes its sign when the ferrimagnet is tuned across CP, providing a way to control heat flux in spincaloritronic devices on the one hand and a feasible way to detect CP of ferrimagnets on the other hand.

Difference in charge and spin dynamics in a quantum dotlead coupled system 
Tomohiro Otsuka, Takashi Nakajima, Matthieu R. Delbecq, Peter Stano, Shinichi Amaha, Jun Yoneda, Kenta Takeda, Giles Allison, Sen Li, Akito Noiri, Takumi Ito, Daniel Loss, Arne Ludwig, Andreas D. Wieck, and Seigo Tarucha Phys. Rev. B 99, 085402 (2019); arXiv:1808.05303.
We analyze time evolution of charge and spin states in a quantum dot coupled to an electric reservoir. Utilizing highspeed singleelectron detection, we focus on dynamics induced by the firstorder tunneling. We find that there is a difference between the spin and the charge relaxation: The former appears slower than the latter. The difference depends on the Fermi occupation factor and the spin relaxation becomes slower when the energy level of the quantum dot is lowered. We explain this behavior by a theory including the firstorder tunneling processes and find a good agreement between the experiment and the theory.

Gatedefined quantum dot in a strong inplane magnetic field: spinorbit and gfactor effects 
Peter Stano, ChenHsuan Hsu, Marcel Serina, Leon C. Camenzind, Dominik M. Zumbuhl, and Daniel Loss Phys. Rev. B 98, 195314 (2018); arXiv:1808.03963.
We analyze orbital effects of an inplane magnetic field on the spin structure of states of a gated quantum dot based in a twodimensional electron gas. Starting with a k⋅p Hamiltonian, we perturbatively calculate these effects for the conduction band of GaAs, up to the third power of the magnetic field. We quantify several corrections to the gtensor and reveal their relative importance. We find that for typical parameters, the Rashba spinorbit term and the isotropic term, H43∝P2B⋅σ, give the largest contributions in magnitude. The inplane anisotropy of the gfactor is, on the other hand, dominated by the Dresselhaus spinorbit term. At zero magnetic field, the total correction to the gfactor is typically 510% of its bulk value. In strong inplane magnetic fields, the corrections are modified appreciably.

Benchmarks for approximate CNOTs based on a 17Qubit Surface Code 
Andreas Peter, Daniel Loss, and James R. Wootton arXiv:1808.03927
Scalable and faulttolerant quantum computation will require error correction. This will demand constant measurement of manyqubit observables, implemented using a vast number of CNOT gates. Indeed, practically all operations performed by a faulttolerant device will be these CNOTs, or equivalent twoqubit controlled operations. It is therefore important to devise benchmarks for these gates that explicitly quantify their effectiveness at this task. Here we develop such benchmarks, and demonstrate their use by applying them to a range of differently implemented controlled gates and a particular quantum error correcting code. Specifically, we consider spin qubits confined to quantum dots that are coupled either directly or via floating gates to implement the minimal 17qubit instance of the surface code. Our results show that small differences in the gate fidelity can lead to large differences in the performance of the surface code. This shows that gate fidelity is not, in general, a good predictor of code performance.

Lifetime of Majorana qubits in Rashba nanowires with nonuniform chemical potential 
Pavel P. Aseev, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 98, 155414 (2018); arXiv:1807.07997.
We study the lifetime of topological qubits based on Majorana bound states hosted in a onedimensional Rashba nanowire (NW) with proximityinduced superconductivity and nonuniform chemical potential needed for manipulation and readout. If nearby gates tune the chemical potential locally so that part of the NW is in the trivial phase, Andreev bound states (ABSs) can emerge which are localized at the interface between topological and trivial phases with energies significantly less than the gap. The emergence of such subgap states strongly decreases the Majorana qubit lifetime at finite temperatures due to local perturbations that can excite the system into these ABSs. Using Keldysh formalism, we study such excitations caused by fluctuating charges in capacitively coupled gates and calculate the corresponding Majorana lifetimes due to thermal noise, which are shown to be much shorter than those in NWs with uniform chemical potential.

Renormalization of quantum dot gfactor in superconducting Rashba nanowires 
Olesia Dmytruk, Denis Chevallier, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 98, 165403 (2018); arXiv:1806.06842.
We study analytically and numerically the renormalization of the gfactor in semiconducting Rashba nanowires (NWs), consisting of a normal and superconducting section. If the potential barrier between the sections is high, a quantum dot (QD) is formed in the normal section. For harmonic (hardwall) confinement, the effective gfactor of all QD levels is suppressed exponentially (powerlaw) in the product of the spinorbit interaction (SOI) wavevector and the QD length. If the barrier between the two sections is removed, the gfactor of the emerging Andreev bound states is suppressed less strongly. In the strong SOI regime and if the chemical potential is tuned to the SOI energy in both sections, the gfactor saturates to a universal constant. Remarkably, the effective gfactor shows a pronounced peak at the SOI energy as function of the chemical potentials. In addition, if the SOI is uniform, the gfactor renormalization as a function of the chemical potential is given by a universal dependence which is independent of the QD size. This prediction provides a powerful tool to determine experimentally whether the SOI in the whole NW is uniform and, moreover, gives direct access to the SOI strengths of the NW via gfactor measurements. In addition, it allows one to find the optimum position of the chemical potential for bringing the NW into the topological phase at large magnetic fields.

Majorana Kramers pairs in higherorder topological insulators 
ChenHsuan Hsu, Peter Stano, Jelena Klinovaja, and Daniel Loss Phys. Rev. Lett. 121, 196801 (2018); arXiv:1805.12146.
We propose a tunefree scheme to realize Kramers pairs of Majorana bound states in recently discovered higherorder topological insulators (HOTIs). We show that, by bringing two hinges of a HOTI into the proximity of an swave superconductor, the competition between local and crossedAndreev pairing leads to formation of Majorana Kramers pairs, when the latter pairing dominates over the former. We demonstrate that such a topological superconductivity is stabilized by moderate electronelectron interactions. The proposed setup avoids the application of a magnetic field or local voltage gates, and requires weaker interactions comparing to nonhelical nanowires.

Proximity effect in a twodimensional electron gas coupled to a thin superconducting layer 
Christopher Reeg, Daniel Loss, and Jelena Klinovaja Beilstein Journal of Nanotechnology 9, 1263 (2018); arXiv:1804.08337.
There have recently been several experiments studying induced superconductivity in semiconducting twodimensional electron gases that are strongly coupled to thin superconducting layers, as well as probing possible topological phases supporting Majorana bound states in such setups. We show that a large band shift is induced in the semiconductor by the superconductor in this geometry, thus making it challenging to realize a topological phase. Additionally, we show that while increasing the thickness of the superconducting layer reduces the magnitude of the band shift, it also leads to a more significant renormalization of the semiconducting material parameters and does not reduce the challenge of tuning into a topological phase.

A fast quantum interface between different spin qubit encodings 
A. Noiri, T. Nakajima, J. Yoneda, M. R. Delbecq, P. Stano, T. Otsuka, K. Takeda, S. Amaha, G. Allison, K. Kawasaki, A. Ludwig, A. D. Wieck, D. Loss, and S. Tarucha Nature Communications 9, 5066 (2018); arXiv:1804.04764.
Singlespin qubits in semiconductor quantum dots hold promise for universal quantum computation with demonstrations of a high singlequbit gate fidelity above 99.9% and twoqubit gates in conjunction with a long coherence time. However, initialization and readout of a qubit is orders of magnitude slower than control, which is detrimental for implementing measurementbased protocols such as errorcorrecting codes. In contrast, a singlettriplet qubit, encoded in a twospin subspace, has the virtue of fast readout with high fidelity. Here, we present a hybrid system which benefits from the different advantages of these two distinct spinqubit implementations. A quantum interface between the two codes is realized by electrically tunable interqubit exchange coupling. We demonstrate a controlledphase gate that acts within 5.5 ns, much faster than the measured dephasing time of 211 ns. The presented hybrid architecture will be useful to settle remaining key problems with building scalable spinbased quantum computers.

Spectroscopy of QuantumDot Orbitals with InPlane Magnetic Fields 
Leon C. Camenzind, Liuqi Yu, Peter Stano, Jeramy Zimmerman, Arthur C. Gossard, Daniel Loss, and Dominik M. Zumbuhl Phys. Rev. Lett. 122, 207701 (2019); Viewpoint; arXiv:1804.00162.
We show that inplane magneticfieldassisted spectroscopy allows extraction of the inplane orientation
and full 3D size parameters of the quantum mechanical orbitals of a single electron GaAs lateral quantum
dot with subnanometer precision. The method is based on measuring the orbital energies in a magnetic field
with various strengths and orientations in the plane of the 2D electron gas. From such data, we deduce the
microscopic confinement potential landscape and quantify the degree by which it differs from a harmonic
oscillator potential. The spectroscopy is used to validate shape manipulation with gate voltages, agreeing
with expectations from the gate layout. Our measurements demonstrate a versatile tool for quantum dots
with one dominant axis of strong confinement.

Orbital effects of a strong inplane magnetic field on a gatedefined quantum dot 
Peter Stano, ChenHsuan Hsu, Leon Camenzind, Liuqi Yu, Dominik Zumbuhl, and Daniel Loss Phys. Rev. B 99, 085308 (2019); arXiv:1804.00128.
We theoretically investigate the orbital effects of an inplane magnetic field on the spectrum of a quantum dot embedded in a twodimensional electron gas (2DEG). We derive an effective twodimensional Hamiltonian where these effects enter in proportion to the flux penetrating the 2DEG. We quantify the latter in detail for harmonic, triangular, and square potential of the heterostructure. We show how the orbital effects allow one to extract a wealth of information, for example, on the heterostructure interface, the quantum dot size and orientation, and the spinorbit fields. We illustrate the formalism by extracting this information from recent measured data [L.~C.~Camenzind, et al., Nat. Commun. 9, 3454 (2018)].

Conductance of fractional Luttinger liquids at finite temperatures 
Pavel P. Aseev, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 98, 045416 (2018); arXiv:1803.07359.
We study the electrical conductance in singlemode quantum wires with Rashba spinorbit interaction subjected to externally applied magnetic fields in the regime in which the ratio of spinorbit momentum to the Fermi momentum is close to an odd integer, so that a combined effect of multielectron interaction and applied magnetic field leads to a partial gap in the spectrum. We study how this partial gap manifests itself in the temperature dependence of the fractional conductance of the quantum wire. We use two complementing techniques based on bosonization: refermionization of the model at a particular value of the interaction parameter and a semiclassical approach within a dilute soliton gas approximation of the functional integral. We show how the lowtemperature fractional conductance can be affected by the finite length of the wire, by the properties of the contacts, and by a shift of the chemical potential, which takes the system away from the resonance condition. We also predict an internal resistivity caused by a dissipative coupling between gapped and gapless modes.

Skyrmions Driven by Intrinsic Magnons 
Christina Psaroudaki and Daniel Loss Phys. Rev. Lett. 120, 237203 (2018); arXiv:1803.04001.
We study the dynamics of a skyrmion in a magnetic insulating nanowire in the presence of timedependent oscillating magnetic field gradients. These ac fields act as a net driving force on the skyrmion via its own intrinsic magnetic excitations. In a microscopic quantum field theory approach we include the unavoidable coupling of the external field to the magnons, which gives rise to timedependent dissipation for the skyrmion. We demonstrate that the magnetic ac field induces a superOhmic to Ohmic crossover behavior for the skyrmion dissipation kernels with timedependent Ohmic terms. The ac driving of the magnon bath at resonance results in a unidirectional helical propagation of the skyrmion in addition to the otherwise periodic bounded motion.

Rashba Sandwiches with Topological Superconducting Phases 
Yanick Volpez, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 97, 195421 (2018); arXiv:1803.00987.
We introduce a versatile heterostructure harboring various topological superconducting phases characterized by the presence of helical, chiral, or unidirectional edge states. Changing parameters, such as an effective Zeeman field or chemical potential, one can tune between these three topological phases in the same setup. Our model relies only on conventional nontopological ingredients. The bilayer setup consists of an swave superconductor sandwiched between two twodimensional electron gas layers with strong Rashba spinorbit interaction. The interplay between two different pairing mechanisms, proximity induced direct and crossed Andreev superconducting pairings, gives rise to multiple topological phases. In particular, helical edge states occur if crossed Andreev superconducting pairing is dominant. In addition, an inplane Zeeman field leads to a 2D gapless topological phase with unidirectional edge states, which were previously predicted to exist only in noncentrosymmetric superconductors. If the Zeeman field is tilted out of the plane, the system is in a topological phase hosting chiral edge states.

Boundary spin polarization as robust signature of topological phase transition in Majorana nanowires 
Marcel Serina, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 98, 035419 (2018); arXiv:1803.00544.
We show that the boundary charge and spin can be used as alternative signatures of the topological phase transition in topological models such as semiconducting nanowires with strong Rashba spinorbit interaction in the presence of a magnetic field and in proximity to an swave superconductor. We identify signatures of the topological phase transition that do not rely on the presence of Majorana zeroenergy modes and, thus, can serve as independent probes of topological properties. The boundary spin component along the magnetic field, obtained by summing contributions from all states below the Fermi level, has a pronounced peak at the topological phase transition point. Generally, such signatures can be observed at boundaries between topological and trivial sections in nanowires and are stable against disorder.

Metallization of Rashba wire by superconducting layer in the strongproximity regime 
Christopher Reeg, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 97, 165425 (2018); arXiv:1801.06509.
Semiconducting quantum wires defined within twodimensional electron gases and strongly coupled to thin superconducting layers have been extensively explored in recent experiments as promising platforms to host Majorana bound states. We study numerically such a geometry, consisting of a quasionedimensional wire coupled to a disordered threedimensional superconducting layer. We find that, in the strongcoupling limit of a sizable proximityinduced superconducting gap, all transverse subbands of the wire are significantly shifted in energy relative to the chemical potential of the wire. For the lowest subband, this band shift is comparable in magnitude to the spacing between quantized levels that arise due to the finite thickness of the superconductor (which typically is ca. 500 meV for a 10nmthick layer of Aluminum); in higher subbands, the band shift is much larger. Additionally, we show that the width of the system, which is usually much larger than the thickness, and moderate disorder within the superconductor have almost no impact on the induced gap or band shift. We provide a detailed discussion of the ramifications of our results, arguing that a huge band shift and significant renormalization of semiconducting material parameters in the strongcoupling limit make it challenging to realize a topological phase in such a setup, as the strong coupling to the superconductor essentially metallizes the semiconductor. This metallization of the semiconductor can be tested experimentally through the measurement of the band shift.

Effects of nuclear spins on the transport properties of the edge of twodimensional topological insulators 
ChenHsuan Hsu, Peter Stano, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 97, 125432 (2018); arXiv:1712.09040.
The electrons in the edge channels of twodimensional topological insulators can be described as a helical TomonagaLuttinger liquid. They couple to nuclear spins embedded in the host materials through the hyperfine interaction, and are therefore subject to elastic spinflip backscattering on the nuclear spins. We investigate the nuclearspininduced edge resistance due to such backscattering by performing a renormalizationgroup analysis. Remarkably, the effect of this backscattering mechanism is stronger in a helical edge than in nonhelical channels, which are believed to be present in the trivial regime of InAs/GaSb quantum wells. In a system with sufficiently long edges, the disordered nuclear spins lead to an edge resistance which grows exponentially upon lowering the temperature. On the other hand, electrons from the edge states mediate an anisotropic RudermanKittelKasuyaYosida nuclear spinspin interaction, which induces a spiral nuclear spin order below the transition temperature. We discuss the features of the spiral order, as well as its experimental signatures. In the ordered phase, we identify two backscattering mechanisms, due to charge impurities and magnons. The backscattering on charge impurities is allowed by the internally generated magnetic field, and leads to an Andersontype localization of the edge states. The magnonmediated backscattering results in a powerlaw resistance, which is suppressed at zero temperature. Overall, we find that in a sufficiently long edge the nuclear spins, whether ordered or not, suppress the edge conductance to zero as the temperature approaches zero.

Direct Rashba spinorbit interaction in Si and Ge nanowires with different growth directions 
Christoph Kloeffel, Marko J. Rancic, and Daniel Loss Phys. Rev. B 97, 235422 (2018); Editor's suggestion; arXiv:1712.03476.
We study theoretically the lowenergy hole states in Si, Ge, and Ge/Si core/shell nanowires (NWs). The NW core in our model has a rectangular cross section, the results for a square cross section are presented in detail. In the case of Ge and Ge/Si core/shell NWs, we obtain very good agreement with previous theoretical results for cylindrically symmetric NWs. In particular, the NWs allow for an unusually strong and electrically controllable spinorbit interaction (SOI) of Rashba type. We find that the dominant contribution to the SOI is the "direct Rashba spinorbit interaction" (DRSOI), which is an important mechanism for systems with heavyholelighthole mixing. Our results for Si NWs depend significantly on the orientation of the crystallographic axes. The numerically observed dependence on the growth direction is consistent with analytical results from a simple model, and we identify a setup where the DRSOI enables spinorbit energies of the order of millielectronvolts in Si NWs. Furthermore, we analyze the dependence of the SOI on the electric field and the cross section of the Ge or Si core. A helical gap in the spectrum can be opened with a magnetic field. We obtain the largest g factors with magnetic fields applied perpendicularly to the NWs.

Majorana Kramers pairs in Rashba double nanowires with interactions and disorder 
Manisha Thakurathi, Pascal Simon, Ipsita Mandal, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 97, 045415 (2018); arXiv:1711.04682.
We analyze the effects of electronelectron interactions and disorder on a Rashba doublenanowire setup coupled to an swave superconductor, which has been recently proposed as a versatile platform to generate Kramers pairs of Majorana bound states in the absence of magnetic fields. We identify the regime of parameters for which these Kramers pairs are stable against interaction and disorder effects. We use bosonization, perturbative renormalization group, and replica techniques to derive the flow equations for various parameters of the model and evaluate the corresponding phase diagram with topological and disorderdominated phases. We confirm aforementioned results by considering a more microscopic approach which starts from the tunneling Hamiltonian between the threedimensional swave superconductor and the nanowires. We find again that the interaction drives the system into the topological phase and, as the strength of the source term coming from the tunneling Hamiltonian increases, strong electronelectron interactions are required to reach the topological phase.

Hyperfinephonon spin relaxation in a singleelectron GaAs quantum dot 
Leon C. Camenzind, Liuqi Yu, Peter Stano, Jeramy Zimmerman, Arthur C. Gossard, Daniel Loss, and Dominik M. Zumbuhl Nature Communications 9, 3454 (2018); arXiv:1711.01474.
Understanding and control of the spin relaxation time T1 is among the key challenges for spin based qubits. A larger T1 is generally favored, setting the fundamental upper limit to the qubit coherence and spin readout fidelity. In GaAs quantum dots at low temperatures and high inplane magnetic fields B, the spin relaxation relies on phonon emission and spinorbit coupling. The characteristic dependence T1 ~1/B^5and pronounced Bfield anisotropy were already confirmed experimentally. However, it has also been predicted 15 years ago that at low enough fields, the spinorbit interaction is replaced by the coupling to the nuclear spins, where the relaxation becomes isotropic, and the scaling changes to T1~1/B^3. We establish these predictions experimentally, by measuring T1 over an unprecedented range of magnetic fields  made possible by lower temperature  and report a maximum T1=57 s at the lowest fields, setting a new record for the electron spin lifetime in a nanostructure.

A repetition code of 15 qubits 
James R. Wootton and Daniel Loss Phys. Rev. A 97, 052313 (2018); arXiv:1709.00990.
The repetition code is an important primitive for the techniques of quantum error correction. Here we implement repetition codes of at most 15 qubits on the 16 qubit \emph{ibmqx3} device. Each experiment is run for a single round of syndrome measurements, achieved using the standard quantum technique of using ancilla qubits and controlled operations. The size of the final syndrome is small enough to allow for lookup table decoding using experimentally obtained data. The results show strong evidence that the logical error rate decays exponentially with code distance, as is expected and required for the development of faulttolerant quantum computers. The results also give insight into the nature of noise in the device.

Topological Phase Detection in Rashba Nanowires with a Quantum Dot 
Denis Chevallier, Pawel Szumniak, Silas Hoffman, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 97, 045404 (2018); arXiv:1710.05576.
We study theoretically the detection of the topological phase transition occurring in Rashba nanowires with proximityinduced superconductivity using a quantum dot. The bulk states lowest in energy of such a nanowire have a spin polarization parallel or antiparallel to the applied magnetic field in the topological or trivial phase, respectively. We show that this property can be probed by the quantum dot created at the end of the nanowire by external gates. By tuning one of the two spinsplit levels of the quantum dot to be in resonance with nanowire bulk states, one can detect the spin polarization of the lowest band via transport measurement. This allows one to determine the topological phase of the Rashba nanowire independently of the presence of Majorana bound states.

DIII Topological Superconductivity with Emergent TimeReversal Symmetry 
Christopher Reeg, Constantin Schrade, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 96, 161407(R) (2017); arXiv:1708.06755.
We find a new class of topological superconductors which possess an emergent timereversal symmetry that is present only after projecting to an effective lowdimensional model. We show that a topological phase in symmetry class DIII can be realized in a noninteracting system coupled to an swave superconductor only if the physical timereversal symmetry of the system is broken, and we provide three general criteria that must be satisfied in order to have such a phase. We also provide an explicit model which realizes the class DIII topological superconductor in 1D. We show that, just as in timereversal invariant topological superconductors, the topological phase is characterized by a Kramers pair of Majorana fermions that are protected by the emergent timereversal symmetry.

Finitesize effects in a nanowire strongly coupled to a thin superconducting shell 
Christopher Reeg, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 96, 125426 (2017); arXiv:1707.08417.
We study the proximity effect in a onedimensional nanowire strongly coupled to a finite superconductor with a characteristic size which is much shorter than its coherence length. Such geometries have become increasingly relevant in recent years in the experimental search for Majorana fermions with the development of thin epitaxial Al shells which form a very strong contact with either InAs or InSb nanowires. So far, however, no theoretical treatment of the proximity effect in these systems has accounted for the finite size of the superconducting film. We show that the finitesize effects become very detrimental when the level spacing of the superconductor greatly exceeds its energy gap. Without any finetuning of the size of the superconductor (on the scale of the Fermi wavelength), the tunneling energy scale must be larger than the level spacing in order to reach the hard gap regime which is seen ubiquitously in the experiments. However, in this regime, the large tunneling energy scale induces a large shift in the effective chemical potential of the nanowire and pushes the topological phase transition to magnetic field strengths which exceed the critical field of Al.

Magnonic topological insulators in antiferromagnets 
Kouki Nakata, Se Kwon Kim (UCLA), Jelena Klinovaja, and Daniel Loss Phys. Rev. B 96, 224414 (2017); arXiv:1707.07427.
Extending the notion of symmetry protected topological phases to insulating antiferromagnets (AFs) described in terms of opposite magnetic dipole moments associated with the magnetic Né el order, we establish a bosonic counterpart of topological insulators in semiconductors. Making use of the AharonovCasher effect, induced by electric field gradients, we propose a magnonic analog of the quantum spin Hall effect (magnonic QSHE) for edge states that carry helical magnons. We show that such up and down magnons form the same Landau levels and perform cyclotron motion with the same frequency but propagate in opposite direction. The insulating AF becomes characterized by a topological ℤ2 number consisting of the Chern integer associated with each helical magnon edge state. Focusing on the topological Hall phase for magnons, we study bulk magnon effects such as magnonic spin, thermal, Nernst, and Ettinghausen effects, as well as the thermomagnetic properties of helical magnon transport both in topologically trivial and nontrivial bulk AFs and establish the magnonic WiedemannFranz law. We show that our predictions are within experimental reach with current device and measurement techniques.

ThreeDimensional Fractional Topological Insulators in Coupled Rashba Layers 
Yanick Volpez, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 96, 085422 (2017); arXiv:1706.09863.
We propose a model of threedimensional topological insulators consisting of weakly coupled electron and holegas layers with Rashba spinorbit interaction stacked along a given axis. We show that in the presence of strong electronelectron interactions the system realizes a fractional strong topological insulator, where the rotational symmetry and condensation energy arguments still allow us to treat the problem as quasionedimensional with bosonization techniques. We also show that if Rashba and Dresselhaus spinorbit interaction terms are equally strong, by doping the system with magnetic impurities, one can bring it into the Weyl semimetal phase.

Lowfield Topological Threshold in Majorana Double Nanowires 
Constantin Schrade, Manisha Thakurathi, Christopher Reeg, Silas Hoffman, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 96, 035306 (2017); arXiv:1705.09364.
A hard proximityinduced superconducting gap has recently been observed in semiconductor nanowire systems at low magnetic fields. However, in the topological regime at high magnetic fields a soft gap reemerges and represents a fundamental obstacle to topologically protected quantum information processing with Majorana bound states. Here we show that this obstacle can be overcome in a setup of double Rashba nanowires which are coupled to an $s$wave superconductor and subjected to an external magnetic field along the wires. Specifically, we demonstrate that the required field strength for the topological threshold can be significantly reduced by the destructive interference of direct and crossedAndreev pairing in this setup; precisely down to the regime in which current experimental technology allows for a hard superconducting gap. We also show that the resulting Majorana bound states exhibit sufficiently short localization lengths which makes them ideal candidates for future braiding experiments.

Spindependent coupling between quantum dots and topological quantum wires 
Silas Hoffman, Denis Chevallier, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 96, 045440 (2017); arXiv:1705.03002.
Considering Rashba quantum wires with a proximityinduced superconducting gap as physical realizations of Majorana fermions and quantum dots, we calculate the overlap of the Majorana wave functions with the local wave functions on the dot. We determine the spindependent tunneling amplitudes between these two localized states and show that we can tune into a fully spin polarized tunneling regime by changing the distance between dot and Majorana fermion. Upon directly applying this to the tunneling model Hamiltonian, we calculate the effective magnetic field on the quantum dot flanked by two Majorana fermions. The direction of the induced magnetic field on the dot depends on the occupation of the nonlocal fermion formed from the two Majorana end states which can be used as a readout for such a Majorana qubit.

Nuclear spininduced localization of the edge states in twodimensional topological insulators 
ChenHsuan Hsu (RIKEN), Peter Stano (RIKEN), Jelena Klinovaja, and Daniel Loss Phys. Rev. B 96, 081405(R) (2017); arXiv:1703.03421.
We investigate the influence of nuclear spins on the resistance of helical edge states of twodimensional topological insulators (2DTIs). Via the hyperfine interaction, nuclear spins allow electron backscattering, otherwise forbidden by timereversal symmetry. We identify two backscattering mechanisms, depending on whether the nuclear spins are ordered or not. Their temperature dependence is distinct but both give resistance, which increases with the edge length, decreasing temperature, and increasing strength of the electronelectron interaction. Overall, we find that the nuclear spins will typically shut down the conductance of the 2DTI edges at zero temperature.

Spin and Charge Signatures of Topological Superconductivity in Rashba Nanowires 
Pawel Szumniak, Denis Chevallier, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 96, 041401(R) (2017); arXiv:1703.00265.
We consider a Rashba nanowire with proximity gap which can be brought into the topological phase by tuning external magnetic field or chemical potential. We study spin and charge of the bulk quasiparticle states when passing through the topological transition for open and closed systems. We show, analytically and numerically, that the spin of bulk states around the topological gap reverses its sign when crossing the transition due to band inversion, independent of the presence of Majorana fermions in the system. This spin reversal can be considered as a bulk signature of topological superconductivity that can be accessed experimentally. We find a similar behaviour for the charge of the bulk quasiparticle states, also exhibiting a sign reversal at the transition. We show that these signatures are robust against random static disorder.

Destructive interference of direct and crossed Andreev pairing in a system of two nanowires coupled via an swave superconductor 
Christopher R. Reeg, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 96, 081301(R) (2017); arXiv:1701.07107.
We consider a system of two onedimensional nanowires coupled via an swave superconducting strip, a geometry that is capable of supporting Kramers pairs of Majorana fermions. By performing an exact analytical diagonalization of a tunneling Hamiltonian describing the proximity effect (via a Bogoliubov transformation), we show that the excitation gap of the system varies periodically on the scale of the Fermi wavelength in the limit where the interwire separation is shorter than the superconducting coherence length. Comparing with the excitation gaps in similar geometries containing only direct pairing, where one wire is decoupled from the superconductor, or only crossed Andreev pairing, where each nanowire is considered as a spinpolarized edge of a quantum Hall state, we find that the gap is always reduced, by orders of magnitude in certain cases, when both types of pairing are present. Our analytical results are further supported by numerical calculations on a tightbinding lattice. Finally, we show that treating the proximity effect by integrating out the superconductor cannot reproduce the results of our exact diagonalization.

Robust SingleShot Spin Measurement with 99.5% Fidelity in a Quantum Dot Array 
Takashi Nakajima, Matthieu R. Delbecq, Tomohiro Otsuka, Peter Stano, Shinichi Amaha, Jun Yoneda, Akito Noiri, Kento Kawasaki, Kenta Takeda, Giles Allison, Arne Ludwig, Andreas D. Wieck, Daniel Loss, Seigo Tarucha Phys. Rev. Lett. 119, 017701 (2017); arXiv:1701.03622.
We demonstrate a new method for projective singleshot measurement of two electron spin states (singlet versus triplet) in an array of gatedefined lateral quantum dots in GaAs. The measurement has very high fidelity and is robust with respect to electric and magnetic fluctuations in the environment. It exploits a longlived metastable charge state, which increases both the contrast and the duration of the charge signal distinguishing the two measurement outcomes. This method allows us to evaluate the charge measurement error and the spintocharge conversion error separately. We specify conditions under which this method can be used, and project its general applicability to scalable quantum dot arrays in GaAs or silicon.

Superconducting gridbus surface code architecture for holespin qubits 
Simon E. Nigg, Andreas Fuhrer (IBM Zurich), and Daniel Loss Phys. Rev. Lett. 118, 147701 (2017); arXiv:1612.07292.
We present a scalable hybrid architecture for the 2D surface code combining superconducting resonators and holespin qubits in nanowires with tunable direct Rashba spinorbit coupling. The backbone of this architecture is a square lattice of capacitively coupled coplanar waveguide resonators each of which hosts a nanowire holespin qubit. Both the frequency of the qubits and their coupling to the microwave field are tunable by a static electric field applied via the resonator center pin. In the dispersive regime, an entangling twoqubit gate can be realized via a third order process, whereby a virtual photon in one resonator is created by a first qubit, coherently transferred to a neighboring resonator, and absorbed by a second qubit in that resonator. Numerical simulations with stateoftheart coherence times yield gate fidelities approaching the 99% fault tolerance threshold.

Quantum dynamics of skyrmions in chiral magnets 
Christina Psaroudaki, Silas Hoffman, Jelena Klinovaja, and Daniel Loss Phys. Rev. X 7, 041045 (2017); arXiv:1612.01885.
We study the quantum propagation of a Skyrmion in chiral magnetic insulators by generalizing the
micromagnetic equations of motion to a finitetemperature path integral formalism, using field theoretic
tools. Promoting the center of the Skyrmion to a dynamic quantity, the fluctuations around the Skyrmionic
configuration give rise to a timedependent damping of the Skyrmion motion. From the frequency
dependence of the damping kernel, we are able to identify the Skyrmion mass, thus providing a
microscopic description of the kinematic properties of Skyrmions. When defects are present or a magnetic
trap is applied, the Skyrmion mass acquires a finite value proportional to the effective spin, even at
vanishingly small temperature. We demonstrate that a Skyrmion in a confined geometry provided by a
magnetic trap behaves as a massive particle owing to its quasionedimensional confinement. An additional
quantum mass term is predicted, independent of the effective spin, with an explicit temperature dependence
which remains finite even at zero temperature.

Finitetemperature conductance of strongly interacting quantum wire with a nuclear spin order 
Pavel Aseev, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 95, 125440 (2017); arXiv:1611.10238.
We study the temperature dependence of the electrical conductance of a clean strongly interacting quantum wire in the presence of a helical nuclear spin order. The nuclear spin helix opens a temperaturedependent partial gap in the electron spectrum. Using a bosonization framework we describe the gapped electron modes by sineGordonlike kinks. We predict an internal resistivity caused by an Ohmiclike friction these kinks experience via interacting with gapless excitations. As a result, the conductance rises from G=e2/h at temperatures below the critical temperature when nuclear spins are fully polarized to G=2e2/h at higher temperatures when the order is destroyed, featuring a relatively wide plateau in the intermediate regime. The theoretical results are compared with the experimental data for GaAs quantum wires obtained recently by Scheller et al. [Phys. Rev. Lett. 112, 066801 (2014)].

Magnonic quantum Hall effect and WiedemannFranz law 
Kouki Nakata, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 95, 125429 (2017); arXiv:1611.09752.
We present a quantum Hall effect of magnons in twodimensional clean insulating magnets at finite temperature. Through the AharonovCasher effect, a magnon moving in an electric field acquires a geometric phase and forms Landau levels in an electric field gradient of sawtooth form. At low temperatures, the lowest energy band being almost flat carries a Chern number associated with a Berry curvature. Appropriately defining the thermal conductance for bosons, we find that the magnon Hall conductances get quantized and show a universal thermomagnetic behavior, i.e., are independent of materials, and obey a WiedemannFranz law for magnon transport. We consider magnons with quadratic and linear (Diraclike) dispersions. Finally, we show that our predictions are within experimental reach for ferromagnets and skyrmion lattices with current device and measurement techniques.

Spin Currents and Magnon Dynamics in Insulating Magnets 
Kouki Nakata, Pascal Simon (Orsay), and Daniel Loss J. Phys. D: Appl. Phys. 50 114004 (2017); arXiv:1610.08901.
NambuGoldstone theorem provides gapless modes to both relativistic and nonrelativistic systems. The NambuGoldstone bosons in insulating magnets are called magnons or spinwaves and play a key role in magnetization transport. We review here our past works on magnetization transport in insulating magnets and also add new insights, with a particular focus on magnon transport. We summarize in detail the magnon counterparts of electron transport, such as the WiedemannFranz law, the Onsager reciprocal relation between the Seebeck and Peltier coefficients, the Hall effects, the superconducting state, the Josephson effects, and the persistent quantized current in a ring to list a few. Focusing on the electromagnetism of moving magnons, i.e., magnetic dipoles, we theoretically propose a way to directly measure magnon currents. As a consequence of the MerminWagnerHohenberg theorem, spin transport is drastically altered in onedimensional antiferromagnetic (AF) spin1/2 chains; where the N\'eel order is destroyed by quantum fluctuations and a quasiparticle magnonlike picture breaks down. Instead, the the lowenergy collective excitations of the AF spin chain are described by a TomonagaLuttinger liquid (TLL) which provides to the spin transport properties in such antiferromagnets some universal features at low enough temperature. Finally, we enumerate open issues and provide a platform to discuss the future directions of magnonics.

Floquet Majorana and ParaFermions in Driven Rashba Nanowires 
Manisha Thakurathi, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 95, 155407 (2017); arXiv:1608.08143.
We study a periodically driven nanowire with Rashbalike conduction and valence bands in the presence of a magnetic field. We identify topological regimes in which the system hosts zeroenergy Majorana fermions. We further investigate the effect of strong electronelectron interactions that give rise to parafermion zero energy modes hosted at the nanowire ends. The first setup we consider allows for topological phases by applying only static magnetic fields without the need of superconductivity. The second setup involves both superconductivity and timedependent magnetic fields and allows one to generate topological phases without finetuning of the chemical potential. Promising candidate materials are graphene nanoribbons due to their intrinsic particlehole symmetry.

Higherorder spin and charge dynamics in a quantum dotlead hybrid system 
Tomohiro Otsuka, Takashi Nakajima, Matthieu R. Delbecq, Shinichi Amaha, Jun Yoneda, Kenta Takeda, Giles Allison, Peter Stano, Akito Noiri, Takumi Ito, Daniel Loss, Arne Ludwig, Andreas D. Wieck, and Seigo Tarucha Scientific Reports 7, 12201 (2017); arXiv:1608.07646.
Understanding the dynamics of open quantum systems is important and challenging in basic physics and
applications for quantum devices and quantum computing. Semiconductor quantum dots offer a good
platform to explore the physics of open quantum systems because we can tune parameters including
the coupling to the environment or leads. Here, we apply the fast singleshot measurement techniques
from spin qubit experiments to explore the spin and charge dynamics due to tunnel coupling to a lead
in a quantum dotlead hybrid system. We experimentally observe both spin and charge time evolution
via first and secondorder tunneling processes, and reveal the dynamics of the spinflip through the
intermediate state. These results enable and stimulate the exploration of spin dynamics in dotlead hybrid
systems, and may offer useful resources for spin manipulation and simulation of open quantum systems.

Proposal for a minimal surface code experiment 
James R. Wootton, Andreas Peter, Janos R. Winkler, and Daniel Loss Phys. Rev. A 96, 032338 (2017); arXiv:1608.05053.
Current quantum technology is approaching the system sizes and fidelities required for quantum error
correction. It is therefore important to determine exactly what is needed for proofofprinciple experiments,
which will be a major step towards faulttolerant quantum computation. Here we propose a surface code based
experiment that is the smallest, both in terms of code size and circuit depth, that would allow errors to be detected
and corrected for both the X and Z bases of a qubit. This requires 17 physical qubits initially prepared in a
product state, on which 16 twoqubit entangling gates are applied before a final measurement of all qubits. A
platform agnostic error model is applied to give some idea of the noise levels required for success. It is found
that a true demonstration of quantum error correction will require fidelities for the preparation and measurement
of qubits and the entangling gates to be above 99%.

Detecting Topological Superconductivity with phi_0 Josephson Junctions 
Constantin Schrade, Silas Hoffman, and Daniel Loss Phys. Rev. B 95, 195421 (2017); arXiv:1607.07794.
The interplay of superconductivity, magnetic fields, and spinorbit interaction lies at the heart of topological superconductivity. Remarkably, the recent experimental discovery of phi_0 Josephson junctions by Szombati et al., Nat. Phys. 12, 568 (2016), characterized by a finite phase offset in the supercurrent, require the same ingredients as topological superconductors, which suggests a profound connection between these two distinct phenomena. Here, we theoretically show that a quantum dot phi_0 Josephson junction can serve as a new qualitative indicator for topological superconductivity: Microscopically, we find that the phase shift in a junction of swave superconductors is due to the spinorbit induced mixing of singly occupied states on the quantum dot, while for a topological superconductor junction it is due to singlettriplet mixing. Because of this important difference, when the spinorbit vector of the quantum dot and the external Zeeman field are orthogonal, the swave superconductors form a pi Josephson junction while the topological superconductors have a finite offset phi_0 by which topological superconductivity can be distinguished from conventional superconductivity. Our prediction can be immediately tested in nanowire systems currently used for Majorana fermion experiments and thus offers a new and realistic approach for detecting topological bound states.

Heavy hole states in Germanium hut wires 
Hannes Watzinger, Christoph Kloeffel, Lada Vukusic, Marta D. Rossell, Violetta Sessi, Josip Kukucka, Raimund Kirchschlager, Elisabeth Lausecker, Alisha Truhlar, Martin Glaser, Armando Rastelli, Andreas Fuhrer, Daniel Loss, and Georgios Katsaros Nano Lett. 16, 6879 (2016); arXiv:1607.02977.
Hole spins have gained considerable interest in the past few years due to their potential for fast electrically controlled qubits. Here, we study holes confined in Ge hut wires, a so far unexplored type of nanostructure. Low temperature magnetotransport measurements reveal a large anisotropy between the inplane and outofplane gfactors of up to 18. Numerical simulations verify that this large anisotropy originates from a confined wave function which is of heavy hole character. A light hole admixture of less than 1% is estimated for the states of lowest energy, leading to a surprisingly large reduction of the outofplane gfactors. However, this tiny light hole contribution does not influence the spin lifetimes, which are expected to be very long, even in non isotopically purified samples.

Role of the electron spin in determining the coherence of the nuclear spins in a quantum dot 
Gunter Wuest, Mathieu Munsch, Franziska Maier, Andreas V. Kuhlmann, Arne Ludwig, Andreas D. Wieck, Daniel Loss, Martino Poggio, and Richard J. Warburton Nature Nanotechnology 11, 885 (2016)
A huge effort is underway to develop semiconductor nanostructures as lownoise qubits. A key source of dephasing for an
electron spin qubit in GaAs and in naturally occurring Si is the nuclear spin bath. The electron spin is coupled to each
nuclear spin by the hyperfine interaction. The same interaction also couples two remote nuclear spins via a common
coupling to the delocalized electron. It has been suggested that this interaction limits both electron and nuclear spin
coherence, but experimental proof is lacking. We show that the nuclear spin decoherence time decreases by two orders of
magnitude on occupying an empty quantum dot with a single electron, recovering to its original value for two electrons.
In the case of one electron, agreement with a model calculation verifies the hypothesis of an electronmediated nuclear
spin–nuclear spin coupling. The results establish a framework to understand the main features of this complex interaction
in semiconductor nanostructures.

Fractional boundary charges in quantum dot arrays with density modulation 
JinHong Park, Guang Yang (Riken), Jelena Klinovaja, Peter Stano (Riken), and Daniel Loss. Phys. Rev. B 94, 075416 (2016); arXiv:1604.05437.
We show that fractional charges can be realized at the boundaries of a linear array of tunnel coupled quantum dots in the presence of a periodically modulated onsite potential. While the charge fractionalization mechanism is similar to the one in polyacetylene, here the values of fractional charges can be tuned to arbitrary values by varying the phase of the onsite potential or the total number of dots in the array. We also find that the fractional boundary charges, unlike the ingap bound states, are stable against static random disorder. We discuss the minimum array size where fractional boundary charges can be observed.

Universal Quantum Computation with Hybrid SpinMajorana Qubits 
Silas Hoffman, Constantin Schrade, Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 94, 045316 (2016); arXiv:1602.06923.
We theoretically propose a set of universal quantum gates acting on a hybrid qubit formed by coupling a quantum dot spin qubit and Majorana fermion qubit. First, we consider a quantum dot tunnelcoupled to two topological superconductors. The effective spinMajorana exchange facilitates a hybrid CNOT gate for which either qubit can be the control or target. The second setup is a modular scalable network of topological superconductors and quantum dots. As a result of the exchange interaction between adjacent spin qubits, a CNOT gate is implemented that acts on neighboring Majorana qubits, and eliminates the necessity of interqubit braiding. In both setups the spinMajorana exchange interaction allows for a phase gate, acting on either the spin or the Majorana qubit, and for a SWAP or hybrid SWAP gate which is sufficient for universal quantum computation without projective measurements.

Majorana bound states in magnetic skyrmions 
Guang Yang (Riken), Peter Stano (Riken), Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 93, 224505 (2016); arXiv:1602.00968.
Magnetic skyrmions are highly mobile nanoscale topological spin textures. We show, both analytically and
numerically, that a magnetic skyrmion of an even azimuthal winding number placed in proximity to an swave
superconductor hosts a zeroenergy Majorana bound state in its core, when the exchange coupling between the
itinerant electrons and the skyrmion is strong. This Majorana bound state is stabilized by the presence of a
spinorbit interaction. We propose the use of a superconducting trijunction to realize nonAbelian statistics of
such Majorana bound states.

Optimal geometry of lateral GaAs and Si/SiGe quantum dots for electrical control of spin qubits 
Ognjen Malkoc (Riken/Lund), Peter Stano (Riken), and Daniel Loss. Phys. Rev. B 93, 235413 (2016); arXiv:1601.05881.
We investigate the effects of the orientation of the magnetic field and the orientation of a quantum dot, with respect to crystallographic coordinates, on the quality of an electrically controlled qubit realized in a gated semiconductor quantum dot. We find that, due to the anisotropy of the spinorbit interactions, varying the two orientations it is possible to tune the qubit in the sense of optimizing the ratio of its couplings to phonons and to a control electric field. We find conditions under which such optimal setup can be reached by solely reorienting the magnetic field, and when a specific positioning of the dot is required. We also find that the knowledge of the relative sign of the spinorbit interactions strengths allows to choose a robust optimal dot geometry, with the dot main axis along [110], or [110], where the qubit can be always optimized by reorienting the magnetic field.

Topological Phases of Inhomogeneous Superconductivity 
Silas Hoffman, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 93, 165418 (2016); arXiv:1601.04270.
We theoretically consider the effect of a spatially periodic modulation of the superconducting order parameter on the formation of Majorana fermions induced by a onedimensional system with magnetic impurities brought into close proximity to an swave superconductor. When the magnetic exchange energy is larger than the interimpurity electron hopping we model the effective system as a chain of coupled Shiba states. While in the opposite regime, the effective system is accurately described by a quantum wire model. Upon including a spatially modulated superconducting pairing, we find, for sufficiently large magnetic exchange energy, the system is able to support a single pair of Majorana fermions with one Majorana fermion on the left end of the system and one on the right end. When the modulation of superconductivity is large compared to the magnetic exchange energy, the Shiba chain returns to a trivially gapped regime while the quantum wire enters a new topological phase capable of supporting two pairs of Majorana fermions.

LongRange Interaction between Charge and Spin Qubits in Quantum Dots 
Marcel Serina, Christoph Kloeffel, and Daniel Loss Phys. Rev. B 95, 245422 (2017); arXiv:1601.03564.
We analyze and give estimates for the longdistance coupling via floating metallic gates between different types of spin qubits in quantum dots made of different commonly used materials. In particular, we consider the hybrid, the singlettriplet, and the spin1/2 qubits, and the pairwise coupling between each type of these qubits with another hybrid qubit in GaAs, InAs, Si, and Si0.9Ge0.1. We show that hybrid qubits can be capacitively coupled strongly enough to implement twoqubit gates, as long as the dimensions of the dots and their distance from the metallic gates are small enough. Thus, hybrid qubits are good candidates for scalable implementations of quantum computing in semiconducting nanostructures.

Chiral and NonChiral Edge States in Quantum Hall Systems with Charge Density Modulation 
Pawel Szumniak, Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 93, 245308 (2016); arXiv:1512.05971.
We consider a system of weakly coupled wires with quantum Hall effect (QHE) and in the presence of a spatially periodic modulation of the chemical potential along the wire, equivalent to a charge density wave (CDW). We investigate the competition between the two effects which both open a gap. We show that by changing the ratio between the amplitudes of the CDW modulation and the tunneling between wires, one can switch between nontopological CDWdominated phase to topological QHEdominated phase. Both phases host edge states of chiral and nonchiral nature robust to onsite disorder. However, only in the topological phase, the edge states are immune to disorder in the phase shifts of the CDWs. We provide analytical solutions for filling factor nu=1 and study numerically effects of disorder as well as present numerical results for higher filling factors.

Phononassisted relaxation and decoherence of singlettriplet qubits in Si/SiGe quantum dots 
Viktoriia Kornich, Christoph Kloeffel, and Daniel Loss. Quantum 2, 70 (2018); arXiv:1511.07369.
We study theoretically the phononinduced relaxation and decoherence of spin states of two electrons in a lateral double quantum dot in a SiGe/Si/SiGe heterostructure. We consider two types of singlettriplet spin qubits and calculate their relaxation and decoherence times, in particular as a function of level hybridization, temperature, magnetic field, spin orbit interaction, and detuning between the quantum dots, using BlochRedfield theory. We show that the magnetic field gradient, which is usually applied to operate the spin qubit, may suppress the relaxation time by more than an order of magnitude. Using this insight, we identify an optimal regime where the magnetic field gradient does not affect the relaxation time significantly, and we propose regimes of longest decay times. We take into account the effects of onephonon and twophonon processes and suggest how our theory can be tested experimentally. The spin lifetimes we find here for Sibased quantum dots are significantly longer than the ones reported for their GaAs counterparts.

Quantum Computing with Parafermions 
Adrian Hutter and Daniel Loss. Phys. Rev. B 93, 125105 (2016); arXiv:1511.02704.
ℤd Parafermions are exotic nonAbelian quasiparticles generalizing Majorana fermions, which correspond to the case d=2. In contrast to Majorana fermions, braiding of parafermions with d>2 allows to perform an entangling gate. This has spurred interest in parafermions and a variety of condensed matter systems have been proposed as potential hosts for them. In this work, we study the computational power of braiding parafermions more systematically. We make no assumptions on the underlying physical model but derive all our results from the algebraical relations that define parafermions. We find a familiy of 2d representations of the braid group that are compatible with these relations. The braiding operators derived this way reproduce those derived previously from physical grounds as special cases. We show that if a dlevel qudit is encoded in the fusion space of four parafermions, braiding of these four parafermions allows to generate the entire singlequdit Clifford group (up to phases), for any d. If d is odd, then we show that in fact the entire manyqudit Clifford group can be generated.

From Coupled Rashba Electron and Hole Gas Layers to 3D Topological Insulators 
Luka Trifunovic, Daniel Loss, and Jelena Klinovaja. Phys. Rev. B 93, 205406 (2016); arXiv:1511.01742.
We introduce a system of stacked twodimensional electron and hole gas layers with Rashba spin orbit interaction and show that the tunnel coupling between the layers induces a strong three dimensional (3D) topological insulator phase. At each of the twodimensional bulk boundaries we find the spectrum consisting of a single anistropic Dirac cone, which we show by analytical and numerical calculations. Our setup has a unitcell consisting of four tunnel coupled Rashba layers and presents a synthetic strong 3D topological insulator and is distinguished by its rather high experimental feasibility.

Josephson Junction through a 3D Topological Insulator with Helical Magnetization 
Alexander Zyuzin, Mohammad Alidoust, and Daniel Loss. Phys. Rev. B 93, 214502 (2016); arXiv:1511.01486.
We study supercurrent and proximity vortices in a Josephson junction made of disordered surface states of a threedimensional topological insulator with a proximity induced inplane helical magnetization. In a regime where the rotation period of helical magnetization is larger than the junction width, we find supercurrent 0{\pi} crossovers as a function of junction thickness, magnetization strength, and parameters inherent to the helical modulation and surface states. The supercurrent reversals are associated with proximity induced vortices, nucleated along the junction width, where the number of vortices and their locations can be manipulated by means of the superconducting phase difference and the parameters mentioned above.

Topological Floquet Phases in Driven Coupled Rashba Nanowires 
Jelena Klinovaja, Peter Stano (Riken), and Daniel Loss. Phys. Rev. Lett. 116, 176401 (2016); arXiv:1510.03640.
We consider periodicallydriven arrays of weakly coupled wires with conduction and valence bands of Rashba type and study the resulting Floquet states. This nonequilibrium system can be tuned into nontrivial phases such as of topological insulators, Weyl semimetals, and dispersionless zeroenergy edge mode regimes. In the presence of strong electronelectron interactions, we generalize these regimes to the fractional case, where elementary excitations have fractional charges e/m with m being an odd integer.

LongDistance Entanglement of Spin Qubits via Quantum Hall Edge States 
Guang Yang (Riken), ChenHsuan Hsu (Riken), Peter Stano (Riken), Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 93, 075301 (2016); arXiv:1509.09006.
The implementation of a functional quantum computer involves entangling and coherent manipulation of a large number of qubits. For qubits based on electron spins confined in quantum dots, which are among the most investigated solidstate qubits at present, architectural challenges are often encountered in the design of quantum circuits attempting to assemble the qubits within the very limited space available. Here, we provide a solution to such challenges based on an approach to realizing entanglement of spin qubits over long distances. We show that longrange RudermanKittelKasuyaYosida interaction of confined electron spins can be established by quantum Hall edge states, leading to an exchange coupling of spin qubits. The coupling is anisotropic and can be either Isingtype or XYtype, depending on the spin polarization of the edge state. Such a property, combined with the dependence of the electron spin susceptibility on the chirality of the edge state, can be utilized to gain valuable insights into the topological nature of various quantum Hall states.

Antiferromagnetic nuclear spin helix and topological superconductivity in 13C nanotubes 
ChenHsuan Hsu (RIKEN), Peter Stano (RIKEN), Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 92, 235435 (2015); arXiv:1509.01685.
We investigate the RKKY interaction arising from the hyperfine coupling between localized nuclear spins and conduction electrons in interacting 13C carbon nanotubes. Using the Luttinger liquid formalism, we show that the RKKY interaction is sublattice dependent, consistent with the spin susceptibility calculation in noninteracting carbon nanotubes, and it leads to an antiferromagnetic nuclear spin helix in finitesize systems. The transition temperature reaches up to tens of millikelvins, due to a strong boost by a positive feedback through the Overhauser field from ordered nuclear spins. Similar to GaAs nanowires, the formation of the helical nuclear spin order gaps out half of the conduction electrons, and is therefore observable as a reduction of conductance by a factor of two in a transport experiment. The nuclear spin helix leads to a density wave combining spin and charge degrees of freedom in the electron subsystem, resulting in synthetic spinorbit interaction, which induces nontrivial topological phases. As a result, topological superconductivity with Majorana fermion bound states can be realized in the system in the presence of proximityinduced superconductivity without the need of fine tuning the chemical potential. We present the phase diagram as function of system parameters, including the pairing gaps, the gap due to the nuclear spin helix, and the Zeeman field perpendicular to the helical plane.

Dephasing due to nuclear spins in largeamplitude electric dipole spin resonance 
LiPing Yang (Beijing), Stefano Chesi (Beijing), and Daniel Loss. Phys. Rev. Lett. 116, 066806 (2016); arXiv:1508.06894.
We analyze effects of the hyperfine interaction on electric dipole spin resonance when the amplitude of
the quantumdot motion becomes comparable or larger than the quantum dot’s size. Away from the wellknown
smalldrive regime, the important role played by transverse nuclear fluctuations leads to a Gaussian
decay with characteristic dependence on drive strength and detuning. A characterization of spinflip gate
fidelity, in the presence of such additional drivedependent dephasing, shows that vanishingly small errors
can still be achieved at sufficiently large amplitudes. Based on our theory, we analyze recent electric dipole
spin resonance experiments relying on spinorbit interactions or the slanting field of a micromagnet. We
find that such experiments are already in a regime with significant effects of transverse nuclear fluctuations
and the form of decay of the Rabi oscillations can be reproduced well by our theory.

Supercurrent Reversal in TwoDimensional Topological Insulators 
Alexander Zyuzin, Mohammad Alidoust, Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 92, 174515 (2015); arXiv:1507.05089.
We theoretically demonstrate that a supercurrent across a twodimensional topological insulator subjected to an external magnetic field unambiguously reveals the existence of edgemode superconductivity. When the edge states of a narrow twodimensional topological insulator are hybridized, an external magnetic field can close the hybridization gap, thus driving a quantum phase transition from insulator to semimetal states of the topological insulator. Importantly, we find a sign reversal of the supercurrent at the quantum phase transition which offers a simple and experimentally feasible way to observe intrinsic properties of topological insulators including edgemode superconductivity.

WiedemannFranz Law for Magnon Transport 
Kouki Nakata, Pascal Simon (Orsay), and Daniel Loss Phys. Rev. B 92, 134425 (2015); arXiv:1507.03807.
One of the main goals of spintronics is to improve transport of information carriers and to achieve new functionalities with ultralow dissipation. A most promising strategy for this holy grail is to use pure magnon currents created and transported in insulating magnets, in the complete absence of any conducting metallic elements. Here we propose a realistic solution to this fundamental challenge by analyzing magnon and heat transport in insulating ferromagnetic junctions. We calculate all transport coefficients for magnon transport and establish Onsager relations between them. We theoretically discover that magnon transport in junctions has a universal behavior, i.e. is independent of material parameters, and establish a magnon analog of the celebrated WiedemannFranz law which governs charge transport at low temperatures. We calculate the Seebeck and Peltier coefficients which are crucial quantities for spin caloritronics and demonstrate that they assume universal values in the low temperature limit. Finally, we show that our predictions are within experimental reach with current device and measurement technologies.

Persistent Skyrmion Lattice of NonInteracting Electrons in SpinOrbit Coupled Double Wells 
Jiyong Fu, Poliana H. Penteado, Marco O. Hachiya, Daniel Loss, and J. Carlos Egues. Phys. Rev. Lett. 117, 226401 (2016); arXiv:1507.00811.
A persistent spin helix (PSH) is a robust helical spindensity pattern arising in disordered 2D electron gases with Rashba a and Dresselhaus b spinorbit (SO) tuned couplings, i.e., a=b. Here, we investigate the emergence of a persistent Skyrmion lattice (PSL) resulting from the coherent superposition of PSHs along orthogonal directions crossed PSHs in wells with two occupied subbands n=1, 2. For realistic GaAs wells, we show that the Rashba a_n and Dresselhaus b_n couplings can be simultaneously tuned to equal strengths but opposite signs, e.g., a_1=b_1 and a_2=b_2. In this regime, and away from band anticrossings, our noninteracting electron gas sustains a topologically nontrivial Skyrmionlattice spindensity excitation, which inherits the robustness against spinindependent disorder and interactions from its underlying crossed PSHs. We find that the spin relaxation rate due to the interband SO coupling is comparable to that of the cubic Dresselhaus term as a mechanism of the PSL decay. Near anticrossings, the interbandinduced spin mixing leads to unusual spin textures along the energy contours beyond those of the RahsbaDresselhaus bands. Our PSL opens up the unique possibility of observing topological phenomena, e.g., topological and Skyrmion Hall effects, in ordinary GaAs wells with noninteracting electrons.

ProximityInduced pi Josephson Junctions in Topological Insulators and Kramers Pairs of Majorana Fermions 
Constantin Schrade, A.A. Zyuzin, Jelena Klinovaja,, and Daniel Loss. Phys. Rev. Lett. 115, 237001 (2015); arXiv:1506.09120.
We study two microscopic models of topological insulators in contact with an swave superconductor.
In the first model the superconductor and the topological insulator are tunnel coupled via a layer of
randomly distributed scalar and of randomly oriented spin impurities. Here, we demonstrate that spinflip
tunneling dominates over the spinconserving one. In the second model the tunnel coupling is realized by a
spatially nonuniform array of singlelevel quantum dots with randomly oriented spins. We find that the
tunnel region forms a \pi junction where the effective order parameter changes sign. Because of the random
spin orientation, effectively both models exhibit timereversal symmetry. The proposed \pi junctions
support topological superconductivity without magnetic fields and can be used to generate and manipulate
Kramers pairs of Majorana fermions by gates.

Probing Atomic Structure and Majorana Wavefunctions in MonoAtomic Fechains on Superconducting PbSurface 
Remy Pawlak, Marcin Kisiel, Jelena Klinovaja, Tobias Meier, Shigeki Kawai, Thilo Glatzel, Daniel Loss, and Ernst Meyer. npj Quantum Information 2, 16035 (2016); arXiv:1505.06078.
Motivated by the striking promise of quantum computation, Majorana bound states (MBSs) in solidstate systems have attracted wide attention in recent years. In particular, the wavefunction localization of MBSs is a key feature and crucial for their future implementation as qubits. Here, we investigate the spatial and electronic characteristics of topological superconducting chains of iron atoms on the surface of Pb(110) by combining scanning tunneling microscopy (STM) and atomic force microscopy (AFM). We demonstrate that the Fe chains are monoatomic, structured in a linear fashion, and exhibit zerobias conductance peaks at their ends which we interprete as signature for a Majorana bound state. Spatially resolved conductance maps of the atomic chains reveal that the MBSs are well localized at the chain ends (below 25 nm), with two localization lengths as predicted by theory. Our observation lends strong support to use MBSs in Fe chains as qubits for quantum computing devices.

Fractional Charge and Spin States in Topological Insulator Constrictions 
Jelena Klinovaja and Daniel Loss. Phys. Rev. B 92, 121410(R) (2015); arXiv:1505.02682.
We investigate theoretically properties of twodimensional topological insulator constrictions both in the integer and fractional regimes. In the presence of a perpedicular magnetic field, the constriction functions as a spin filter with nearperfect efficiency and can be switched by electric fields only. Domain walls between different topological phases can be created in the constriction as an interface between tunneling, magnetic fields, charge density wave, or electronelectron interactions dominated regions. These domain walls host nonAbelian bound states with fractional charge and spin and result in degenerate ground states with parafermions. If a proximity gap is induced bound states give rise to an exotic Josephson current with 8πperidiodicity.

Parafermions in a Kagome lattice of qubits for topological quantum computation 
Adrian Hutter, James R. Wootton, and Daniel Loss. Phys. Rev. X 5, 041040 (2015); arXiv:1505.01412.
Engineering complex nonAbelian anyon models with simple physical systems is crucial for topological quantum computation. Unfortunately, the simplest systems are typically restricted to Majorana zero modes (Ising anyons). Here, we go beyond this barrier, showing that the ℤ4 parafermion model of nonAbelian anyons can be realized on a qubit lattice. Our system additionally contains the Abelian D(ℤ4) anyons as lowenergetic excitations. We show that braiding of these parafermions with each other and with the D(ℤ4) anyons allows the entire d=4 Clifford group to be generated. The errorcorrection problem for our model is also studied in detail, guaranteeing fault tolerance of the topological operations. Crucially, since the nonAbelian anyons are engineered through defect lines rather than as excitations, nonAbelian error correction is not required. Instead, the errorcorrection problem is performed on the underlying Abelian model, allowing high noise thresholds to be realized.

Impurity Induced Quantum Phase Transitions and Magnetic Order in Conventional Superconductors: Competition between Bound and Quasiparticle states 
Silas Hoffman, Jelena Klinovaja, Tobias Meng (TU Dresden), and Daniel Loss. Phys. Rev. B 92, 125422 (2015); arXiv:1503.08762.
We theoretically study bound states generated by magnetic impurities within conventional swave superconductors, both analytically and numerically. In determining the effect of the hybridization of two such bound states on the energy spectrum as a function of magnetic exchange coupling, relative angle of magnetization, and distance between impurities, we find that quantum phase transitions can be modulated by each of these parameters. Accompanying such transitions, there is a change in the preferred spin configuration of the impurities. Although the interaction between the impurity spins is overwhelmingly dominated by the quasiparticle contribution, the ground state of the system is determined by the bound state energies. Selfconsistently calculating the superconducting order parameter, we find a discontinuity when the system undergoes a quantum phase transition as indicated by the bound state energies.

Voltage induced conversion of helical to uniform nuclear spin polarization in a quantum wire 
Viktoriia Kornich, Peter Stano (Tokyo), Alexander A. Zyuzin, and Daniel Loss. Phys. Rev. B 91, 195423 (2015); arXiv:1503.06950.
We study the effect of bias voltage on the nuclear spin polarization of a ballistic wire, which contains electrons and nuclei interacting via hyperfine interaction. In equilibrium, the localized nuclear spins are helically polarized due to the electronmediated RudermanKittelKasuyaYosida (RKKY) interaction. Focusing here on nonequilibrium, we find that an applied bias voltage induces a uniform polarization, from both helically polarized and unpolarized spins available for spin flips. Once a macroscopic uniform polarization in the nuclei is established, the nuclear spin helix rotates with frequency proportional to the uniform polarization. The uniform nuclear spin polarization monotonically increases as a function of both voltage and temperature, reflecting a thermal activation behavior. Our predictions offer specific ways to test experimentally the presence of a nuclear spin helix polarization in semiconducting quantum wires.

Fielddependent superradiant quantum phase transition of molecular magnets in microwave cavities 
Dimitrije Stepanenko (Belgrade), Mircea Trif (Paris/Beijing), Oleksandr Tsyplyatyev (Frankfurt), and Daniel Loss. Semicond. Sci. Technol. 31, 094003 (2016); arXiv:1502.04075.
We study a superradiant quantum phase transition in the model of triangular molecular magnets
coupled to the electric component of a microwave cavity field. The transition occurs when the
coupling strength exceeds a critical value, dc, which, in sharp contrast to the standard twolevel
emitters, can be tuned by an external magnetic field. In addition to emitted radiation, the
molecules develop an inplane electric dipole moment at the transition. We estimate that the
transition can be detected in stateoftheart microwave cavities if their electric field couples to a
crystal containing a sufficient number of oriented molecules.

Magnon transport through microwave pumping 
Kouki Nakata, Pascal Simon (Paris), and Daniel Loss. Phys. Rev. B 92, 014422 (2015); arXiv:1502.03865.
We present a microscopic theory ofmagnon transport in ferromagnetic insulators (FIs).Using magnon injection
through microwave pumping, we propose a way to generate magnon dc currents and show how to enhance
their amplitudes in hybrid ferromagnetic insulating junctions. To this end, focusing on a single FI, we first
revisit microwave pumping at finite (room) temperature from the microscopic viewpoint of magnon injection.
Next, we apply it to two kinds of hybrid ferromagnetic insulating junctions. The first is the junction between
a quasiequilibrium magnon condensate and magnons being pumped by microwave, while the second is the
junction between such pumped magnons and noncondensed magnons. We show that quasiequilibrium magnon
condensates generate ac and dc magnon currents, while noncondensed magnons produce essentially a dc magnon
current. The ferromagnetic resonance (FMR) drastically increases the density of the pumped magnons and
enhances such magnon currents. Lastly, using microwave pumping in a single FI, we discuss the possibility that
a magnon current through an AharonovCasher phase flows persistently even at finite temperature.We show that
such a magnon current arises even at finite temperature in the presence of magnonmagnon interactions. Due to
FMR, its amplitude becomes much larger than the condensed magnon current.

Superconducting Gap Renomalization around two Magnetic Impurities: From Shiba to Andreev Bound States 
Tobias Meng (Dresden), Jelena Klinovaja, Silas Hoffman, Pascal Simon (Paris), and Daniel Loss. Phys. Rev. B 92, 064503 (2015); arXiv:1501.07901.
We study the renormalization of the gap of an swave superconductor in the presence of two magnetic
impurities. For weakly bound Shiba states, we analytically calculate the part of the gap renormalization that is
sensitive to the relative orientation of the two impurity spins. For impurities with a strong exchange coupling to
the conduction electrons, we solve the gap equation selfconsistently by numerics and find that the subgap Shiba
state turns into a supragap Andreev state when the local gap parameter changes sign under the impurities.

LongDistance Entanglement of Soliton Spin Qubits in Gated Nanowires 
Pawel Szumniak, Jaroslaw Pawlowski (Krakow), Stanislaw Bednarek (Krakow), and Daniel Loss. Phys. Rev. B 92, 035403 (2015); arXiv:1501.01932.
We investigate numerically charge, spin, and entanglement dynamics of two electrons confined in a gated semiconductor nanowire. The electrostatic coupling between electrons in the nanowire and the charges in the metal gates leads to a selftrapping of the electrons which results in solitonlike properties. We show that the interplay of an allelectrically controlled coherent transport of the electron solitons and of the exchange interaction can be used to realize ultrafast SWAP and entangling SQRTofSWAP gates for distant spin qubits. We demonstrate that the latter gate can be used to generate a maximally entangled spin state of spatially separated electrons. The results are obtained by quantum mechanical timedependent calculations with exact inclusion of electronelectron correlations.

Electricallytunable hole gfactor of an opticallyactive quantum dot for fast spin rotations 
Jonathan H. Prechtel, Franziska Maier, Julien Houel, Andreas V. Kuhlmann, Arne Ludwig, Andreas D. Wieck, Daniel Loss, and Richard J. Warburton. Phys. Rev. B 91, 165304 (2015); arXiv:1412.4238.
We report a large gfactor tunability of a single hole spin in an InGaAs quantum dot via an electric field. The magnetic field lies in the inplane direction x, the direction required for a coherent hole spin. The electrical field lies along the growth direction z and is changed over a large range, 100 kV/cm. Both electron and hole gfactors are determined by high resolution laser spectroscopy with resonance fluorescence detection. This, along with the low electricalnoise environment, gives very high quality experimental results. The hole gfactor g_xh depends linearly on the electric field Fz, dg_xh/dFz = (8.3 +/ 1.2)* 10^4 cm/kV, whereas the electron gfactor g_xe is independent of electric field, dg_xe/dFz = (0.1 +/ 0.3)* 10^4 cm/kV (results averaged over a number of quantum dots). The dependence of g_xh on Fz is well reproduced by a 4x4 k.p model demonstrating that the electric field sensitivity arises from a combination of soft hole confining potential, an In concentration gradient and a strong dependence of material parameters on In concentration. The electric field sensitivity of the hole spin can be exploited for electricallydriven hole spin rotations via the gtensor modulation technique and based on these results, a hole spin coupling as large as ~ 1 GHz is expected to be envisaged.

Integer and Fractional Quantum Anomalous Hall Effect in a Strip of Stripes Model 
Jelena Klinovaja, Yaroslav Tserkovnyak (UCLA), and Daniel Loss. Phys. Rev. B 91, 085426 (2015); arXiv:1412.0548.
We study the quantum anomalous Hall effect in a strip of stripes model coupled to a magnetic texture with zero total magnetization and in the presence of strong electronelectron interactions. A helical magnetization along the stripes and a spinselective coupling between the stripes gives rise to a bulk gap and chiral edge modes. Depending on the ratio between the period of the magnetic structure and the Fermi wavelength, the system can exhibit the integer or fractional quantum anomalous Hall effect. In the fractional regime, the quasiparticles have fractional charges and nontrivial Abelian braid statistics.

Quantum Memories at Finite Temperature 
Benjamin J. Brown (Imperial London), Daniel Loss, Jiannis K. Pachos (Leeds), Chris N. Self (Leeds), and James R. Wootton. Rev. Mod. Phys. 88, 045005 (2016); arXiv:1411.6643.
To use quantum systems for technological applications we first need to preserve their coherence for macroscopic timescales, even at finite temperature. Quantum error correction has made it possible to actively correct errors that affect a quantum memory. An attractive scenario is the construction of passive storage of quantum information with minimal active support. Indeed, passive protection is the basis of robust and scalable classical technology, physically realized in the form of the transistor and the ferromagnetic hard disk. The discovery of an analogous quantum system is a challenging open problem, plagued with a variety of nogo theorems. Several approaches have been devised to overcome these theorems by taking advantage of their loopholes. Here we review the stateoftheart developments in this field in an informative and pedagogical way. We give the main principles of selfcorrecting quantum memories and we analyze several milestone examples from the literature of two, three and higherdimensional quantum memories.

Spin and Orbital Magnetic Response on the Surface of a Topological Insulator 
Yaroslav Tserkovnyak (UCLA), D. A. Pesin (Univ. of Utah), and Daniel Loss. Phys. Rev. B 91, 041121(R) (2015); arXiv:1411.2070.
Coupling of the spin and orbital degrees of freedom on the surface of a strong threedimensional insulator, on the one hand, and textured magnetic configuration in an adjacent ferromagnetic film, on the other, is studied using a combination of transport and thermodynamic considerations. Expressing exchange coupling between the localized magnetic moments and Dirac electrons in terms of the electrons' outofplane orbital and spin magnetizations, we relate the thermodynamic properties of a general ferromagnetic spin texture to the physics in the zeroth Landau level. Persistent currents carried by Dirac electrons endow the magnetic texture with a DzyaloshinskiMoriya interaction, which exhibits a universal scaling form as a function of electron temperature, chemical potential, and the timereversal symmetry breaking gap. In addition, the orbital motion of electrons establishes a direct magnetoelectric coupling between the unscreened electric field and local magnetic order, which furnishes complex longranged interactions within the magnetic film.

Improved HDRG decoders for qudit and nonAbelian quantum error correction 
Adrian Hutter, Daniel Loss, and James R. Wootton. New J. Phys. 17, 035017 (2015); arXiv:1410.4478.
Harddecision renormalization group (HDRG) decoders are an important class of decoding algorithms for topological quantum error correction. Due to their versatility, they have been used to decode systems with fractal logical operators, color codes, qudit topological codes, and nonAbelian systems. In this work, we develop a method of performing HDRG decoding which combines strenghts of existing decoders and further improves upon them. In particular, we increase the minimal number of errors necessary for a logical error in a system of linear size L from Θ(L2/3) to Ω(L1−ϵ) for any ϵ>0. We apply our algorithm to decoding D(ℤd) quantum double models and a nonAbelian anyon model with Fibonaccilike fusion rules, and show that it indeed significantly outperforms previous HDRG decoders. Furthermore, we provide the first study of continuous error correction with imperfect syndrome measurements for the D(ℤd) quantum double models. The parallelized runtime of our algorithm is poly(logL) for the perfect measurement case. In the continuous case with imperfect syndrome measurements, the averaged runtime is O(1) for Abelian systems, while continuous error correction for nonAbelian anyons stays an open problem.

Majorana Fermions in Ge/Si Hole Nanowires 
Franziska Maier, Jelena Klinovaja (Harvard), and Daniel Loss. Phys. Rev. B 90, 195421 (2014); arXiv:1409.8645.
We consider Ge/Si core/shell nanowires with hole states coupled to an swave superconductor in the presence of electric and magnetic fields. We employ a microscopic model that takes into account materialspecific details of the band structure such as strong and electrically tunable Rashbatype spinorbit interaction and g factor anisotropy for the holes. In addition, the proximityinduced superconductivity Hamiltonian is derived starting from a microscopic model. In the topological phase, the nanowires host Majorana fermions with localization lengths that depend strongly on both the magnetic and electric fields. We identify the optimal regime in terms of the directions and magnitudes of the fields in which the Majorana fermions are the most localized at the nanowire ends. In short nanowires, the Majorana fermions hybridize and form a subgap fermion whose energy is split away from zero and oscillates as a function of the applied fields. The period of these oscillations could be used to measure the dependence of the spinorbit interaction on the applied electric field and the g factor anisotropy.

Fast LongDistance Control of Spin Qubits by Photon Assisted Cotunneling 
Peter Stano (Riken), Jelena Klinovaja (Harvard), Floris R. Braakman, Lieven M. K. Vandersypen (Delft), and Daniel Loss. Phys. Rev. B 92, 075302 (2015); arXiv:1409.4852.
We investigate theoretically the longdistance coupling and spin exchange in an array of quantum dot spin qubits
in the presence of microwaves.We find that photonassisted cotunneling is boosted at resonances between photon
and energies of virtually occupied excited states and showhowtomake it spin selective.We identify configurations
that enable fast switching and spin echo sequences for efficient and nonlocal manipulation of spin qubits.We devise
configurations in which the nearresonantly boosted cotunneling provides nonlocal coupling which, up to certain
limit, does not diminish with distance between themanipulated dots before it decays weakly with inverse distance.

Highefficiency resonant amplification of weak magnetic fields for single spin magnetometry 
Luka Trifunovic, Fabio L. Pedrocchi, Silas Hoffman, Patrick Maletinsky, Amir Yacoby (Harvard), and Daniel Loss. Nature Nanotechnology 10, 541 (2015); arXiv:1409.1497.
Magnetic resonance techniques not only provide powerful imaging tools that have revolutionized medicine, but they have a
wide spectrum of applications in other fields of science such as biology, chemistry, neuroscience and physics. However,
current stateoftheart magnetometers are unable to detect a single nuclear spin unless the tiptosample separation is
made sufficiently small. Here, we demonstrate theoretically that by placing a ferromagnetic particle between a nitrogen–
vacancy magnetometer and a target spin, the magnetometer sensitivity is improved dramatically. Using materials and
techniques that are already experimentally available, our proposed setup is sensitive enough to detect a single nuclear
spin within ten milliseconds of data acquisition at room temperature. The sensitivity is practically unchanged when the
ferromagnet surface to the target spin separation is smaller than the ferromagnet lateral dimensions; typically about a
tenth of a micrometre. This scheme further benefits when used for nitrogen–vacancy ensemble measurements, enhancing
sensitivity by an additional three orders of magnitude.

Fermionic and Majorana Bound States in Hybrid Nanowires with NonUniform SpinOrbit Interaction 
Jelena Klinovaja (Harvard) and Daniel Loss. Eur. Phys. J. B 88, 62 (2015); arXiv:1408.3366.
We study intragap bound states in the topological phase of a Rashba nanowire in the presence of a magnetic field and with nonuniform spin orbit interaction (SOI) and proximityinduced superconductivity gap. We show that fermionic bound states (FBS) can emerge inside the proximity gap. They are localized at the junction between two wire sections characterized by different directions of the SOI vectors, and they coexist with Majorana bound states (MBS) localized at the nanowire ends. The energy of the FBS is determined by the angle between the SOI vectors and the lengthscale over which the SOI changes compared to the Fermi wavelength and the localization length. We also consider doublejunctions and show that the two emerging FBSs can hybridize and form a double quantum dotlike structure inside the gap. We find explicit analytical solutions of the bound states and their energies for certain parameter regimes such as weak and strong SOI. The analytical results are confirmed and complemented by an independent numerical tightbinding model approach. Such FBS can act as quasiparticle traps and thus can have implications for topological quantum computing schemes based on braiding MBSs.

Spontaneous Helical Order of Electron and Nuclear Spins in a Luttinger Liquid 
Christian P Scheller, Bernd Braunecker, Daniel Loss, and Dominik M Zumbuhl Progress in Physics (44), COMMUNICATIONS of the Swiss Physical Society (SPS) 9, 23 (2014)
In a onedimensional (1D) conductor, electrons are confined to move along a single
direction, occupying only the quantum mechanical ground state orbital of the transverse
dimensions of the wire. What is the electrical conductance of such a quantum wire? This
fundamental question was answered by Rolf Landauer many years ago for noninteracting
electrons in a clean, ballistic conductor: each spin species carries the quantum of
conductance, e2/h [1], with e the electron charge and h the Planck constant. For a spin
degenerate 1D conductor with a single subband, the conductance is therefore 2e2/h. If the
spin degeneracy is broken and transport of one spin direction is blocked, the conductance is
thus reduced to 1e2/h. Similar to spin, other degeneracies such as valley degeneracies or
multiple 1D subbands due to weaker confinement can also open additional conductance …

NMR Response of Nuclear Spin Helix in Quantum Wires with Hyperfine and SpinOrbit Interaction 
Peter Stano (RIKEN) and Daniel Loss. Phys. Rev. B 90, 195312 (2014); arXiv:1408.2353.
We calculate the nuclear magnetic resonance (NMR) response of a quantum wire where at low temperature a selfsustained electronnuclear spin order is created. Our model includes the electron mediated RudermanKittelKasuyaYosida (RKKY) exchange, electron spinorbit interactions, nuclear dipolar interactions, and the static and oscillating NMR fields, all of which play an essential role. The paramagnet to helimagnet transition in the nuclear system is reflected in an unusual response: it absorbs at a frequency given by the internal RKKY exchange field, rather than the external static field, whereas the latter leads to a splitting of the resonance peak.

Strongly Interacting Holes in Ge/Si Nanowires 
Franziska Maier, Tobias Meng, and Daniel Loss. Phys. Rev. B 90, 155437 (2014); arXiv:1408.0631.
We consider holes confined to Ge/Si core/shell nanowires subject to strong Rashba spinorbit interaction and screened Coulomb interaction. Such wires can, for instance, serve as host systems for Majorana bound states. Starting from a microscopic model, we find that the Coulomb interaction strongly influences the properties of experimentally realistic wires. To show this, a Luttinger liquid description is derived based on a renormalization group analysis. This description in turn allows to calculate the scaling exponents of various correlation functions as a function of the microscopic system parameters. It furthermore permits to investigate the effect of Coulomb interaction on a small magnetic field, which opens a strongly anisotropic partial gap.

Conductance behavior in nanowires with spinorbit interaction  A numerical study 
Diego Rainis and Daniel Loss. Phys. Rev. B 90, 235415 (2014); arXiv:1407.8239.
We consider electronic transport through semiconducting nanowires (W) with spinorbit interaction (SOI), in a hybrid NWN setup where the wire is contacted by normalmetal leads (N). We investigate the conductance behavior of the system as a function of gate and bias voltage, magnetic field, wire length, temperature, and disorder. The transport calculations are performed numerically and are based on standard recursive Green's function techniques. In particular, we are interested in understanding if and how it is possible to deduce the strength of the SOI from the transport behavior. This is a very relevant question since so far no clear experimental observation in that direction has been produced. We find that the smoothness of the electrostatic potential profile between the contacts and the wire plays a crucial role, and we show that in realistic regimes the NWN setup may mask the effects of SOI, and a trivial behavior with apparent vanishing SOI is observed. We identify an optimal parameter regime, with neither too smooth nor too abrupt potentials, where the signature of SOI is best visible, with and without FabryPerot oscillations, and is most resilient to disorder and temperature effects.

RKKY Interaction On Surfaces of Topological Insulators With Superconducting Proximity Effect 
Alexander A. Zyuzin and Daniel Loss. Phys. Rev. B 90, 125443 (2014); arXiv:1407.6632.
We consider the RudermanKittelKasuyaYosida (RKKY) interaction between magnetic impurities on the surface of a threedimensional topological insulator with proximity induced superconductivity. A superconductor placed on the top of the topological insulator induces a gap in the surface electron states and gives rise to a longranged inplane antiferromagnetic RKKY interaction. This interaction is frustrated due to strong spinorbit coupling, decays as 1/r for r < xi, where r is the distance between two magnetic impurities and xi the superconducting coherence length, and dominates over the ferromagnetic and DzyaloshinskiiMoriya type interactions for r > xi. We find the condition for the YuShibaRusinov intragap states that are bound to the magnetic impurities.

Helical nuclear spin order in a strip of stripes in the Quantum Hall regime 
Tobias Meng, Peter Stano (RIKEN), Jelena Klinovaja (Harvard), and Daniel Loss. Eur. Phys. J. B 87, 203 (2014); arXiv:1407.3726.
We investigate nuclear spin effects in a twodimensional electron gas in the quantum Hall regime modeled by a weakly coupled array of interacting quantum wires. We show that the presence of hyperfine interaction between electron and nuclear spins in such wires can induce a phase transition, ordering electrons and nuclear spins into a helix in each wire. Electronelectron interaction effects, pronounced within the onedimensional stripes, boost the transition temperature up to tens to hundreds of millikelvins in GaAs. We predict specific experimental signatures of the existence of nuclear spin order, for instance for the resistivity of the system at transitions between different quantum Hall plateaus.

Nuclear Spin Relaxation in Rashba Nanowires 
Alexander A. Zyuzin, Tobias Meng, Viktoriia Kornich, and Daniel Loss. Phys. Rev. B 90, 195125 (2014); arXiv:1407.2582.
We study the nuclear spin relaxation in a ballistic nanowire with hyperfine and Rashba spinorbit interactions (SOI) and in the presence of magnetic field and electron interactions. The relaxation rate shows pronounced peaks as function of magnetic field and chemical potential due to van Hove singularities in the Rashba bands. As a result, the regimes of weak and strong SOIs can be distinguished by the number of peaks in the rate. The relaxation rate increases with increasing magnetic field if both Rashba subbands are occupied, whereas it decreases if only the lowest one is occupied.

Josephson and Persistent Spin Currents in BoseEinstein Condensates of Magnons 
Kouki Nakata, Kevin A. van Hoogdalem, Pascal Simon (Paris), and Daniel Loss. Phys. Rev. B 90, 144419 (2014); arXiv:1406.7004.
Using the AharonovCasher (AC) phase, we present a microscopic theory of the Josephson and persistent spin currents in quasiequilibrium BoseEinstein condensates (BECs) of magnons in ferromagnetic insulators. Starting from a microscopic spin model that we map onto a GrossPitaevskii Hamiltonian, we derive a twostate model for the Josephson junction between the weakly coupled magnonBECs. We then show how to obtain the alternatingcurrent (ac) Josephson effect with magnons as well as macroscopic quantum selftrapping in a magnonBEC. We next propose how to control the directcurrent (dc) Josephson effect electrically using the AC phase, which is the geometric phase acquired by magnons moving in an electric field. Finally, we introduce a magnonBEC ring and show that persistent magnonBEC currents flow due to the AC phase. Focusing on the feature that the persistent magnonBEC current is a steady flow of magnetic dipoles that produces an electric field, we propose a method to directly measure it experimentally.

Singlespin manipulation in a double quantum dot with micromagnet 
Stefano Chesi (Beijing), YingDan Wang (Beijing), Jun Yoneda (Tokyo), Tomohiro Otsuka (Tokyo), Seigo Tarucha (Tokyo), and Daniel Loss. Phys. Rev. B 90, 235311 (2014); arXiv:1405.7618.
The manipulation of single spins in double quantum dots by making use of the exchange interaction and a highly inhomogeneous magnetic field was discussed in [W. A. Coish and D. Loss, Phys. Rev. B 75, 161302 (2007)]. However, such large inhomogeneity is difficult to achieve through the slanting field of a micromagnet in current designs of lateral double dots. Therefore, we examine an analogous spin manipulation scheme directly applicable to realistic GaAs double dot setups. We estimate that typical gate times, realized at the singlettriplet anticrossing induced by the inhomogeneous micromagnet field, can be a few nanoseconds. We discuss the optimization of initialization, readout, and singlespin gates through suitable choices of detuning pulses and an improved geometry. We also examine the effect of nuclear dephasing and charge noise. The latter induces fluctuations of both detuning and tunneling amplitude. Our results suggest that this scheme is a promising approach for the realization of fast singlespin operations.

LongDistance Entanglement of Spin Qubits via Ferromagnet 
Luka Trifunovic, Fabio L. Pedrocchi, and Daniel Loss. Phys. Rev. X 3, 041023 (2013); arXiv:1305.2451.
We propose a mechanism of a longrange coherent interaction between two
singlettriplet qubits dipolarly coupled to a dogboneshaped ferromagnet. An
effective qubitqubit interaction Hamiltonian is derived and the coupling
strength is estimated. Furthermore we derive the effective coupling between two
spin1/2 qubits that are coupled via dipolar interaction to the ferromagnet and
that lie at arbitrary positions and deduce the optimal positioning. We consider
hybrid systems consisting of spin1/2 and ST qubits and derive the effective
Hamiltonian for this case. We then show that operation times vary between 1MHz
and 100MHz and give explicit estimates for GaAs, Silicon, and NVcenter based
spin qubits. Finally, we explicitly construct the required sequences to
implement a CNOT gate. The resulting quantum computing architecture retains all
the single qubit gates and measurement aspects of earlier approaches, but
allows qubit spacing at distances of order 1$\,\mu$m for twoqubit gates,
achievable with current semiconductor technology.

Quantum charge pumping through fractional Fermions in charge density modulated quantum wires and Rashba nanowires 
Arijit Saha, Diego Rainis, Rakesh P. Tiwari, and Daniel Loss. Phys. Rev. B 90, 035422 (2014); arXiv:1405.5719.
We study the phenomenon of adiabatic quantum charge pumping in systems supporting fractionally charged fermionic bound states, in two different setups. The first quantum pump setup consists of a chargedensitymodulated quantum wire, and the second one is based on a semiconducting nanowire with Rashba spinorbit interaction, in the presence of a spatially oscillating magnetic field. In both these quantum pumps transport is investigated in a NXN geometry, with the system of interest (X) connected to two normalmetal leads (N), and the two pumping parameters are the strengths of the effective wirelead barriers. Pumped charge is calculated within the scattering matrix formalism. We show that quantum pumping in both setups provides a unique signature of the presence of the fractionalfermion bound states, in terms of asymptotically quantized pumped charge. Furthermore, we investigate shot noise arising due to quantum pumping, verifying that quantized pumped charge corresponds to minimal shot noise.

Acoustic phonons and strain in core/shell nanowires 
Christoph Kloeffel, Mircea Trif (Paris), and Daniel Loss. Phys. Rev. B 90, 115419 (2014); arXiv:1405.4834.
We study theoretically the lowenergy phonons and the static strain in cylindrical core/shell nanowires (NWs). Assuming pseudomorphic growth, isotropic media, and a forcefree wire surface, we derive algebraic expressions for the dispersion relations, the displacement fields, and the stress and strain components from linear elasticity theory. Our results apply to NWs with arbitrary radii and arbitrary elastic constants for both core and shell. The expressions for the static strain are consistent with experiments, simulations, and previous analytical investigations; those for phonons are consistent with known results for homogeneous NWs. Among other things, we show that the dispersion relations of the torsional, longitudinal, and flexural modes change differently with the relative shell thickness, and we identify new terms in the corresponding strain tensors that are absent for uncapped NWs. We illustrate our results via the example of Ge/Si core/shell NWs and demonstrate that shellinduced strain has large effects on the hole spectrum of these systems.

Characterization of spinorbit interactions of GaAs heavy holes using a quantum point contact 
Fabrizio Nichele (ETHZ), Stefano Chesi (RIKEN), Szymon Hennel (ETHZ), Angela Wittmann (ETHZ), Christian Gerl (ETHZ), Werner Wegscheider (ETHZ), Daniel Loss, Thomas Ihn (ETHZ), and Klaus Ensslin (ETHZ). Phys. Rev. Lett. 113, 046801 (2014); arXiv:1405.2981.
We present transport experiments performed in high quality quantum point contacts embedded in a GaAs twodimensional hole gas. The strong spinorbit interaction results in peculiar transport phenomena, including the previously observed anisotropic Zeeman splitting and leveldependent effective gfactors. Here we find additional effects, namely the crossing and the anticrossing of spinsplit levels depending on subband index and magnetic field direction. Our experimental observations are reconciled in an heavy hole effective spinorbit Hamiltonian where cubic and quadraticinmomentum terms appear. The spinorbit components, being of great importance for quantum computing applications, are characterized in terms of magnitude and spin structure. In the light of our results, we explain the level dependent effective gfactor in an inplane field. Through a tilted magnetic field analysis, we show that the QPC outofplane gfactor saturates around the predicted 7.2 bulk value.

Kramers Pairs of Majorana Fermions and Parafermions in Fractional Topological Insulators 
Jelena Klinovaja (Harvard), Amir Yacoby (Harvard), and Daniel Loss. Phys. Rev. B 90, 155447 (2014); arXiv:1403.4125.
We propose a scheme based on topological insulators to generate Kramers pairs of Majorana fermions or parafermions in the complete absence of magnetic fields. Our setup consists of two topological insulators whose edge states are brought close to an swave superconductor. The resulting proximity effect leads to an interplay between a nonlocal crossed Andreev pairing, which is dominant in the strong electronelectron interaction regime, and usual superconducting pairing, which is dominant at large separation between the two topological insulator edges. As a result, there are zeroenergy bound states localized at interfaces between spatial regions dominated by the two different types of pairing. Due to the preserved timereversal symmetry, the bound states come in Kramers pairs. If the topological insulators carry fractional edge states, the zeroenergy bound states are parafermions, otherwise, they are Majorana fermions.

Renormalization of anticrossings in interacting quantum wires with Rashba and Dresselhaus spinorbit couplings 
Tobias Meng, Jelena Klinovaja (Harvard), and Daniel Loss. Phys. Rev. B 89, 205133 (2014); arXiv:1403.2759.
We discuss how electronelectron interactions renormalize the spinorbit induced anticrossings between different subbands in ballistic quantum wires. Depending on the ratio of spinorbit coupling and subband spacing, electronelectron interactions can either increase or decrease anticrossing gaps. When the anticrossings are closing due to a special combination of Rashba and Dresselhaus spinorbit couplings, their gap approaches zero as an interaction dependent power law of the spinorbit couplings, which is a consequence of Luttinger liquid physics. Monitoring the closing of the anticrossings allows to directly measure the related renormalization group scaling dimension in an experiment. If a magnetic field is applied parallel to the spinorbit coupling direction, the anticrossings experience different renormalizations. Since this difference is entirely rooted in electronelectron interactions, unequally large anticrossings also serve as a direct signature of Luttinger liquid physics. Electronelectron interactions furthermore increase the sensitivity of conductance measurements to the presence of anticrossing.

Inhibition of dynamic nuclear polarization by heavyhole noncollinear hyperfine interactions 
Hugo Ribeiro (McGill), Franziska Maier, and Daniel Loss. Phys. Rev. B 92, 075421 (2015); arXiv:1403.0490.
We show that the effective hyperfine interaction for heavyhole states (or any particle described with a plike
Bloch function) can induce nontrivial dynamics of nuclear spins. Experimental evidence can be found, e.g., in
selfassembled quantum dots by measuring the saturation of nuclear spin polarization with different orientations
of an external magnetic field.

Breakdown of surfacecode error correction due to coupling to a bosonic bath 
Adrian Hutter and Daniel Loss. Phys. Rev. A 89, 042334 (2014); arXiv:1402.3108.
We consider a surface code suffering decoherence due to coupling to a bath of bosonic modes at finite temperature and study the time available before the unavoidable breakdown of error correction occurs as a function of coupling and bath parameters. We derive an exact expression for the error rate on each individual qubit of the code, taking spatial and temporal correlations between the errors into account. We investigate numerically how different kinds of spatial correlations between errors in the surface code affect its threshold error rate. This allows us to derive the maximal duration of each quantum errorcorrection period by studying when the singlequbit error rate reaches the corresponding threshold. At the time when error correction breaks down, the error rate in the code can be dominated by the direct coupling of each qubit to the bath, by mediated subluminal interactions, or by mediated superluminal interactions. For a twodimensional Ohmic bath, the time available per quantum errorcorrection period vanishes in the thermodynamic limit of a large code size L due to induced superluminal interactions, although it does so only like 1/√lnL. For all other bath types considered, this time remains finite as L→∞.

A quantum magnetic RC circuit 
Kevin A. van Hoogdalem, Mathias Albert (Orsay), Pascal Simon (Orsay), and Daniel Loss. Phys. Rev. Lett. 113, 037201 (2014); arXiv:1401.5712.
We propose a setup that is the spin analog of the chargebased quantum RC circuit. We define and compute the spin capacitance and the spin resistance of the circuit for both ferromagnetic (FM) and antiferromagnetic (AF) systems. We find that the antiferromagnetic setup has universal properties, but the ferromagnetic setup does not. We discuss how to use the proposed setup as a quantum source of spin excitations, and put forward a possible experimental realization using ultracold atoms in optical lattices.

TimeReversal Invariant Parafermions in Interacting Rashba Nanowires 
Jelena Klinovaja (Harvard) and Daniel Loss. Phys. Rev. B 90, 045118 (2014); arXiv:1312.1998.
We propose a scheme to generate pairs of timereversal invariant parafermions. Our setup consists of two quantum wires with Rashba spinorbit interactions coupled to an swave superconductor, in the presence of electronelectron interactions. The zeroenergy bound states localized at the wire ends arise from the interplay between two types of proximityinduced superconductivity: the usual intrawire superconductivity and the interwire superconductivity due to crossed Andreev reflections. If the latter dominates, which is the case for strong electronelectron interactions, the system supports Kramers pair of parafermions. Moreover, the scheme can be extended to a twodimensional sea of timereversal invariant parafermions.

Parafermions in Interacting Nanowire Bundle 
Jelena Klinovaja (Harvard) and Daniel Loss. Phys. Rev. Lett. 112, 246403 (2014); arXiv:1311.3259.
We propose a scheme to induce Z3 parafermion modes, exotic zeroenergy bound states that possess nonAbelian statistics. We consider a minimal setup consisting of a bundle of four tunnel coupled nanowires hosting spinless electrons that interact strongly with each other. The hallmark of our setup is that it relies only on simple onedimensional wires, uniform magnetic fields, and strong interactions, but does not require the presence of superconductivity or exotic quantum Hall phases.

PhononMediated Decay of SingletTriplet Qubits in Double Quantum Dots 
Viktoriia Kornich, Christoph Kloeffel, and Daniel Loss. Phys. Rev. B 89, 085410 (2014); arXiv:1311.2197.
We study theoretically the phononinduced decoherence and relaxation of singlettriplet qubits in lateral GaAs double quantum dots. For typical setups, the decoherence and relaxation rates due to onephonon processes are proportional to the temperature T, whereas the rates due to twophonon processes reveal a transition from T^2 to higher powers as T is decreased. In contrast to previous calculations of the phononlimited lifetimes T_1 (relaxation) and T_2 (decoherence), we find T_2 \neq 2 T_1 in this system. Remarkably, both T_1 and T_2 exhibit a maximum when the external magnetic field is applied along a certain axis within the plane of the twodimensional electron gas. We compare our results with recent experiments and analyze the dependence of T_1 and T_2 on system properties such as the detuning, the spinorbit parameters, and the orientation of the double quantum dot and the applied magnetic field with respect to the main crystallographic axes.

Anisotropic g factor in InAs selfassembled quantum dots 
Robert Zielke, Franziska Maier, and Daniel Loss. Phys. Rev. B 89, 115438 (2014); arXiv:1311.0908.
We investigate the wavefunctions, spectrum, and g factor anisotropy of lowenergy electrons confined to selfassembled, pyramidal InAs quantum dots (QDs) subject to external magnetic and electric fields. We present the construction of trial wavefunctions for a pyramidal geometry with hardwall confinement. We explicitly find the ground and first excited states and show the associated probability distributions and energies. Subsequently, we use these wavefunctions and 8band k⋅p theory to derive a Hamiltonian describing the QD states close to the valence band edge. Using a perturbative approach, we find an effective conduction band Hamiltonian describing lowenergy electronic states in the QD. From this, we further extract the magnetic field dependent eigenenergies and associated g factors. We examine the g factors regarding anisotropy and behavior under small electric fields. In particular, we find strong anisotropies, with the specific shape depending strongly on the considered subband. Our results are in good agreement with recent measurements [Takahashi et al., Phys. Rev. B 87, 161302 (2013)] and support the possibility to control a spin qubit by means of g tensor modulation.

Error Correction for NonAbelian Topological Quantum Computation 
James R. Wootton, Jan Burri, Sofyan Iblisdir, and Daniel Loss. Phys. Rev. X 4, 011051 (2014); Popular Summary; arXiv:1310.3846.
The possibility of quantum computation using nonAbelian anyons has been considered for over a decade. However, the question of how to obtain and process information about what errors have occurred in order to negate their effects has not yet been considered. This is in stark contrast with quantum computation proposals for Abelian anyons, for which decoding algorithms have been tailormade for many topological errorcorrecting codes and error models. Here, we address this issue by considering the properties of nonAbelian error correction, in general. We also choose a specific anyon model and error model to probe the problem in more detail. The anyon model is the charge submodel of D(S3). This shares many properties with important models such as the Fibonacci anyons, making our method more generally applicable. The error model is a straightforward generalization of those used in the case of Abelian anyons for initial benchmarking of error correction methods. It is found that error correction is possible under a threshold value of 7% for the total probability of an error on each physical spin. This is remarkably comparable with the thresholds for Abelian models.

Transport signature of fractional Fermions in Rashba nanowires 
Diego Rainis, Arijit Saha, Jelena Klinovaja (Harvard), Luka Trifunovic, and Daniel Loss. Phys. Rev. Lett. 112, 196803 (2014); arXiv:1309.3738.
We theoretically study transport through a semiconducting Rashba nanowire (NW) in the presence of
uniform and spatially modulated magnetic fields. The system is fully gapped, and the interplay between the
spin orbit interaction and the magnetic fields leads to fractionally charged fermion (FF) bound states of the
JackiwRebbi type at each end of the nanowire. We investigate the transport and noise behavior of a
N=NW=N system, where the wire is contacted by two normal leads (N), and we look for possible signatures
that could help in the experimental detection of such states. We find that the differential conductance and
the shot noise exhibit a subgap structure which fully reveals the presence of the FF state. Alternatively,
another confirmation of the presence of the FFs is provided by a conductance measurement in an
AharonovBohm setup, where the FFs are responsible for oscillations with double period. Our predictions
can be tested in InSb or InAs nanowires and are within reach of the present technology.

Structure factor of interacting onedimensional helical systems 
Suhas Gangadharaiah, Thomas L. Schmidt, and Daniel Loss. Phys. Rev. B 89, 035131 (2014); arXiv:1308.5982.
We calculate the dynamical structure factor S(q, {\omega}) of a weakly
interacting helical edge state in the presence of a magnetic field B. The
latter opens a gap of width 2B in the singleparticle spectrum, which becomes
strongly nonlinear near the Dirac point. For chemical potentials {\mu} > B,
the system then behaves as a nonlinear helical Luttinger liquid, and a
mobileimpurity analysis reveals interactiondependent powerlaw singularities
in S(q,{\omega}). For {\mu} < B, the lowenergy excitations are gapped, and
we determine S(q,{\omega}) by using an analogy to exciton physics.

Topological Superconductivity and Majorana Fermions in RKKY Systems 
Jelena Klinovaja, Peter Stano (Riken), Ali Yazdani (Princeton), and Daniel Loss. Phys. Rev. Lett. 111, 186805 (2013); arXiv:1307.1442.
We consider quasi onedimensional RKKY systems in proximity to an swave
superconductor. We show that a $2k_F$peak in the spin susceptibility of the
superconductor in the onedimensional limit supports helical order of localized
magnetic moments via RKKY interaction, where $k_F$ is the Fermi wavevector. The
magnetic helix is equivalent to a uniform magnetic field and very strong
spinorbit interaction (SOI) with an effective SOI length $1/2k_F$. We find the
conditions to establish such a magnetic state in atomic chains and
semiconducting nanowires with magnetic atoms or nuclear spins. Generically,
these systems are in a topological phase with Majorana fermions. The inherent
selftuning of the helix to $2k_F$ eliminates the need to tune the chemical
potential.

Enhanced thermal stability of the toric code through coupling to a bosonic bath 
Fabio L. Pedrocchi, Adrian Hutter, James R. Wootton, and Daniel Loss. Phys. Rev. A 88, 062313 (2013); arXiv:1309.0621.
We propose and study a model of a quantum memory that features
selfcorrecting properties and a lifetime growing arbitrarily with system size
at nonzero temperature. This is achieved by locally coupling a 2D L x L toric
code to a 3D bath of bosons hopping on a cubic lattice. When the stabilizer
operators of the toric code are coupled to the displacement operator of the
bosons, we solve the model exactly via a polaron transformation and show that
the energy penalty to create anyons grows linearly with L. When the stabilizer
operators of the toric code are coupled to the bosonic density operator, we use
perturbation theory to show that the energy penalty for anyons scales with
ln(L). For a given error model, these energy penalties lead to a lifetime of
the stored quantum information growing respectively exponentially and
polynomially with L. Furthermore, we show how to choose an appropriate coupling
scheme in order to hinder the hopping of anyons (and not only their creation)
with energy barriers that are of the same order as the anyon creation gaps. We
argue that a toric code coupled to a 3D Heisenberg ferromagnet realizes our
model in its lowenergy sector. Finally, we discuss the delicate issue of the
stability of topological order in the presence of perturbations. While we do
not derive a rigorous proof of topological order, we present heuristic
arguments suggesting that topological order remains intact when perturbative
operators acting on the toric code spins are coupled to the bosonic
environment.

Lowenergy properties of fractional helical Luttinger liquids 
Tobias Meng, Lars Fritz (Koeln), Dirk Schuricht (Aachen), and Daniel Loss. Phys. Rev. B 89, 045111 (2014); arXiv:1308.3169.
We investigate the lowenergy properties of (quasi) helical and fractional
helical Luttinger liquids. In particular, we calculate the Drude peak of the
optical conductivity, the density of states, as well as charge transport
properties of the interacting system with and without attached Fermi liquid
leads at small and large (compared to the gap) frequencies. For fractional
wires, we find that the low energy tunneling density of states vanishes. The
conductance of a fractional helical Luttinger liquid is noninteger. It is
independent of the Luttinger parameters in the wire, despite the intricate
mixing of charge and spin degrees of freedom, and only depends on the relative
locking of charge and spin degrees of freedom.

Circuit QED with HoleSpin Qubits in Ge/Si Nanowire Quantum Dots 
Christoph Kloeffel, Mircea Trif (UCLA), Peter Stano (RIKEN), and Daniel Loss. Phys. Rev. B 88, 241405(R) (2013); arXiv:1306.3596.
We propose a setup for universal and electrically controlled quantum
information processing with hole spins in Ge/Si core/shell nanowire quantum
dots (NW QDs). Singlequbit gates can be driven through electricdipoleinduced
spin resonance, with spinflip times shorter than 100 ps. Longdistance
qubitqubit coupling can be mediated by the cavity electric field of a
superconducting transmission line resonator, where we show that operation times
below 20 ns seem feasible for the entangling squarerootofiSWAP gate. The
absence of Dresselhaus spinorbit interaction (SOI) and the presence of an
unusually strong Rashbatype SOI enable precise control over the transverse
qubit coupling via an externally applied, perpendicular electric field. The
latter serves as an onoff switch for quantum gates and also provides control
over the g factor, so that single and twoqubit gates can be operated
independently. Remarkably, we find that idle states are insensitive to charge
noise and phonons, and we discuss strategies for enhancing noiselimited gate
fidelities.

Vortex Loops and Majorana Fermions 
Stefano Chesi (Riken), Arthur Jaffe (Harvard), Daniel Loss, and Fabio L. Pedrocchi. J. Math. Phys. 54, 112203 (2013); arXiv:1305.6270.
We investigate the role that vortex loops play in characterizing eigenstates
of certain systems of halfinteger spins with nearestneighbor interaction on a
trivalent lattice. In particular we focus on ground states (and other lowlying
states). We test our ideas on a "spin ladder" In certain cases we show how the
vortex configuration of the ground state is determined by the relative signs of
the coupling constants. Two methods yield exact results: i.) We utilize the
equivalence of spin Hamiltonians with quartic interactions of Majorana
fermions, and analyze that fermionic Hamiltonian. ii) We use reflection
positivity for Majorana fermions to characterize vortices in ground states for
reflectionsymmetric couplings. Two additional methods suggest potential wider
applicability of these results: iii.) Numerical evidence suggests similar
behavior for certain systems without reflection symmetry. iv.) A perturbative
analysis also suggests similar behavior without the assumption of reflection
symmetry.

Integer and Fractional Quantum Hall Effect in a Strip of Stripes 
Jelena Klinovaja and Daniel Loss. Eur. Phys. J. B (2014) 87: 171; arXiv:1305.1569.
We study anisotropic stripe models of interacting electrons in the presence
of magnetic fields in the quantum Hall regime with integer and fractional
filling factors. The model consists of an infinite strip of finite width that
contains periodically arranged stripes (forming supercells) to which the
electrons are confined and between which they can hop with associated magnetic
phases. The interacting electron system within the onedimensional stripes are
described by Luttinger liquids and shown to give rise to charge and spin
density waves that lead to periodic structures within the stripe with a
reciprocal wavevector 8k_F. This wavevector gives rise to Umklapp scattering
and resonant scattering that results in gaps and chiral edge states at all
known integer and fractional filling factors \nu. The integer and odd
denominator filling factors arise for a uniform distribution of stripes,
whereas the even denominator filling factors arise for a nonuniform stripe
distribution. We calculate the Hall conductance via the Streda formula and show
that it is given by \sigma_H=\nu e^2/h for all filling factors. We show that
the composite fermion picture follows directly from the condition of the
resonant Umklapp scattering.

MagneticallyDefined Qubits on 3D Topological Insulators 
Gerson J. Ferreira and Daniel Loss. Phys. Rev. Lett. 111, 106802 (2013); arXiv:1305.5003.
We explore timereversalsymmetrybreaking potentials to confine the surface
states of 3D topological insulators into quantum wires and quantum dots. A
magnetic domain wall on a ferromagnet insulator cap layer provides interfacial
states predicted to show the quantum anomalous Hall effect (QAHE). Here we show
that confinement can also occur at magnetic domain heterostructures, with
states extended in the inner domain, as well as interfacial QAHE states at the
surrounding domain walls. The proposed geometry allows the isolation of the
wire and dot from spurious circumventing surface states. For the quantum dots,
we find that highly spinpolarized quantized QAHE states at the dot edge
constitute a promising candidate for quantum computing qubits.

Correlations between Majorana fermions through a superconductor 
A.A. Zyuzin, Diego Rainis, Jelena Klinovaja, and Daniel Loss. Phys. Rev. Lett. 111, 056802 (2013); arXiv:1305.4187.
We consider a model of ballistic quasione dimensional semiconducting wire
with intrinsic spinorbit interaction placed on the surface of a bulk swave
superconductor (SC), in the presence of an external magnetic field. This setup
has been shown to give rise to a topological superconducting state in the wire,
characterized by a pair of Majoranafermion (MF) bound states formed at the two
ends of the wire. Here we demonstrate that, besides the wellknown direct
overlapinduced energy splitting, the two MF bound states may hybridize via
elastic correlated tunneling processes through virtual quasiparticles states in
the SC, giving rise to an additional energy splitting between MF states from
the same as well as from different wires.

Spintronics in MoS_2 monolayer quantum wires 
Jelena Klinovaja and Daniel Loss. Phys. Rev. B 88, 075404 (2013); arXiv:1304.4542.
We study analytically and numerically spin effects in MoS_2 monolayer
armchair quantum wires and quantum dots. The interplay between intrinsic and
Rashba spin orbit interactions induced by an electric field leads to helical
modes, giving rise to spin filtering in timereversal invariant systems. The
Rashba spin orbit interaction can also be generated by spatially varying
magnetic fields. In this case, the system can be in a helical regime with
nearly perfect spin polarization. If such a quantum wire is brought into
proximity to an swave superconductor, the system can be tuned into a
topological phase, resulting in midgap Majorana fermions localized at the wire
ends.

Strongly anisotropic spin response as a signature of the helical regime in Rashba nanowires 
Tobias Meng and Daniel Loss. Phys. Rev. B 88, 035437 (2013); arXiv:1303.6994.
Rashba nanowires in a magnetic field exhibit a helical regime when the
spinorbit momentum is close to the Fermi momentum, k_F \approx k_{SO}. We show
that this regime is characterized by a strongly anisotropic electron spin
susceptibility, with an exponentially suppressed signal along one direction in
spin space, and that there are no low frequency spin fluctuations along this
direction. Since the spin response in the gapless regime k_F \not \approx
k_{SO} has a power law behavior in all three directions, spin measurements
provide a signature of the helical regime that complements spininsensitive
conductance measurements.

Helical nuclear spin order in twosubband quantum wires 
Tobias Meng and Daniel Loss. Phys. Rev. B 87, 235427 (2013); arXiv:1303.1542.
In quantum wires, the hyperfine coupling between conduction electrons and
nuclear spins can lead to a (partial) ordering of both of them at low
temperatures. By an interactionenhanced mechanism, the nuclear spin order,
caused by RKKY exchange, acts back onto the electrons and gaps out part of
their spectrum. In wires with two subbands characterized by distinct Fermi
momenta kF1 and kF2, the nuclear spins form a superposition of two helices with
pitches {\pi}/kF1 and {\pi}/kF2, thus exhibiting a beating pattern. This order
results in a reduction of the electronic conductance in two steps upon lowering
the temperature.

Local Spin Susceptibilities of LowDimensional Electron Systems 
Peter Stano, Jelena Klinovaja, Amir Yacoby (Harvard), and Daniel Loss. Phys. Rev. B 88, 045441 (2013); arXiv:1303.1151.
We investigate, assess, and suggest possibilities for a measurement of the
local spin susceptibility of a conducting lowdimensional electron system. The
basic setup of the experiment we envisage is a sourceprobe one. Locally
induced spin density (e.g. by a magnetized atomic force microscope tip) extends
in the medium according to its spin susceptibility. The induced magnetization
can be detected as a dipolar magnetic field, for instance, by an
ultrasensitive nitrogenvacancy center based detector, from which the spatial
structure of the spin susceptibility can be deduced. We find that
onedimensional systems, such as semiconducting nanowires or carbon nanotubes,
are expected to yield a measurable signal. The signal in a twodimensional
electron gas is weaker, though materials with high enough $g$factor (such as
InGaAs) seem promising for successful measurements.

Topological Edge States and Fractional Quantum Hall Effect from Umklapp Scattering 
Jelena Klinovaja and Daniel Loss. Phys. Rev. Lett. 111, 196401 (2013); arXiv:1302.6132.
We study anisotropic lattice strips in the presence of a magnetic field in
the quantum Hall effect regime. At specific magnetic fields, causing resonant
Umklapp scattering, the system is gapped in the bulk and supports chiral edge
states in close analogy to topological insulators. These gaps result in
plateaus for the Hall conductivity exactly at the known fillings n/m (both
positive integers and m odd) for the integer and fractional quantum Hall
effect. For double strips we find topological phase transitions with phases
that support midgap edge states with flat dispersion. The topological effects
predicted here could be tested directly in optical lattices.

Tunable g factor and phononmediated hole spin relaxation in Ge/Si nanowire quantum dots 
Franziska Maier, Christoph Kloeffel, and Daniel Loss. Phys. Rev. B 87, 161305(R) (2013); arXiv:1302.5027.
We theoretically consider g factor and spin lifetimes of holes in a
longitudinal Ge/Si core/shell nanowire quantum dot that is exposed to external
magnetic and electric fields. For the ground states, we find a large anisotropy
of the g factor which is highly tunable by applying electric fields. This
tunability depends strongly on the direction of the electric field with respect
to the magnetic field. We calculate the singlephonon hole spin relaxation
times T1 for zero and small electric fields and propose an optimal setup in
which very large T1 of the order of tens of milliseconds can be reached.
Increasing the relative shell thickness or the longitudinal confinement length
prolongs T1 further. In the absence of electric fields, the dephasing vanishes
and the decoherence time T2 is determined by T2 = 2 T1.

Dynamic Generation of Topologically Protected SelfCorrecting Quantum Memory 
Daniel Becker, Tetsufumi Tanamoto (Toshiba), Adrian Hutter, Fabio L. Pedrocchi, and Daniel Loss. Phys. Rev. A 87, 042340 (2013); arXiv:1302.3998.
We propose a scheme to dynamically realize a thermally stable quantum memory
based on the toric code. The code is generated from qubit systems with typical
twobody interactions (Ising, XY, Heisenberg) using periodic, NMRlike, pulse
sequences. It allows one to encode the logical qubits without measurements and
to protect them dynamically against the time evolution of the physical qubits.
Thermal stability is achieved by weakly coupling the qubits to additional
cavity modes that mediate longrange attractive interactions between the
stabilizer operators of the toric code. We investigate how the fidelity, with
which the toric code is realized, depends on the period length T of the pulse
sequence and the magnitude of possible pulse errors. We derive an optimal
period T_opt that maximizes the fidelity.

An efficient decoding algorithm for stabilizer codes 
Adrian Hutter, James R. Wootton, and Daniel Loss. Phys. Rev. A 89, 022326 (2014); arXiv:1302.2669.
To date, the best classical algorithm for performing error correction in the
surface code has been minimumweight perfect matching. However, in this work we
present a Markov chain Monte Carlo algorithm that achieves significantly lower
logical error rates. It therefore allows any target logical error rate to be
obtained using a significantly smaller code. This increase in performance does
come at the cost of an increased runtime complexity, but only by a polynomial
factor $O(L^\eps)$ for $\eps<2$. Our algorithm is based on an analytically
exact rewriting of the probability of each logical equivalence class, which
also suggests that for arbitrary stabilizer codes error correction can be
performed to arbitrary accuracy in a runtime $O(\m{poly}(L))$. It is applicable
to any stabilizer code, allows for parallelization, and can be used to correct
in the case of imperfect stabilizer measurements.

LongRange Interaction of SpinQubits via Ferromagnets 
Luka Trifunovic, Fabio L. Pedrocchi, and Daniel Loss. Phys. Rev. X 3, 041023 (2013); arXiv:1302.4017.
We propose a mechanism of longrange coherent coupling between spins coupled
to a ferromagnet by exchnage or dipolar coupling. An effective twospin
interaction Hamiltonian is derived and the coupling strength is estimated. We
also discuss mechanisms of decoherence and consider possibilities for gate
control of the interaction between neighboring spinqubits. The resulting
quantum computing architecture retains all the single qubit gates and
measurement aspects of earlier approaches, but allows qubit spacing at
distances of order 1$\,\mu$m for twoqubit gates, achievable with current
semiconductor technology. The clock speed depends strongly on the
dimensionality of the ferromagnet and is between MHz and GHz.

Fractional Fermions with NonAbelian Statistics 
Jelena Klinovaja and Daniel Loss. Phys. Rev. Lett. 110, 126402 (2013); arXiv:1301.5822.
We introduce a novel class of lowdimensional topological tightbinding
models that allow for bound states that are fractionally charged fermions and
exhibit nonAbelian braiding statistics. The proposed model consists of a
double (single) ladder of spinless (spinful) fermions in the presence of
magnetic fields. We study the system analytically in the continuum limit as
well as numerically in the tightbinding representation. We find a topological
phase transition with a topological gap that closes and reopens as a function
of system parameters and chemical potential. The topological phase is of the
type BDI and carries two degenerate midgap bound states that are localized at
opposite ends of the ladders. We show numerically that these bound states are
robust against a wide class of perturbations.

RKKY interaction in carbon nanotubes and graphene nanoribbons 
Jelena Klinovaja and Daniel Loss. Phys. Rev. B 87, 045422 (2013); arXiv:1211.3067.
We study RudermannKittelKasuyaYosida (RKKY) interaction in carbon nanotubes (CNTs) and graphene nanoribbons in the presence of spinorbit interactions and magnetic fields. For this, we evaluate the static spin susceptibility tensor in real space in various regimes at zero temperature. In metallic CNTs, the RKKY interaction depends strongly on the sublattice and, at the Dirac point, is purely ferromagnetic (antiferromagnetic) for the localized spins on the same (different) sublattice, whereas in semiconducting CNTs, the spin susceptibility depends only weakly on the sublattice and is dominantly ferromagnetic. The spinorbit interactions break the SU(2) spin symmetry of the system, leading to an anisotropic RKKY interaction of Ising and DzyaloshinskiiMoriya form, aside from the usual isotropic Heisenberg interaction. All these RKKY terms can be made of comparable magnitude by tuning the Fermi level close to the gap induced by the spinorbit interaction. We further calculate the spin susceptibility also at finite frequencies and thereby obtain the spin noise in real space via the fluctuationdissipation theorem.

Giant spin orbit interaction due to rotating magnetic fields in graphene nanoribbons 
Jelena Klinovaja and Daniel Loss. Phys. Rev. X 3, 011008 (2013); arXiv:1211.2739.
We theoretically study graphene nanoribbons in the presence of spatially
varying magnetic fields produced e.g. by nanomagnets. We show both analytically
and numerically that an exceptionally large Rashba spin orbit interaction (SOI)
of the order of 10 meV can be produced by the nonuniform magnetic field. As a
consequence, helical modes exist in armchair nanoribbons that exhibit nearly
perfect spin polarization and are robust against boundary defects. This paves
the way to realizing spin filter devices in graphene nanoribbons in the
temperature regime of a few Kelvins. If a nanoribbon in the helical regime is
in proximity contact to an swave superconductor, the nanoribbon can be tuned
into a topological phase sustaining Majorana fermions.

Effective quantummemory Hamiltonian from local twobody interactions 
Adrian Hutter, Fabio L. Pedrocchi, James R. Wootton, and Daniel Loss. Phys. Rev. A 90, 012321 (2014); arXiv:1209.5289.
In Phys. Rev. A 88, 062313 (2013) we proposed and studied a model for a selfcorrecting quantum memory in which the energetic cost for introducing a defect in the memory grows without bounds as a function of system size. This positive behavior is due to attractive longrange interactions mediated by a bosonic field to which the memory is coupled. The crucial ingredients for the implementation of such a memory are the physical realization of the bosonic field as well as local fivebody interactions between the stabilizer operators of the memory and the bosonic field. Here, we show that both of these ingredients appear in a lowenergy effective theory of a Hamiltonian that involves only twobody interactions between neighboring spins. In particular, we consider the lowenergy, longwavelength excitations of an ordered Heisenberg ferromagnet (magnons) as a realization of the bosonic field. Furthermore, we present perturbative gadgets for generating the required fivespin operators. Our Hamiltonian involving only local twobody interactions is thus expected to exhibit selfcorrecting properties as long as the noise affecting it is in the regime where the effective lowenergy description remains valid.

Ultrafast magnontransistor at room temperature 
Kevin A. van Hoogdalem and Daniel Loss. Phys. Rev. B 88, 024420 (2013); arXiv:1209.5594; See News and Views, SPINTRONICS: An insulatorbased transistor, by Yaroslav Tserkovnyak; Nature Nanotechnology 8, 706 (2013).
We study sequential tunneling of magnetic excitations in nonitinerant systems
(either magnons or spinons) through triangular molecular magnets. It is known
that the quantum state of such molecular magnets can be controlled by
application of an electric or a magnetic field. Here, we use this fact to
control the flow of a spin current through the molecular magnet by electric or
magnetic means. This allows us to design a system that behaves as a
magnontransistor. We show how to combine three magnontransistors to form a
NANDgate, and give several possible realizations of the latter, one of which
could function at room temperature using transistors with a 11 ns switching
time.

Magnetic textureinduced thermal Hall effects 
Kevin A. van Hoogdalem, Yaroslav Tserkovnyak (UCLA), and Daniel Loss. Phys. Rev. B 87, 024402 (2013); arXiv:1208.1646.
Magnetic excitations in ferromagnetic systems with a noncollinear ground
state magnetization experience a fictitious magnetic field due to the
equilibrium magnetic texture. Here, we investigate how such fictitious fields
lead to thermal Hall effects in twodimensional insulating magnets in which the
magnetic texture is caused by spinorbit interaction. We find that, besides the
wellknown geometric texture contribution to the fictitious magnetic field in
such systems, there exists also an equally important but often neglected
contribution due to the original spinorbit term in the free energy. We
consider the different possible ground states in the phase diagram of a
twodimensional ferromagnet with spinorbit interaction: The spiral state and
the skyrmion lattice, and find that thermal Hall effects can occur in certain
domain walls as well as the skyrmion lattice.

Helical States in Curved Bilayer Graphene 
Jelena Klinovaja, Gerson J. Ferreira, and Daniel Loss. Phys. Rev. B 86, 235416 (2012); arXiv:1208.2601.
We study spin effects of quantum wires formed in bilayer graphene by
electrostatic confinement. With a proper choice of the confinement direction,
we show that in the presence of magnetic field, spinorbit interaction induced
by curvature, and intervalley scattering, bound states emerge that are helical.
The localization length of these helical states can be modulated by the gate
voltage which enables the control of the tunnel coupling between two parallel
wires. Allowing for proximity effect via an swave superconductor, we show that
the helical modes give rise to Majorana fermions in bilayer graphene.

Transition from fractional to Majorana fermions in Rashba nanowires 
Jelena Klinovaja, Peter Stano, and Daniel Loss. Phys. Rev. Lett. 109, 236801 (2012); arXiv:1207.7322.
We study hybrid superconductingsemiconducting nanowires in the presence of
Rashba spinorbit interaction as well as helical magnetic fields. We show that
the interplay between them leads to a competition of phases with two
topological gaps closing and reopening, resulting in unexpected reentrance
behavior. Besides the topological phase with localized Majorana fermions (MFs)
we find new phases characterized by fractionally charged fermion (FF) bound
states of JackiwRebbi type. The system can be fully gapped by the magnetic
fields alone, giving rise to FFs that transmute into MFs upon turning on
superconductivity. We find explicit analytical solutions for MF and FF bound
states and determine the phase diagram numerically by determining the
corresponding Wronskian null space. We show by renormalization group arguments
that electronelectron interactions enhance the Zeeman gaps opened by the
fields.

Realistic transport modeling for a superconducting nanowire with Majorana fermions 
Diego Rainis, Luka Trifunovic, Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 87, 024515 (2013); arXiv:1207.5907.
Motivated by recent experiments searching for Majorana fermions (MFs) in hybrid semiconductingsuperconducting nanostructures, we consider a realistic tightbinding model and analyze its transport behavior numerically. In particular, we take into account the presence of a superconducting contact, used in real experiments to extract the current, which is usually not included in theoretical calculations. We show that important features emerge that are absent in simpler models, such as the shift in energy of the proximity gap signal, and the enhanced visibility of the topological gap for increased spinorbit interaction. We find oscillations of the zero bias peak as a function of the magnetic field and study them analytically. We argue that many of the experimentally observed features hint at an actual spinorbit interaction larger than the one typically assumed. However, even taking into account all the known ingredients of the experiments and exploring many parameter regimes for MFs, we are not able to reach full agreement with the reported data. Thus, a different physical origin for the observed zerobias peak cannot be excluded.

Exchangebased CNOT gates for singlettriplet qubits with spin orbit interaction 
Jelena Klinovaja, Dimitrije Stepanenko, Bertrand I. Halperin (Harvard), and Daniel Loss. Phys. Rev. B 86, 085423 (2012); arXiv:1206.2579.
We propose a scheme for implementing the CNOT gate over qubits encoded in a
pair of electron spins in a double quantum dot. The scheme is based on exchange
and spin orbit interactions and on local gradients in Zeeman fields. We find
that the optimal device geometry for this implementation involves effective
magnetic fields that are parallel to the symmetry axis of the spin orbit
interaction. We show that the switching times for the CNOT gate can be as fast
as a few nanoseconds for realistic parameter values in GaAs semiconductors.
Guided by recent advances in surface codes, we also consider the perpendicular
geometry. In this case, leakage errors due to spin orbit interaction occur but
can be suppressed in strong magnetic fields.

Selfcorrecting quantum memory with a boundary 
Adrian Hutter, James R. Wootton, Beat Roethlisberger, and Daniel Loss. Phys. Rev. A 86, 052340 (2012); arXiv:1206.0991.
We study the twodimensional toriccode Hamiltonian with effective longrange interactions between its anyonic excitations induced by coupling the toric code to external fields. It has been shown that such interactions allow an arbitrary increase in the lifetime of the stored quantum information by making L, the linear size of the memory, larger [ Chesi et al. Phys. Rev. A 82 022305 (2010)]. We show that for these systems the choice of boundary conditions (open boundaries as opposed to periodic boundary conditions) is not a mere technicality; the influence of anyons produced at the boundaries becomes in fact dominant for large enough L. This influence can be either beneficial or detrimental. In particular, we study an effective Hamiltonian proposed by Pedrocchi et al. [ Phys. Rev. B 83 115415 (2011)] that describes repulsion between anyons and anyon holes. For this system, we find a lifetime of the stored quantum information that grows exponentially in L2 for both periodic and open boundary conditions, although the exponent in the latter case is found to be less favorable. However, L is upper bounded through the breakdown of the perturbative treatment of the underlying Hamiltonian.

Decoherence of Majorana qubits by noisy gates 
Manuel J. Schmidt (Aachen), Diego Rainis, and Daniel Loss. Phys. Rev. B 86, 085414 (2012); arXiv:1206.0743.
We propose and study a realistic model for the decoherence of topological
qubits, based on Majorana fermions in onedimensional topological
superconductors. The source of decoherence is the fluctuating charge on a
capacitively coupled gate, modeled by noninteracting electrons. In this
context, we clarify the role of quantum fluctuations and thermal fluctuations
and find that quantum fluctuations do not lead to decoherence, while thermal
fluctuations do. We explicitly calculate decay times due to thermal noise and
give conditions for the gap size in the topological superconductor and the gate
temperature. Based on this result, we provide simple rules for gate geometries
and materials optimized for reducing the negative effect of thermal charge
fluctuations on the gate.

Composite Majorana Fermion Wavefunctions in Nanowires 
Jelena Klinovaja and Daniel Loss. Phys. Rev. B 86, 085408 (2012); arXiv:1205.7054.
We consider Majorana fermions (MFs) in quasionedimensional nanowire systems containing normal and superconducting sections where the topological phase based on Rashba spinorbit interaction can be tuned by magnetic fields. We derive explicit analytic solutions of the MF wave function in the weak and strong spin orbit interaction regimes. We find that the wave function for one single MF is a composite object formed by superpositions of different MF wave functions which have nearly disjoint supports in momentum space. These contributions are coming from the extrema of the spectrum, one centered around zero momentum and the other around the two Fermi points. As a result, the various MF wave functions have different localization lengths in real space and interference among them leads to pronounced oscillations of the MF probability density. For a transparent normalsuperconducting junction we find that in the topological phase the MF leaks out from the superconducting into the normal section of the wire and is delocalized over the entire normal section, in agreement with numerical results obtained in previous studies.

Hyperfineinduced decoherence in triangular spincluster qubits 
Filippo Troiani (Modena), Dimitrije Stepanenko, and Daniel Loss. Phys. Rev. B 86, 161409 (2012); arXiv:1205.5629.
We investigate hyperfineinduced decoherence in a triangular spincluster for
different qubit encodings. Electrically controllable eigenstates of spin
chirality (C_z) show decoherence times that approach milliseconds, two orders
of magnitude longer than those estimated for the eigenstates of the total spin
projection (S_z) and of the partial spin sum (S_{12}). The robustness of
chirality is due to its decoupling from both the total and individualspin
components in the cluster. This results in a suppression of the effective
interaction between C_z and the nuclear spin bath.

Prospects for SpinBased Quantum Computing 
Christoph Kloeffel and Daniel Loss. Annu. Rev. Condens. Matter Phys. 4, 51 (2013); arXiv:1204.5917.
Experimental and theoretical progress toward quantum computation
with spins in quantum dots (QDs) is reviewed, with particular focus
on QDs formed in GaAs heterostructures, on nanowirebased QDs,
and on selfassembled QDs. We report on a remarkable evolution
of the field, where decoherence—one of the main challenges for
realizing quantum computers—no longer seems to be the stumbling
block it had originally been considered. General concepts, relevant
quantities, and basic requirements for spinbased quantum computing
are explained; opportunities and challenges of spinorbit interaction
and nuclear spins are reviewed. We discuss recent achievements,
present current theoretical proposals, and make several suggestions
for further experiments.

Cotunneling in the 5/2 fractional quantum Hall regime 
Robert Zielke, Bernd Braunecker, and Daniel Loss. Phys. Rev. B 86, 235307 (2012); arXiv:1204.4400.
We show that cotunneling in the 5/2 fractional quantum Hall regime allows to
test the MooreRead wave function, proposed for this regime, and to probe the
nature of the fractional charge carriers. We calculate the cotunneling current
for electrons that tunnel between two quantum Hall edge states via a quantum
dot and for quasiparticles with fractional charges e/4 and e/2 that tunnel via
an antidot. While electron cotunneling is strongly suppressed, the
quasiparticle tunneling shows signatures characteristic for the MooreRead
state. For comparison, we also consider cotunneling between Laughlin states,
and find that electrontransport between MooreRead states and the one between
Laughlin states at filling factor 1/3 have identical voltage dependences.

Nonabelian Majoranas and braiding in inhomogeneous spin ladders 
Fabio L. Pedrocchi, Stefano Chesi (Montreal), Suhas Gangadharaiah (Bhopal), and Daniel Loss. Phys. Rev. B 86, 205412 (2012); arXiv:1204.3044.
We propose an inhomogeneous open spin ladder, related to the Kitaev honeycomb model, which can be tuned between topological and nontopological phases. In extension of Lieb's theorem, we show numerically that the ground state of the spin ladder is either vortex free or vortex full. We study the robustness of Majorana end states (MES) which emerge at the boundary between sections in different topological phases and show that while the MES in the homogeneous ladder are destroyed by singlebody perturbations, in the presence of inhomogeneities at least twobody perturbations are required to destabilize MES. Furthermore, we prove that x, y, or z inhomogeneous magnetic fields are not able to destroy the topological degeneracy. Finally, we present a trijunction setup where MES can be braided. A network of such spin ladders provides thus a promising platform for realization and manipulation of MES.

Majorana qubit decoherence by quasiparticle poisoning 
Diego Rainis and Daniel Loss. Phys. Rev. B 85, 174533 (2012); arXiv:1204.3326.
We consider the problem of quasiparticle poisoning in a nanowirebased realization of a Majorana qubit, where a spinorbitcoupled semiconducting wire is placed on top of a (bulk) superconductor. By making use of recent experimental data exhibiting evidence of a lowtemperature residual nonequilibrium quasiparticle population in superconductors, we show by means of analytical and numerical calculations that the dephasing time due to the tunneling of quasiparticles into the nanowire may be problematically short to allow for qubit manipulation.

Effect of strain on hyperfineinduced holespin decoherence in quantum dots 
Franziska Maier and Daniel Loss. Phys. Rev. B 85, 195323 (2012); arXiv:1203.3876.
We theoretically consider the effect of strain on the spin dynamics of a single heavy hole (HH) confined to a selfassembled quantum dot and interacting with the surrounding nuclei via hyperfine interaction. Confinement and strain hybridize the HH states, which show an exponential decay for a narrowed nuclear spin bath. For different strain configurations within the dot, the dependence of the spin decoherence time T2 on external parameters is shifted and the nonmonotonic dependence of the peak is altered. Application of external strain yields considerable shifts in the dependence of T2 on external parameters. We find that external strain affects mostly the effective hyperfine coupling strength of the conduction band (CB), indicating that the CB admixture of the hybridized HH states plays a crucial role in the sensitivity of T2 on strain.

Frequency dependent transport through a spin chain 
Kevin A. van Hoogdalem and Daniel Loss. Phys. Rev. B 85, 054413 (2012); arXiv:1111.4803.
Motivated by potential applications in spintronics, we study frequency
dependent spin transport in nonitinerant onedimensional spin chains. We
propose a system that behaves as a capacitor for the spin degree of freedom. It
consists of a spin chain with two impurities a distance $d$ apart. We find that
at low energy (frequency) the impurities flow to strong coupling, thereby
effectively cutting the chain into three parts, with the middle island
containing a discrete number of spin excitations. At finite frequency spin
transport through the system increases. We find a strong dependence of the
finite frequency characteristics both on the anisotropy of the spin chain and
the applied magnetic field. We propose a method to measure the finitefrequency
conductance in this system.

High threshold error correction for the surface code 
James R. Wootton and Daniel Loss. Phys. Rev. Lett. 109, 160503 (2012); arXiv:1202.4316.
An algorithm is presented for error correction in the surface code quantum
memory. This is shown to correct depolarizing noise up to a threshold error
rate of 18.5%, exceeding previous results and coming close to the upper bound
of 18.9%. The time complexity of the algorithm is found to be subexponential,
offering a significant speedup over brute force methods and allowing efficient
error correction for codes of realistic sizes.

ElectricField Induced Majorana Fermions in Armchair Carbon Nanotubes 
Jelena Klinovaja, Suhas Gangadharaiah, and Daniel Loss. Phys. Rev. Lett. 108, 196804 (2012); arXiv:1201.0159.
We consider theoretically an armchair carbon nanotube (CNT) in the presence of an electric field and in contact with an swave superconductor. We show that the proximity effect opens up superconducting gaps in the CNT of different strengths for the exterior and interior branches of the two Dirac points. For strong proximity induced superconductivity the interior gap can be of the pwave type, while the exterior gap can be tuned by the electric field to be of the swave type. Such a setup supports a single Majorana bound state at each end of the CNT. In the case of a weak proximity induced superconductivity, the gaps in both branches are of the pwave type. However, the temperature can be chosen in such a way that the smallest gap is effectively closed. Using renormalization group techniques we show that the Majorana bound states exist even after taking into account electronelectron interactions.

ThinFilm Magnetization Dynamics on the Surface of a Topological Insulator 
Yaroslav Tserkovnyak (UCLA) and Daniel Loss. Phys. Rev. Lett. 108, 187201 (2012); arXiv:1112.5884.
We theoretically study the magnetization dynamics of a thin ferromagnetic
film exchangecoupled with a surface of a strong threedimensional topological
insulator. We focus on the role of electronic zero modes associated with domain
walls (DW's) and other topological textures in the magnetic film.
Thermodynamically reciprocal hydrodynamic equations of motion are derived for
the DW responding to electronic spin torques, on the one hand, and fictitious
electromotive forces in the electronic chiral mode fomented by the DW, on the
other. An experimental realization illustrating this physics is proposed based
on a ferromagnetic strip, which cuts the topological insulator surface into two
gapless regions. In the presence of a ferromagnetic DW, a chiral mode
transverse to the magnetic strip acts as a dissipative interconnect, which is
itself a dynamic object that controls (and, inversely, responds to) the
magnetization dynamics.

Ferromagnetic order of nuclear spins coupled to conduction electrons: a combined effect of the electronelectron and spinorbit interactions 
Robert Andrzej Zak, Dmitrii L. Maslov (Gainesville), and Daniel Loss. Phys. Rev. B 85, 115424 (2012); viewpoint; arXiv:1112.4786.
We analyze the ordered state of nuclear spins embedded in an interacting twodimensional electron gas (2DEG) with Rashba spinorbit interaction (SOI). Stability of the ferromagnetic nuclearspin phase is governed by nonanalytic dependences of the electron spin susceptibility $\chi^{ij}$ on the momentum ($\tilde{\mathbf{q}}$) and on the SOI coupling constant ($\alpha$). The uniform ($\tq=0$) spin susceptibility is anisotropic (with the outofplane component, $\chi^{zz}$, being larger than the inplane one, $\chi^{xx}$, by a term proportional to $U^2(2k_F)\alpha$, where $U(q)$ is the electronelectron interaction). For $\tq \leq 2m^*\alpha$, corrections to the leading, $U^2(2k_F)\alpha$, term scale linearly with $\tq$ for $\chi^{xx}$ and are absent for $\chi^{zz}$. This anisotropy has important consequences for the ferromagnetic nuclearspin phase: $(i)$ the ordered stateif achievedis of an Ising type and $(ii)$ the spinwave dispersion is gapped at $\tq=0$. To second order in $U(q)$, the dispersion a decreasing function of $\tq$, and anisotropy is not sufficient to stabilize longrange order. However, renormalization in the Cooper channel for $\tq\ll2m^*\alpha$ is capable of reversing the sign of the $\tq$dependence of $\chi^{xx}$ and thus stabilizing the ordered state. We also show that a combination of the electronelectron and SO interactions leads to a new effect: longwavelength Friedel oscillations in the spin (but not charge) electron density induced by local magnetic moments. The period of these oscillations is given by the SO length $\pi/m^*\alpha$.

Singlettriplet splitting in double quantum dots due to spin orbit and hyperfine interactions 
Dimitrije Stepanenko, Mark Rudner (Harvard), Bertrand I. Halperin (Harvard), and Daniel Loss. Phys. Rev. B 85, 075416 (2012); arXiv:1112.1644.
We analyze the lowenergy spectrum of a twoelectron double quantum dot under a potential bias in the presence of an external magnetic field. We focus on the regime of spin blockade, taking into account the spinorbit interaction and hyperfine coupling of electron and nuclear spins. Starting from a model for two interacting electrons in a double dot, we derive an effective twolevel Hamiltonian in the vicinity of an avoided crossing between singlet and triplet levels, which are coupled by the spinorbit and hyperfine interactions. We evaluate the level splitting at the anticrossing, and show that it depends on a variety of parameters including the spinorbit coupling strength, the orientation of the external magnetic field relative to an internal spinorbit axis, the potential detuning of the dots, and the difference between hyperfine fields in the two dots. We provide a formula for the splitting in terms of the spinorbit length, the hyperfine fields in the two dots, and the double dot parameters such as tunnel coupling and Coulomb energy. This formula should prove useful for extracting spinorbit parameters from transport or charge sensing experiments in such systems. We identify a parameter regime where the spinorbit and hyperfine terms can become of comparable strength, and discuss how this regime might be reached.

Incoherent dynamics in the toric code subject to disorder 
Beat Roethlisberger, James R. Wootton, Robert M. Heath (Leeds), Jiannis K. Pachos (Leeds), and Daniel Loss. Phys. Rev. A 85, 022313 (2012); arXiv:1112.1613.
We numerically study the effects of two forms of quenched disorder on the
anyons of the toric code. Firstly, a new class of codes based on random
lattices of stabilizer operators is presented, and shown to be superior to the
standard square lattice toric code for certain forms of biased noise. It is
further argued that these codes are close to optimal, in that they tightly
reach the upper bound of error thresholds beyond which no correctable CSS codes
can exist. Additionally, we study the classical motion of anyons in toric codes
with randomly distributed onsite potentials. In the presence of repulsive
longrange interaction between the anyons, a surprising increase with disorder
strength of the lifetime of encoded states is reported and explained by an
entirely incoherent mechanism. Finally, the coherent transport of the anyons in
the presence of both forms of disorder is investigated, and a significant
suppression of the anyon motion is found.

Localized end states in density modulated quantum wires and rings 
Suhas Gangadharaiah, Luka Trifunovic, and Daniel Loss. Phys. Rev. Lett. 108, 136803 (2012); arXiv:1111.5273.
We study finite quantum wires and rings in the presence of a chargedensity wave gap induced by a periodic modulation of the chemical potential. We show that the TammShockley bound states emerging at the ends of the wire are stable against weak disorder and interactions, for discrete open chains and for continuum systems. The lowenergy physics can be mapped onto the JackiwRebbi equations describing massive Dirac fermions and bound end states. We treat interactions via the continuum model and show that they increase the charge gap and further localize the end states. The electrons placed in the two localized states on the opposite ends of the wire can interact via exchange interactions and this setup can be used as a double quantum dot hosting spin qubits. The existence of these states could be experimentally detected through the presence of an unusual 4\pi AharonovBohm periodicity in the spectrum and persistent current as a function of the external flux.

Rashba spin orbit interaction in a quantum wire superlattice 
Gunnar Thorgilsson (Reykjavik), J. Carlos Egues (Sao Carlos), Daniel Loss, and Sigurdur I. Erlingsson (Reykjavik). Phys. Rev. B 85, 045306 (2012); Phys. Rev. B 85, 039904(E) (2012); arXiv:1111.1534.
In this work we study the effects of a longitudinal periodic potential on a
parabolic quantum wire defined in a twodimensional electron gas with Rashba
spinorbit interaction. For an infinite wire superlattice we find, by direct
diagonalization, that the energy gaps are shifted away from the usual Bragg
planes due to the Rashba spinorbit interaction. Interestingly, our results
show that the location of the band gaps in energy can be controlled via the
strength of the Rashba spinorbit interaction. We have also calculated the
charge conductance through a periodic potential of a finite length via the
nonequilibrium Green's function method combined with the Landauer formalism.
We find dips in the conductance that correspond well to the energy gaps of the
infinite wire superlattice. From the infinite wire energy dispersion, we derive
an equation relating the location of the conductance dips as a function of the
(gate controllable) Fermi energy to the Rashba spinorbit coupling strength. We
propose that the strength of the Rashba spinorbit interaction can be extracted
via a charge conductance measurement.

Longdistance spinspin coupling via floating gates 
Luka Trifunovic, Oliver Dial (Harvard), Mircea Trif, James R. Wootton, Rediet Abebe (Harvard), Amir Yacoby (Harvard), and Daniel Loss. Synopsis; Phys. Rev. X 2, 011006 (2012); arXiv:1110.1342.
The electron spin is a natural two level system that allows a qubit to be
encoded. When localized in a gate defined quantum dot, the electron spin
provides a promising platform for a future functional quantum computer. The
essential ingredient of any quantum computer is entanglementbetween electron
spin qubitscommonly achieved via the exchange interaction. Nevertheless,
there is an immense challenge as to how to scale the system up to include many
qubits. Here we propose a novel architecture of a large scale quantum computer
based on a realization of longdistance quantum gates between electron spins
localized in quantum dots. The crucial ingredients of such a longdistance
coupling are floating metallic gates that mediate electrostatic coupling over
large distances. We show, both analytically and numerically, that distant
electron spins in an array of quantum dots can be coupled selectively, with
coupling strengths that are larger than the electron spin decay and with
switching times on the order of nanoseconds.

Crossed Andreev Reflection in Quantum Wires with Strong SpinOrbit Interaction 
Koji Sato (UCLA), Daniel Loss, and Yaroslav Tserkovnyak (UCLA). Phys. Rev. B 85, 235433 (2012); arXiv:1109.6357.
We theoretically study tunneling of Cooper pairs from an swave
superconductor into two semiconductor quantum wires with strong spinorbit
interaction under magnetic field, which approximate helical Luttinger liquids.
The entanglement of electrons within a Cooper pair can be detected by the
electric current cross correlations in the wires. By controlling the relative
orientation of the wires, either lithographically or mechanically, on the
substrate, the current correlations can be tuned, as dictated by the initial
spin entanglement. This proposal of a spintocharge readout of quantum
correlations is alternative to a recently proposed utilization of the quantum
spin Hall insulator.

Strong SpinOrbit Interaction and Helical Hole States in Ge/Si Nanowires 
Christoph Kloeffel, Mircea Trif, and Daniel Loss. Phys. Rev. B 84, 195314 (2011); arXiv:1107.4870.
We study theoretically the lowenergy hole states of Ge/Si core/shell
nanowires. The lowenergy valence band is quasidegenerate, formed by two
doublets of different orbital angular momentum, and can be controlled via the
relative shell thickness and via external fields. We find that direct (dipolar)
coupling to a moderate electric field leads to an unusually large spinorbit
interaction of Rashbatype on the order of meV which gives rise to pronounced
helical states enabling electrical spincontrol. The system allows for quantum
dots and spinqubits with energy levels that can vary from nearly zero to
several meV, depending on the relative shell thickness.

libCreme: An optimization library for evaluating convexroof entanglement measures 
Beat Roethlisberger, Joerg Lehmann (ABB Baden), and Daniel Loss. Comput. Phys. Comm. 183, 155 (2012); arXiv:1107.4497.
We present the software library libCreme which we have previously used to
successfully calculate convexroof entanglement measures of mixed quantum
states appearing in realistic physical systems. Evaluating the amount of
entanglement in such states is in general a nontrivial task requiring to solve
a highly nonlinear complex optimization problem. The algorithms provided here
are able to achieve to do this for a large and important class of entanglement
measures. The library is mostly written in the Matlab programming language, but
is fully compatible to the free and opensource Octave platform. Some
inefficient subroutines are written in C/C++ for better performance. This
manuscript discusses the most important theoretical concepts and workings of
the algorithms, focussing on the actual implementation and usage within the
library. Detailed examples in the end should make it easy for the user to apply
libCreme to specific problems.

Absence of spontaneous magnetic order of lattice spins coupled to itinerant interacting electrons in one and two dimensions 
Daniel Loss, Fabio L. Pedrocchi, and Anthony J. Leggett (Urbana). Phys. Rev. Lett. 107, 107201 (2011); arXiv:1107.1223; Synopsis.
We extend the MerminWagner theorem to a system of lattice spins which are spin coupled to itinerant and interacting charge carriers. We use the Bogoliubov inequality to rigorously prove that neither (anti) ferromagnetic nor helical longrange order is possible in one and two dimensions at any finite temperature. Our proof applies to a wide class of models including any form of electronelectron and singleelectron interactions that are independent of spin. In the presence of Rashba or Dresselhaus spinorbit interactions (SOI) magnetic order is not excluded and intimately connected to equilibrium spin currents. However, in the special case when Rashba and Dresselhaus SOIs are tuned to be equal, magnetic order is excluded again. This opens up a new possibility to control magnetism electrically.

Carbon nanotubes in electric and magnetic fields 
Jelena Klinovaja, Manuel J. Schmidt, Bernd Braunecker, and Daniel Loss. Phys. Rev. B 84, 085452 (2011); arXiv:1106.3332.
We derive an effective lowenergy theory for metallic (armchair and nonarmchair) singlewall nanotubes in the presence of an electric field perpendicular to the nanotube axis, and in the presence of magnetic fields, taking into account spinorbit interactions and screening effects on the basis of a microscopic tightbinding model. The interplay between electric field and spinorbit interaction allows us to tune armchair nanotubes into a helical conductor in both Dirac valleys. Metallic nonarmchair nanotubes are gapped by the surface curvature, yet helical conduction modes can be restored in one of the valleys by a magnetic field along the nanotube axis. Furthermore, we discuss electric dipole spin resonance in carbon nanotubes, and find that the Rabi frequency shows a pronounced dependence on the momentum along the nanotube.

Low Bias Negative Differential Resistance in Graphene Nanoribbon Superlattices 
Gerson J. Ferreira (S. Carlos), Michael N. Leuenberger (Orlando), Daniel Loss, and J. Carlos Egues (S. Carlos). Phys. Rev. B 84, 125453 (2011); arXiv:1105.4850.
We theoretically investigate negative differential resistance (NDR) for
ballistic transport in semiconducting armchair graphene nanoribbon (aGNR)
superlattices (5 to 20 barriers) at low bias voltages $V_{SD} < 500$ mV. We
combine the graphene Dirac hamiltonian with the LandauerB\"uttiker formalism
to calculate the current $I_{SD}$ through the system. We find three distinct
transport regimes in which NDR occurs: (i) a "classical" regime for wide
layers, through which the transport across bandgaps is strongly suppressed,
leading to alternating regions of nearly unity and zero transmission
probabilities as a function of $V_{SD}$ due to crossing of bandgaps from
different layers. (ii) a quantum regime dominated by superlattice miniband
conduction, with current suppression arising from the misalignment of miniband
states with increasing $V_{SD}$; and (iii) a WannierStark ladder regime with
current peaks occurring at the crossings of WannierStark rungs from distinct
ladders. We observe NDR at voltage biases as low as 10 mV with a high current
density, making the aGNR superlattices attractive for device applications.

Physical solutions of the Kitaev honeycomb model 
Fabio L. Pedrocchi, Stefano Chesi (McGill Univ.), and Daniel Loss. Phys. Rev. B 84, 165414 (2011); arXiv:1105.4573.
We investigate the exact solution of the honeycomb model proposed by Kitaev and derive an explicit formula for the projector onto the physical subspace. The physical states are simply characterized by the parity of the total occupation of the fermionic eigenmodes. We consider a general lattice on a torus and show that the physical fermion parity depends in a nontrivial way on the vortex configuration and the choice of boundary conditions. In the vortexfree case with a constant gauge field we are able to obtain an analytical expression of the parity. For a general configuration of the gauge field the parity can be easily evaluated numerically, which allows the exact diagonalization of large spin models. We consider physically relevant quantities, as in particular the vortex energies, and show that their true value and associated states can be substantially different from the one calculated in the unprojected space, even in the thermodynamic limit.

SchriefferWolff transformation for quantum manybody systems 
Sergey Bravyi (IBM Yorktown), David DiVincenzo (Julich), and Daniel Loss. Annals of Physics 326, 27932826 (2011); arXiv:1105.0675.
The SchriefferWolff (SW) method is a version of degenerate perturbation
theory in which the lowenergy effective Hamiltonian H_{eff} is obtained from
the exact Hamiltonian by a unitary transformation decoupling the lowenergy and
highenergy subspaces. We give a selfcontained summary of the SW method with a
focus on rigorous results. We begin with an exact definition of the SW
transformation in terms of the socalled direct rotation between linear
subspaces. From this we obtain elementary proofs of several important
properties of H_{eff} such as the linked cluster theorem. We then study the
perturbative version of the SW transformation obtained from a Taylor series
representation of the direct rotation. Our perturbative approach provides a
systematic diagram technique for computing highorder corrections to H_{eff}.
We then specialize the SW method to quantum spin lattices with shortrange
interactions. We establish unitary equivalence between effective lowenergy
Hamiltonians obtained using two different versions of the SW method studied in
the literature. Finally, we derive an upper bound on the precision up to which
the ground state energy of the nth order effective Hamiltonian approximates
the exact ground state energy.

Universal quantum computation with topological spinchain networks 
Yaroslav Tserkovnyak (UCLA) and Daniel Loss. Phys. Rev. A 84, 032333 (2011); arXiv:1104.1210.
It is shown that anisotropic spin chains with gapped bulk excitations and magnetically ordered ground states offer a promising platform for quantum computation, which bridges the conventional singlespinbased qubit concept with recently developed topological Majoranabased proposals. We show how to realize the singlequbit Hadamard, phase, and π/8 gates as well as the twoqubit controllednot (cnot) gate, which together form a faulttolerant universal set of quantum gates. The gates are implemented by judiciously controlling Ising exchange and magnetic fields along a network of spin chains, with each individual qubit furnished by a spinchain segment. A subset of singlequbit operations is geometric in nature, relying on control of anisotropy of spin interactions rather than their strength. We contrast topological aspects of the anisotropic spinchain networks to those of pwave superconducting wires discussed in the literature.

Rectification of spin currents in spin chains 
Kevin A. van Hoogdalem and Daniel Loss. Phys. Rev. B 84, 024402 (2011); arXiv:1102.4801.
We study spin transport in nonitinerant onedimensional quantum spin chains.
Motivated by possible applications in spintronics, we consider rectification
effects in both ferromagnetic and antiferromagnetic systems. We find that the
crucial ingredients in designing a system that displays a nonzero
rectification current are an anisotropy in the exchange interaction of the spin
chain combined with an offset magnetic field. For both ferromagnetic and
antiferromagnetic systems we can exploit the gap in the excitation spectrum
that is created by a bulk anisotropy to obtain a measurable rectification
effect at realistic magnetic fields. For antiferromagnetic systems we also find
that we can achieve a similar effect by introducing a magnetic impurity,
obtained by altering two neighboring bonds in the spin Hamiltonian.

Helical modes in carbon nanotubes generated by strong electric fields 
Jelena Klinovaja, Manuel J. Schmidt, Bernd Braunecker, and Daniel Loss. Phys. Rev. Lett. 106, 156809 (2011); arXiv:1011.3630.
Helical modes, conducting opposite spins in opposite directions, are shown to
exist in metallic armchair nanotubes in an allelectric setup. This is a
consequence of the interplay between spinorbit interaction and strong electric
fields. The helical regime can also be obtained in chiral metallic nanotubes by
applying an additional magnetic field. In particular, it is possible to obtain
helical modes at one of the two Dirac points only, while the other one remains
gapped. Starting from a tightbinding model we derive the effective lowenergy
Hamiltonian and the resulting spectrum.

Majorana edge states in interacting onedimensional systems 
Suhas Gangadharaiah, Bernd Braunecker, Pascal Simon (Orsay, Paris), and Daniel Loss. Phys. Rev. Lett. 107, 036801 (2011); arXiv:1101.0094.
We show that onedimensional electron systems in the proximity of a superconductor that support Majorana edge states are extremely susceptible to electronelectron interactions. Strong interactions generically destroy the induced superconducting gap that stabilizes the Majorana edge states. For weak interactions, the renormalization of the gap is nonuniversal and allows for a regime in which the Majorana edge states persist. We present strategies of how this regime can be reached.

Spectrum of an electron spin coupled to an unpolarized bath of nuclear spins 
Oleksandr Tsyplyatyev (University of Sheffield) and Daniel Loss. Phys. Rev. Lett. 106, 106803 (2011); arXiv:1102.2426.
The main source of decoherence for an electron spin confined to a quantum dot
is the hyperfine interaction with nuclear spins. To analyze this process
theoretically we diagonalize the central spin Hamiltonian in the high magnetic
Bfield limit. Then we project the eigenstates onto an unpolarized state of the
nuclear bath and find that the resulting density of states has Gaussian tails.
The level spacing of the nuclear sublevels is exponentially small in the middle
of each of the two electron Zeeman levels but increases superexponentially
away from the center. This suggests to select states from the wings of the
distribution when the system is projected on a single eigenstate by a
measurement to reduce the noise of the nuclear spin bath. This theory is valid
when the external magnetic field is larger than a typical Overhauser field at
high nuclear spin temperature.

Quantum memory coupled to cavity modes 
Fabio L. Pedrocchi, Stefano Chesi, and Daniel Loss. Phys. Rev. B.83, 115415 (2011); arXiv:1011.3762.
Inspired by spinelectric couplings in molecular magnets, we introduce in the Kitaev honeycomb model a linear modification of the Ising interactions due to the presence of quantized cavity fields. This allows to control the properties of the lowenergy toric code Hamiltonian, which can serve as a quantum memory, by tuning the physical parameters of the cavity modes, like frequencies, photon occupations, and coupling strengths. We study the properties of the model perturbatively by making use of the SchriefferWolff transformation and show that, depending on the specific setup, the cavity modes can be useful in several ways. They allow to detect the presence of anyons through frequency shifts and to prolong the lifetime of the memory by enhancing the anyon excitation energy or mediating longrange anyonanyon interactions with tunable sign. We consider both resonant and largely detuned cavity modes.

Controlling the electron spinnuclear spin interaction of a quantum dot in the tunneling regime 
C. Kloeffel, P. A. Dalgarno (Edinburgh), B. Urbaszek (Toulouse), B. D. Gerardot (Edinburgh), D. Brunner (Edinburgh), P. M. Petroff (UC Santa Barbara), D. Loss, and R. J. Warburton. Phys. Rev. Lett. 106, 046802 (2011); arXiv:1010.3330.
We present a technique for manipulating the nuclear spins and the emission
polarization from a single opticallyactive quantum dot. When the quantum dot
is tunnel coupled to a Fermi sea, we have discovered a natural cycle in which
an electron spin is repeatedly created with resonant optical excitation. The
spontaneous emission polarization and the nuclear spin polarization exhibit a
bistability. For a sigma(+) pump, the emission switches from sigma(+) to
sigma() at a particular detuning of the laser. Simultaneously, the nuclear
spin polarization switches from positive to negative. Away from the
bistability, the nuclear spin polarization can be changed continuously from
negative to positive, allowing precise control via the laser wavelength.

Hybridization and spin decoherence in heavyhole quantum dots 
Jan Fischer and Daniel Loss. Phys. Rev. Lett. 105, 266603 (2010); arXiv:1009.5195.
We theoretically investigate the spin dynamics of a heavy hole confined to a IIIV semiconductor quantum dot interacting with a narrowed nuclearspin bath. We show that band hybridization leads to an exponential decay of holespin superpositions due to hyperfinemediated nuclear pair flips, and that the accordant singleholespin decoherence time T2 can be tuned over many orders of magnitude by changing external parameters. In particular, we show that, under experimentally accessible conditions, it is possible to suppress hyperfinemediated nuclearpairflip processes so strongly that holespin quantum dots may be operated beyond the `ultimate limitation' set by the hyperfine interaction which is present in other spinqubit candidate systems.

Geometric Correlations and Breakdown of Mesoscopic Universality in Spin Transport 
I. Adagideli (Istanbul), Ph. Jacquod (Tucson, AZ), M. Scheid (Regensburg), M. Duckheim (Berlin), D. Loss, and K. Richter (Regensburg). Phys. Rev. Lett. 105, 246807 (2010); arXiv:1008.4656.
We construct a unified semiclassical theory of charge and spin transport in chaotic ballistic and
disordered diffusive mesoscopic systems with spinorbit interaction. Neglecting dynamic effects of spinorbit
interaction, we reproduce the random matrix theory results that the spin conductance fluctuates
universally around zero average. Incorporating these effects into the theory, we show that geometric
correlations generate finite average spin conductances, but that they do not affect the charge conductance
to leading order. The theory, which is confirmed by numerical transport calculations, allows us to
investigate the entire range from the weak to the previously unexplored strong spinorbit regime, where
the spin rotation time is shorter than the momentum relaxation time.

Energy spectra for quantum wires and 2DEGs in magnetic fields with Rashba and Dresselhaus spinorbit interactions 
Sigurdur I. Erlingsson (Reykjavik), J. Carlos Egues (Sao Carlos), and Daniel Loss. Phys. Rev. B82, 155456 (2010); arXiv:1008.1317.
We introduce an analytical approximation scheme to diagonalize parabolically
confined two dimensional electron systems with both the Rashba and Dresselhaus
spinorbit interactions. The starting point of our perturbative expansion is a
zerothorder Hamiltonian for an electron confined in a quantum wire with an
effective spinorbit induced magnetic field along the wire, obtained by
properly rotating the usual spinorbit Hamiltonian. We find that the
spinorbitrelated transverse coupling terms can be recast into two parts W and
V, which couple crossing and noncrossing adjacent transverse modes,
respectively. Interestingly, the zerothorder Hamiltonian together with W can
be solved exactly, as it maps onto the JaynesCummings model of quantum optics.
We treat the V coupling by performing a SchriefferWolff transformation. This
allows us to obtain an effective Hamiltonian to third order in the coupling
strength k_Rl of V, which can be straightforwardly diagonalized via an
additional unitary transformation. We also apply our approach to other types of
effective parabolic confinement, e.g., 2D electrons in a perpendicular magnetic
field. To demonstrate the usefulness of our approximate eigensolutions, we
obtain analytical expressions for the n^th Landaulevel g_nfactors in the
presence of both Rashba and Dresselhaus couplings. For small values of the bulk
gfactors, we find that spinorbit effects cancel out entirely for particular
values of the spinorbit couplings. By solving simple transcendental equations
we also obtain the band minima of a Rashbacoupled quantum wire as a function
of an external magnetic field. These can be used to describe Shubnikovde Haas
oscillations. This procedure makes it easier to extract the strength of the
spinorbit interaction in these systems via proper fitting of the data.

Poor man's derivation of the BetheAnsatz equations for the Dicke model 
Oleksandr Tsyplyatyev, Jan von Delft (LMU Munich), and Daniel Loss. Phys. Rev. B 82, 092203 (2010); arXiv:1008.1844.
We present an elementary derivation of the exact solution (BetheAnsatz
equations) of the Dicke model, using only commutation relations and an informed
Ansatz for the structure of its eigenstates.

Spin susceptibility of interacting twodimensional electrons in the presence of spinorbit coupling 
Robert Andrzej Zak, Dmitrii L. Maslov (Gainesville, FL), and Daniel Loss. Phys. Rev. B 82, 115415 (2010); arXiv:1005.1913.
A longrange interaction via virtual particlehole pairs between Fermiliquid
quasiparticles leads to a nonanalytic behavior of the spin susceptibility
$\chi$ as a function of the temperature ($T$), magnetic field ($\mathbf{B}$),
and wavenumber. In this paper, we study the effect of the Rashba spinorbit
interaction (SOI) on the nonanalytic behavior of $\chi$ for a twodimensional
electron liquid. Although the SOI breaks the $SU(2)$ symmetry, it does not
eliminate nonanalyticity but rather makes it anisotropic: while the linear
scaling of $\chi_{zz}$ with $T$ and $\mathbf{B}$ saturates at the energy
scale set by the SOI, that of $\chi_{xx}$ ($=\chi_{yy}$) continues through this
energy scale, until renormalization of the electronelectron interaction in the
Cooper channel becomes important. We show that the Renormalization Group flow
in the Cooper channel has a nontrivial fixed point, and study the consequences
of this fixed point for the nonanalytic behavior of $\chi$. An immediate
consequence of SOIinduced anisotropy in the nonanalytic behavior of $\chi$ is
a possible instability of a secondorder ferromagnetic quantum phase transition
with respect to a firstorder transition to an $XY$ ferromagnetic state.

Tunable edge magnetism at graphene/graphane interfaces 
Manuel J. Schmidt and Daniel Loss. Phys. Rev. B 82, 085422 (2010); arXiv:1004.4363.
We study the magnetic properties of graphene edges and graphene/graphane
interfaces under the influence of electrostatic gates. For this, an effective
lowenergy theory for the edge states, which is derived from the Hubbard model
of the honeycomb lattice, is used. We first study the edge state model in a
meanfield approximation for the Hubbard Hamiltonian and show that it
reproduces the results of the extended 2D lattice theory. Quantum fluctuations
around the meanfield theory of the effective onedimensional model are treated
by means of the bosonization technique in order to check the stability of the
meanfield solution. We find that edge magnetism at graphene/graphane
interfaces can be switched on and off by means of electrostatic gates. We
describe a quantum phase transition between an ordinary and a ferromagnetic
Luttinger liquid  a realization of itinerant onedimensional ferromagnetism.
This mechanism may provide means to experimentally discriminate between edge
magnetism or disorder as the reason for a transport gap in very clean graphene
nanoribbons.

RKKY interaction in a disordered twodimensional electron gas with Rashba and Dresselhaus spinorbit couplings 
Stefano Chesi and Daniel Loss. Phys. Rev. B.82,165303 (2010); arXiv:1007.3506.
We study theoretically the statistical properties of the
RudermanKittelKasuyaYosida (RKKY) interaction between localized magnetic
moments in a disordered twodimensional electron gas with both Rashba and
Dresselhaus spinorbit couplings. Averaging over disorder, the static spin
susceptibility tensor is evaluated diagrammatically in the mesoscopic
(phasecoherent) regime. The disorderaveraged susceptibility leads to a
twisted exchange interaction suppressed exponentially with distance, whereas
the secondorder correlations, which determine the fluctuations (variance) of
the RKKY energy, decay with the same powerlaw as in the clean case. We obtain
analytic expressions in the limits of large/small spin orbit interactions and
for equal Rashba and Dresselhaus couplings. Beside these limiting cases, we
study numerically the variance of the RKKY interaction in the presence of pure
Rashba spinorbit coupling. Our results are relevant for magnetic or nuclear
moments embedded in IIIV twodimensional heterostructures or in contact with
surface states of metals and metal alloys, which can display a sizable Rashba
spinorbit coupling.

Spinselective Peierls transition in interacting onedimensional conductors with spinorbit interaction 
Bernd Braunecker, George I. Japaridze (Tbilisi, Georgia), Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 82, 045127 (2010); arXiv:1004.0467.
Interacting onedimensional conductors with Rashba spinorbit coupling are
shown to exhibit a spinselective Peierlslike transition into a mixed
spincharge density wave state. The transition leads to a gap for onehalf of
the conducting modes, which is strongly enhanced by electronelectron
interactions. The other half of the modes remains in a strongly renormalized
gapless state and conducts opposite spins in opposite directions, thus
providing a perfect spin filter. The transition is driven by magnetic field and
by spinorbit interactions. As an example we show for semiconducting quantum
wires and carbon nanotubes that the gap induced by weak magnetic fields or
intrinsic spinorbit interactions can get renormalized by one order of
magnitude up to 1030 Kelvins.

CooperPair Injection into Topological Insulators 
Koji Sato (UCLA), Daniel Loss, and Yaroslav Tserkovnyak (UCLA). Phys. Rev. Lett. 105, 226401 (2010); arXiv:1003.4316.
We theoretically study tunneling of Cooper pairs (CP's) from a superconductor
spanning a twodimensional topological insulator strip into its helical edge
states. The coherent lowenergy electronpair tunneling sets off positive
nonlocal current crosscorrelations along the edges, which reflect an interplay
of two quantumentanglement mechanisms. First of all, superconducting spin
pairing dictates a CP partitioning into the helical edge liquids, which
transport electrons in the opposite directions for opposite spin orientations.
Luttingerliquid (LL) correlations for the electrondensity fluctuations are,
furthermore, forcing paired electrons to enter into opposite insulatorstrip
edges, revealing CP spin entanglement in the interedge current correlations.
At the same time, the LL behavior, in the absence of Fermiliquid leads,
fractionalizes electrons injected at a given edge into counterpropagating
charge pulses carrying definite fractions of the elementary electron charge.
The superconductivity as well as LL correlations thus introduce positive
current crosscorrelations, which reveal a wealth of information about both
subsystems.

The classical and quantum dynamics of the inhomogeneous Dicke model and its Ehrenfest time 
Oleksandr Tsyplyatyev and Daniel Loss. Phys. Rev. B 82, 024305 (2010); arXiv:1002.3932.
We show that in the fewexcitation regime the classical and quantum timeevolution of the inhomogeneous Dicke model for N twolevel systems coupled to a single boson mode agree for N>>1. In the presence of a single excitation only, the leading term in an 1/Nexpansion of the classical equations of motion reproduces the result of the Schroedinger equation. For a small number of excitations, the numerical solutions of the classical and quantum problems become equal for N sufficiently large. By solving the Schroedinger equation exactly for two excitations and a particular inhomogeneity we obtain 1/Ncorrections which lead to a significant difference between the classical and quantum solutions at a new time scale which we identify as an Ehrenferst time, given by tau_E=sqrt{N

Spin electric effects in molecular antiferromagnets 
Mircea Trif, Filippo Troiani (Modena), Dimitrije Stepanenko, and Daniel Loss. Phys. Rev. B 82, 045429 (2010); arXiv:1001.3584.
Molecular nanomagnets show clear signatures of coherent behavior and have a wide variety of effective
lowenergy spin Hamiltonians suitable for encoding qubits and implementing spinbased quantum information
processing. At the nanoscale, the preferred mechanism for the control of a quantum systems is the application
of electric fields, which are strong, can be locally applied, and rapidly switched. In this work, we provide the
theoretical tools for identifying molecular nanomagnets suitable for electric control. By grouptheoretical
symmetry analysis we find that the spinelectric coupling in triangular molecules is governed by the modification
of the exchange interaction and is possible even in the absence of spinorbit coupling. In pentagonal
molecules the spinelectric coupling can exist only in the presence of spinorbit interaction. This kind of
coupling is allowed for both s=1/2 and s=3/2 spins at the magnetic centers. Within the Hubbard model, we
find a relation between the spinelectric coupling and the properties of the chemical bonds in a molecule,
suggesting that the best candidates for strong spinelectric coupling are molecules with nearly degenerate bond
orbitals. We also investigate the possible experimental signatures of spinelectric coupling in nuclear magnetic
resonance and electron spin resonance spectroscopy, as well as in the thermodynamic measurements of magnetization,
electric polarization, and specific heat of the molecules.

Magnetic order in nuclear spin twodimensional lattices due to electronelectron interactions. 
P. Simon, B. Braunecker, and D. Loss, Physica E 42 (2010) 634638
We focus on nuclear spins embedded in a twodimensional (2D) electron gas. The nuclear spins interact with each other through the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, which is carried by the electron gas. We show that a nuclear magnetic order at finite temperature relies on the anomalous behaviour of the 2D static electron spin susceptibility due to electron–electron interactions. This provides a connection between lowdimensional magnetism and nonanalyticities in interacting 2D electron systems. We discuss the conditions for nuclear magnetism, and show that the associated Curie temperature increases with the electron–electron interactions and may reach up into the millikelvin regime. We also shortly discussed what happens when the dimensionality is further reduced to one dimension.

Freeinduction decay and envelope modulations in a narrowed nuclear spin bath 
W. A. Coish (Waterloo), Jan Fischer, and Daniel Loss. Phys. Rev. B 81, 165315 (2010); arXiv:0911.4149.
We evaluate freeinduction decay for the transverse components of a localized electron spin coupled to a bath of nuclear spins via the Fermi contact hyperfine interaction. Our perturbative treatment is valid for special (narrowed) bath initial conditions and when the Zeeman energy of the electron $b$ exceeds the total hyperfine coupling constant $A$: $b>A$. Using one unified and systematic method, we recover previous results reported at short and long times using different techniques. We find a new and unexpected modulation of the freeinductiondecay envelope, which is present even for a purely isotropic hyperfine interaction without spin echoes and for a single nuclear species. We give subleading corrections to the decoherence rate, and show that, in general, the decoherence rate has a nonmonotonic dependence on electron Zeeman splitting, leading to a pronounced maximum. These results illustrate the limitations of methods that make use of leadingorder effective Hamiltonians and reexponentiation of shorttime expansions for a stronglyinteracting system with nonMarkovian (historydependent) dynamics.

Edge states and enhanced spinorbit interaction at graphene/graphane interfaces 
Manuel J. Schmidt and Daniel Loss. Phys. Rev. B 81, 165439 (2010); arXiv:0910.5333.
We study interfaces between graphene and graphane. If the interface is oriented along a zigzag direction, edge states are found which exhibit a strong amplification of effects related to the spinorbit interaction. The enhanced spin splitting of the edge states allows a conversion between valley polarization and spin polarization at temperatures near one Kelvin. We show that these edge states give rise to quantum spin and/or valley Hall effects.

Onestep multiqubit GHZ state generation in a circuit QED system 
YingDan Wang, Stefano Chesi, Daniel Loss, and Christoph Bruder. Phys. Rev. B 81, 104524 (2010); arXiv:0911.1396.
We propose a onestep scheme to generate GHZ states for superconducting flux qubits or charge qubits in a circuit QED setup. The GHZ state can be produced within the coherence time of the multiqubit system. Our scheme is independent of the initial state of the transmission line resonator and works in the presence of higher harmonic modes. Our analysis also shows that the scheme is robust to various operation errors and environmental noise.

Spin Accumulation in Diffusive Conductors with Rashba and Dresselhaus SpinOrbit Interaction 
Mathias Duckheim, Daniel Loss, Matthias Scheid (Regensburg), Klaus Richter (Regensburg), Inanc Adagideli (Istanbul), and Philippe Jacquod (Tucson). Physical Review B 81 085303 (2010); arXiv:0909.4253.
We calculate the electrically induced spin accumulation in diffusive systems due to both Rashba (with strength $\alpha)$ and Dresselhaus (with strength $\beta)$ spinorbit interaction. Using a diffusion equation approach we find that magnetoelectric effects disappear and that there is thus no spin accumulation when both interactions have the same strength, $\alpha=\pm \beta$. In thermodynamically large systems, the finite spin accumulation predicted by Chaplik, Entin and Magarill, [Physica E 13, 744 (2002)] and by Trushin and Schliemann [Phys. Rev. B 75, 155323 (2007)] is recovered an infinitesimally small distance away from the singular point $\alpha=\pm \beta$. We show however that the singularity is broadened and that the suppression of spin accumulation becomes physically relevant (i) in finitesized systems of size $L$, (ii) in the presence of a cubic Dresselhaus interaction of strength $\gamma$, or (iii) for finite frequency measurements. We obtain the parametric range over which the magnetoelectric effect is suppressed in these three instances as (i) $\alpha\beta \lesssim 1/mL$, (ii)$\alpha\beta \lesssim \gamma p_{\rm F}^2$, and (iii) $\alpha\beta \lesssiM \sqrt{\omega/m p_{\rm F}\ell}$ with $\ell$ the elastic mean free path and $p_{\rm F}$ the Fermi momentum. We attribute the absence of spin accumulation close to $\alpha=\pm \beta$ to the underlying U (1) symmetry. We illustrate and confirm our predictions numerically.

Dynamic spinHall effect and driven spin helix for linear spinorbit interactions 
Mathias Duckheim, Dmitrii L. Maslov (Gainesville, FL), and Daniel Loss. Phys. Rev. B 80, 235327 (2009); arXiv:0909.1892.
We derive boundary conditions for the electrically induced spin accumulation in a finite 2D semiconductor channel. While for DC electric fields these boundary conditions select spatially constant spin profiles equivalent to a vanishing spinHall effect, we show that an inplane ac electric field results in a nonzero ac spinHall effect, i.e., it generates a spatially nonuniform outofplane polarization even for linear intrinsic spinorbit interactions. Analyzing different geometries in [001] and [110]grown quantum wells, we find that although this outofplane polarization is typically confined to within a few spinorbit lengths from the channel edges, it is also possible to generate spatially oscillating spin profiles which extend over the whole channel. The latter is due to the excitation of a driven spinhelix mode in the transverse direction of the channel. We show that while finite frequencies suppress this mode, it can be amplified by a magnetic field tuned to resonance with the frequency of the electric field. In this case, finite size effects at equal strengths of Rashba and Dresselhaus SOI lead to an enhancement of the magnitude of this helix mode. We comment on the relation between spin currents and boundary conditions. http://prb.aps.org/kaleidoscope

A SelfCorrecting Quantum Memory in a Thermal Environment 
Stefano Chesi, Beat Röthlisberger, and Daniel Loss. Phys. Rev. A 82, 022305 (2010); arXiv:0908.4264.
The ability to store information is of fundamental importance to any computer, be it classical or quantum. To identify systems for quantum memories, which rely, analogously to classical memories, on passive error protection (“selfcorrection”), is of greatest interest in quantum information science. While systems with topological ground states have been considered to be promising candidates, a large class of them was recently proven unstable against thermal fluctuations. Here, we propose twodimensional (2D) spin models unaffected by this result. Specifically, we introduce repulsive longrange interactions in the toric code and establish a memory lifetime polynomially increasing with the system size. This remarkable stability is shown to originate directly from the repulsive longrange nature of the interactions. We study the time dynamics of the quantum memory in terms of diffusing anyons and support our analytical results with extensive numerical simulations. Our findings demonstrate that selfcorrecting quantum memories can exist in 2D at finite temperatures.

Holonomic Quantum Computation with Electron Spins in Quantum Dots 
Vitaly N. Golovach (LMU Munich), Massoud Borhani (Buffalo), and Daniel Loss. Phys. Rev. A 81, 022315 (2010); arXiv:0908.2800.
With the help of the spinorbit interaction, we propose a scheme to perform holonomic single qubit gates on the electron spin confined to a quantum dot. The manipulation is done in the absence (or presence) of an applied magnetic field. By adiabatic changing the position of the confinement potential, one can rotate the spin state of the electron around the Bloch sphere in semiconductor heterostructures. The dynamics of the system is equivalent to employing an effective nonAbelian gauge potential whose structure depends on the type of the spinorbit interaction. As an example, we find an analytic expression for the electron spin dynamics when the dot is moved around a circular path (with radius R) on the two dimensional electron gas (2DEG), and show that all single qubit gates can be realized by tuning the radius and orientation of the circular paths. Moreover, using the Heisenberg exchange interaction, we demonstrate how one can generate twoqubit gates by bringing two quantum dots near each other, yielding a scalable scheme to perform quantum computing on arbitrary N qubits. This proposal shows a way of realizing holonomic quantum computers in solidstate systems.

Nuclear magnetism and electron order in interacting onedimensional conductors 
Bernd Braunecker, Pascal Simon (Orsay, Paris), and Daniel Loss. Phys. Rev. B 80, 165119 (2009); arXiv:0908.0904; News in Physics World, Feb 21 (2014).
The interaction between localized magnetic moments and the electrons of a onedimensional conductor can lead to an ordered phase in which the magnetic moments and the electrons are tightly bound to each other. We show here that this occurs when a lattice of nuclear spins is embedded in a Luttinger liquid. Experimentally available examples of such a system are single wall carbon nanotubes grown entirely from 13C and GaAsbased quantum wires. In these systems the hyperfine interaction between the nuclear spin and the conduction electron spin is very weak, yet it triggers a strong feedback reaction that results in an ordered phase consisting of a nuclear helimagnet that is inseparably bound to an electronic density wave combining charge and spin degrees of freedom. This effect can be interpreted as a strong renormalization of the nuclear Overhauser field and is a unique signature of Luttinger liquid physics. Through the feedback the order persists up into the millikelvin range. A particular signature is the reduction of the electric conductance by the universal factor 2.

Spin Hall effect due to intersubbandinduced spinorbit interaction in symmetric quantum well 
Minchul Lee (Yongin, S. Korea), Marco O. Hachiya (Sao Carlos, Brazil), E. Bernardes (Sao Carlos, Brazil), J. Carlos Egues (Sao Carlos, Brazil), and Daniel Loss. Phys. Rev. B 80, 155314 (2009); arXiv:0907.4078.
We investigate the intrinsic spin Hall effect in twodimensional electron gases in quantum wells with two subbands, where a new intersubbandinduced spinorbit coupling is operative. The bulk spin Hall conductivity $\sigma^z_{xy}$ is calculated in the ballistic limit within the standard Kubo formalism in the presence of a magnetic field $B$ and is found to remain finite in the B=0 limit, as long as only the lowest subband is occupied. Our calculated $\sigma^z_{xy}$ exhibits a nonmonotonic behavior and can change its sign as the Fermi energy (the carrier areal density $n_{2D}$) is varied between the subband edges. We determine the magnitude of $\sigma^z_{xy}$ for realistic InSb quantum wells by performing a selfconsistent calculation of the intersubbandinduced spinorbit coupling.

Thermodynamic stability criteria for a quantum memory based on stabilizer and subsystem codes 
Stefano Chesi, Daniel Loss, Sergey Bravyi (IBM Yorktown), and Barbara M. Terhal (IBM Yorktown). New J. Phys. 12 025013 (2010); arXiv:0907.2807.
We discuss and review several thermodynamic criteria that have been introduced to characterize the thermal stability of a selfcorrecting quantum memory. We first examine the use of symmetrybreaking fields in analyzing the properties of selfcorrecting quantum memories in the thermodynamic limit: we show that the thermal expectation values of all logical operators vanish for any stabilizer and any subsystem code in any spatial dimension. On the positive side, we generalize the results in [R. Alicki et al., arXiv:0811.0033] to obtain a general upper bound on the relaxation rate of a quantum memory at nonzero temperature, assuming that the quantum memory interacts via a Markovian master equation with a thermal bath. This upper bound is applicable to quantum memories based on either stabilizer or subsystem codes.

Dicke model: entanglement as a finite size effect 
Oleksandr Tsyplyatyev and Daniel Loss. J. Phys.: Conf. Ser. 193, 012134(2009); arXiv:0907.2553v1.
We analyze Dicke model at zero temperature by matrix diagonalization to determine the entanglement in the ground state. In the infinite system limit the mean field approximation predicts a quantum phase transition from a noninteracting state to a BoseEinstein condensate at a threshold coupling. We show that in a finite system the spin part of the ground state is a bipartite entangled state, which can be tested by probing two parts of the spin system separately, but only in a narrow regime around the threshold coupling. Around the resonance, the size of this regime is inversely proportional to the number of spins and shrinks down to zero for infinite systems. This spin entanglement is a nonperturbative effect and is also missed by the meanfield approximation.

Quantum Computing with Electron Spins in Quantum Dots 
Robert Andrzej Żak, Beat Röhlisberger, Stefano Chesi, and Daniel Loss. Lecture notes for Course CLXXI "Quantum Coherence in Solid State Systems" Int. School of Physics "Enrico Fermi", Varenna, July 2008. Rivista del Nuovo Cimento 033 (Issue 07), 345399 (2010); arXiv:0906.4045.
Several topics on the implementation of spin qubits in quantum dots are reviewed. We first provide an introduction to the standard model of quantum computing and the basic criteria for its realization. Other alternative formulations such as measurementbased and adiabatic quantum computing are briefly discussed. We then focus on spin qubits in single and double GaAs electron quantum dots and review recent experimental achievements with respect to initialization, coherent manipulation and readout of the spin states. We extensively discuss the problem of decoherence in this system, with particular emphasis on its theoretical treatment and possible ways to overcome it.

Hyperfine interaction and electronspin decoherence in graphene and carbon nanotube quantum dots 
Jan Fischer, Bjoern Trauzettel (Wuerzburg), and Daniel Loss. Phys. Rev. B 80, 155401 (2009); arXiv:0906.2800.
We analytically calculate the nuclearspin interactions of a single electron confined to a carbon nanotube or graphene quantum dot. While the conductionband states in graphene are ptype, the accordant states in a carbon nanotube are sphybridized due to curvature. This leads to an interesting interplay between isotropic and anisotropic hyperfine interactions. By using only analytical methods, we are able to show how the interaction strength depends on important physical parameters, such as curvature and isotope abundances. We show that for the investigated carbon structures, the 13C hyperfine coupling strength is less than 1 mueV, and that the associated electronspin decoherence time can be expected to be as long as several tens of microseconds, depending on the abundance of spincarrying 13C nuclei. Furthermore, we find an unusual hyperfineinduced alignment of the 13C nuclear spins in nanotubes of any chirality.

Dealing with Decoherence 
Jan Fischer and Daniel Loss. Perspective article Science 324, 1277 (2009)
The dream of building computers that work according to the rules of quantum mechanics has strongly driven research over the past decade in many fields of basic and applied sciences, including physics, chemistry, and computer science. About 10 years ago, it was shown mathematically that the direct use of quantum phenomena such as interference and entanglement could crucially speed up data searching and prime factorization for encryption. To turn quantum computers into reality, however, many issues in engineering and in basic physics need to be addressed.

Numerical evaluation of convexroof entanglement measures with applications to spin rings 
Beat Röthlisberger, Jörg Lehmann (ABB Baden), and Daniel Loss. Phys. Rev. A 80, 042301 (2009); arXiv:0905.3106.
We present two readytouse numerical algorithms to evaluate convexroof extensions of arbitrary purestate entanglement monotones. Their implementation leaves the user merely with the task of calculating derivatives of the respective purestate measure. We provide numerical tests of the algorithms and demonstrate their good convergence properties. We further employ them in order to investigate the entanglement in particular fewspins systems at finite temperature. Namely, we consider ferromagnetic Heisenberg exchangecoupled spin1/2 rings subject to an inhomogeneous inplane field geometry obeying full rotational symmetry around the axis perpendicular to the ring through its center. We demonstrate that highly entangled states can be obtained in these systems at sufficiently low temperatures and by tuning the strength of a magnetic field configuration to an optimal value which is identified numerically.

Nanotubes: Carbon surprises again 
Björn Trauzettel (Wuerzburg) and Daniel Loss. News and Views Nature Physics 5, 317 (2009)
Experiments in 13C nanotubes reveal surprisingly strong nuclear spin effects that, if properly harnessed, could provide a mechanism for manipulation and storage of quantum information.

Undoing a quantum measurement 
Christoph Bruder and Daniel Loss. Physics 1, 34 (2008)
Quantum measurements are conventionally thought of as irretrievably "collapsing" a wave function to the observed state. However, experiments with superconducting qubits show that the partial collapse resulting from a weak continuous measurement can be restored.

Spin interactions, relaxation and decoherence in quantum dots 
Jan Fischer, Mircea Trif, W. A. Coish (Waterloo), and Daniel Loss. Solid State Communications 149, 1443 (2009); arXiv:0903.0527.
We review recent studies on spin decoherence of electrons and holes in quasitwodimensional quantum dots, as well as electronspin relaxation in nanowire quantum dots. The spins of confined electrons and holes are considered major candidates for the realization of quantum information storage and processing devices, provided that sufficently long coherence and relaxation times can be achieved. The results presented here indicate that this prerequisite might be realized in both electron and hole quantum dots, taking one large step towards quantum computation with spin qubits.

Spin orbitinduced anisotropic conductivity of a disordered 2DEG 
Oleg Chalaev and Daniel Loss. Phys. Rev. B 80, 035305 (2009); arXiv:0902.3277.
We present a semiautomated computerassisted method to generate and calculate diagrams in the disorder averaging approach to disordered 2D conductors with intrinsic spinorbit interaction (SOI). As an application, we calculate the effect of the SOI on the charge conductivity for disordered 2D systems and rings in the presence of Rashba and Dresselhaus SOI. In an infinitesize 2D system, anisotropic corrections to the conductivity tensor arise due to phasecoherence and the interplay of Rashba and Dresselhaus SOI. The effect is more pronounced in the quasionedimensional case, where the conductivity becomes anisotropic already in the presence of only one type of SOI. The anisotropy further increases if the timereversal symmetry of the Hamiltonian is broken.

Relaxation of hole spins in quantum dots via twophonon processes 
Mircea Trif, Pascal Simon (Orsay), and Daniel Loss. Phys. Rev. Lett. 103, 106601 (2009); arXiv:0902.2457.
We investigate theoretically spin relaxation in heavy hole quantum dots in low external magnetic fields. We demonstrate that twophonon processes and spinorbit interaction are experimentally relevant and provide an explanation for the recently observed saturation of the spin relaxation rate in heavy hole quantum dots with vanishing magnetic fields. We propose further experiments to identify the relevant spin relaxation mechanisms in low magnetic fields.

Interference of heavy holes in an AharonovBohm ring 
Dimitrije Stepanenko, Minchul Lee (Marseille), Guido Burkard (Konstanz), and Daniel Loss. Phys. Rev. B 79, 235301 (2009); arXiv:0811.4566.
We study the coherent transport of heavy holes through a onedimensional ring in the presence of spinorbit coupling. Spinorbit interaction of holes, cubic in the inplane components of momentum, gives rise to an angular momentum dependent spin texture of the eigenstates and influences transport. We analyze the dependence of the resulting differential conductance of the ring on hole polarization of the leads and the signature of the textures in the AharonovBohm oscillations when the ring is in a perpendicular magnetic field. We find that the polarizationresolved conductance reveals whether the dominant spinorbit coupling is of Dresselhaus or Rashba type, and that the cubic spinorbit coupling can be distinguished from the conventional linear coupling by observing the fourpeak structure in the AharonovBohm oscillations.

Exact quantum dynamics of the inhomogeneous Dicke model 
Oleksandr Tsyplyatyev and Daniel Loss. Phys. Rev. A 80, 023803 (2009); arXiv:0811.2386.
We study the time dynamics of a single boson coupled to a bath of twolevel systems (spins 1/2) with different excitation energies, described by an inhomogeneous Dicke model. Solving the timedependent Schroedinger equation exactly we find that at resonance the boson decays in time to an oscillatory state characterized by a single Rabi frequency. In the limit of small inhomogeneity, the decay is suppressed and exhibits a complex (mainly Gaussianlike) behavior, whereas the decay is complete and of exponential form in the opposite limit. For intermediate inhomogeneity, the boson decay is partial and governed by a combination of exponential and power laws.

Momentum dependence of the spinsusceptibility in two dimensions: nonanalytic corrections in the Cooper channel 
Stefano Chesi, Robert Andrzej Żak, Pascal Simon (Orsay), and Daniel Loss. Phys. Rev. B 79, 115445 (2009); arXiv:0811.0996.
We consider the effect of rescattering of pairs of quasiparticles in the Cooper channel resulting in the strong renormalization of second order corrections to the spin susceptibility in a twodimensional electron system. We use the Fourier expansion of the scattering potential in the vicinity of the Fermi surface to find that each harmonic becomes renormalized independently. Since some of those harmonics are negative, the slope of the spin susceptibility is bound to be negative at small momenta, in contrast to the lowest order perturbation theory result, which predicts a positive slope. We present in detail an effective method to calculate diagrammatically corrections to the spin susceptibility to infinite order.

Semiconductor spintronics: Snapshots of spins separating 
Mathias Duckheim and Daniel Loss. News and Views Nature Physics 4, 836837 (2008)
Theories of the spin Hall effect suggest that spin currents generated by electric fields accumulate spin polarization at the sample edges. Now an experiment has observed this conversion in real time.

Magnetic Order in KondoLattice Systems due to ElectronElectron Interactions 
Bernd Braunecker, Pascal Simon (Orsay), and Daniel Loss. AIP Conf. Proc. 1074, 62 (2008); arXiv:0808.4063.
The hyperfine interaction between the electron spin and the nuclear spins is one of the main sources of decoherence for spin qubits when the nuclear spins are disordered. An ordering of the latter largely suppresses this source of decoherence. Here we show that such an ordering can occur through a thermodynamic phase transition in twodimensional (2D) Kondolattice type systems. We specifically focus on nuclear spins embedded in a 2D electron gas. The nuclear spins interact with each other through the RKKY interaction, which is carried by the electron gas. We show that a nuclear magnetic order at finite temperature relies on the anomalous behavior of the 2D static electron spin susceptibility due to electronelectron interactions. This provides a connection between lowdimensional magnetism and nonanalyticities in interacting 2D electron systems. We discuss the conditions for nuclear magnetism, and show that the associated Curie temperature increases with the electronelectron interactions and may reach up into the millikelvin regime. The further reduction of dimensionality to one dimension is shortly discussed.

Nuclear Magnetism and Electronic Order in 13C Nanotubes 
Bernd Braunecker, Pascal Simon, and Daniel Loss. Phys. Rev. Lett. 102, 116403 (2009); arXiv.org:0808.1685.
Nuclear spins embedded in correlated metals offer an ideal platform to investigate the effect of electronelectron interactions on magnetism of localized moments. Here we focus on a onedimensional realization of such a system: Single wall carbon nanotubes grown entirely from the 13C isotope. With this isotope the nanotube forms a nuclear spin lattice which couples through the hyperfine interaction to a Luttinger liquid if the nanotube is in the metallic regime. Even though the hyperfine interaction is very weak, the system is driven into an ordered phase which combines electron and nuclear spin degrees of freedom. This phase persists up into the millikelvin regime and leads to a reduction of the nanotube conductance by a universal factor of 2, allowing for an experimental detection by standard transport measurements.

Intersubbandinduced spinorbit interaction in quantum wells 
Rafael S. Calsaverini, Esmerindo Bernardes, J. Carlos Egues, and Daniel Loss. Phys. Rev. B 78, 155313 (2008); arXiv:0807.0771.
Recently, we have found an additional spinorbit (SO) interaction in quantum wells with two subbands [Phys. Rev. Lett. 99, 076603 (2007)]. This new SO term is nonzero even in symmetric geometries, as it arises from the intersubband coupling between confined states of distinct parities, and its strength is comparable to that of the ordinary Rashba. Starting from the $8 \times 8$ Kane model, here we present a detailed derivation of this new SO Hamiltonian and the corresponding SO coupling. In addition, within the selfconsistent Hartree approximation, we calculate the strength of this new SO coupling for realistic symmetric modulationdoped wells with two subbands. We consider gated structures with either a constant areal electron density or a constant chemical potential. In the parameter range studied, both models give similar results. By considering the effects of an external applied bias, which breaks the structural inversion symmetry of the wells, we also calculate the strength of the resulting induced Rashba couplings within each subband. Interestingly, we find that for double wells the Rashba couplings for the first and second subbands interchange signs abruptly across the zero bias, while the intersubband SO coupling exhibits a resonant behavior near this symmetric configuration. For completeness we also determine the strength of the Dresselhaus couplings and find them essentially constant as function of the applied bias.

Spin decoherence of a heavy hole coupled to nuclear spins in a quantum dot 
Jan Fischer, W. A. Coish (Waterloo), D. V. Bulaev (Chernogolovka), and Daniel Loss. Phys. Rev. B 78, 155329 (2008); arXiv:0807.0368.
We theoretically study the interaction of a heavy hole with nuclear spins in a quasitwodimensional IIIV semiconductor quantum dot and the resulting dephasing of heavyhole spin states. It has frequently been stated in the literature that heavy holes have a negligible interaction with nuclear spins. We show that this is not the case. In contrast, the interaction can be rather strong and will be the dominant source of decoherence in some cases. We also show that the form of the interaction is Isinglike, resulting in unique and interesting decoherence properties, which might provide a crucial advantage to using dotconfined hole spins for quantum information processing, as compared to electron spins.

Mesoscopic fluctuations in the spinelectric susceptibility due to Rashba spinorbit interaction 
Mathias Duckheim and Daniel Loss. Phys. Rev. Lett. 101, 226602 (2008); arXiv:0805.4143.
We investigate mesoscopic fluctuations in the spin polarization generated by a static electric field and by Rashba spinorbit interaction in a disordered 2D electron gas. In a diagrammatic approach we find that the outofplane polarization  while being zero for selfaveraging systems  exhibits large sampletosample fluctuations which are shown to be well within experimental reach. We evaluate the disorderaveraged variance of the susceptibility and find its dependence on magnetic field, spinorbit interaction, dephasing, and chemical potential difference.

SpinElectric Coupling in Molecular Magnets 
Mircea Trif, Filippo Troiani (Modena), Dimitrije Stepanenko, and Daniel Loss. Phys. Rev. Lett. 101, 217201 (2008); arXiv:0805.1158.
We study the triangular antiferromagnet Cu$_3$ in external electric fields, using symmetry group arguments and a Hubbard model approach. We identify a spinelectric coupling caused by an interplay between spin exchange, spinorbit interaction, and the chirality of the underlying spin texture of the molecular magnet. This coupling allows for the electric control of the spin (qubit) states, e.g. by using an STM tip or a microwave cavity. We propose an experimental test for identifying molecular magnets exhibiting spinelectric effects.

AC magnetization transport and power absorption in nonitinerant spin chains 
Bjoern Trauzettel (Wuerzburg), Pascal Simon (Grenoble), and Daniel Loss. Phys. Rev. Lett. 101, 017202 (2008); arXiv:0804.3697.
We investigate the ac transport of magnetization in nonitinerant quantum systems such as spin chains described by the XXZ Hamiltonian. Using linear response theory, we calculate the ac magnetization current and the power absorption of such magnetic systems. Remarkably, the difference in the exchange interaction of the spin chain itself and the bulk magnets (i.e. the magnetization reservoirs), to which the spin chain is coupled, strongly influences the absorbed power of the system. This feature can be used in future spintronic devices to control power dissipation. Our analysis allows to make quantitative predictions about the power absorption and we show that magnetic systems are superior to their electronic counter parts.

Quantum Hall ferromagnetic states and spinorbit interactions in the fractional regime 
Stefano Chesi and Daniel Loss. Phys. Rev. Lett. 101, 146803 (2008); arXiv:0804.3332.
The competition between the Zeeman energy and the Rashba and Dresselhaus spinorbit couplings is studied for fractional quantum Hall states by including correlation effects. A transition of the direction of the spinpolarization is predicted at specific values of the Zeeman energy. We show that these values can be expressed in terms of the paircorrelation function, which thus provides a way to obtain experimental access to the corresponding ground state. As specific examples, we consider the Laughlin wavefunctions and the 5/2Pfaffian state and find indications of nonanalytic features around the fractional states. We also include effects of the nuclear bath, becoming relevant in the mKregime.

Nuclear spin dynamics and Zeno effect in quantum dots and defect centers 
D. Klauser, W. A. Coish (Waterloo), and Daniel Loss. Phys. Rev. B 78, 205301 (2008); arXiv:0802.2463.
We analyze nuclear spin dynamics in quantum dots and defect centers with a bound electron under electronmediated coupling between nuclear spins due to the hyperfine interaction ("Jcoupling" in NMR). Our analysis shows that the Overhauser field generated by the nuclei at the position of the electron has shorttime dynamics quadratic in time for an initial nuclear spin state without transverse coherence. The quadratic shorttime behavior allows for an extension of the Overhauser field lifetime through a sequence of projective measurements (quantum Zeno effect). We analyze the requirements on the repetition rate of measurements and the measurement accuracy to achieve such an effect. Further, we calculate the longtime behavior of the Overhauser field for effective electron Zeeman splittings larger than the hyperfine coupling strength and find, both in a Dyson series expansion and a generalized master equation approach, that for a nuclear spin system with a sufficiently smooth polarization the electronmediated interaction alone leads only to a partial decay of the Overhauser field by an amount on the order of the inverse number of nuclear spins interacting with the electron.

Simulation of ManyBody Hamiltonians using Perturbation Theory with BoundedStrength Interactions 
Sergey Bravyi, David P. DiVincenzo, Daniel Loss, and Barbara M. Terhal. Phys. Rev. Lett. 101, 070503 (2008); arXiv:0803.2686; News & Views, M.M. Wolf, Nature Physics 4, 834 (2008).
We show how to map a given nqubit target Hamiltonian with boundedstrength kbody interactions onto a simulator Hamiltonian with twobody interactions, such that the groundstate energy of the target and the simulator Hamiltonians are the same up to an extensive error O(epsilon n) for arbitrary small epsilon. The strength of interactions in the simulator Hamiltonian depends on epsilon and k but does not depend on n. We accomplish this reduction using a new way of deriving an effective lowenergy Hamiltonian which relies on the SchriefferWolff transformation of manybody physics.

Spinorbit interaction and anomalous spin relaxation in carbon nanotube quantum dots 
Denis V. Bulaev, Bjoern Trauzettel (Wuerzburg), and Daniel Loss. Phys. Rev. B 77, 235301 (2008); arXiv:0712.3767 [condmat.meshall].
We study spin relaxation and decoherence caused by electronlattice and spinorbit interaction and predict striking effects induced by magnetic fields B. For particular values of B, destructive interference occurs resulting in ultralong spin relaxation times T1 exceeding tens of seconds. For small phonon frequencies \omega, we find a 1/\sqrt{\omega} spinphonon noise spectrum a novel dissipation channel for spins in quantum dots which can reduce T1 by many orders of magnitude. We show that nanotubes exhibit zerofield level splitting caused by spinorbit interaction. This enables an allelectrical and phasecoherent control of spin.

Exponential decay in a spin bath 
W. A. Coish (Waterloo), Jan Fischer, and Daniel Loss. Phys. Rev. B 77, 125329 (2008); arXiv:0710.3762 [condmat.meshall].
We show that the coherence of an electron spin interacting with a bath of nuclear spins can exhibit a welldefined purely exponential decay for special (`narrowed') bath initial conditions in the presence of a strong applied magnetic field. This is in contrast to the typical case, where spinbath dynamics have been investigated in the nonMarkovian limit, giving superexponential or powerlaw decay of correlation functions. We calculate the relevant decoherence time T_2 explicitly for freeinduction decay and find a simple expression with dependence on bath polarization, magnetic field, the shape of the electron wave function, dimensionality, total nuclear spin I, and isotopic concentration for experimentally relevant heteronuclear spin systems.

Anisotropic conductivity of disordered 2DEGs due to spinorbit interactions 
Oleg Chalaev and Daniel Loss. Phys. Rev. B 77, 115352 (2008); arXiv:0708.3504.
We show that the conductivity tensor of a disordered twodimensional electron gas becomes anisotropic in the presence of both Rashba and Dresselhaus spinorbit interactions (SOI). This anisotropy is a mesoscopic effect and vanishes with vanishing charge dephasing time. Using a diagrammatic approach including zero, one, and twoloop diagrams, we show that a consistent calculation needs to go beyond a Boltzmann equation approach. In the absence of charge dephasing and for zero frequency, a finite anisotropy \sigma_{xy}~ e^2/lhpf arises even for infinitesimal SOI.

Magnetic Ordering of Nuclear Spins in an Interacting 2D Electron Gas 
Pascal Simon, Bernd Braunecker, and Daniel Loss. Phys. Rev. B 77, 045108 (2008); arXiv:0709.0164.
We investigate the magnetic behavior of nuclear spins embedded in a 2D interacting electron gas using a Kondo lattice model description. We derive an effective magnetic Hamiltonian for the nuclear spins which is of the RKKY type and where the interactions between the nuclear spins are strongly modified by the electronelectron interactions. We show that the nuclear magnetic ordering at finite temperature relies on the (anomalous) behavior of the 2D static electron spin susceptibility, and thus provides a connection between lowdimensional magnetism and nonanalyticities in interacting 2D electron systems. Using various perturbative and nonperturbative approximation schemes in order to establish the general shape of the electron spin susceptibility as function of its wave vector, we show that the nuclear spins locally order ferromagnetically, and that this ordering can become global in certain regimes of interest. We demonstrate that the associated Curie temperature for the nuclear system increases with the electronelectron interactions up to the millikelvin range.

Spin dynamics in InAsnanowire quantumdots coupled to a transmission line 
Mircea Trif, Vitaly N. Golovach (LMU Munich), and Daniel Loss. Phys. Rev. B 77, 045434 (2008); arXiv:0708.2091v1.
We study theoretically electron spins in nanowire quantum dots placed inside a transmission line resonator. Because of the spinorbit interaction, the spins couple to the electric component of the resonator electromagnetic field and enable coherent manipulation, storage, and readout of quantum information in an allelectrical fashion. Coupling between distant quantumdot spins, in one and the same or different nanowires, can be efficiently performed via the resonator mode either in real time or through virtual processes. For the latter case we derive an effective spinentangling interaction and suggest means to turn it on and off. We consider both transverse and longitudinal types of nanowire quantumdots and compare their manipulation timescales against the spin relaxation times. For this, we evaluate the rates for spin relaxation induced by the nanowire vibrations (phonons) and show that, as a result of phonon confinement in the nanowire, this rate is a strongly varying function of the spin operation frequency and thus can be drastically reduced compared to lateral quantum dots in GaAs. Our scheme is a step forward to the formation of hybrid structures where qubits of different nature can be integrated in a single device.

CNOT for MultiParticle Qubits and Topological Quantum Computation based on Parity Measurements 
Oded Zilberberg, Bernd Braunecker, and Daniel Loss. Phys. Rev. A 77, 012327 (2008); arXiv:0708.1062v1.
We discuss a measurementbased implementation of a ControlledNOT (CNOT) quantum gate. Such a gate has recently been discussed for free electron qubits. Here we extend this scheme for qubits encoded in product states of two (or more) spins1/2 or in equivalent systems. The key to such an extension is to find a feasible qubitparity meter. We present a general scheme for reducing this meter to a local spinparity measurement performed on two spins, one from each qubit. Two possible realizations of a multiparticle CNOT are further discussed: Electron spins in double quantum dots in the singlettriplet encoding and nu=5/2 Ising nonAbelian anyons using topological quantum computation braiding operations and nontopological charge measurements.

Theory of spin qubits in nanostructures 
B. Trauzettel, M. Borhani, M. Trif, and D. Loss. J. Phys. Soc. Jpn. 77 (2008) 031012; arXiv:0707.4622v1.
We review recent advances on the theory of spin qubits in nanostructures. We focus on four selected topics. First, we show how to form spin qubits in the new and promising material graphene. Afterwards, we discuss spin relaxation and decoherence in quantum dots. In particular, we demonstrate how charge fluctations in the surrounding environment cause spin decay via spinorbit coupling. We then turn to a brief overview of how one can use electrondipole spin resonance (EDSR) to perform single spin rotations in quantum dots using an oscillating electric field. The final topic we cover is the spinspin coupling via spinorbit interaction which is an alternative to the usual spinspin coupling via the Heisenberg exchange interaction.

Electron and hole spin dynamics and decoherence in quantum dots 
D. Klauser, D. V. Bulaev, W. A. Coish, and Daniel Loss. Chapter 10 in Semiconductor Quantum Bits, eds. O. Benson and F. Henneberger, World Scientific, 2008. ISBN 9789814241052 arXiv:0706.1514v1
In this article we review our work on the dynamics and decoherence of electron and hole spins in single and double quantum dots. The first part, on electron spins, focuses on decoherence induced via the hyperfine interaction while the second part covers decoherence and relaxation of heavyhole spins due to spinorbit interaction as well as the manipulation of heavyhole spin using electric dipole spin resonance.

Polynomialtime algorithm for simulation of weakly interacting quantum spin systems 
Sergey Bravyi (IBM Yorktown), David DiVincenzo (IBM Yorktown), and Daniel Loss. Commun. Math. Phys. 284, 481 (2008); arXiv:0707.1894.
We describe an algorithm that computes the ground state energy and correlation functions for 2local Hamiltonians in which interactions between qubits are weak compared to singlequbit terms. The running time of the algorithm is polynomial in the number of qubits and the required precision. Specifically, we consider Hamiltonians of the form $H=H_0+\epsilon V$, where H_0 describes noninteracting qubits, V is a perturbation that involves arbitrary twoqubit interactions on a graph of bounded degree, and $\epsilon$ is a small parameter. The algorithm works if $\epsilon$ is below a certain threshold value that depends only upon the spectral gap of H_0, the maximal degree of the graph, and the maximal norm of the twoqubit interactions. The main technical ingredient of the algorithm is a generalized KirkwoodThomas ansatz for the ground state. The parameters of the ansatz are computed using perturbative expansions in powers of $\epsilon$. Our algorithm is closely related to the coupled cluster method used in quantum chemistry.

Highly Entangled Ground States in Tripartite Qubit Systems 
Beat Röthlisberger, Jörg Lehmann, D. S. Saraga, Philipp Traber, and Daniel Loss. Phys. Rev. Lett. 100, 100502 (2008); arXiv:0705.1710v1 [quantph].
We investigate the creation of highly entangled ground states in a system of three exchangecoupled qubits arranged in a ring geometry. Suitable magnetic field configurations yielding approximate GHZ and exact W ground states are identified. The entanglement in the system is studied at finite temperature in terms of the mixedstate tangle tau. By adapting a steepestdescent optimization algorithm we demonstrate that tau can be evaluated efficiently and with high precision. We identify the parameter regime for which the equilibrium entanglement of the tripartite system reaches its maximum.

Observation of extremely slow hole spin relaxation in selfassembled quantum dots 
D. Heiss (1), S. Schaeck (1), H. Huebl (1), M. Bichler (1), G. Abstreiter (1), J. J. Finley (1), D. V. Bulaev, Daniel Loss ((1), and Technische Universität München). Phys. Rev. B 76, 241306 (2007); condmat/0705.1466.
We report the measurement of extremely slow hole spin relaxation dynamics in selfassembled InGaAs quantum dots. Individual spin orientated holes are optically created in the lowest orbital state of each dot and read out after a defined storage time using spin memory devices. The hole spin relaxation time (T_1h) is measured as a function of the external magnetic field and lattice temperature. As predicted by theory, hole spin relaxation can occur over remarkably long timescales in strongly confined quantum dots (up to T_1h ~270 \mus) comparable to the corresponding time for electrons. Our findings are supported by calculations that reproduce both the observed magnetic field and temperature dependencies. The results show that hole spin relaxation in strongly confined quantum dots is governed by spinlattice interaction, in marked contrast to higher dimensional nanostructures where it is limited by spinorbit coupling between valence bands.

Spin qubits with electrically gated polyoxometalate molecules 
Jörg Lehmann, Alejandro GaitaAriño, Eugenio Coronado (Valencia), and Daniel Loss. Nature Nanotech. 2, 312 (2007); News and Views, Nature Nanotech. 2, 271 (2007); condmat/0703501.
Spin qubits offer one of the most promising routes to the implementation of quantum computers. Very recent results in semiconductor quantum dots show that electricallycontrolled gating schemes are particularly wellsuited for the realization of a universal set of quantum logical gates. Scalability to a larger number of qubits, however, remains an issue for such semiconductor quantum dots. In contrast, a chemical bottomup approach allows one to produce identical units in which localized spins represent the qubits. Molecular magnetism has produced a wide range of systems with tailored properties, but molecules permitting electrical gating have been lacking. Here we propose to use the polyoxometalate [PMo12O40(VO)2]q, where two localized spins1/2 can be coupled through the electrons of the central core. Via electrical manipulation of the molecular redox potential, the charge of the core can be changed. With this setup, twoqubit gates and qubit readout can be implemented.

Universal phase shift and nonexponential decay of driven singlespin oscillations 
F.H.L. Koppens (TU Delft), D. Klauser, W. A. Coish, K. C. Nowack (TU Delft), L.P. Kouwenhoven (TU Delft), D. Loss, and L.M.K. Vandersypen (TU Delft). Phys. Rev. Lett. 99, 106803 (2007); condmat/0703640.
We study, both theoretically and experimentally, driven Rabi oscillations of a single electron spin coupled to a nuclear spin bath. Due to the long correlation time of the bath, two unusual features are observed in the oscillations. The decay follows a power law, and the oscillations are shifted in phase by a universal value of ~pi/4. These properties are well understood from a theoretical expression that we derive here in the static limit for the nuclear bath. This improved understanding of the coupled electronnuclear system is important for future experiments using the electron spin as a qubit.

Spin densities in parabolic quantum wires with Rashba spinorbit interaction 
S. I. Erlingsson (Reykjavik), J. C. Egues (Sao Paulo), and D. Loss. Phys. Stat. Sol. (c) 3, 4317 (2006) (PASPS IV Proceedings); condmat/0701564.
Using canonical transformations we diagonalize approximately the Hamiltonian of a gaussian wire with Rashba spinorbit interaction. This proceedure allows us to obtain the energy dispersion relations and the wavefunctions with good accuracy, even in systems with relatively strong Rashba coupling. With these eigenstates one can calculate the nonequilibrium spin densities induced by applying bias voltages across the sample. We focus on the $z$component of the spin density, which is related to the spin Hall effect.

Spin orbit interaction and zitterbewegung in symmetric wells 
E Bernardes (São Paulo), J. Schliemann (Regensburg), J. C. Egues (São Paulo), and D. Loss. Phys. Stat. Sol. (c) 3, No. 12, 4330  4333 (2006); (PASPS IV Proceedings).
Recently, we have introduced a novel intersubbandinduced spinorbit (so) coupling (condmat/0607218) arising in symmetric wells with at least two subbands. This new so coupling gives rise to an usual zitterbewegung  i.e. the semiconductor analog to the relativistic trembling motion of electrons  with cycloidal motion without magnetic fields. Here we complement these findings by explicitly deriving expressions for the corresponding zitterbewegung in spin space.

Direct Measurement of the SpinOrbit Interaction in a TwoElectron InAs Nanowire Quantum Dot 
C. Fasth, A. Fuhrer, L. Samuelson (Lund University), Vitaly N. Golovach (LMU Munich), and Daniel Loss. Phys. Rev. Lett. 98, 266801 (2007); condmat/0701161.
We demonstrate control of the electron number down to the last electron in tunable fewelectron quantum dots defined in catalytically grown InAs nanowires. Using low temperature transport spectroscopy in the Coulomb blockade regime we propose a simple method to directly determine the magnitude of the spinorbit interaction in a twoelectron artificial atom with strong spinorbit coupling. Due to a large effective gfactor g*=8+/1 the transition from singlet S to triplet T+ groundstate with increasing magnetic field is dominated by the Zeeman energy rather than by orbital effects. We find that the spinorbit coupling mixes the T+ and S states and thus induces an avoided crossing with magnitude $\Delta_{SO}$=0.25+/0.05 meV. This allows us to calculate the spinorbit length $\lambda_{SO}\approx$127 nm in such systems using a simple model.

Resonant spin polarization and spin current in a twodimensional electron gas 
Mathias Duckheim and Daniel Loss. Phys. Rev. B 75, 201305 (2007); condmat/0701559.
We study the spin polarization and its associated spinHall current due to EDSR in disordered twodimensional electron systems. We show that the disorder induced damping of the resonant spin polarization can be strongly reduced by an optimal field configuration that exploits the interference between Rashba and Dresselhaus spinorbit interaction. This leads to a striking enhancement of the spin susceptibility while the spinHall current vanishes at the same time. We give an interpretation of the spin current in geometrical terms which are associated with the trajectories the polarization describes in spin space.

Spin relaxation at the singlettriplet transition in a quantum dot 
Vitaly N. Golovach (LMU Munich), Alexander Khaetskii (Chernogolovka), and Daniel Loss. Phys. Rev. B 77, 045328 (2008); condmat/0703427.
We study spin relaxation in a twoelectron quantum dot in the vicinity of the singlettriplet transition. The spin relaxation occurs due to a combined effect of the spinorbit, Zeeman, and electronphonon interactions. The singlettriplet relaxation rates exhibit strong variations as a function of the singlettriplet splitting. We show that the Coulomb interaction between the electrons has two competing effects on the singlettriplet spin relaxation. One effect is to enhance the relative strength of spinorbit coupling in the quantum dot, resulting in larger spinorbit splittings and thus in a stronger coupling of spin to charge. The other effect is to make the charge density profiles of the singlet and triplet look similar to each other, thus diminishing the ability of charge environments to discriminate between singlet and triplet states. We thus find essentially different channels of singlettriplet relaxation for the case of strong and weak Coulomb interaction. Finally, for the linear in momentum Dresselhaus and Rashba spinorbit interactions, we calculate the singlettriplet relaxation rates to leading order in the spinorbit interaction, and find that they are proportional to the second power of the Zeeman energy, in agreement with recent experiments on triplettosinglet relaxation in quantum dots.

Nuclear spin ferromagnetic phase transition in an interacting 2D electron gas 
Pascal Simon and Daniel Loss. Phys. Rev. Lett. 98, 156401 (2007); condmat/0611292.
Electrons in a twodimensional semiconducting heterostructure interact with nuclear spins via the hyperfine interaction. Using a a Kondo lattice formulation of the electronnuclear spin interaction, we show that the nuclear spin system within an interacting twodimensional electron gas undergoes a ferromagnetic phase transition at finite temperatures. We find that electronelectron interactions and nonFermi liquid behavior substantially enhance the nuclear spin Curie temperature into the $mK$ range with decreasing electron density.

Spin qubits in graphene quantum dots 
B. Trauzettel, Denis V. Bulaev, Daniel Loss, and Guido Burkard. Nature Physics 3, 192 (2007); News and Views; Research Highlights; condmat/0611252.
We propose how to form spin qubits in graphene. A crucial requirement to achieve this goal is to find quantum dot states where the usual valley degeneracy in bulk graphene is lifted. We show that this problem can be avoided in quantum dots based on ribbons of graphene with semiconducting armchair boundaries. For such a setup, we find the energies and the exact wave functions of bound states, which are required for localized qubits. Additionally, we show that spin qubits in graphene can not only be coupled between nearest neighbor quantum dots via Heisenberg exchange interaction but also over long distances. This remarkable feature is a direct consequence of the quasirelativistic spectrum of graphene. See also http://www.nature.com/nnano/reshigh/2006/1106/full/nnano.2006.167.html

Quantum vs. classical hyperfineinduced dynamics in a quantum dot 
W. A. Coish, E. A. Yuzbashyan (Rutgers University), B. L. Altshuler (Columbia University), and Daniel Loss. J. Appl. Phys. 101, 081715 (2007); condmat/0610633.
In this article we analyze spin dynamics for electrons confined to semiconductor quantum dots due to the contact hyperfine interaction. We compare meanfield (classical) evolution of an electron spin in the presence of a nuclear field with the exact quantum evolution for the special case of uniform hyperfine coupling constants. We find that (in this special case) the zeromagneticfield dynamics due to the meanfield approximation and quantum evolution are similar. However, in a finite magnetic field, the quantum and classical solutions agree only up to a certain time scale t<\tau_c, after which they differ markedly.

Exchangecontrolled singleelectronspin rotations in quantum dots 
W. A. Coish and Daniel Loss. Phys. Rev. B 75, 161302 (2007) (R); condmat/0610443.
We show theoretically that arbitrary coherent rotations can be performed quickly (with a gating time ~1 ns) and with high fidelity on the spin of a single electron confined to a quantum dot using exchange. These rotations can be performed using experimentally proven techniques for rapid exchange control, without the need for spinorbit interaction or ac electromagnetic fields. We expect that implementations of this scheme would achieve gate error rates on the order of $\eta\lesssim 10^{3}$, within reach of several known errorcorrection schemes.

Transport through a quantum dot with SU(4) Kondo entanglement 
Karyn Le Hur (Yale), Pascal Simon, and Daniel Loss. Phys. Rev. B 75, 035332 (2007); condmat/0609298.
We investigate a mesoscopic setup composed of a small electron droplet (dot) coupled to a larger quantum dot (grain) also subject to Coulomb blockade as well as two macroscopic leads used as source and drain. An exotic Kondo ground state other than the standard SU(2) Fermi liquid unambiguously emerges: an SU(4) Kondo correlated liquid. The transport properties through the small dot are analyzed for this regime, through boundary conformal field theory, and allow a clear distinction with other regimes such as a twochannel spin state or a twochannel orbital state.

Measurement, control, and decay of quantumdot spins 
W. A. Coish, Vitaly N. Golovach, J. Carlos Egues, and Daniel Loss. Physica Status Solidi (b) 243, 3658 (2006); condmat/0606782.
In this review we discuss a recent proposal to perform partial Bellstate (parity) measurements on twoelectron spin states for electrons confined to quantum dots. The realization of this proposal would allow for a physical implementation of measurementbased quantum computing. In addition, we consider the primary sources of energy relaxation and decoherence which provide the ultimate limit to all proposals for quantum information processing using electron spins in quantum dots. We give an account of the Hamiltonians used for the most important interactions (spinorbit and hyperfine) and survey some of the recent work done to understand dynamics, control, and decoherence under the action of these Hamiltonians. We conclude the review with a table of important decay times found in experiment, and relate these time scales to the potential viability of measurementbased quantum computing.

Sequential Tunneling through Molecular Spin Rings 
Jörg Lehmann and Daniel Loss. Phys. Rev. Lett. 98, 117203 (2007); condmat/0608642.
We consider electrical transport through molecules with Heisenbergcoupled spins arranged in a ring structure in the presence of an easyaxis anisotropy. The molecules are coupled to two metallic leads and a gate. In the charged state of the ring, a Zener doubleexchange mechanism links transport properties to the underlying spin structure. This leads to a remarkable contactsite dependence of the current, which for an antiferromagnetic coupling of the spins can lead to a total suppression of the zerobias conductance when the molecule is contacted at adjacent sites.

Spinspin coupling in electrostatically coupled quantum dots 
Mircea Trif, Vitaly N. Golovach, and Daniel Loss. Phys. Rev. B 75, 085307 (2007); condmat/0608512.
We study the spinspin coupling between two singleelectron quantum dots due to the Coulomb and spinorbit interactions, in the absence of tunneling between the dots. We find an anisotropic XY spinspin interaction that is proportional to the Zeeman splitting produced by the external magnetic field. This interaction is studied both in the limit of weak and strong Coulomb repulsion with respect to the level spacing of the dot. The interaction is found to be a nonmonotonic function of interdot distance $a_0$ and external magnetic field, and, moreover, vanishes for some special values of $a_0$ and/or magnetic field orientation. This mechanism thus provides a new way to generate and tune spin interaction between quantum dots. We propose a scheme to measure this spinspin interaction based on the spinrelaxationmeasurement technique.

Electric Dipole Spin Resonance for Heavy Holes in Quantum Dots 
Denis V. Bulaev and Daniel Loss. Phys. Rev. Lett. 98, 097202 (2007); condmat/0608410.
We propose and analyze a new method for manipulation of a heavy hole spin in a quantum dot. Due to spinorbit coupling between states with different orbital momenta and opposite spin orientations, an applied rf electric field induces transitions between spinup and spindown states. This scheme can be used for detection of heavyhole spin resonance signals, for the control of the spin dynamics in twodimensional systems, and for determining important parameters of heavyholes such as the effective $g$factor, mass, spinorbit coupling constants, spin relaxation and decoherence times.

Spinorbit interaction in symmetric wells and cycloidal orbits without magnetic fields 
Esmerindo S. Bernardes, John Schliemann (Regensburg), J. Carlos Egues, and Daniel Loss. Phys. Rev. Lett. 99, 076603 (2007); condmat/0607218.
We investigate the spinorbit (so) interaction in twodimensional electron gases (2DEGs) in quantum wells with two subbands. From the $8\times 8$ Kane model, we derive a new intersubbandinduced so coupling which resembles the functional form of the Rashba so  but is nonzero even in \emph{symmetric} structures. This follows from the distinct parity of the confined states (even/odd) which obliterates the need for asymmetric potentials. Interestingly, our so interaction gives rise to an unusual \emph{zitterbewegung} of spinpolarized electrons with cycloidal trajectories \textit{without} magnetic fields. We also predict a sizable effectivemass renormalization due to the soinduced subband warping.

Quantum computing with spins in solids 
W. A. Coish and Daniel Loss. Handbook of Magnetism and Advanced Magnetic Materials. Helmut Kronmüller (editor) and Stuart Parkin (editor). Volume 5: Spintronics and Magnetoelectronics. ; 2007 John Wiley & Sons, Ltd. ISBN: 9780470022177; condmat/0606550.
The ability to perform highprecision one and twoqubit operations is sufficient for universal quantum computation. For the LossDiVincenzo proposal to use single electron spins confned to quantum dots as qubits, it is therefore sufficient to analyze only single and coupled doubledot structures, since the strong Heisenberg exchange coupling between spins in this proposal falls off exponentially with distance and longranged dipolar coupling mechanisms can be made significantly weaker. This scalability of the LossDiVincenzo design is both a practical necessity for eventual applications of multiqubit quantum computing and a great conceptual advantage, making analysis of the relevant components relatively transparent and systematic. We review the LossDiVincenzo proposal for quantumdotconfned electron spin qubits, and survey the current state of experiment and theory regarding the relevant single and double quantum dots, with a brief look at some related alternative schemes for quantum computing.

Quantumdot spin qubit and hyperfine interaction 
D. Klauser, W. A. Coish, and Daniel Loss. Advances in Solid State Physics 46, p.1729, (2007); condmat/0604252.
We review our investigation of the spin dynamics for two electrons confined to a double quantum dot under the influence of the hyperfine interaction between the electron spins and the surrounding nuclei. Further we propose a scheme to narrow the distribution of difference in polarization between the two dots in order to suppress hyperfine induced decoherence.

Electricdipoleinduced spin resonance in disordered semiconductors (Article) 
Mathias Duckheim and Daniel Loss. Nature Physics 2, 195199 (2006); Supplementary Information; condmat/0605735.
One of the hallmarks of spintronics is the control of magnetic moments by electric fields enabled by strong spinorbit interaction (SOI) in semiconductors. A powerful way of manipulating spins in such structures is electricdipoleinduced spin resonance (EDSR), where the radiofrequency fields driving the spins are electric, not magnetic as in standard paramagnetic resonance. Here, we present a theoretical study of EDSR for a twodimensional electron gas in the presence of disorder, where random impurities not only determine the electric resistance but also the spin dynamics through SOI. Considering a specific geometry with the electric and magnetic fields parallel and inplane, we show that the magnetization develops an outofplane component at resonance that survives the presence of disorder. We also discuss the spin Hall current generated by EDSR. These results are derived in a diagrammatic approach, with the dominant effects coming from the spin vertex correction, and the optimal parameter regime for observation is identified. See also, 'Semiconductor physics: Electric fields drive spins', by Emmanuel I. Rashba, Nature Physics 2, 149150 (01 Mar 2006) News and Views

Molecular states in carbon nanotube double quantum dots 
M.R. Graeber, W.A. Coish, C. Hoffmann, M. Weiss, J. Furer, S. Oberholzer, D. Loss, and C. Schoenenberger. Phys. Rev. B 74, 075427 (2006); condmat/0603367.
We report electrical transport measurements through a semiconducting singlewalled carbon nanotube (SWNT) with three additional topgates. At low temperatures the system acts as a double quantum dot with large interdot tunnel coupling allowing for the observation of tunnelcoupled molecular states extending over the whole doubledot system. We precisely extract the tunnel coupling and identify the molecular states by the sequentialtunneling line shape of the resonances in differential conductance.

Electric Dipole Induced Spin Resonance in Quantum Dots 
Vitaly N. Golovach, Massoud Borhani, and Daniel Loss. Phys. Rev. B 74, 165319 (2006); condmat/0601674.
An alternating electric field, applied to a ``spin 1/2'' quantum dot, couples to the electron spin via the spinorbit interaction. We analyze different types of spinorbit couplings known in the literature and find that an electric dipole spin resonance (EDSR) scheme for spin manipulation can be realized with the uptodate experimental setups. In particular, for the Rashba and Dresselhaus spinorbit couplings, a fully transverse effective magnetic field arises in the presence of a Zeeman splitting in the lowest order of spinorbit interaction. Spin manipulation and measurement of the spin decoherence time $T_2$ are straightforward in lateral GaAs quantum dots through the use of EDSR.

Dynamics of Coupled Qubits Interacting with an OffResonant Cavity 
Oliver Gywat (UCSB), Florian Meier (UCSB), Daniel Loss, and D. D. Awschalom (UCSB). Phys. Rev. B 73, 125336 (2006); condmat/0511592.
We study a model for a pair of qubits which interact with a single offresonant cavity mode and, in addition, exhibit a direct interqubit coupling. Possible realizations for such a system include coupled superconducting qubits in a line resonator as well as exciton states or electron spin states of quantum dots in a cavity. The emergent dynamical phenomena are strongly dependent on the relative energy scales of the interqubit coupling strength, the coupling strength between qubits and cavity mode, and the cavity mode detuning. We show that the cavity mode dispersion enables a measurement of the state of the coupledqubit system in the perturbative regime. We discuss the effect of the direct interqubit interaction on a cavitymediated twoqubit gate. Further, we show that for asymmetric coupling of the two qubits to the cavity, the direct interqubit coupling can be controlled optically via the ac Stark effect.

Spin Decay in a Quantum Dot Coupled to a Quantum Point Contact 
Massoud Borhani, Vitaly N. Golovach, and Daniel Loss. Phys. Rev. B 73, 155311 (2006); condmat/0510758.
We consider a mechanism of spin decay for an electron spin in a quantum dot due to coupling to a nearby quantum point contact (QPC) with and without an applied bias voltage. The coupling of spin to charge is induced by the spinorbit interaction in the presence of a magnetic field. We perform a microscopic calculation of the effective Hamiltonian coupling constants to obtain the QPCinduced spin relaxation and decoherence rates in a realistic system. This rate is shown to be proportional to the shot noise of the QPC in the regime of large bias voltage and scales as $a^{6}$ where $a$ is the distance between the quantum dot and the QPC. We find that, for some specific orientations of the setup with respect to the crystallographic axes, the QPCinduced spin relaxation and decoherence rates vanish, while the charge sensitivity of the QPC is not changed. This result can be used in experiments to minimize QPCinduced spin decay in readout schemes.

Zitterbewegung of electrons and holes in IIIV semiconductor quantum wells 
John Schliemann (Regensburg), Daniel Loss, and R.M. Westervelt (Harvard). Phys. Rev. B 73, 085323 (2006); condmat/0512148.
The notion of zitterbewegung is a longstanding prediction of relativistic quantum mechanics. Here we extend earlier theoretical studies on this phenomenon for the case of IIIV zincblende semiconductors which exhibit particularly strong spinorbit coupling. This property makes nanostructures made of these materials very favorable systems for possible experimental observations of zitterbewegung. Our investigations include electrons in ndoped quantum wells under the influence of both Rashba and Dresselhaus spinorbit interaction, and also the twodimensional hole gas. Moreover, we give a detailed anaysis of electron zitterbewegung in quantum wires which appear to be particularly suited for experimentally observing this effect.

A Mesoscopic Resonating Valence Bond system on a triple dot 
Karyn Le Hur (Sherbrooke), Patrik Recher (Stanford), Emilie Dupont (Sherbrooke), and Daniel Loss. Phys. Rev. Lett. 96, 106803 (2006); condmat/0510450.
We theoretically introduce a mesoscopic pendulum from a triple dot. The pendulum is fastened through a singlyoccupied dot (spin qubit). Two other strongly capacitively coupled islands form a doubledot charge qubit with one electron in excess oscillating between the two lowenergy charge states (1, 0) and (0, 1). The triple dot is placed between two superconducting leads. Under realistic conditions, the main proximity effect stems from the injection of resonating singlet (valence) bonds on the triple dot. This gives rise to a Josephson current that is charge and spindependent and, as a consequence, exhibits a distinct resonance as a function of the superconducting phase difference.

Nuclear spin state narrowing via gatecontrolled Rabi oscillations in a double quantum dot 
D. Klauser, W.A. Coish, and Daniel Loss. Phys. Rev. B 73, 205302 (2006); condmat/0510177.
We study spin dynamics for two electrons confined to a double quantum dot under the influence of an oscillating exchange interaction. This leads to driven Rabi oscillations between the $\ket{\uparrow\downarrow}$state and the $\ket{\downarrow\uparrow}$state of the twoelectron system. The width of the Rabi resonance is proportional to the amplitude of the oscillating exchange. A measurement of the Rabi resonance allows one to narrow the distribution of nuclear spin states and thereby to prolong the spin decoherence time. Further, we study decoherence of the twoelectron states due to the hyperfine interaction and give requirements on the parameters of the system in order to initialize in the $\ket{\uparrow\downarrow}$state and to perform a $\sqrt{\mathrm{SWAP}}$ operation with unit fidelity.

Cotunneling current through quantum dots with phononassisted spinflip processes 
Jörg Lehmann and Daniel Loss. Phys. Rev. B 73, 045328 (2006); condmat/0509420.
We consider cotunneling through a quantum dot in the presence of spinflip processes induced by the coupling to acoustic phonons of the surrounding. An expression for the phononassisted cotunneling current is derived by means of a generalized SchriefferWolff transformation. The influence of the spinphonon coupling on the heating of the dot is considered. The result is evaluated for the case of a parabolic semiconductor quantum dot with Rashba and Dresselhaus spinorbit coupling and a novel method for the determination of the spinphonon relaxation rate is proposed.

Shot noise and spinorbit coherent control of entangled and spin polarized electrons 
J. Carlos Egues (Sao Paulo), Guido Burkard, D. Saraga, John Schliemann, and Daniel Loss. Phys. Rev. B 72, 235326 (2005); condmat/0509038.
We extend our previous work on shot noise for entangled and spin polarized electrons in a beamsplitter geometry with spinorbit (\textit{so}) interaction in one of the incoming leads (lead 1). Besides accounting for both the Dresselhaus and the Rashba spinorbit terms, we present general formulas for the shot noise of singlet and triplets states derived within the scattering approach. We determine the full scattering matrix of the system for the case of leads with \textit{two} orbital channels coupled via weak \textit{so} interactions inducing channel anticrossings. We show that this interband coupling coherently transfers electrons between the channels and gives rise to an additional modulation angle  dependent on both the Rashba and Dresselhaus interaction strengths  which allows for further independent coherent control of the electrons traversing the incoming leads. We derive explicit shot noise formulas for a variety of correlated pairs (e.g., Bell states) and lead spin polarizations. Interestingly, the singlet and \textit{each} of the triplets defined along the quantization axis perpendicular to lead 1 (with the local \textit{so} interaction) and in the plane of the beam splitter display distinctive shot noise for injection energies near the channel anticrossings; hence, one can tell apart all the triplets, in addition to the singlet, through noise measurements. We also find that spinorbit induced backscattering within lead 1 reduces the visibility of the noise oscillations, due to the additional partition noise in this lead. Finally, we consider injection of twoparticle wavepackets into leads with multiple discrete states and find that twoparticle entanglement can still be observed via noise bunching and antibunching.

Phase coherence in the inelastic cotunneling regime 
M. Sigrist, T. Ihn, K. Ensslin (ETH Zurich), D. Loss, M. Reinwald, and W. Wegscheider (Regensburg). Phys. Rev. Lett. 96, 036804 (2006); condmat/0508757.
Two quantum dots with tunable mutual tunnel coupling have been embedded in a twoterminal AharonovBohm geometry. AharonovBohm oscillations are investigated in the cotunneling regime. Visibilities of more than 0.8 are measured indicating that phasecoherent processes are involved in the elastic and inelastic cotunneling. An oscillationphase change of pi is detected as a function of bias voltage at the inelastic cotunneling onset.

Fermionic BellState Analyzer for Spin Qubits 
HansAndreas Engel (Harvard Univ.) and Daniel Loss. Science 309, 586 (2005)
We propose a protocol and physical implementation for partial Bellstate measurements of Fermionic qubits, allowing for deterministic quantum computing in solidstate systems without the need for twoqubit gates. Our scheme consists of two spin qubits in a double quantum dot where the two dots have different Zeeman splittings and resonant tunneling between the dots is only allowed when the spins are antiparallel. This converts spin parity into charge information by means of a projective measurement and can be implemented with established technologies. This measurementbased qubit scheme greatly simplifies the experimental realization of scalable quantum computers in electronic nanostructures.
[See also Science Perspective, Fingerprinting Spin Qubits, J. Carlos Egues, Science , Vol 309, Issue 5734, 565 (2005); Philip Ball, Nature news: Quantum computers go for a spin. ]

Singlettriplet decoherence due to nuclear spins in a double quantum dot 
W. A. Coish and Daniel Loss. Phys. Rev. B 72, 125337 (2005); condmat/0506090.
We have evaluated hyperfineinduced electron spin dynamics for two electrons confined to a double quantum dot. Our quantum solution accounts for decay of a singlettriplet correlator even in the presence of a fully static nuclear spin system, with no ensemble averaging over initial conditions. In contrast to an earlier semiclassical calculation, which neglects the exchange interaction, we find that the singlettriplet correlator shows a longtime saturation value that differs from 1/2, even in the presence of a strong magnetic field. Furthermore, we find that the form of the longtime decay undergoes a transition from a rapid Gaussian to a slow power law ($\sim 1/t^{3/2}$) when the exchange interaction becomes nonzero and the singlettriplet correlator acquires a phase shift given by a universal (parameter independent) value of $3\pi/4$ at long times. The oscillation frequency and timedependent phase shift of the singlettriplet correlator can be used to perform a precision measurement of the exchange interaction and Overhauser field fluctuations in an experimentally accessible system. We also address the effect of orbital dephasing on singlettriplet decoherence, and find that there is an optimal operating point where orbital dephasing becomes negligible.

Fermi liquid parameters in 2D with spinorbit interaction 
D. S. Saraga and Daniel Loss. Phys. Rev. B 72, 195319 (2005); condmat/0504661.
We derive analytical expressions for the quasiparticle lifetime \tau, the effective mass m^*, and the Green's function renormalization factor Z for a 2D Fermi liquid with electronelectron interaction in the presence of the Rashba spinorbit interaction. For the lifetime \tau and the renormalization Z, we find that the modification is independent of the Rashba band index \rho, and occurs in second order of the spinorbit coupling \alpha . On the contrary, the modification of the effective mass m^* is linear in \alpha and is different for the two Rashba bands, yielding a spindependent effective mass. In the derivation of these results, we also discuss the screening of the Coulomb interaction, and the susceptibility.

Determining the spin Hall conductance via charge transport 
Sigurdur I. Erlingsson and Daniel Loss. Phys. Rev. B 72, 121310 (2005); condmat/0503605.
We propose a scheme where transport measurements of charge current and its noise can be used to determine the spin Hall conductance in a fourterminal setup. Starting from the scattering formalism we express the spin current and spin Hall conductance in terms of spindependent transmission coefficients. These coefficients are then expressed in terms of charge current and noise. We use the scheme to characterize the spin injection efficiency of a ferromagnetic/semiconductor interface.

Zitterbewegung of electronic wave packets in semiconductor nanostructures 
John Schliemann, Daniel Loss, and R.M. Westervelt (Harvard Univ.). Phys. Rev. Lett. 94, 206801 (2005); condmat/0410321.
We study the zitterbewegung of electronic wave packets in IIIV zincblende semiconductor quantum wells due to spinorbit coupling. Our results suggest a direct experimental proof of this fundamental effect, confirming a longstanding theoretical prediction. For electron motion in a harmonic quantum wire, we numerically and analytically find a resonance condition maximizing the zitterbewegung. See also, 'Dirac gets the jitters', M. Buchanan, Nature Physics 1, 5 (2005); http://www.nature.com/nphys/journal/v1/n1/full/nphys132.html

Phonon Bottleneck Effect Leads to Observation of Quantum Tunneling of the Magnetization and Butterfly Hysteresis Loops in (Et4N)3Fe2F9 
Ralph Schenker, Michael N. Leuenberger (UC San Diego), Gregory Chaboussant (Uni Bern), Daniel Loss, and Hans U. Gudel (Uni Bern). Phys. Rev. B 72, 184403 (2005); condmat/0502548.
A detailed investigation of the unusual dynamics of the magnetization of (Et4N)3Fe2F9 (Fe2), containing isolated [Fe2F9]3 dimers, is presented and discussed. Fe2 possesses an S=5 ground state with an energy barrier of 2.40 K due to an axial anisotropy. Poor thermal contact between sample and bath leads to a phonon bottleneck situation, giving rise to butterflyshaped hysteresis loops below 5 K concomitant with slow decay of the magnetization for magnetic fields Hz applied along the FeFe axis. The butterfly curves are reproduced using a microscopic model based on the interaction of the spins with resonant phonons. The phonon bottleneck allows for the observation of resonant quantum tunneling of the magnetization at 1.8 K, far above the blocking temperature for spinphonon relaxation. The latter relaxation is probed by AC magnetic susceptibility experiments at various temperatures and bias fields. At H=0, no outofphase signal is detected, indicating that at T smaller than 1.8 K Fe2 does not behave as a singlemolecule magnet. At 1 kG, relaxation is observed, occurring over the barrier of the thermally accessible S=4 first excited state that forms a combined system with the S=5 state.

Spin relaxation and decoherence of holes in quantum dots 
Denis V. Bulaev and Daniel Loss. Phys. Rev. Lett. 95, 076805 (2005); condmat/0503181.
We investigate heavyhole spin relaxation and decoherence in quantum dots in perpendicular magnetic fields. We show that at low temperatures the spin decoherence time is two times longer than the spin relaxation time. We find that the spin relaxation time for heavy holes can be comparable to or even longer than that for electrons in strongly twodimensional quantum dots. We discuss the difference in the magneticfield dependence of the spin relaxation rate due to Rashba or Dresselhaus spinorbit coupling for systems with positive (i.e., GaAs quantum dots) or negative (i.e., InAs quantum dots) $g$factor.

Cluster States From Heisenberg Interaction 
Massoud Borhani and Daniel Loss. Phys. Rev. A 71, 034308 (2005); quantph/0410145.
We show that a special type of entangled states, cluster states, can be created with Heisenberg interactions and local rotations in 2d steps where d is the dimension of the lattice. We find that, by tuning the coupling strengths, anisotropic exchange interactions can also be employed to create cluster states. Finally, we propose electron spins in quantum dots as a possible realization of a oneway quantum computer based on cluster states.

Recipes for spinbased quantum computing 
Veronica Cerletti, W. A. Coish, Oliver Gywat, and Daniel Loss. Nanotechnology 16, R27 (2005); condmat/0412028.
Technological growth in the electronics industry has historically been measured by the number of transistors that can be crammed onto a single microchip. Unfortunately, all good things must come to an end; spectacular growth in the number of transistors on a chip requires spectacular reduction of the transistor size. For electrons in semiconductors, the laws of quantum mechanics take over at the nanometre scale, and the conventional wisdom for progress (transistor cramming) must be abandoned. This realization has stimulated extensive research on ways to exploit the spin (in addition to the orbital) degree of freedom of the electron, giving birth to the field of spintronics. Perhaps the most ambitious goal of spintronics is to realize complete control over the quantum mechanical nature of the relevant spins. This prospect has motivated a race to design and build a spintronic device capable of complete control over its quantum mechanical state, and ultimately, performing computations: a quantum computer.
In this tutorial we summarize past and very recent developments which point the way to spinbased quantum computing in the solidstate. After introducing a set of basic requirements for any quantum computer proposal, we offer a brief summary of some of the many theoretical proposals for solidstate quantum computers. We then focus on the LossDiVincenzo proposal for quantum computing with the spins of electrons confined to quantum dots. There are many obstacles to building such a quantum device. We address these, and survey recent theoretical, and then experimental progress in the field. To conclude the tutorial, we list some asyet unrealized experiments, which would be crucial for the development of a quantumdot quantum computer. 
Entanglement transfer from electron spins to photons 
Veronica Cerletti, Oliver Gywat, and Daniel Loss. Phys. Rev. B 72, 115316 (2005); condmat/0411235.
We show that electron recombination in spin lightemitting diodes provides an efficient method to transfer entanglement from electron spins onto pairs of polarizationentangled photons. Because of the interplay of quantum mechanical selection rules and interference, maximally entangled electron pairs are converted into maximally entangled photon pairs with unity fidelity for a continuous set of observation directions. We describe the dynamics of the conversion process using a masterequation approach and show that the implementation of our scheme is feasible with current experimental techniques.

Double Occupancy Errors in Quantum Computing Operations: Corrections to Adiabaticity 
Ryan Requist (SUNY), John Schliemann, Alexander G. Abanov (SUNY), and Daniel Loss. Phys. Rev. B 71, 115315 (2005); condmat/0409096.
We study the corrections to adiabatic dynamics of two coupled quantum dot spinqubits, each dot singly occupied with an electron, in the context of a quantum computing operation. Tunneling can lead to a double occupancy at the conclusion of an operation and constitutes a processing error. Our model for the dynamics of an effective twolevel system is integrable and possesses three independent parameters. We confirm the accuracy of Dykhne's formula, a nonperturbative estimate of transitions, and discuss physically intuitive conditions for its validity. Our semiclassical results are in excellent agreement with numerical simulations of the exact time evolution. A similar approach applies to twolevel systems in different contexts.

Spin Relaxation and Anticrossing in Quantum Dots: Rashba versus Dresselhaus SpinOrbit Coupling 
Denis V. Bulaev and Daniel Loss. Phys. Rev. B 71, 205324 (2005); condmat/0409614.
The spinorbit splitting of the electron levels in a twodimensional quantum dot in a perpendicular magnetic field is studied. It is shown that at the point of an accidental degeneracy of the two lowest levels above the ground state the Rashba spinorbit coupling leads to a level anticrossing and to mixing of spinup and spindown states, whereas there is no mixing of these levels due to the Dresselhaus term. We calculate the relaxation and decoherence times of the three lowest levels due to phonons. We find that the spin relaxation rate as a function of a magnetic field exhibits a cusplike structure for Rashba but not for Dresselhaus spinorbit interaction.

SpinHall conductivity due to Rashba spinorbit interaction in disordered systems 
Oleg Chalaev and Daniel Loss. Phys. Rev. B 71, 245318 (2005); condmat/0407342.
We consider the spinHall current in a disordered twodimensional electron gas in the presence of Rashba spinorbit interaction. We derive a generalized KuboGreenwood formula for the spinHall conductivity $\sigma$ and evaluate it in an systematic way using standard diagrammatic techniques for disordered systems. We find that in the diffusive regime both Boltzmann and the weak localization contributions to the $\sigma$ vanish in the zero frequency limit. We show that the uniform spin current is given by the total time derivative of the magnetization from which we can conclude that the spin current vanishes exactly in the stationary limit. This conclusion is valid for arbitrary spinindependent disorder, external electric field strength, and also for interacting electrons.

Controlling Spin Qubits in Quantum Dots 
HansAndreas Engel, L.P. Kouwenhoven (Delft), Daniel Loss, and C.M. Marcus (Harvard). Quantum Information Processing, 3, 115132 (2004); condmat/0409294.
We review progress on the spintronics proposal for quantum computing where the quantum bits (qubits) are implemented with electron spins. We calculate the exchange interaction of coupled quantum dots and present experiments, where the exchange coupling is measured via transport. Then, experiments on single spins on dots are described, where long spin relaxation times, on the order of a millisecond, are observed. We consider spinorbit interaction as sources of spin decoherence and find theoretically that also long decoherence times are expected. Further, we describe the concept of spin filtering using quantum dots and show data of successful experiments. We also show an implementation of a read out scheme for spin qubits and define how qubits can be measured with high precision. Then, we propose new experiments, where the spin decoherence time and the Rabi oscillations of single electrons can be measured via charge transport through quantum dots. Finally, all these achievements have promising applications both in conventional and quantum information processing.

Asymmetric Quantum Shot Noise in Quantum Dots 
HansAndreas Engel and Daniel Loss. Phys. Rev. Lett. 93, 136602 (2004); condmat/0312107.
We analyze the frequencydependent noise of a current through a quantum dot which is coupled to Fermi leads and which is in the Coulomb blockade regime. We show that the asymmetric shot noise as function of frequency shows steps and becomes superPoissonian. This provides experimental access to the quantum fluctuations of the current. We present an exact calculation for a single dot level and a perturbative evaluation of the noise in Born approximation (sequential tunneling regime but without Markov approximation) for the general case of many levels with charging interaction.

Reduced Visibility of Rabi Oscillations in Superconducting Qubits 
Florian Meier (UC Santa Barbara) and Daniel Loss. Phys. Rev. B 71, 094519 (2005); condmat/0408594.
Coherent Rabi oscillations between quantum states of superconducting microcircuits have been observed in a number of experiments, albeit with a visibility which is typically much smaller than unity. Here, we show that the coherent coupling to background charge fluctuators [R.W. Simmonds et al., Phys. Rev. Lett. 93, 077003 (2004)] leads to a significantly reduced visibility if the Rabi frequency is comparable to the coupling energy of microcircuit and fluctuator. For larger Rabi frequencies, transitions to the second excited state of the superconducting microcircuit become dominant in suppressing the Rabi oscillation visibility. We also calculate the probability for Bogoliubov quasiparticle excitations in typical Rabi oscillation experiments.

Probing SingleElectron Spin Decoherence in Quantum Dots using Charged Excitons 
Oliver Gywat, HansAndreas Engel, and Daniel Loss. Journal of Superconductivity 18 (2), 175  183 ( 2005); condmat/0408451.
We propose to use optical detection of magnetic resonance (ODMR) to measure the decoherence time T_{2} of a single electron spin in a semiconductor quantum dot. The electron is in one of the spin 1/2 states and a circularly polarized laser can only create an optical excitation for one of the electron spin states due to Pauli blocking. An applied electron spin resonance (ESR) field leads to Rabi spin flips and thus to a modulation of the photoluminescence or, alternatively, of the photocurrent. This allows one to measure the ESR linewidth and the coherent Rabi oscillations, from which the electron spin decoherence can be determined. We study different possible schemes for such an ODMR setup, including cw or pulsed laser excitation.

Coulomb scattering crosssection in a 2D electron gas and production of entangled electrons 
D. S. Saraga, B. L. Altshuler (Princeton), Daniel Loss, and R. M. Westervelt (Harvard). Phys. Rev. B 71, 045338 (2005); condmat/0408362.
We calculate the Coulomb scattering amplitude for two electrons injected with opposite momenta in an interacting 2DEG. We include the effect of the Fermi liquid background by solving the 2D BetheSalpeter equation for the twoparticle Green function vertex, in the ladder and random phase approximations. This result is used to discuss the feasibility of producing spin EPR pairs in a 2DEG by collecting electrons emerging from collisions at a pi/2 scattering angle, where only the entangled spinsinglets avoid the destructive interference resulting from quantum indistinguishability. Furthermore, we study the effective 2D electronelectron interaction due to the exchange of virtual acoustic and optical phonons, and compare it to the Coulomb interaction. Finally, we show that the 2D KohnLuttinger pairing instability for the scattering electrons is negligible in a GaAs 2DEG.

Creation and detection of mobile and nonlocal spinentangled electrons 
Patrik Recher (Stanford), D.S. Saraga, and Daniel Loss. pp. 179202, in Fundamental Problems of Mesoscopic Physics, eds. I.V. Lerner et al., NATO Science Ser. II, Vol. 154 (Kluwer, Dordrecht, 2004); condmat/0408526.
We present electron spin entanglersdevices creating mobile spinentangled electrons that are spatially separatedwhere the spinentanglement in a superconductor present in form of Cooper pairs and in a single quantum dot with a spin singlet groundstate is transported to two spatially separated leads by means of a correlated twoparticle tunneling event. The unwanted process of both electrons tunneling into the same lead is suppressed by strong Coulomb blockade effects caused by quantum dots, Luttinger liquid effects or by resistive outgoing leads. In this review we give a transparent description of the different setups, including discussions of the feasibility of the subsequent detection of spinentanglement via charge noise measurements. Finally, we show that quantum dots in the spin filter regime can be used to perform Belltype measurements that only require the measurement of zero frequency charge noise correlators.

Spin susceptibilities, spin densities and their connection to spincurrents 
Sigurdur I. Erlingsson, John Schliemann, and Daniel Loss. Phys. Rev. B 71, 035319 (2005); condmat/0406531.
We calculate the frequency dependent spin susceptibilities for a twodimensional electron gas with both Rashba and Dresselhaus spinorbit interaction. The resonances of the susceptibilities depends on the relative values of the Rashba and Dresselhaus spinorbit constants, which could be manipulated by gate voltages. We derive exact continuity equations, with source terms, for the spin density and use those to connect the spin current to the spin density. In the free electron model the susceptibilities play a central role in the spin dynamics since both the spin density and the spin current are proportional to them.

Rigorous Born Approximation and beyond for the SpinBoson Model 
D. P. DiVincenzo (IBM) and D. Loss. The Mathematica file can be obtained directly here. Phys. Rev. B 71, 035318 (2005); condmat/0405525.
Within the lowestorder Born approximation, we present an exact calculation of the time dynamics of the spinboson model in the ohmic regime. We observe nonMarkovian effects at zero temperature that scale with the systembath coupling strength and cause qualitative changes in the evolution of coherence at intermediate times of order of the oscillation period. These changes could significantly affect the performance of these systems as qubits. In the biased case, we find a prompt loss of coherence at these intermediate times, whose decay rate is set by $\sqrt{\alpha}$, where $\alpha$ is the coupling strength to the environment. We also explore the calculation of the next order Born approximation: we show that, at the expense of very large computational complexity, interesting physical quantities can be rigorously computed at fourth order using computer algebra, presented completely in an accompanying Mathematica file. We compute the $O(\alpha)$ corrections to the long time behavior of the system density matrix; the result is identical to the reduced density matrix of the equilibrium state to the same order in $\alpha$. All these calculations indicate precision experimental tests that could confirm or refute the validity of the spinboson model in a variety of systems.

Hyperfine interaction in a quantum dot: NonMarkovian electron spin dynamics 
W. A. Coish and Daniel Loss. Phys. Rev. B 70, 195340 (2004); condmat/0405676.
We have performed a systematic calculation for the nonMarkovian dynamics of a localized electron spin interacting with an environment of nuclear spins via the Fermi contact hyperfine interaction. This work applies to an electron in the s type orbital ground state of a quantum dot or bound to a donor impurity, and is valid for arbitrary polarization p of the nuclear spin system, and arbitrary nuclear spin I in high magnetic fields. In the limit of p=1 and I=1/2, the Born approximation of our perturbative theory recovers the exact electron spin dynamics. We have found the form of the generalized master equation (GME) for the longitudinal and transverse components of the electron spin to all orders in the electron spinnuclear spin flipflop terms. Our perturbative expansion is regular, unlike standard timedependent perturbation theory, and can be carriedout to higher orders. We show this explicitly with a fourthorder calculation of the longitudinal spin dynamics. In zero magnetic field, the fraction of the electron spin that decays is bounded by the smallness parameter \delta=1/p^{2}N, where N is the number of nuclear spins within the extent of the electron wave function. However, the form of the decay can only be determined in a high magnetic field, much larger than the maximum Overhauser field. In general the electron spin shows rich dynamics, described by a sum of contributions with nonexponential decay, exponential decay, and undamped oscillations. There is an abrupt crossover in the electron spin asymptotics at a critical dimensionality and shape of the electron envelope wave function. We propose a scheme that could be used to measure the nonMarkovian dynamics using a standard spinecho technique, even when the fraction that undergoes nonMarkovian dynamics is small.

SpinHall transport of heavy holes in IIIV semiconductor quantum wells 
John Schliemann and Daniel Loss. Phys. Rev. B 71, 085308 (2005); condmat/0405436.
We investigate spin transport of heavy holes in IIIV semiconductor quantum wells in the presence of spinorbit coupling of the Rashba type due to structureinversion asymmetry. Similarly to the case of electrons, the longitudinal spin conductivity vanishes, whereas the offdiagonal elements of the spinconductivity tensor are finite giving rise to an intrinsic spinHall effect. For a clean system we find a closed expression for the spinHall conductivity depending on the length scale of the Rashba coupling and the hole density. In this limit the spinHall conductivity is enhanced compared to its value for electron systems, and it vanishes with increasing strength of the impurity scattering. As an aside, we also derive explicit expressions for the Fermi momenta and the densities of holes in the different dispersion branches as a function of the spinorbit coupling parameter and the total hole density. These results are of relevance for the interpretation of possible Shubnikovde Haas measurements detecting the Rashba spin splitting.

Grover algorithm with large nuclear spins in semiconductors 
Michael N. Leuenberger and Daniel Loss. Phys. Rev. B 68, 165317 (2003); condmat/0304674.
We show a possible way to implement the Grover algorithm in large nuclear spins 1/2<I<9/2 in semiconductors. The Grover sequence is performed by means of multiphoton transitions that distribute the spin amplitude between the nuclear spin states. They are distinguishable due to the quadrupolar splitting, which makes the nuclear spin levels nonequidistant. We introduce a generalized rotating frame for an effective Hamiltonian that governs the nonperturbative time evolution of the nuclear spin states for arbitrary spin lengths I. The larger the quadrupolar splitting, the better the agreement between our approximative method using the generalized rotating frame and exact numerical calculations.

Spin injection across magnetic/nonmagnetic interfaces with finite magnetic layers 
Alexander Khaetskii, J. Carlos Egues, Daniel Loss, Charles Gould, Georg Schmidt, and Laurens W. Molenkamp (Wuerzburg). Phys. Rev. B 71, 235327 (2005); condmat/0312705.
We have reconsidered the relevant problem of spin injection across ferromagnet/nonmagneticsemiconductor (FM/NMS) and dilutemagneticsemiconductor/nonmagneticsemiconductor interfaces, for structures with \textit{finite} magnetic layers (FM or DMS). By using appropriate physical boundary conditions, we find new expressions for the resistances of these structures which are in general different from previous results in the literature. The results obtained can be important for the interpretation of the experimental data in the case of DMS/NMS structures.

Molecular spintronics: Coherent spin transfer in coupled quantum dots 
Florian Meier (UCSB), Veronica Cerletti, Oliver Gywat, Daniel Loss, and D. D. Awschalom (UCSB). Phys. Rev. B 69, 195315 (2004); condmat/0401397.
Timeresolved Faraday rotation has recently demonstrated coherent transfer of electron spin between quantum dots coupled by conjugated molecules. Using a transfer Hamiltonian ansatz for the coupled quantum dots, we calculate the Faraday rotation signal as a function of the probe frequency in a pumpprobe setup using neutral quantum dots. Additionally, we study the signal of one spinpolarized excess electron in the coupled dots. We show that, in both cases, the Faraday rotation angle is determined by the spin transfer probabilities and the Heisenberg spin exchange energy. By comparison of our results with experimental data, we find that the transfer matrix element for electrons in the conduction band is of order 0.08 eV and the spin transfer probabilities are of order 10%.

Towards Quantum Communication with Electron Spins 
D.S. Saraga, G. Burkard, J.C. Egues, H.A. Engel, P. Recher, and D. Loss. Turk J Phys 27, 427 (2003) Proceedings of the Quantum Computation at the Atomic Scale Conference, (Istanbul, 111 June, 2003)
We review our recent work towards quantum communication in a solidstate environment with qubits carried by electron spins. We propose three schemes to produce spinentangled electrons, where the required separation of the partner electrons is achieved via Coulomb interaction. The nonproduct spinstates originate either from the Cooper pairs found in a superconductor, or in the ground state of a quantum dot with an even number of electrons. In a second stage, we show how spinentanglement carried by a singlet can be detected in a beamsplitter geometry by an increased (bunching) or decreased (antibunching) noise signal. We also discuss how a local spinorbit interaction can be used to provide a continuous modulation of the noise as a signature of entanglement. Finally, we review how one can use a quantum dot as a spin lter, a spinmemory readout, a probe for singlespin decoherence and ultimately, a singlespin measurement apparatus.

Electron spin dynamics in quantum dots and related nanostructures due to hyperfine interaction with nuclei 
John Schliemann, Alexander Khaetskii, and Daniel Loss. J. Phys.: Condens. Matter 15, R1809R1833 (2003); condmat/0311159.
We review and summarize recent theoretical and experimental work on electron spin dynamics in quantum dots and related nanostructures due to hyperfine interaction with surrounding nuclear spins. This topic is of particular interest with respect to several proposals for quantum information processing in solid state systems. Specifically, we investigate the hyperfine interaction of an electron spin confined in a quantum dot in an stype conduction band with the nuclear spins in the dot. This interaction is proportional to the square modulus of the electron wave function at the location of each nucleus leading to an inhomogeneous coupling, i.e. nuclei in different locations are coupled with different strength. In the case of an initially fully polarized nuclear spin system an exact analytical solution for the spin dynamics can be found. For not completely polarized nuclei, approximationfree results can only be obtained numerically in sufficiently small systems. We compare these exact results with findings from several approximation strategies.

Phononinduced decay of the electron spin in quantum dots 
Vitaly N. Golovach, Alexander Khaetskii, and Daniel Loss. Phys. Rev. Lett. 93, 016601 (2004); condmat/0310655.
We study spin relaxation and decoherence in a
GaAs quantum dot due to spinorbit interaction. We derive an effective Hamiltonian which couples the electron spin to phonons or any other fluctuation of the dot potential. We show that the spin decoherence time $T_2$ is as large as the spin relaxation time $T_1$, under realistic conditions. For the Dresselhaus and Rashba spinorbit couplings, we find that, in leading order, the effective magnetic field can have only fluctuations transverse to the applied magnetic field. As a result, $T_2=2T_1$ for arbitrarily large Zeeman splittings, in contrast to the naively expected case $T_2\ll T_1$. We show that the spin decay is drastically suppressed for certain magnetic field directions and values of the Rashba coupling constant. Finally, for the spin coupling to acoustic phonons, we show that $T_2=2T_1$ for all spinorbit mechanisms in leading order in the electronphonon interaction. 
Coulomb scattering in a 2D interacting electron gas and production of EPR pairs 
D.S. Saraga, B.L. Altshuler (Princeton), Daniel Loss, and R.M. Westervelt (Harvard). Phys. Rev. Lett. 92, 246803 (2004); condmat/0310421.
We propose a setup to generate nonlocal spinEPR pairs via pair collisions in a 2D interacting electron gas, based on constructive twoparticle interference in the spin singlet channel at the pi/2 scattering angle. We calculate the scattering amplitude via the BetheSalpeter equation in the ladder approximation and small r_s limit, and find that the Fermi sea leads to a substantial renormalization of the bare scattering process. From the scattering length we estimate the current of spinentangled electrons and show that it is within experimental reach.

Dissipation effects in spinHall transport of electrons and holes 
John Schliemann and Daniel Loss. Phys. Rev. B 69, 165315 (2004); condmat/0310108.
We investigate the spinHall effect of both electrons and holes in semiconductors using the Kubo formula in the correct zerofrequency limit taking into account the finite momentum relaxation time of carriers in real semiconductors. This approach allows to analyze the range of validity of recent theoretical findings. In particular, the spinHall conductivity vanishes for vanishing spinorbit coupling if the correct zerofrequency limit is performed.

Measurement efficiency and nshot read out of spin qubits 
HansAndreas Engel, Vitaly Golovach, Daniel Loss, L.M.K. Vandersypen (TU Delft), J.M. Elzerman (TU Delft), R. Hanson (TU Delft), and L.P. Kouwenhoven (TU Delft). Phys. Rev. Lett. 93, 106804 (2004); condmat/0309023.
We consider electron spin qubits in quantum dots and define a measurement efficiency e to characterize reliable measurements via nshot read outs. We propose various implementations based on a double dot and quantum point contact (QPC) and show that the associated efficiencies e vary between 50% and 100%, allowing singleshot read out in the latter case. We model the read out microscopically and derive its time dynamics in terms of a generalized master equation, calculate the QPC current and show that it allows spin read out under realistic conditions.

Transport through a double quantum dot in the sequential and co tunneling regimes 
Vitaly N. Golovach and Daniel Loss. Phys. Rev. B 69, 245327 (2004); condmat/0308241.
We study transport through a double quantum dot, both in the sequential tunneling and cotunneling regimes. Using a master equation approach, we find that, in the sequential tunneling regime, the differential conductance
$G$ as a function of the bias voltage $\Delta\mu$ has a number of satellite peaks with respect to the main peak of the Coulomb blockade diamond. The position of these peaks is related to the interdot tunnel splitting and the singlettriplet splitting. We find satellite peaks with both {\em positive} and {\em negative} values of differential conductance for realistic parameter regimes. Relating our theory to a microscopic (HundMulliken) model for the double dot, we find a temperature regime for which the Hubbard ratio (=tunnel coupling over onsite Coulomb repulsion) can be extracted from $G(\Delta\mu)$ in the cotunneling regime. In addition, we consider a combined effect of cotunneling and sequential tunneling, which leads to new peaks (dips) in $G(\Delta\mu)$ inside the Coulomb blockade diamond below some temperature scales, which we specify. 
Optical Detection of SingleElectron Spin Decoherence in a Quantum Dot 
Oliver Gywat, HansAndreas Engel, Daniel Loss, R. J. Epstein, F. Mendoza, and D. D. Awschalom (UC Santa Barbara). Phys. Rev. B 69, 205303 (2004); condmat/0307669.
We propose a method based on optically detected magnetic resonance (ODMR) to measure the decoherence time $T_{2}$ of a single electron spin in a semiconductor quantum dot. The electron spin resonance (ESR) of a single excess electron on a quantum dot is probed by circularly polarized laser excitation. The photoluminescence is modulated due to the ESR which enables the measurement of electron spin decoherence. We study different possible schemes for such an ODMR setup.

Dynamical Coulomb blockade and spinentangled electrons 
Patrik Recher and Daniel Loss. Phys. Rev. Lett. 91, 267003 (2003); condmat/0307444.
We consider the production of mobile and nonlocal pairwise spinentangled electrons from tunneling of a BCSsuperconductor (SC) to two normal Fermi liquid leads. The necessary mechanism to separate the two electrons coming from the same Cooper pair (spinsinglet) is achieved by coupling the SC to leads with a finite resistance. The resulting dynamical Coulomb blockade effect, which we describe phenomenologically in terms of an electromagnetic environment, is shown to be enhanced for tunneling of two spinentangled electrons into the same lead compared to the process where the pair splits and each electron tunnels into a different lead. On the other hand in the pairsplit process, the spatial correlation of a Cooper pair leads to a current suppression as a function of distance between the two tunnel junctions which is weaker for effectively lower dimensional SCs.

Hyperfine interactions and electron spin dynamics in a quantum dot 
A. Khaetskii, D. Loss, and L. Glazman (Minnesota). Journal of Superconductivity: Incorporating Novel Magnetism 16, 221 (2003)
We show that a wide range of spin clusters with antiferromagnetic intracluster exchange interaction allows one to define a qubit. For these spin cluster qubits, initialization, quantum gate operation, and readout are possible using the same techniques as for single spins. Quantum gate operation for the spin cluster qubit does not require control over the intracluster exchange interaction. Electric and magnetic fields necessary to effect quantum gates need only be controlled on the length scale of the spin cluster rather than the scale for a single spin. Here, we calculate the energy gap separating the logical qubit states from the next excited state and the matrix elements which determine quantum gate operation times. We discuss spin cluster qubits formed by one and twodimensional arrays of s=1/2 spins as well as clusters formed by spins s>1/2. We illustrate the advantages of spin cluster qubits for various suggested implementations of spin qubits and analyze the scaling of decoherence time with spin cluster size.

Anisotropic transport in the twodimensional electron gas in the presence of spinorbit coupling 
John Schliemann and Daniel Loss. Phys. Rev. B 68, 165311 (2003); condmat/0306528.
In a twodimensional electron gas as realized by a semiconductor quantum well, the presence of spinorbit coupling of both the Rashba and Dresselhaus type leads to anisotropic dispersion relations and Fermi contours. We study the effect of this anisotropy on the electrical conductivity in the presence of fixed impurity scatterers. The conductivity also shows in general an anisotropy which can be tuned by varying the Rashba coefficient. This effect provides a method of detecting and investigating spinorbit coupling by measuring spinunpolarized electrical currents in the diffusive regime. Our approach is based on an exact solution of the twodimensional Boltzmann equation and provides also a natural framework for investigating other transport effects including the anomalous Hall effect.

Noise of SpinPolarized Currents at a Beam Splitter with Local SpinOrbit Interaction 
G. Burkard, J. C. Egues (Sao Paulo), and D. Loss. J. Supercond. 16, 237 (2003)
An electronic beam splitter with a local Rashba spinorbit coupling can serve as a detector for spinpolarized currents. The spinorbit coupling plays the role of a tunable spin rotator and can be controlled via a gate electrode on top of the conductor. We use spinresolved scattering theory to calculate the zerotemperature current fluctuations (shot noise) for such a fourterminal device and show that the shot noise is proportional to the spin polarization of the source. Moreover, we analyze the effect of spinorbitinduced intersubband coupling, leading to an additional spin rotation.

Quantum computing with antiferromagnetic spin clusters 
Florian Meier, Jeremy Levy (Pittsburgh), and Daniel Loss. Phys. Rev. B 68, 134417 (2003); condmat/0304296.
We show that a wide range of spin clusters with antiferromagnetic intracluster exchange interaction allows one to define a qubit. For these spin cluster qubits, initialization, quantum gate operation, and readout are possible using the same techniques as for single spins. Quantum gate operation for the spin cluster qubit does not require control over the intracluster exchange interaction. Electric and magnetic fields necessary to effect quantum gates need only be controlled on the length scale of the spin cluster rather than the scale for a single spin. Here, we calculate the energy gap separating the logical qubit states from the next excited state and the matrix elements which determine quantum gate operation times. We discuss spin cluster qubits formed by one and twodimensional arrays of s = 1/2 spins as well as clusters formed by spins s > 1/2. We illustrate the advantages of spin cluster qubits for various suggested implementations of spin qubits and analyze the scaling of decoherence time with spin cluster size.

Exact Born Approximation for the SpinBoson Model 
Daniel Loss and David P. DiVincenzo (IBM Yorktown). condmat/0304118
Within the lowestorder Born approximation, we present an exact calculation of the time dynamics of the spinboson model in the Ohmic regime. We observe nonMarkovian effects at zero temperature that scale with the systembath coupling strength and cause qualitative changes in the evolution of coherence at intermediate times of order of the oscillation period. These changes could significantly affect the performance of these systems as qubits. In the biased case, we find a prompt loss of coherence at these intermediate times, whose decay rate is set by $\sqrt{\alpha}$, where $\alpha$ is the coupling strength to the environment. These calculations indicate precision experimental tests that could confirm or refute the validity of the spinboson model in a variety of systems.

Shot Noise of Cotunneling Current 
Eugene Sukhorukov (Geneva), Guido Burkard (IBM Yorktown), and Daniel Loss. in "Quantum Noise in Mesoscopic Physics", ed. Y.V. Nazarov, pp 149172, Kluwer, 2003, The Netherlands; condmat/0211024.
We study the noise of the cotunneling current through one or several tunnelcoupled quantum dots in the Coulomb blockade regime. The various regimes of weak and strong, elastic and inelastic cotunneling are analyzed for quantumdot systems (QDS) with fewlevel, nearlydegenerate, and continuous electronic spectra. In the case of weak cotunneling we prove a nonequilibrium fluctuationdissipation theorem, which leads to a universal expression for the noisetocurrent ratio (Fano factor). The noise of strong inelastic cotunneling can be superPoissonian due to switching between QDS states carrying currents of different strengths. The transport through a doubledot (DD) system shows an AharonovBohm effect both in noise and current. In the case of cotunneling through a QDS with a continuous energy spectrum the Fano factor is very close to one.

Lower bound for electron spin entanglement from beamsplitter current correlations 
Guido Burkard (IBM Yorktown Heights) and Daniel Loss. Phys. Rev. Lett. 91, 087903 (2003); condmat/0303209.
We determine a lower bound for the entanglement of pairs of electron spins injected into a mesoscopic conductor. The bound can be expressed in terms of experimentally accessible quantities, the zerofrequency current correlators (shot noise power or crosscorrelators) after transmission through an electronic beam splitter. The effect of spin relaxation (T_1 processes) and decoherence (T_2 processes) during the ballistic coherent transmission of the carriers in the wires is taken into account within Bloch theory. The presence of a variable inhomogeneous magnetic field allows the determination of a useful lower bound for the entanglement of arbitrary entangled states. The decrease in entanglement due to thermally mixed states is studied. Both the entanglement of the output of a source (entangler) and the relaxation (T_1) and decoherence (T_2) times can be determined.

Discrete Fourier Transform in Nanostructures using Scattering 
Michael N. Leuenberger (Iowa), Michael E. Flatte (Iowa), Daniel Loss, and D. D. Awschalom (UCSB). J. Appl. Phys. 95, 8167 (2004); condmat/0302279.
In this paper we show that the discrete Fourier transform can be performed by scattering a coherent particle or laser beam off a twodimensional potential that has the shape of rings or peaks. After encoding the initial vector into the twodimensional potential, the Fouriertransformed vector can be read out by detectors surrounding the potential. The wavelength of the laser beam determines the necessary accuracy of the 2D potential, which makes our method very faulttolerant.

SpinOrbit Coupling and TimeReversal Symmetry in Quantum Gates 
D. Stepanenko (Florida State), N. E. Bonesteel (Florida State), D.P. DiVincenzo (IBM Yorktown), G. Burkard (IBM Yorktown), and D. Loss. Phys. Rev. B 68, 115306 (2003); condmat/0303474.
We study the effect of spinorbit coupling on quantum gates produced by pulsing the exchange interaction between two single electron quantum dots. Spinorbit coupling enters as a small spin precession when electrons tunnel between dots. For adiabatic pulses the resulting gate is described by a unitary operator acting on the fourdimensional Hilbert space of two qubits. If the precession axis is fixed, timesymmetric pulsing constrains the set of possible gates to those which, when combined with single qubit rotations, can be used in a simple CNOT construction. Deviations from timesymmetric pulsing spoil this construction. The effect of time asymmetry is studied by numerically integrating the Schr\"odinger equation using parameters appropriate for GaAs quantum dots. Deviations of the implemented gate from the desired form are shown to be proportional to dimensionless measures of both spinorbit coupling and time asymmetry of the pulse.

Spin Stiffness of Mesoscopic Quantum Antiferromagnets 
Daniel Loss and Dmitrii L. Maslov. Phys. Rev. Lett. 74, 178 (1995)
We study the spin stiffness of a onedimensional quantum antiferromagnet in the whole range of system sizes L and temperatures T. We show that for integer and halfodd integer spin cases the stiffness differs fundamentally in its L and T dependences, and that in the latter case the stiffness exhibits a striking dependence on the parity of the number of sites. Integer spin chains are treated in terms of the nonlinear sigma model, while halfodd integer spin chains are discussed in a renormalization group approach leading to a Luttinger liquid with AharonovBohmtype boundary conditions.

Quantum Information is Physical 
David P. DiVincenzo (IBM Yorktown) and Daniel Loss. Superlattices and Microstructures 23, 419 (1998); condmat/9710259.
We discuss a few current developments in the use of quantum mechanically coherent systems for information processing. In each of these developments, Rolf Landauer has played a crucial role in nudging us and other workers in the field into asking the right questions, some of which we have been lucky enough to answer. A general overview of the key ideas of quantum error correction is given. We discuss how quantum entanglement is the key to protecting quantum states from decoherence in a manner which, in a theoretical sense, is as effective as the protection of digital data from bit noise. We also discuss five general criteria which must be satisfied to implement a quantum computer in the laboratory, and we illustrate the application of these criteria by discussing our ideas for creating a quantum computer out of the spin states of coupled quantum dots. [Special Issue on the Occasion of Rolf Landauer's 70th Birthday.]

Quantum information processing and communication  Strategic report on current status, visions and goals for research in Europe 
Zoller P, Beth T, Binosi D, Blatt R, Briegel H, Bruss D, Calarco T, Cirac JI, Deutsch D, Eisert J, Ekert A, Fabre C, Gisin N, Grangiere P, Grassl M, Haroche S, Imamoglu A, Karlson A, Kempe J, Kouwenhoven L, Kroll S, Leuchs G, Lewenstein M, Loss D, Lutkenhaus N, Massar S, Mooij JE, Plenio MB, Polzik E, Popescu S, Rempe G, Sergienko A, Suter D, Twamley J, Wendin G, Werner R, Winter A, Wrachtrup J, and Zeilinger A. Eur. Phys. J. D 36, 203 (2005)
We present an excerpt of the document "Quantum Information Processing and Communication: Strategic report on current status, visions and goals for research in Europe", which has been recently published in electronic form at the website of FET (the Future and Emerging Technologies Unit of the Directorate General Information Society of the European Commission, http://qurope.eu/content/Roadmap). This document has been elaborated, following a former suggestion by FET, by a committee of QIPC scientists to provide input towards the European Commission for the preparation of the Seventh Framework Program. Besides being a document addressed to policy makers and funding agencies (both at the European and national level), the document contains a detailed scientific assessment of the stateoftheart, main research goals, challenges, strengths, weaknesses, visions and perspectives of all the most relevant QIPC subfields, that we report here.

Probing entanglement via Rashbainduced shot noise oscillations 
J. Carlos Egues, Guido Burkard, and Daniel Loss. J. Superconductivity, 16, 711 (2003); condmat/0207392.
We have recently calculated shot noise for entangled and spinpolarized electrons in novel beamsplitter geometries with a local Rashba so interaction in the incoming leads. This interaction allows for a gatecontrolled rotation of the incoming electron spins. Here we present an alternate simpler route to the shot noise calculation in the above work and focus on only electron pairs. Shot noise for these shows continuous bunching and antibunching behaviors. In addition, entangled and unentangled triplets yield distinctive shot noise oscillations. Besides allowing for a direct way to identify triplet and singlet states, these oscillations can be used to extract so coupling constants through noise measurements. Incoming leads with spinorbit interband mixing give rise an additional modulation of the current noise. This extra rotation allows the design of a spin transistor with enhanced spin control.

Shot noise for entangled and spinpolarized electrons 
J. C. Egues, P. Recher, D. S. Saraga, V. N. Golovach, G. Burkard, E. V. Sukhorukov, and D. Loss. Quantum Noise in Mesoscopic Physics, NATO ASI Series II, Vol. 97 (Kluwer, 2003), pp 241274; Proceedings of the NATO Advanced Research Workshop (Delft, The Netherlands, 24 June 2002); condmat/0210498.
We review our recent contributions on shot noise for entangled electrons and spinpolarized currents in novel mesoscopic geometries. We first discuss some of our recent proposals for electron entanglers involving a superconductor coupled to a double dot in the Coulomb blockade regime, a superconductor tunnelcoupled to Luttingerliquid leads, and a tripledot setup coupled to Fermi leads. We calculate current and shot noise for spinpolarized currents and entangled/unentangled electron pairs in a beamsplitter geometry with a \textit{local} Rashba spinorbit (so) interaction in the incoming leads. We find \textit{continuous} bunching and antibunching behaviors for the \textit{entangled} pairs  triplet and singlet  as a function of the Rashba rotation angle. In addition, we find that unentangled triplets and the entangled one exhibit distinct shot noise. Shot noise for spinpolarized currents shows sizable oscillations as a function of the Rashba phase. This happens only for electrons injected perpendicular to the Rashba rotation axis; spinpolarized carriers along the Rashba axis are noiseless. We find an additional spin rotation for electrons with energies near the crossing of the bands where so induced interband coupling is relevant. This gives rise to an additional modulation of the noise for both electron pairs and spinpolarized currents. Finally, we briefly discuss shot noise for a double dot near the Kondo regime.

Semiconductor Spintronics and Quantum Computation 
eds. D.D. Awschalom, D. Loss, and N. Samarth. Springer, Berlin, 2002.
The manipulation of electric charge in bulk semiconductors and their heterostructures forms the basis of virtually all contemporary electronic and optoelectronic devices. Recent studies of spindependent phenomena in semiconductors have now opened the door to technological possibilities that harness the spin of the electron in semiconductor devices. In addition to providing spindependent analogies that extend existing electronic devices into the realm of semiconductor "spintronics" the spin degree of freedom also offers prospects for fundamentally new functionality within the quantum domain, ranging from storage to computation. It is anticipated that the spin degree of freedom in semiconductors will play a crucial role in the development of information technologies in the 21st century. This book brings together a team of experts to provide an overview of emerging concepts in this rapidly developing field. The topics range from spin transport and injection in semiconductors and their heterostructures to coherent processes and quantum computation in semiconductor quantum structures and microcavities.

Dynamics of entanglement between quantum dot spinqubits 
John Schliemann and Daniel Loss. Proceedings of the E. Fermi School, "Quantum Phenomena of Mesoscopic Systems", 9  19 July 2002, Varenna, Italy; condmat/0212141.
We briefly review the physics of gate operations between quantum dot spinqubits and analyze the dynamics of quantum entanglement in such processes. The indistinguishable character of the electrons whose spins realize the qubits gives rise to further entanglementlike quantum correlations that go beyond simple antisymmetrization effects. We also summarize further recent results concerning this type of quantum correlations of indistinguishable particles. Finally we discuss decoherence properties of spinqubits when coupled to surrounding nuclear spins in a semiconductor nanostructure

Electron spin evolution induced by interaction with nuclei in a quantum dot 
Alexander Khaetskii, Daniel Loss, and Leonid Glazman (Minnesota). Phys. Rev. B 67, 195329 (2003); condmat/0211678.
We study the decoherence of a single electron spin in an isolated quantum dot induced by hyperfine interaction with nuclei for times smaller than the nuclear spin relaxation time. The decay is caused by the spatial variation of the electron envelope wave function within the dot, leading to a nonuniform hyperfine coupling $A$. We show that the usual treatment of the problem based on the Markovian approximation is impossible because the correlation time for the nuclear magnetic field seen by the electron spin is itself determined by the flipflop processes.
The decay of the electron spin correlation function is not exponential but rather power (inverse logarithm) lawlike. For polarized nuclei we find an exact solution and show that the precession amplitude and the decay behavior can be tuned by the magnetic field. The decay time is given by $\hbar N/A$, where $N$ is the number of nuclei inside the dot. The amplitude of precession, reached as a result of the decay, is finite. We show that there is a striking difference between the decoherence time for a single dot and the dephasing time for an ensemble of dots. 
Coherent spin quantum dynamics in antiferromagnetic molecular rings 
Florian Meier and Daniel Loss. Physica B 329, 1140 (2003)
Molecular magnetic clusters with antiferromagnetic exchange interaction and easy axis anisotropy belong to the most promising candidate systems for the observation of coherent spin quantum tunneling on the mesoscopic scale. We point out that both nuclear magnetic resonance and electron spin resonance on doped rings are adequate experimental techniques for the detection of coherent spin quantum tunneling in antiferromagnetic molecular rings. Although challenging, the experiments are feasible with present day techniques.

Nonballistic spin fieldeffect transistor 
John Schliemann, J. Carlos Egues, and Daniel Loss. Phys. Rev. Lett. 90, 146801 (2003); condmat/0211603.
We propose a spin fieldeffect transistor based on spinorbit (so) coupling of both the Rashba and the Dresselhaus types. Differently from earlier proposals, spin transport through our device is tolerant against spinindependent scattering processes. Hence the requirement of strictly ballistic transport can be relaxed. This follows from a unique interplay between the Dresselhaus and the (gatecontrolled) Rashba interactions; these can be tuned to have equal strengths thus yielding kindependent eigenspinors even in two dimensions. We discuss implementations with twodimensional devices and quantum wires. In the latter, our setup presents strictly parabolic dispersions which avoids complications arising from anticrossings of different bands.

Spin qubits in solidstate structures 
G. Burkard and D. Loss. Europhysics News 33 5 (2002) 166170
It is remarkable that today's computers, after the tremendous development during the last 50 years, are still essentially described by the mathematical model formulated by Alan Turing in the 1930's. Turing's model describes computers which operate according to the laws of classical physics. What would happen if a computer was operating according to the quantum laws? Physicists and computer scientists have been interested in this question since the early 1980's, but research in quantum computation really started to flourish after 1994 when Peter Shor discovered a quantum algorithm to find prime factors of large integers efficiently, a problem which is intrinsically hard for any classical computer (see [1] for an introduction into quantum computation). The lack of an algorithm for efficient factoring on a classical machine is actually the basis of the widely used RSA encryption scheme. Phase coherence needs to be maintained for a sufficiently long time in the memory of a quantum computer. This may sound like a harmless requirement, but in fact it is the main reason why the physical implementation of quantum computation is so difficult. Usually, a quantum memory is thought of as a set of twolevel systems, named quantum bits, or qubits for short. In analogy to the classical bit, two orthogonal computational basis states 0> and 1> are defined. The textbook example of a quantum twolevel system is the spin 1/2 of, say, an electron, where one can identify the "spin up" state with 0> and the "spin down" state with 1>. While several other twolevel systems have been proposed for quantum computing, we will devote the majority of our discussion to the potential use of electron spins in nanostructures (such as quantum dots) as qubits.

Magnetization transport and quantized spin conductance 
Florian Meier and Daniel Loss. Phys. Rev. Lett. 90, 167204 (2003); condmat/0209521.
We analyze transport of magnetization in insulating systems described by a spin Hamiltonian. The magnetization current through a quasi onedimensional magnetic wire of finite length suspended between two bulk magnets is determined by the spin conductance which remains finite in the ballistic limit due to contact resistance. For ferromagnetic systems, magnetization transport can be viewed as transmission of magnons and the spin conductance depends on the temperature T. For antiferromagnetic isotropic spin1/2 chains, the spin conductance is quantized in units of order $(g \mu_B)^2/h$ at T=0. Magnetization currents produce an electric field and hence can be measured directly. For magnetization transport in electric fields phenomena analogous to the Hall effect emerge.

A DattaDas transistor with enhanced spin control 
J. Carlos Egues, Guido Burkard, and Daniel Loss. Appl. Phys. Lett. 82, 2658 (2003); condmat/0209682.
We consider a twochannel spin transistor with weak spinorbit induced interband coupling. We show that the coherent transfer of carriers between the coupled channels gives rise to an additional spin rotation. We calculate the corresponding spinresolved current in a DattaDas geometry and show that a weak interband mixing leads to enhanced spin control.

Variational study of the nu=1 quantum Hall ferromagnet in the presence of spinorbit interaction 
John Schliemann, J. Carlos Egues, and Daniel Loss. Phys. Rev. B 67, 085302 (2003); condmat/0209185.
We investigate the nu=1 quantum Hall ferromagnet in the presence of spinorbit coupling of the Rashba or Dresselhaus type by means of HartreeFocktyped variational states. In the presence of Rashba (Dresselhaus) spinorbit coupling the fully spinpolarized quantum Hall state is always unstable resulting in a reduction of the spin polarization if the product of the particle charge $q$ and the effective $g$factor is positive (negative). In all other cases an alternative variational state with O(2) symmetry and finite inplane spin components is lower in energy than the fully spinpolarized state for large enough spinorbit interaction. The phase diagram resulting from these considerations differs qualitatively from earlier studies.

Spin decay and quantum parallelism 
John Schliemann, Alexander V. Khaetskii, and Daniel Loss. Phys. Rev. B 66, 245303 (2002); condmat/0207195.
We study the time evolution of a single spin coupled inhomogeneously to a spin environment. Such a system is realized by a single electron spin bound in a semiconductor nanostructure and interacting with surrounding nuclear spins. We find striking dependencies on the type of the initial state of the nuclear spin system. Simple product states show a profoundly different behavior than randomly correlated states whose time evolution provides an illustrative example of quantum parallelism and entanglement in a decoherence phenomenon.

Quantum computing with spin cluster qubits 
Florian Meier, Jeremy Levy (Pittsburgh), and Daniel Loss. Phys. Rev. Lett. 90, 047901 (2003); condmat/0206310.
We study the low energy states of finite spin chains with isotropic (Heisenberg) and anisotropic (XY and Isinglike) exchange interaction with uniform and nonuniform coupling constants. We show that for an odd number of sites a spin cluster qubit can be defined in terms of the ground state doublet. This qubit is remarkably insensitive to the placement and coupling anisotropy of spins within the cluster. One and twoqubit quantum gates can be generated by magnetic fields and intercluster exchange, and leakage during quantum gate operation is small. Spin cluster qubits inherit the long decoherence times and short gate operation times of single spins. Control of single spins is hence not necessary for the realization of universal quantum gates.

Quantum Spin Dynamics in Molecular Magnets 
Michael N. Leuenberger, Florian Meier, and Daniel Loss. Monatshefte für Chem. 134, 217(2003); condmat/0205457.
The detailed theoretical understanding of quantum spin dynamics in various molecular magnets is an important step on the roadway to technological applications of these systems. Quantum effects in both ferromagnetic and antiferromagnetic molecular clusters are, by now, theoretically well understood. Ferromagnetic molecular clusters allow one to study the interplay of incoherent quantum tunneling and thermally activated transitions between states with different spin orientation. The Berry phase oscillations found in Fe_8 are signatures of the quantum mechanical interference of different tunneling paths. Antiferromagnetic molecular clusters are promising candidates for the observation of coherent quantum tunneling on the mesoscopic scale. Although challenging, applications of molecular magnetic clusters for data storage and quantum data processing are within experimental reach already with present day technology.

Spinentangled currents created by a triple quantum dot 
Daniel S. Saraga and Daniel Loss. Phys. Rev. Lett. 90, 166803 (2003); condmat/0205553.
We propose a simple setup of three coupled quantum dots in the Coulomb blockade regime as a source for spatially separated currents of spinentangled electrons. The entanglement originates from the singlet ground state of a quantum dot with an even number of electrons. To preserve the entanglement of the electron pair during its extraction to the drain leads, the electrons are transported through secondary dots. This prevents oneelectron transport by energy mismatch, while joint transport is resonantly enhanced by conservation of the total twoelectron energy.

Rashba spinorbit interaction and shot noise for spinpolarized and entangled electrons 
J. Carlos Egues, Guido Burkard, and Daniel Loss. Phys. Rev. Lett. 89, 176401 (2002); condmat/0204639.
We study shot noise for spinpolarized currents and entangled electron pairs in a fourprobe (beam splitter) geometry with a local Rashba spinorbit (so) interaction in the incoming leads. Within the scattering formalism we find that shot noise exhibits Rashbainduced oscillations with continuous bunching and antibunching. We show that entangled states as well as triplet states can be identified via their Rashba phase in noise measurements. For twochannel leads we find an additional spin rotation due to so induced interband coupling which provides additional spin control. We show that the so interaction determines the Fano factor which provides a direct way to measure the Rashba coupling constant via noise.

Quantum information processing with large nuclear spins in GaAs semiconductors 
Michael N. Leuenberger, Daniel Loss, Martino Poggio (UCSB), and David D. Awschalom (UCSB). Phys. Rev. Lett. 89, 207601 (2002); condmat/0204355.
We propose an implementation for quantum information processing based on coherent manipulations of nuclear spins I=3/2 in GaAs semiconductors. We describe theoretically an NMR method which involves multiphoton transitions and which exploits the nonequidistance of nuclear spin levels due to quadrupolar splittings. Starting from known spin anisotropies we derive effective Hamiltonians in a generalized rotating frame, valid for arbitrary I, which allow us to describe the nonperturbative time evolution of spin states generated by magnetic rf fields. We identify an experimentally accessible regime where multiphoton Rabi oscillations are observable. In the nonlinear regime, we find Berry phase interference effects.

Superconductor coupled to two Luttinger liquids as an entangler for electron spins 
Patrik Recher and Daniel Loss. Phys. Rev. B 65, 165327 (2002); condmat/0112298.
We consider an swave superconductor (SC) which is tunnelcoupled to two spatially separated Luttinger liquid (LL) leads. We demonstrate that such a setup acts as an entangler, i.e. it creates spinsinglets of two electrons which are spatially separated, thereby providing a source of electronic EinsteinPodolskyRosen pairs. We show that in the presence of a bias voltage, which is smaller than the energy gap in the SC, a stationary current of spinentangled electrons can flow from the SC to the LL leads due to Andreev tunneling events. We discuss two competing transport channels for Cooper pairs to tunnel from the SC into the LL leads. On the one hand, the coherent tunneling of two electrons into the same LL lead is shown to be suppressed by strong LL correlations compared to singleelectron tunneling into a LL. On the other hand, the tunneling of two spinentangled electrons into different leads is suppressed by the initial spatial separation of the two electrons coming from the same Cooper pair. We show that the latter suppression depends crucially on the effective dimensionality of the SC. We identify a regime of experimental interest in which the separation of two spinentangled electrons is favored. We determine the decay of the singlet state of two electrons injected into different leads caused by the LL correlations. Although the electron is not a proper quasiparticle of the LL, the spin information can still be transported via the spin density fluctuations produced by the injected spinentangled electrons.

Electron spin decoherence in quantum dots due to interaction with nuclei 
Alexander Khaetskii, Daniel Loss, and Leonid Glazman (Univ. of Minnesota). Phys. Rev. Lett. 88, 186802 (2002); condmat/0201303.
We study the decoherence of a single electron spin in an isolated quantum dot induced by hyperfine interaction with nuclei for times smaller than the nuclear spin relaxation time. The decay is caused by the spatial variation of the electron envelope wave function within the dot, leading to a nonuniform hyperfine coupling.
We evaluate the spin correlation function with and without magnetic fields and find that the decay of the spin precession amplitude is not exponential but rather power (inverse logarithm) lawlike. For fully polarized nuclei we find an exact solution and show that the precession amplitude and the decay behavior can be tuned by the magnetic field. The corresponding decay time is given by $\hbar N/A$, where $A$ is a hyperfine interaction constant and $N$ the number of nuclei inside the dot. The amplitude of precession, reached as a result of the decay, is finite. We show that there is a striking difference between the decoherence time for a single dot and the dephasing time for an ensemble of dots. 
Creation of Nonlocal SpinEntangled Electrons via Andreev Tunneling, Coulomb Blockade, and Resonant Transport 
Patrik Recher and Daniel Loss. Journal of Superconductivity: Incorporating Novel Magnetism 15: 4965 (2002); arXiv:condmat/0205484.
We discuss several scenarios for the creation of nonlocal spinentangled electrons which provide a source of electronic EinsteinPodolskyRosen (EPR) pairs. Such EPR pairs can be used to test nonlocality of electrons in solid state systems, and they form the basic resources for quantum information processing. The central idea is to exploit the spin correlations naturally present in superconductors in form of Cooper pairs possessing spinsinglet wavefunctions. We show that nonlocal spinentanglement in form of an effective Heisenberg spin interaction is induced between electron spins residing on two quantum dots with no direct coupling between them, but each of them being tunnelcoupled to the same superconductor. We then discuss a nonequilibrium setup with an applied bias where mobile and nonlocal spinentanglement can be created by coherent injection of two electrons, in a pair (Andreev) tunneling process, into two spatially separated quantum dots and subsequently into two Fermi liquid leads. The current for injecting two spinentangled electrons into different leads shows a resonance and allows the injection of electrons at the same orbital energy, which is a crucial requirement for the detection of spinentanglement via the current noise. On the other hand, tunneling via the same dot into the same lead is suppressed by the Coulomb blockade effect of the quantum dots. We discuss AharonovBohm oscillations in the current and show that they contain h/e and h/2e periods, which provides an experimental means to test the nonlocality of the entangled pair. Finally, we discuss a structure consisting of a superconductor weakly coupled to two separate onedimensional leads with Luttinger liquid properties. We show that strong correlations again suppress the coherent subsequent tunneling of two electrons into the same lead, thus generating again nonlocal spinentangled electrons in the Luttinger liquid leads.

Electron Spins in Artificial Atoms and Molecules for Quantum Computing 
Vitaly N. Golovach and Daniel Loss. Semicond. Sci. Technol. 17, 355 366 (2002); condmat/0201437.
Achieving control over the electron spin in quantum dots (artificial atoms) or real atoms promises access to new technologies in conventional and in quantum information processing. Here we review our proposal for quantum computing with spins of electrons confined to quantum dots. We discuss the basic requirements for implementing spinqubits, and describe a complete set of quantum gates for single and twoqubit operations. We show how a quantum dot attached to leads can be used for spin filtering and spin readout, and as a spinmemory device. Finally, we focus on the experimental characterization of the quantum dot systems, and discuss transport properties of a doubledot and show how Kondo correlations can be used to measure the Heisenberg exchange interaction between the spins of two dots.

Entanglement and Quantum Gate Operations with SpinQubits in Quantum Dots 
John Schliemann and Daniel Loss. ``Future Trends in Microelectronics: The Nano Millenium", eds. S. Luryi, J. Xu, and A. Zaslavsky, Wiley, 2002, pp. 319334; condmat/0110150.
We give an elementary introduction to the notion of quantum entanglement between distinguishable parties and review a recent proposal about solid state quantum computation with spinqubits in quantum dots. The indistinguishable character of the electrons whose spins realize the qubits gives rise to further entanglementlike quantum correlations. We summarize recent results concerning this type of quantum correlations of indistinguishable particles.

Single Spin Dynamics and Decoherence in a Quantum Dot via Charge Transport 
HansAndreas Engel and Daniel Loss. Phys. Rev. B 65, 1953211 (2002); condmat/0109470.
We investigate the spin dynamics of a quantum dot with a spin1/2 ground state in the Coulomb blockade regime and in the presence of a magnetic rf field leading to electron spin resonances (ESR). We show that by coupling the dot to leads, spin properties on the dot can be accessed via the charge current in the stationary and nonstationary limit. We present a microscopic derivation of the current and the master equation of the dot using superoperators, including contributions to decoherence and energy shifts due to the tunnel coupling. We give a detailed analysis of sequential and cotunneling currents, for linearly and circularly oscillating ESR fields, applied in cw and pulsed mode. We show that the sequential tunneling current exhibits a spin satellite peak whose linewidth gives a lower bound on the decoherence time T_2 of the dotspin. Similarly, the spin decoherence can be accessed also in the cotunneling regime via ESR induced spin flips. We show that the conductance ratio of the spin satellite peak and the conventional peak due to sequential tunneling saturates at the universal conductance ratio of 0.71 for strong ESR fields. We describe a doubledot setup which generates spin dependent tunneling and acts as a current pump (at zero bias), and as a spin inverter which inverts the spinpolarization of the current. We show that Rabi oscillations of the dotspin induce coherent oscillations in the timedependent current. These oscillations are observable in the timeaveraged current as function of ESR pulseduration, and they allow one to access the spin coherence directly in the time domain. We analyze the measurement and readout process of the dotspin via currents in spinpolarized leads and identify measurement time and efficiency by calculating the counting statistics, noise, and the Fano factor.

Biexcitons in Coupled Quantum Dots as a Source for Entangled Photons 
Oliver Gywat, Guido Burkard, and Daniel Loss. Phys. Rev. B 65, 205329 (2002); condmat/0109223.
We study biexcitonic states in two tunnelcoupled semiconductor quantum dots and show that they provide a source for entangled photons which are spatially separated at production. We distinguish between the various spin configurations and calculate the lowenergy biexciton spectrum using the HeitlerLondon approximation as a function of magnetic and electric fields. We calculate the oscillator strengths for the biexciton recombination involving the sequential emission of two photons with entangled polarizations corresponding to the spin configuration in the biexciton states.

Quantum coherent dynamics in molecular magnetic rings 
A. Honecker, F. Meier, Daniel Loss, and B. Normand. Eur. Phys. J. B 27, 487 (2002); condmat/0109201.
We present detailed calculations of lowenergy spin dynamics in the ``ferric wheel'' systems Na:Fe_6 and Cs:Fe_8 in a magnetic field. We compute by exact diagonalization the lowenergy spectra and matrix elements for totalspin and N'eelvector components, and thus the timedependent correlation functions of these operators. We compare our results with semiclassical tunneling descriptions, and discuss their implications for mesoscopic quantum coherence, as well as for the experimental techniques to observe it, in molecular magnetic rings.

Kondo effect and singlettriplet splitting in coupled quantum dots in a magnetic field 
Vitaly N. Golovach and Daniel Loss. Europhys. Lett. 62, 83 (2003); condmat/0109155.
We study two tunnelcoupled quantum dots each with a spin 1/2 and attached to leads in the Coulomb blockade regime. We study the interplay between Kondo correlations and the singlettriplet exchange splitting $K$ between the two spins. We calculate the cotunneling current with elastic and inelastic contributions and its renormalization due to Kondo correlations, away and at the degeneracy point K=0. We show that these Kondo correlations induce pronounced peaks in the conductance as function of magnetic field $B$, interdot coupling $t_0$, and temperature. Moreover, the longrange part of the Coulomb interaction becomes visibile due to Kondo correlations resulting in an additional peak in the conductance vs $t_0$ with a strong $B$field dependence. These conductance peaks thus provide direct experimental access to $K$, and thus to a crucial control parameter for spinbased qubits and entanglement.

Cancellation of spinorbit effects in quantum gates based on the exchange coupling 
Guido Burkard and Daniel Loss. Phys. Rev. Lett. 88, 047903 (2002); condmat/0108101.
We study the effect of the spinorbit interaction on quantum gate operations based on the spin exchange coupling where the qubit is represented by the electron spin in a quantum dot or a similar nanostructure. Our main result is the exact cancellation of the spinorbit effects in the sequence producing the quantum XOR gate for the ideal case where the pulse shapes of the exchange and spinorbit interactions are identical. For the nonideal case, we show that the two pulse shapes can be made almost identical and that the gate error is strongly suppressed by two small parameters, the spinorbit interaction constant and the smallness of the deviation of the two pulse shapes. Similarly, we show that the dipoledipole interaction leads only to very small errors in the XOR gate.

Electron and Nuclear Spin Dynamics in Ferric Wheels 
Florian Meier and Daniel Loss. Phys. Rev. Lett. 86, 5373 (2001); condmat/0101073.
We study theoretically the spin dynamics of the ferric wheel, an antiferromagnetic molecular ring. For a single nuclear or impurity spin coupled to one of the electron spins of the ring, we calculate nuclear and electronic spin correlation functions and show that nuclear magnetic resonance (NMR) and electron spin resonance (ESR) techniques can be used to detect coherent tunneling of the Neel vector in these rings. The location of the NMR/ESR resonances gives the tunnel splitting and its linewidth an upper bound on the decoherence rate of the electron spin dynamics. We illustrate the experimental feasibility of our proposal with estimates for Fe_10 molecules.

Thermodynamics and Spin Tunneling Dynamics in Ferric Wheels with Excess Spin 
Florian Meier and Daniel Loss. Phys. Rev. B 64, 224411 (2001); condmat/0107025.
We study theoretically the thermodynamic properties and spin dynamics of a class of magnetic rings closely related to ferric wheels, antiferromagnetic ring systems, in which one of the Fe (III) ions has been replaced by a dopant ion to create an excess spin. Using a coherentstate spin path integral formalism, we derive an effective action for the system in the presence of a magnetic field. We calculate the functional dependence of the magnetization and tunnel splitting on the magnetic field and show that the parameters of the spin Hamiltonian can be inferred from the magnetization curve. We study the spin dynamics in these systems and show that quantum tunneling of the Neel vector also results in tunneling of the total magnetization. Hence, the spin correlation function shows a signature of Neel vector tunneling, and electron spin resonance (ESR) techniques or AC susceptibility measurements can be used to measure both the tunneling and the decoherence rate. We compare our results with exact diagonalization studies on small ring systems. Our results can be easily generalized to a wide class of nanomagnets, such as ferritin.

Quantum Correlations in TwoFermion Systems 
John Schliemann (Univ. of Texas), J. I. Cirac (Univ. of Innsbruck), M. Kus (Polish Academy of Science, Warsaw), M. Lewenstein (Univ. of Hannover), and Daniel Loss. Phys. Rev. A 64, 0223039 (2001); quantph/0012094.
We characterize and classify quantum correlations in twofermion systems having 2K singleparticle states. For pure states we introduce the Slater decomposition and rank (in analogy to Schmidt decomposition and rank), i.e. we decompose the state into a combination of elementary Slater determinants formed by mutually orthogonal singleparticle states. Mixed states can be characterized by their Slater number which is the minimal Slater rank required to generate them. For K=2 we give a necessary and sufficient condition for a state to have a Slater number of 1. We introduce a correlation measure for mixed states which can be evaluated analytically for K=2. For higher K, we provide a method of constructing and optimizing Slater number witnesses, i.e. operators that detect Slater number for some states.

Quantum Computing in Molecular Magnets 
Michael N. Leuenberger and Daniel Loss. Nature 410, 789 (2001); condmat/0011415.
We propose the implementation of Grover's algorithm with molecular magnets with spin s>>1 in a unary representation. Shor and Grover demonstrated that a quantum computer can outperform any classical computer in factoring numbers[1] and in searching a database[2] by exploiting the parallelism of quantum mechanics. Whereas Shor's algorithm requires both superposition and entanglement of a manyparticle system[3], the superposition of singleparticle quantum states is sufficient for Grover's algorithm[4]. Recently, the latter has been successfully implemented[5] using Rydberg atoms. Here we propose an implementation of Grover's algorithm that uses molecular magnets[6,7,8,9,10], which are solidstate systems with a large spin; their spin eigenstates make them natural candidates for singleparticle systems. We show theoretically that molecular magnets can be used to build dense and efficient memory devices based on the Grover algorithm. In particular, one single crystal can serve as a storage unit of a dynamic random access memory device. Fast electron spin resonance pulses can be used to decode and read out stored numbers of up to 105, with access times as short as 1010 seconds. We show that our proposal should be feasible using the molecular magnets Fe8 and Mn12.

Detection of Single Spin Decoherence in a Quantum Dot via Charge Currents 
HansAndreas Engel and Daniel Loss. Phys. Rev. Lett. 86, 4648 (2001); condmat/0011193.
We consider a quantum dot attached to leads in the Coulomb blockade regime which has a spin 1/2 ground state. We show that by applying an ESR field to the dotspin the stationary current in the sequential tunneling regime exhibits a resonance whose line width is determined by the singlespin decoherence time T_2. The Rabi oscillations of the dotspin are shown to induce coherent current oscillations from which T_2 can be deduced in the time domain. We describe a spininverter which can be used to pump current through a doubledot via spin flips generated by ESR.

Magnetization in Molecular Iron Rings 
B. Normand, X. Wang, X. Zotos, and Daniel Loss. Phys. Rev. B 63, 184409 (2001); condmat/0011403.
The organometallic ring molecules Fe_6 and Fe_10 are leading examples of a class of nanoscopic molecular magnets, which have been of intense recent interest both for their intrinsic magnetic properties and as candidates for the observation of macroscopic quantum coherent phenomena. Torque magnetometry experiments have been performed to measure the magnetization in single crystals of both systems. We provide a detailed interpretation of these results, with a view to full characterization of the material parameters. We present both the most accurate numerical simulations performed to date for ring molecules, using Exact Diagonalization and Density Matrix Renormalization Group techniques, and a semiclassical description for purposes of comparison. The results permit quantitative analysis of the variation of critical fields with angle, of the nature and height of magnetization and torque steps, and of the width and rounding of the plateau regions in both quantities.

Noise of a QuantumDot System in the Cotunneling Regime 
Eugene V. Sukhorukov, Guido Burkard, and Daniel Loss. Phys. Rev. B 63, 125315 (2001); condmat/0010458.
We study the noise of the cotunneling current through one or several tunnelcoupled quantum dots in the Coulomb blockade regime. The various regimes of weak and strong, elastic and inelastic cotunneling are analyzed for quantumdot systems (QDS) with fewlevel, nearlydegenerate, and continuous electronic spectra. We find that in contrast to sequential tunneling where the noise is either Poissonian (due to uncorrelated tunneling events) or subPoissonian (suppressed by charge conservation on the QDS), the noise in inelastic cotunneling can be superPoissonian due to switching between QDS states carrying currents of different strengths. In the case of weak cotunneling we prove a nonequilibrium fluctuationdissipation theorem which leads to a universal expression for the noisetocurrent ratio (Fano factor). In order to investigate strong cotunneling we develop a microscopic theory of cotunneling based on the densityoperator formalism and using the projection operator technique. The master equation for the QDS and the expressions for current and noise in cotunneling in terms of the stationary state of the QDS are derived and applied to QDS with a nearly degenerate and continuous spectrum.

AndreevTunneling, Coulomb Blockade, and Resonant Transport of NonLocal SpinEntangled Electrons 
Patrik Recher, Eugene V. Sukhorukov, and Daniel Loss. Phys. Rev. B 63, 165314 (2001); condmat/0009452.
We propose and analyze a spinentangler for electrons based on an swave superconductor coupled to two quantum dots each of which is tunnelcoupled to normal Fermi leads. We show that in the presence of a voltage bias and in the Coulomb blockade regime two correlated electrons provided by the Andreev process can coherently tunnel from the superconductor via different dots into different leads. The spinsinglet coming from the Cooper pair remains preserved in this process, and the setup provides a source of mobile and nonlocal spinentangled electrons. The transport current is calculated and shown to be dominated by a twoparticle BreitWigner resonance which allows the injection of two spinentangled electrons into different leads at exactly the same orbital energy, which is a crucial requirement for the detection of spin entanglement via noise measurements. The coherent tunneling of both electrons into the same lead is suppressed by the onsite Coulomb repulsion and/or the superconducting gap, while the tunneling into different leads is suppressed through the initial separation of the tunneling electrons. In the regime of interest the particlehole excitations of the leads are shown to be negligible. The AharonovBohm oscillations in the current are shown to contain single and twoelectron periods with amplitudes that both vanish with increasing Coulomb repulsion albeit differently fast.

Spintronics and Quantum Computing: Switching Mechanisms for Qubits 
Michael N. Leuenberger and Daniel Loss. Physica E 10 452457 (2001); condmat/0010434.
Quantum computing and quantum communication are remarkable examples of new information processing technologies that arise from the coherent manipulation of spins in nanostructures. We review our theoretical proposal for using electron spins in quantumconfined nanostructures as qubits. We present single and twoqubit gate mechanisms in laterally as well as vertically coupled quantum dots and discuss the possibility to couple spins in quantum dots via exchange or superexchange. In addition, we propose a new stationary wave switch, which allows to perform quantum operations with quantum dots or spin1/2 molecules placed on a 1D or 2D lattice.

DoubleOccupancy Errors, Adiabaticity, and Entanglement of SpinQubits in Quantum Dots 
John Schliemann (University of Texas, USA), Daniel Loss, and A.H. MacDonald (University of Texas, USA). Phys. Rev. B 63, 085311 (2001); condmat/0009083.
Quantum gates that temporarily increase singlettriplet splitting in order to swap electronic spins in coupled quantum dots, lead inevitably to a finite doubleoccupancy probability for both dots. By solving the timedependent Schröodinger equation for a coupled dot model, we demonstrate that this does not necessarily lead to quantum computation errors. Instead, the coupled dot ground state evolves quasiadiabatically for typical system parameters so that the doubleoccupancy probability at the completion of swapping is negligibly small. We introduce a measure of entanglement which explicitly takes into account the possibilty of double occupancies and provides a necessary and sufficient criterion for entangled states.

Quantum Dot as Spin Filter and Spin Memory 
Patrik Recher, Eugene V. Sukhorukov, and Daniel Loss. Phys. Rev. Lett. 85, 1962 (2000); condmat/0003089].
We consider a quantum dot in the Coulomb blockade regime weakly coupled to current leads and show that in the presence of a magnetic field the dot acts as an efficient spinfilter (at the singlespin level) which produces a spinpolarized current. Conversely, if the leads are fully spinpolarized the up or down state of the spin on the dot results in a large sequential or small cotunneling current, and thus, together with ESR techniques, the setup can be operated as a singlespin memory.

Spintronics and Quantum Computing with Quantum Dots 
Patrik Recher, Daniel Loss, and Jeremy Levy (University of Pittsburgh, USA). p.293306, in Macroscopic Quantum Coherence and Quantum Computing", eds. D.V. Averin, B. Ruggiero, and P. Silvestrini, Kluwer Academic/Plenum Publishers, New York, 2001; condmat/0009270.
The creation, coherent manipulation, and measurement of spins in nanostructures open up completely new possibilities for electronics and information processing, among them quantum computing and quantum communication. We review our theoretical proposal for using electron spins in quantum dots as quantum bits. We present single and two qubit gate mechanisms in laterally as well as vertically coupled quantum dots and discuss the possibility to couple spins in quantum dots via superexchange. We further present the recently proposed schemes for using a single quantum dot as spinfilter and spin readout/memory device.

Spin tunneling and topological selection rules for integer spins 
Michael N. Leuenberger and Daniel Loss. Phys. Rev. B 63, 054414 (2001); condmat/0006075.
We present new topological interference effects for the tunneling of a single large spin, which are caused by the symmetry of a general class of magnetic anisotropies. The interference results from spin Berry phases associated with different tunneling paths exposed to the same dynamics. Introducing a generalized path integral for coherent spin states we evaluate transition amplitudes between ground as well as lowlying excited states. We show that these interference effects lead to topological selection rules and spin parity effects for integer spins which agree with quantum selection rules and which thus provide a generalization of the Kramers degeneracy to integer spins.

Coulomb Blockade in the Fractional Quantum Hall Effect Regime 
Michael R. Geller (Athens, Georgia, USA) and Daniel Loss. Phys. Rev. B 62, 16298 (2000) (Rapid Communication); condmat/0003318.
We use chiral Luttinger liquid theory to study transport through a quantum dot in the fractional quantum Hall effect regime and find rich nonFermiliquid tunneling characteristics. In particular, we predict a remarkable Coulombblockadetype energy gap that is quantized in units of the noninteracting level spacing, new powerlaw tunneling exponents for voltages beyond threshold, and a line shape as a function of gate voltage that is dramatically different than that for a Fermi liquid. We propose experiments to use these unique spectral properties as a new probe of the fractional quantum Hall effect.

Superconductors, Quantum Dots, and Spin Entanglement 
MahnSoo Choi, Christoph Bruder, and Daniel Loss Proceedings of 236. WEHeraeusSeminar: Interacting Electrons in Nanostructures, Lecture Notes in Physics, Vol. 579, eds. R. Haug and H. Schoeller, Berlin, 2001, pp. 4666.
In this paper, we review a double quantum dot each dot of which is tunnelcoupled to superconducting leads. In the Coulomb blockade regime, a spindependent Josephson coupling between two superconductors is induced, as well as an antiferromagnetic Heisenberg exchange coupling between the spins on the double dot which can be tuned by the superconducting phase difference. We show that the correlated spin states—singlet or triplets—on the double dot can be probed via the Josephson current in a dcSQUID setup. We also briefly review the Andreev entangler, a nonequilibrium setup that provides a source of pairwise entangled electrons.

SpinDependent Josephson Current through Double Quantum Dots and Measurement of Entangled Electron States 
MahnSoo Choi, C. Bruder, and Daniel Loss. Phys. Rev. B 62, 13569 (2000)
We study a double quantum dot each dot of which is tunnelcoupled to superconducting leads. In the Coulomb blockade regime, a spindependent Josephson coupling between two superconductors is induced, as well as an antiferromagnetic Heisenberg exchange coupling between the spins on the double dot which can be tuned by the superconducting phase difference. We show that the correlated spin statessinglet or tripletson the double dot can be probed via the Josephson current in a dcSQUID setup.

Spintronics and Quantum Dots for Quantum Computing and Quantum Communication 
Guido Burkard, HansAndreas Engel, and Daniel Loss. Fortschr. Phys. 48, 911, 965986 (2000); special issue on Experimental Proposals for Quantum Computation, eds. H.K. Lo and S. Braunstrein; condmat/0004182.
Control over electronspin states, such as coherent manipulation, filtering and measurement promises access to new technologies in conventional as well as in quantum computation and quantum communication. We review our proposal of using electron spins in quantum confined structures as qubits and discuss the requirements for implementing a quantum computer. We describe several realizations of one and twoqubit gates and of the readin and readout tasks. We discuss recently proposed schemes for using a single quantum dot as spinfilter and spinmemory device. Considering electronic EPR pairs needed for quantum communication we show that their spin entanglement can be detected in mesoscopic transport measurements using metallic as well as superconducting leads attached to the dots.

Conductance fluctuations in diffusive rings: Berry phase effects and criteria for adiabaticity 
HansAndreas Engel and Daniel Loss. Phys. Rev. B 62, 1023810254 (2000); condmat/0002396.
We study Berry phase effects on conductance properties of diffusive mesoscopic conductors, which are caused by an electron spin moving through an orientationally inhomogeneous magnetic field. Extending previous work, we start with an exact, i.e. not assuming adiabaticity, calculation of the universal conductance fluctuations in a diffusive ring within the weak localization regime, based on a differential equation which we derive for the diffuson in the presence of Zeeman coupling to a magnetic field texture. We calculate the field strength required for adiabaticity and show that this strength is reduced by the diffusive motion. We demonstrate that not only the phases but also the amplitudes of the h/2e AharonovBohm oscillations are strongly affected by the Berry phase. In particular, we show that these amplitudes are completely suppressed at certain magic tilt angles of the external fields, and thereby provide a useful criterion for experimental searches. We also discuss Berry phaselike effects resulting from spinorbit interaction in diffusive conductors and derive exact formulas for both magnetoconductance and conductance fluctuations. We discuss the power spectra of the magnetoconductance and the conductance fluctuations for inhomogeneous magnetic fields and for spinorbit interaction.

Quantum Computers and Quantum Coherence 
David P. DiVincenzo (IBM Yorktown Heights, NY, USA) and Daniel Loss. J. of Magnetism and Magnetic Matls. 200, 202 (1999); condmat/9901137.
If the states of spins in solids can be created, manipulated, and measured at the singlequantum level, an entirely new
form of information processing, quantum computing, will be possible. We first give an overview of quantum information
processing, showing that the famous Shor speedup of integer factoring is just one of a host of important applications for
qubits, including cryptography, counterfeit protection, channel capacity enhancement, distributed computing, and
others. We review our proposed spinquantum dot architecture for a quantum computer, and we indicate a variety of first
generation materials, optical, and electrical measurements which should be considered. We analyze the efficiency of
a twodot device as a transmitter of quantum information via the propagation of qubit carriers (i.e. electrons) in a Fermi
sea.

Transport and Noise of Entangled Electrons 
Eugene V. Sukhorukov, Daniel Loss, and Guido Burkard. arXiv:condmat/9909348
We consider a scattering setup with an entangler and beam splitter where the current noise exhibits bunching behavior for electronic singlet states and antibunching behavior for triplet states. We show that the entanglement of two electrons in the doubledot can be detected in mesoscopic transport measurements. In the cotunneling regime the singlet and triplet states lead to phasecoherent current contributions of opposite signs and to AharonovBohm and Berry phase oscillations in response to magnetic fields. We analyze the Fermi liquid effects in the transport of entangled electrons.

Electron Spins in Quantum Dots as Quantum Bits 
Daniel Loss, Guido Burkard, and David P. DiVincenzo. JOURNAL OF NANOPARTICLE RESEARCH 2, 401411 (2000)
The creation, coherent manipulation, and measurement of spins in nanostructures open up completely new possibilities for electronics and information processing, among them quantum computing and quantum communication. We review our theoretical proposal for using electron spins in quantum dots as quantum bits, explaining why this scheme satisfies all the essential requirements for quantum computing. We include a discussion of the recent measurements of surprisingly long spin coherence times in semiconductors. Quantum gate mechanisms in laterally and vertically tunnelcoupled quantum dots and methods for singlespin measurements are introduced. We discuss detection and transport of electronic EPR pairs in normal and superconducting systems.

Spin interactions and switching in vertically tunnelcoupled quantum dots 
Guido Burkard, Georg Seelig, and Daniel Loss. Phys. Rev. B 62, 2581 (2000); condmat/9910105.
We determine the spin exchange coupling J between two electrons located in two vertically tunnelcoupled quantum dots, and its variation when magnetic (B) and electric (E) fields (both inplane and perpendicular) are applied. We predict a strong decrease of J as the inplane B field is increased, mainly due to orbital compression. Combined with the Zeeman splitting of the triplet, this leads to a singlettriplet crossing, which can be observed as a pronounced jump in the magnetization, occurring at inplane fields of a few Tesla, and perpendicular fields of the order of 10 Tesla for typical selfassembled dots. We use harmonic potentials to model the confining of electrons, and calculate the exchange J using the HeitlerLondon and HundMulliken technique, including the longrange Coulomb interaction. With our results we provide experimental criteria for the distinction of singlet and triplet states and therefore for a microscopic spin measurement. In the case where quantum dots of different size are coupled, we present a simple method to switch on and off the spin coupling with exponential sensitivity using an inplane electric field. Switching the spin coupling is essential for quantum computation using electronic spins as qubits.

Probing Entanglement and Nonlocality of Electrons in a DoubleDot via Transport and Noise 
Daniel Loss and Eugene V. Sukhorukov. Phys. Rev. Lett. 84 10351038 (2000); condmat/9907129.
Addressing the feasibilty of quantum communication with electrons we consider entangled spin states of electrons in a doubledot which is weakly coupled to inand outgoing leads. We show that the entanglement of two electrons in the doubledot can be detected in mesoscopic transport and noise measurements. In the Coulomb blockade and cotunneling regime the singlet and triplet states lead to phasecoherent current and noise contributions of opposite signs and to AharonovBohm and Berry phase oscillations in response to magnetic fields. These oscillations are a genuine twoparticle effect and provide a direct measure of nonlocality in entangled states. We show that the ratio of zerofrequency noise to current (Fano factor) is universal and equal to the electron charge.

Quantum Computation and Spin Electronics 
D. P. DiVincenzo, G. Burkard, D. Loss, and E. Sukhorukov. condmat/9911245. in Quantum Mesoscopic Phenomena and Mesoscopic Devices in Microelectronics, eds. I. O. Kulik and R. Ellialtoglu (NATO ASI), p. 399428, 2000, Kluwer, Netherlands
In this chapter we explore the connection between mesoscopic physics and quantum computing. After giving a bibliography providing a general introduction to the subject of quantum information processing, we review the various approaches that are being considered for the experimental implementation of quantum computing and quantum communication in atomic physics, quantum optics, nuclear magnetic resonance, superconductivity, and, especially, normalelectron solid state physics. We discuss five criteria for the realization of a quantum computer and consider the implications that these criteria have for quantum computation using the spin states of singleelectron quantum dots. Finally, we consider the transport of quantum information via the motion of individual electrons in mesoscopic structures; specific transport and noise measurements in coupled quantum dot geometries for detecting and characterizing electronstate entanglement are analyzed.

Spin tunneling and phononassisted relaxation in Mn12acetate 
Michael Leuenberger and Daniel Loss. Phys. Rev. B 61, 1286 (2000); condmat/9907154.
We present a comprehensive theory of the magnetization relaxation in a Mn_{12}acetate crystal in the high temperature regime (T>1K), which is based on phononassisted spin tunneling induced by quartic magnetic anisotropy and weak transverse magnetic fields. The overall relaxation rate as function of the longitudinal magnetic field is calculated and shown to agree well with experimental data including all resonance peaks measured so far. The Lorentzian shape of the resonances, which we obtain via a generalized master equation that includes spin tunneling, is also in good agreement with recent data. We derive a general formula for the tunnel splitting energy of these resonances. We show that fourthorder diagonal terms in the Hamiltonian lead to satellite peaks. A derivation of the effective linewidth of a resonance peak is given and shown to agree well with experimental data. In addition, previously unknown spinphonon coupling constants are calculated explicitly. The values obtained for these constants and for the sound velocity are also in good agreement with recent data. We show that the spin relaxation in Mn_{12}acetate takes place via several transition paths of comparable weight. These transition paths are expressed in terms of intermediate relaxation times, which are calculated and which can be tested experimentally.

Noise of Entangled Electrons: Bunching and Antibunching 
Guido Burkard, Daniel Loss, and Eugene V. Sukhorukov. Phys. Rev. B 61, R16303 (2000); condmat/9906071].
Addressing the feasibility of quantum communication with entangled electrons in an interacting manybody environment, we propose an interference experiment using a scattering setup with an entangler and a beam splitter. It is shown that, due to electronelectron interaction, the delity of the entangled singlet and triplet states is reduced by z_F^2 in a conductor described by Fermi liquid theory. We calculate the quasiparticle weight factor z_F for a twodimensional electron system. The current noise for electronic singlet states turns out to be enhanced (bunching behavior), while it is reduced for triplet states (antibunching). Within standard scattering theory, we nd that the Fano factor (noisetocurrent ratio) for singlets is twice as large as for independent classical particles and is reduced to zero for triplets.

Quantum Dynamics of Pseudospin Solitons in DoubleLayer Quantum Hall, Systems 
J. Kyriakidis, D. Loss, and A. MacDonald (Bloomington, IN, USA). Phys. Rev. Lett. 83, 14111414 (1999); condmat/9904185.
Pseudospin solitons in doublelayer quantum Hall systems can be introduced by a magnetic field component coplanar with the electrons and can be pinned by applying voltages to external gates. We estimate the temperature below which depinning occurs predominantly via tunneling and calculate lowtemperature depinning rates for realistic geometries. We discuss the local changes in charge and current densities and in spectral functions that can be used to detect solitons and observe their temporal evolution.

Observing the Berry phase in diffusive conductors: Necessary conditions for adiabaticity 
D. Loss, H. Schoeller (Karlsruhe, D), and P.M. Goldbart (Urbana, IL, USA). Phys. Rev. B 59 (1999) 1332813337
We investigate Berry phase effects in the magnetoconductance of diffusive systems and determine the precise criterion for adiabaticity within the weak localization formalism. We show that the exact solution of the Cooperon propagator for the special case of a cylindrically symmetric texture agrees with the adiabatic approximation in the adiabatic limit characterized by $\tau \gg {1\over d} {l^{2}\over L^{2}}$. We point out that orientational inhomogeneities in the magnetic field induce dephasing effects that can mask the Berry phase (and any other phase coherent phenomena) for certain parameter values of system and field.

Incoherent Zener tunneling and its application to molecular magnets 
Michael Leuenberger and Daniel Loss. Phys. Rev. B 61, 12200 (2000); condmat/9911065.
We generalize the LandauZener theory of coherent tunneling transitions by taking thermal relaxation into account. The evaluation of a new generalized master equation containing a dynamic tunneling rate that includes the interaction between the relevant system and its environment leads to an incoherent Zener transition probability with an exponent that is twice as large as the one of the coherent Zener probability in the limit T>0. We apply our results to molecular clusters, in particular to recent measurements of the tunneling transition of spins in Fe_{8} crystals performed by Wernsdorfer and Sessoli [Science 284, 133 (1999)].

Excess Spin and the Dynamics of Antiferromagnetic Ferritin 
J.G.E. Harris (UC Santa Barbara, CA, USA), J.E. Grimaldi (UCSB), D.D. Awschalom (UCSB), A. Chiolero, and D. Loss. Phys. Rev. B 60, 34533456 (1999); condmat/9904051.
Temperaturedependent magnetization measurements on a series of synthetic ferritin proteins containing from 100 to 3000 Fe(III) ions are used to determine the uncompensated moment of these antiferromagnetic particles. The results are compared with recent theories of macroscopic quantum coherence which explicitly include the effect of this excess moment. The scaling of the excess moment with protein size is consistent with a simple model of finite size effects and sublattice noncompensation.

Physical Optimization of Quantum Error Correction Circuits 
Guido Burkard, Daniel Loss, David P. DiVincenzo (IBM Yorktown Heights, NY, USA), and John A. Smolin (IBM). Phys. Rev. B 60, 11404 (1999); condmat/9905230.
Quantum error correcting codes have been developed to protect a quantum computer from decoherence due to a noisy environment. In this paper, we present two methods for optimizing the physical implementation of such error correction schemes. First, we discuss an optimal quantum circuit implementation of the smallest errorcorrecting code (the three bit code). Quantum circuits are physically implemented by serial pulses, i.e. by switching on and off external parameters in the Hamiltonian one after another. In contrast to this, we introduce a new parallel switching method which allows faster gate operation by switching all external parameters simultaneously, and which has potential applications for arbitrary quantum computer architectures. We apply both serial and parallel switching to electron spins in coupled quantum dots subject to a Heisenberg coupling $H=J(t) S_{1}· S_{2}$. We provide a list of steps that can be implemented experimentally and used as a test for the functionality of quantum error correction.

Quantum information processing using electron spins and cavityQED 
A. Imamoglu (UC Santa Barbara, CA, USA), D. D. Awschalom (UCSB), G. Burkard, D. P. DiVincenzo (IBM Yorktown), D. Loss, M. Sherwin (UC Santa Barbara), and A. Small (UC Santa Barbara). Phys. Rev. Lett. 83, 4204 (1999); quantph/9904096.
The electronic spin degrees of freedom in semiconductors typically have decoherence times that are several orders of magnitude longer than other relevant timescales. A solidstate quantum computer based on localized electron spins as qubits is therefore of potential interest. Here, a scheme that realizes controlled interactions between two distant quantum dot spins is proposed. The effective longrange interaction is mediated by the vacuum field of a high finesse microcavity. By using conductionbandhole Raman transitions induced by classical laser fields and the cavitymode, arbitrary single qubit rotations and controllednot operations can be realized. Optical techniques can also be used to measure the spinstate of each quantum dot.

Coupled quantum dots as quantum gates 
G. Burkard, D. Loss, and D.P. DiVincenzo (IBM Yorktown Heights, NY, USA). Phys. Rev. B 59, 2070 (1999); condmat/9808026.
We consider a new quantum gate mechanism based on electron spins in coupled semiconductor quantum dots. Such gates provide a general source of spin entanglement and can be used for quantum computers. We determine the exchange coupling $J$ in the effective Heisenberg model as a function of magnetic ($B$) and electric fields, and of the interdot distance $a$ within the HeitlerLondon approximation of molecular physics. This result is refined by using sphybridization, and by the HundMulliken molecularorbit approach which leads to an extended Hubbard description for the twodot system that shows a remarkable dependence on $B$ and $a$ due to the longrange Coulomb interaction. We find that the exchange $J$ changes sign at a finite field (leading to a pronounced jump in the magnetization) and then decays exponentially. The magnetization and the spin susceptibilities of the coupled dots are calculated. We show that the dephasing due to nuclear spins in GaAs can be strongly suppressed by dynamical nuclear spin polarization and/or by magnetic fields.

Nonlinear sigma Model Treatment of Quantum Antiferromagnets in a Magnetic Field 
Bruce Normand, Jordan Kyriakidis, and Daniel Loss Ann. Phys. (Leipzig) 9, 133 (2000); arXiv:condmat/9902104; .
We present a theoretical analysis of the properties of lowdimensional quantum antiferromagnets in applied magnetic fields. In a nonlinear sigma model description, we use a spin stiffness analysis, a 1/N expansion, and a renormalization group approach to describe the brokensymmetry regimes of finite magnetization, and, in cases of most interest, a lowfield regime where symmetry is restored by quantum fluctuations. We compute the magnetization, critical fields, spin correlation functions, and decay exponents accessible by nuclear magnetic resonance experiments. The model is relevant to many systems exhibiting Haldane physics, and provides good agreement with data for the twochain spin ladder compound CuHpCl.

Noise in Multiterminal Diffusive Conductors: Universality, Nonlocality and Exchange Effects 
Eugene V. Sukhorukov and Daniel Loss. Phys. Rev. Lett. 80, 4959 (1998); condmat/9802050; Phys. Rev. B 59, 1305413066 (1999); condmat/9809239.
We study noise and transport in multiterminal diffusive conductors. Using a BoltzmannLangevin equation approach we reduce the calculation of shotnoise correlators to the solution of diffusion equations. Within this approach we prove the universality of shot noise in multiterminal diffusive conductors of arbitrary shape and dimension for purely elastic scattering as well as for hot electrons. We show that shot noise in multiterminal conductors is a nonlocal quantity and that exchange effects can occur in the absence of quantum phase coherence even at zero electron temperature. It is also shown that the exchange effect measured in one contact is always negative  in agreement with the Pauli principle. We discuss a new phenomenon in which current noise is induced by thermal transport. We propose a possible experiment to measure locally the effective noise temperature. Concrete numbers for shot noise are given that can be tested experimentally.

Quantum computation with quantum dots 
D. Loss and D.P. DiVincenzo (IBM Yorktown Heights, NY, USA). Phys. Rev. A 57 120126 (1998); Milestone paper of Phys. Rev. A (19702020); condmat/9701055.
We propose a new implementation of a universal set of one and twoqubit gates for quantum computation using the spin states of coupled singleelectron quantum dots. Desired operations are effected by the gating of the tunneling barrier between neighboring dots. Several measures of the gate quality are computed within a newly derived spin master equation incorporating decoherence caused by a prototypical magnetic environment. Dotarray experiments which would provide an initial demonstration of the desired nonequilibrium spin dynamics are proposed.

Quantum dynamics in mesoscopic magnetism 
D. Loss, in Dynamical Properties of, and Unconventional Magnetic Systems. A.T. Skjeltorp and D. Sherrington (eds.), NATO ASI Series E: Applied Sciences, Vol. 349, 1998 Kluwer Academic Publishers, p. 2975
A review of quantum coherence effects in mesoscopic spin systems is presented. We begin with a general introduction to the topic of mesoscopic effects in magnetism and give some specific examples of current interest. We review then theoretical results in single domain magnetism of superparamagnetic type and mention recent measurements on antiferromagnetic grains (ferritin) and their interpretation in terms of macroscopic quantum coherence. Introducing the effects originating from spin parity in the context of ferromagnetic grains, we discuss antiferromagnetic particles with excess spins and molecular magnets such as the ferric wheel. It is shown that tunneling in such magnets can be tuned by external magnetic fields and is directly observable via the magnetization and the Schottky anomaly in the specific heat. The main part of the review will be devoted to nonuniform magnets and specifically to the quantum dynamics of domain walls or magnetic solitons. In a semiclassical analysis based on coherent spinstate pathintegrals an effective model for the domain wall dynamics is derived which includes the effects of spinwave dissipation and of quantum spin phases (Berry phases). In the presence of a Peierls potential (e.g. due to the discrete lattice) the soliton center can tunnel coherently between the lattice sites and form a Bloch band. Integer and halfodd integer spins have different energy dispersions resulting from interference between soliton states of opposite chiralitythe internal rotation sense of the soliton. These effects occur in ferro and antiferromagnets due to the presence of a topological spin phase. For antiferromagnetic chains, this spin phase occurs in addition to the Pontryaginindex phase.We will discuss experimental consequences of this Bloch band structure and show that in analogy to Bloch oscillations of crystal electrons static magnetic fields induce large oscillations in the sample magnetization. We will also discuss the extreme quantum limit of spin1/2 chains in the Ising regime, and show that, quite remarkably, the semiclassical analysis is valid even in this regime. In particular, for antiferromagnetic Ising chains the lowenergy excitations are solitons (Villain modes) which have been observed in neutron scattering experiments on CsCoCl_{3}. We show that the prediction of chirality effects could be tested via the measurement of the offdiagonal components of the dynamical structure factor. The concept of chirality is shown to be of universal character in a variety of magnetic systems, a notable example being the motion of a hole in a 2D antiferromagnetic background. A common thread in the discussion of quantum dynamics in magnets is provided by the Berry phases and their associated interference effects which can lead to surprising spin parity effects.

Macroscopic quantum coherence in molecular magnets 
A. Chiolero and D. Loss. Phys. Rev. Lett. 80, 169172 (1998)
We study macroscopic quantum coherence in antiferromagnetic molecular magnets in the presence of magnetic fields. Such fields generate artificial tunnel barriers with externally tunable strength. We give detailed semiclassical predictions for the tunnel splitting in various regimes for low and high magnetic fields. We show that the tunneling dynamics of the N\'eel vector can be directly measured via the static magnetization and the specific heat. We also report on a new quantum phase arising from fluctuations. The analytic results are complemented by numerical simulations.

Bloch oscillations of magnetic solitons in anisotropic spin1/2 chains 
Jordan Kyriakidis and Daniel Loss. Phys. Rev. B 58 55685583 (1998); condmat/9803156.
We study the quantum dynamics of solitonlike domain walls in anisotropic spin1/2 chains in the presence of magnetic fields. In the absence of fields, domain walls form a Bloch band of delocalized quantum states while a static field applied along the easy axis localizes them into Wannier wave packets and causes them to execute Bloch oscillations, i.e.\ the domain walls oscillate along the chain with a finite Bloch frequency and amplitude. In the presence of the field, the Bloch band, with a continuum of extended states, breaks up into the WannierZeeman laddera discrete set of equally spaced energy levels. We calculate the dynamical structure factor $S^{zz} (q,\omega)$ in the onesoliton sector at finite frequency, wave vector, and temperature, and find sharp peaks at frequencies which are integer multiples of the Bloch frequency. We further calculate the uniform magnetic susceptibility and find that it too exhibits peaks at the Bloch frequency. We identify several candidate materials where these Bloch oscillations should be observable, for example, via neutron scattering measurements. For the particular compound ${\rm CoCl_{2} \! · \! 2H_{2}O}$ we estimate the Bloch amplitude to be on the order of a few lattice constants, and the Bloch frequency on the order of 100\,GHz for magnetic fields in the Tesla range and at temperatures of about 18\,Kelvin.

Macroscopic quantum coherence in ferrimagnets 
A. Chiolero and D. Loss. Phys. Rev. B 56, 738 (1997)
We study macroscopic quantum coherence (MQC) in small magnetic particles where the magnetization (in ferromagnets) or the N\'eel vector (in antiferromagnets) can tunnel between energy minima. We consider here the more general case of MQC in ferrimagnets by studying a model for a mesoscopic antiferromagnet with an uncompensated magnetic moment. Through semiclassical calculations we show that even a small moment has a drastic effect on MQC. In particular, there is a rapid crossover to a regime where the MQC tunnel splitting is equal to that obtained for a ferromagnet, even though the system is still an antiferromagnet for all other aspects. We calculate this tunnel splitting via instanton methods and compare it with numerical evaluations. As an application we reexamine the experimental evidence for MQC in ferritin and show that even though the uncompensated moment of ferritin is small it greatly modifies the MQC behavior. The excess spin allows us to extract values for experimental parameters without making any assumption about the classical attempt frequency, in contrast to previous fits. Finally, we also discuss the implications of our results for MQC in molecular magnets.

AharonovBohm effect in the chiral Luttinger liquid 
M. R. Geller (Athens, Georgia, USA) and D. Loss. Phys. Rev. B 56 96929706 (1997)
Edge states of the quantum Hall fluid provide an almost unparalled opportunity to study mesoscopic effects in a highly correlated electron system. In this paper we develop a bosonization formalism for the finitesize edge state, as described by chiral Luttinger liquid theory, and use it to study the AharonovBohm effect. We study tunneling through an edge state formed around an antidot in the fractional quantum Hall regime using Wen's chiral Luttinger liquid theory extended to include mesoscopic effects. We identify a new regime where the AharonovBohm oscillation amplitude exhibits a distinctive crossover from Luttinger liquid powerlaw behavior to Fermiliquidlike behavior as the temperature is increased. Near the crossover temperature the amplitude has a pronounced maximum. This nonmonotonic behavior and novel hightemperature nonlinear phenomena that we also predict provide new ways to distinguish experimentally between Luttinger and Fermi liquids. Finally, we predict new mesoscopic edgecurrent oscillations, which are similar to the persistent currentoscillations in a mesoscopic ring, except that they are not reduced in amplitude by weak disorder. In the fractional quantum Hall regime, these ``chiral persistent currents'' have a universal nonFermiliquid temperature dependence and may be another ideal system to observe a chiral Luttinger liquid.

Chirality correlation of spin solitons: Bloch walls, spin1/2 solitons and holes in a 2d antiferromagnetic background 
HansBenjamin Braun and Daniel Loss. Int. J. Mod. Phys. 10, 21 (1996)
We consider the quantum dynamics of spin solitons in a variety of lowdimensional magnetic systems in the semiclassical and the extreme quantum limit. Introducing the concept of chirality of the soliton we derive the dispersion of spin solitons moving through a periodic pinning potential and show that for halfodd integer spin the topological part of the Berry phase induces a halving of the Brillouin zone as well as chirality correlations between subsequent band minima. We demonstrate that these chirality and spin parity effects are universal by considering quasionedimensional ferromagnets and antiferromagnets with local anisotropies and large spins, as well as spin½ ferromagnetic and antiferromagnetic Heisenberg chains in the Ising limit. For large spin systems, the tunneling rate between states of opposite chiralities is derived and shown to provide a novel scenario for macroscopic quantum phenomena. The results are extended to solitons moving as holes in a twodimensional antiferromagnetic background, leading to a hole spectrum which is in remarkable agreement with recent ARPES measurements on highTc compounds.

Macroscopic quantum tunneling of ferromagnetic domain walls 
H.B. Braun (Paul Scherrer Institut, CH), J. Kyriakidis, and D. Loss. Phys. Rev. B 56, 81298137 (1997); condmat/9710064.
Quantum tunneling of domain walls out of an impurity potential in a mesoscopic ferromagnetic sample is investigated. Using improved expressions for the domain wall mass and for the pinning potential, we find that the crossover temperature between thermal activation and quantum tunneling is of a different functional form than found previously. In materials like Ni or YIG, the crossover temperatures are around 5 mK. We also find that the WKB exponent is typically two orders of magnitude larger than current estimates. The sources for these discrepancies are discussed, and precise estimates for the transition from threedimensional to onedimensional magnetic behavior of a wire are given. The crossover temperatures from thermal to quantum transitions and tunneling rates are calculated for various materials and sample sizes.

Mesoscopic Effects in the Fractional Quantum Hall Regime: Chiral Luttinger versus Fermi Liquid 
Michael R. Geller, Daniel Loss, and George Kirczenow. Phys. Rev. Lett. 77, 51105113(1996); condmat/9606070.
We study tunneling through an edge state formed around an antidot in the fractional quantum Hall regime using Wen's chiral Luttinger liquid theory extended to include mesoscopic effects. We identify a new regime where the AharonovBohm oscillation amplitude exhibits a distinctive crossover from Luttinger liquid powerlaw behavior to Fermiliquidlike behavior as the temperature is increased. Near the crossover temperature the amplitude has a pronounced maximum. This nonmonotonic behavior and novel hightemperature nonlinear phenomena that we also predict provide new ways to distinguish experimentally between Luttinger and Fermi liquids.

Berry's phase and quantum dynamics of ferromagnetic solitons 
HansBenjamin Braun and Daniel Loss. Phys. Rev. B 53, 3237 (1996)
We study spin parity effects and the quantum propagation of solitons "Bloch walls" in quasionedimensional ferromagnets. Within a coherent state path integral approach we derive a quantum ﬁeld theory for nonuniform spin conﬁgurations. The effective action for the soliton position is shown to contain a gauge potential due to the Berry phase and a damping term caused by the interaction between soliton and spin waves. For tempera tures below the anisotropy gap this dissipation reduces to a pure soliton mass renormalization. The quantum dynamics of the soliton in a periodic lattice or pinning potential reveals remarkable consequences of the Berry phase. For halfinteger spin, destructive interference between opposite chiralities suppresses nearestneighbor hopping. Thus the Brillouin zone is halved, and for small mixing of the chiralities the dispersion reveals a surprising dynamical correlation: Two subsequent band minima belong to different chirality states of the soliton. For integer spin the Berry phase is inoperative and a simple tightbinding dispersion is obtained. Finally it is shown that external ﬁelds can be used to interpolate continuously between the Bloch wall disper sions for halfinteger and integer spin.

Absence of spontaneous persistent current for interacting fermions in a onedimensional mesoscopic ring 
Daniel Loss and Thierry Martin. Phys. Rev. B 47, 4619 (1993)
We address the issue of whether a system of interacting electrons confined to a onedimensional ring can sustain a spontaneous persistent current in the absence of an externally applied flux. The currentcurrent coupling between electrons, describing radiative backaction effects, is exactly treated in the formalism of quantum electrodynamics, where the electrons interact via the exchange of virtual photons. In addition, the instantaneous screened Coulomb potential is taken into account using the Luttinger liquid model including finitesize parity effects. The partition function is calculated exactly, with the result that the system does not possess a spontaneous persistent current. We show that, in the presence of an external flux, the amplitude of the (conventional) persistent current is reduced by quantum fluctuations of the internal transverse electromagnetic field. These corrections can be expressed in terms of the selfinduction of the ring and are shown to be of first and higher order in the small dimensionless parameter αvF*/c, where α is the finestructure constant and vF* the Fermi velocity renormalized through Coulomb interactions.

Suppression of tunneling due to interference in spin sytems 
Daniel Loss, David P. DiVincenzo (IBM Watson, NY), and Goeffrey Grinstein (IBM Watson, NY). Phys. Rev. Lett. 69, 3232 (1992)
Within a wide class of ferromagnetic and antiferromagnetic systems, quantum tunneling of magnetization direction is spinparity dependent: it vanishes for magnetic particles with halfinteger spin, but is allowed for integer spin. A coherentstate pathintegral calculation shows that this topological effect results from interference between tunneling paths.

Macroscopic Quantum Tunneling in Magnetic Proteins 
D. D. Awschalom, J. F. Smyth, G. Grinstein, D. P. DiVincenzo, and D. Loss. Phys. Rev. Lett. 68, 3092 (1992); Phys. Rev. Lett. 71, 4279 (1993); Phys. Rev. Lett. 70, 2199 (1993).
We report lowtemperature measurements of the frequencydependent magnetic noise and magnetic susceptibility of nanometerscale antiferromagnetic horsespleen ferritin particles, using an integrated dc SQUID microsusceptometer. A sharply defined resonance near 1 MHz develops below T∼0.2 K. The behavior of this resonance as a function of temperature, applied magnetic field, and particle concentration indicates that it results from macroscopic quantum tunneling of the Néel vector of the antiferromagnets.

Josephson current and proximity effect in Luttinger liquids 
D. L. Maslov, M. Stone, P. M. Goldbart, and D. Loss. Phys. Rev. B53, 1548 (1996)
A theory describing a onedimensional Luttinger liquid in contact with a superconductor is developed. Boundary conditions for the fermion fields describing Andreev reflection at the contacts are derived and used to construct a bosonic representation of the fermions. The Josephson current through a superconductor/Luttinger liquid/superconductor junction is considered for both perfectly and poorly transmitting interfaces. In the former case, the Josephson current at low temperatures is found to be essentially unaffected by electronelectron interactions. In the latter case, significant renormalization of the Josephson current occurs. The profile of the (induced) condensate wave function in a semiinfinite Luttinger liquid in contact with a superconductor is shown to decay as a power law, the exponent depending on the sign and strength of the interactions. In the case of repulsive (attractive) interactions the decay is faster (slower) than in their absence. An equivalent method of calculating the Josephson current through a Luttinger liquid, which employs the bosonization of the system as a whole (i.e., superconductor, as well as Luttinger liquid) is developed and shown to give the results equivalent to those obtained via boundary condition describing Andreev reflection.

Onset of superconducting fluctuations for interacting fermions coupled to acoustic phonons in one dimension 
Thierry Martin (Marseille) and Daniel Loss. Int. J. Mod. Phys. B 9, 495 (1995)
We consider a onedimensional system consisting of electrons with shortranged repulsive interactions and coupled to smallmomentum transfer acoustic phonons. The interacting electrons are bosonized and described in terms of a Luttinger liquid which allows us to calculate exactly the one and twoelectron Green function. For noninteracting electrons, the coupling to phonons alone induces a singularity at the Fermi surface which is analogous to that encountered for electrons with an instantaneous attractive interaction. The exponents which determine the presence of singlet/triplet superconducting pairing fluctuations, and spin/charge density wave fluctuations are strongly affected by the presence of the WentzelBardeen singularity, resulting in the favoring of superconducting fluctuations. For the Hubbard model the equivalent of a phase diagram is established, as a function of: the electronphonon coupling, the electron filling factor, and the onsite repulsion between electrons. The WentzelBardeen singularity can be reached for arbitrary values of the electronphonon coupling constant by varying the filling factor. This provides an effective mechanism to push the system from the antiferromagnetic into the metallic phase, and finally into the superconducting phase as the electron filling factor is increased towards halffilling.

WentzelBardeen singularity and phase diagram for interacting electrons coupled to acoustic phonons in one dimension 
Daniel Loss and Thierry Martin (Marseille). Phys. Rev. B 50, 12160 (1994)
We consider strongly correlated electrons coupled to lowenergy acoustic phonons in one dimension. Using a Luttinger liquid description we calculate the exponents of various response functions and discuss their remarkable sensitivity to the WentzelBardeen singularity induced by the presence of phonons. For the Hubbard model plus phonons the equivalent of a phase diagram is established. By increasing the filling factor towards half filling the WentzelBardeen singularity is approached. This in turn triggers a simple and efficient mechanism to suppress antiferromagnetic fluctuations and to drive the system via a normal metallic state towards a superconducting phase.

Parity effects in a Luttinger liquid: Diamagnetic and paramagnetic ground states 
Daniel Loss Phys. Rev. Lett. 69, 343 (1992)
The concept of a Luttinger liquid in 1D is extended to include twisted boundary conditions on a ring and mesoscopic parity effects due to evenness and oddness of the particle number N0. Using Haldane’s notion of topological excitations, a proof of Leggett’s conjecture is presented, stating that the ground state of interacting 1D quantum systems is diamagnetic or paramagnetic depending on the parity of N0. The persistent currents produced by an AharonovBohm flux are calculated and shown to have period and amplitude that are in agreement with recent experiments.

Experimental consequences of persistent currents due to the Berry phase 
Daniel Loss and Paul Goldbart. Phys. Lett. A 215, 197 (1996)
It has recently been proposed that a mesoscopic ring of normal metal or semiconductor should exhibit equilibrium persistent currents of charge and spin, when embedded in an inhomogeneous magnetic field. The origin of these phenomena lies in the coupling between spin and orbital motion due to the Zeeman interaction, and the resulting geometric phase acquired during orbital motion. We present expressions for the ground state charge and spin currents for systems of many spinl /2 fermions moving in a onedimensional ring, and give numerical estimates of the magnitudes of the currents, and also ol‘ the electric field which results from the spin current.

Period and amplitude halving in mesoscopic rings with spin 
Daniel Loss and Paul Goldbart. Phys. Rev. B 43, 13762 (1991)
We consider the flux dependence of persistent currents in mesoscopic rings threaded by magnetic flux, and extend wellknown arguments to include particles of spin 1/2. We find several interesting consequences of spin, such as period and amplitude halving in a single ring, without including electronelectron interactions, transverse channels, or disorder. These consequences depend sensitively on the fixed number (modulo 4) of particles on a given ring, and lead to strong fluctuations between samples containing a small number of rings.

Persistent currents from Berry's phase in mesoscopic systems 
Daniel Loss and Paul M. Goldbart (UrbanaChampaign). Phys. Rev. B 45, 13544 (1992)
The quantum orbital motion of electrons in mesoscopic normalmetal rings threaded by a magnetic flux produces striking interference phenomena such as persistent currents due to the AharonovBohm effect. Similarly, when a quantum spin adiabatically follows a magnetic field that rotates slowly in time, the phase of its state vector acquires an additional contribution known as the Berry phase. We explore the combination of these two quantum phenomena by examining the interplay between orbital and spin degrees of freedom for a charged spin1/2 particle moving in a mesoscopic ring embedded in a classical, static inhomogeneous magnetic field, i.e, a texture. As a consequence of its orbital motion through the texture, the spin experiences, via the Zeeman interaction, a varying magnetic field. This results in a Berry—or geometric—phase, leading to persistent (i.e., equilibrium) currents of charge and spin. These mesoscopic phenomena are related to (but should be distinguished from) the conventional persistent currents that result from magnetic flux through a ring. We develop a pathintegral approach to decouple the orbital and spin motion and, by using an adiabatic approximation, we compute the equilibrium expectation values of the persistent charge and spin currents and the magnetization. We find that the persistent currents depend on the texture in a striking manner through a geometric phase (related to a surface area characterizing the texture) and a geometric vector (related to the projections of this area). In the special case of a cylindrically symmetric texture we use a spectrum obtained by Kuratsuji and Iida to obtain exact results that confirm, independently, the validity of the pathintegral approach in the adiabatic limit. We discuss the connection between the geometric vector and quantummechanical correlations, and examine quantum fluctuations and the zeropoint energy.

Weaklocalization effects and conductance fluctuations: Implications of inhomogeneous magnetic fields 
Paul M. Goldbart, Herbert Schoeller, and Daniel Loss. Phys. Rev. B 48, 15218 (1993)
Lowtemperature transport in disordered conductors exhibits a variety of fascinating quantummechanical interference effects associated with the phenomenon of weak localization. Such effects are typically isolated and probed by virtue of their sensitivity to applied homogeneous magnetic fields, which introduce AharonovBohm phase factors into quantummechanical amplitudes. Analogous interference effects have been proposed in the context of the quantum transport of (possibly electrically neutral) particles with spin in the presence of inhomogeneous magnetic fields, which have the effect of introducing Berry phases. Thus, the possibility is raised of isolating and probing quantum interference effects through their sensitivity to the inhomogeneity of applied magnetic fields. In this appear we develop an approach to the study of quantum transport in disordered conductors in the presence of almost arbitrarily inhomogeneous magnetic fields, which is based on diagrammatic and semiclassical pathintegral techniques and a subsequent adiabatic approximation. We illustrate these ideas with applications to three examples: anomalous weakfield magnetoconductance, conductance oscillations in mesoscopic multiply connected structures, and sampledependent mesoscopic conductance fluctuations. Among other things, we find that while in the context of the disorderaveraged conductance it is accurate to regard systems as being composed of two independent subsystems (having spins aligned or antialigned with the local external magnetic field) a more interesting and refined structure emerges in the context of conductance fluctuations.

Berry's phase and persistent charge and spin currents in textured mesoscopic rings 
Daniel Loss, Paul Goldbart (UrbanaChampaign), and A. V. Balatsky (Los Alamos). Phys. Rev. Lett. 65, 1655 (1990)
We consider the motion of electrons through a mesoscopic ring in the presence of a classical, static, inhomogeneous, magnetic field. Zeeman interaction between the electron spin and this texture couples spin and orbital motion, and results in a Berry phase. As a consequence, the system supports persistent equilibrium spin and charge currents, even in the absence of conventional electromagnetic flux through the ring. We mention the possibility of analogous persistent mass and spin currents in normal 3He.

Commutation relations for periodic operators. 
D. Loss and K. Mullen J. Phys. A 25 (1992) L235L239
Although periodic variables are common in quantum systems, there is still some question of their proper commutation relations. The authors show that the standard commutation relations, when applied carefully, do not lead to inconsistencies. They discuss other approaches to the problem in the literature.

The effect of dissipation on phase periodicity and the quantum dynamics of Josephson junctions. 
D. Loss and K. Mullen. Phys. Rev. A43 (1991) 21292138
We consider systems described by compact, periodic variables, such as Josephson junctions and small normal metal rings. We examine how they can be coupled to a heat bath, via either momentum or position coupling, and how the two descriptions can be related by a unitary transformation. We show how it is critical to transform not only the Hamiltonian, but also the initial conditions. We then demonstrate that for certain types of initial conditions, paths of different winding number can interfere. Still other, ostensibly reasonable initial conditions, have no such interference. We conclude with a discussion of appropriate models to describe periodic systems.

Dephasing by a dynamic environment. 
D. Loss and K. Mullen. Phys. Rev. B43 (1991) 1325213261.
We investigate the manner in which quantum interference is suppressed when a particle interacts with a spatially localized, dynamical environment. To do so we examine a model with two classical paths along which an electron can travel and allow it to interact with a bath of harmonic oscillators on one path and travel freely on the other. In particular, we show that the quantum fluctuations of the path of the particle can couple to the environment and thus lead to dephasing and calculate the dephasing time in the hightemperature limit. We compare this result to other views of how propagating electrons lose phase coherence.

Hopping conductivity for localized electronic statesLiouville space formalism. 
D. Loss and P. C. W. Holdsworth. Physica B 176 (1992) 319326.
We calculate the conductivity for a system of localized electrons, with conduction via phonon assisted transitions, using a Liouville space formalism. The result is in agreement with lowest order rate equation calculations, leading to an equivalent random resistance network. Our work is in contradiction with a similar work of Capek. We illustrate the differences between the two calculations and show that ours is the correct procedure to follow. We also comment on the criticism that the predictions of the random resistance network model do not agree with experimental data.

Linear quantum Enskog equation. I. Homogeneous quantum uids. 
Daniel Loss J. Stat. Phys. 59 (1990) 691723.
The quantumstatistical generalization of the wellknown classical, linear revised Enskog equation is derived for spatially uniform systems. This new quantum kinetic equation allows the study of equilibrium time correlation functions and their associated transport coefficients of normal quantum fluids where static correlations and degeneracy effects due to particle statistics (both are treated exactly) are important. Furthermore, we derive the quantumstatistical analog of the classical ring operator. These microscopic and systematic derivations are based on a recently developed superoperator formalism (including cluster expansion techniques) that, as a main feature, allows a clear distinction between static and dynamic correlations, which is crucial in the discussion of the Enskog approximation.

Linear quantum Enskog equation. II. Inhomogeneous quantum fluids. 
D. Loss. J. Stat. Phys. 61(1990) 463493.
This is the second part of a work concerned with the quantumstatistical generalization of classical Enskog theory, whereby the first part is extended to spatially inhomogeneous fluids. In particular, working with Liouville operators and using cluster expansions and projection operators, we derive the inhomogeneous linear quantum Enskog equation and express the dynamic structure factor and the nonlocal mobility tensor in terms of the corresponding quantum Enskog collision operator. Thereby static correlations due to excluded volume effects and quantumstatistical correlations due to the fermionic (bosonic) character of the pairwise strongly interacting particles are treated exactly. When static correlations are neglected, this Enskog equation reduces to the inhomogeneous linear quantum Boltzmann equation (containing an exchangemodifiedtmatrix). In the classical limit, the wellknown linear revised Enskog theory is recovered for hard spheres.

Quantum BoltzmannLorentz model approach to the line shape problem. 
D. Loss, A. Thellung and T.A. Turski. Phys. Rev. A41 (1990) 30053015.
Following our earlier work on quantum kinetic equations and the use of the BoltzmannLorentz model to describe the collisional broadening of spectra, we develop a new approach to that problem in which the centerofmass motion of the emitter as well as the interaction during the collision between the emitter and the buffer gas are treated quantum mechanically. Possible further extensions of our model are discussed.

Comparison between different Markov approximations for open spin systems. 
M. Celio and D. Loss. Physica A158 (1989) 769785.
Various Markov approximations obtained from the ZwanzigNakajima and the memoryless master equations are investigated. In particular, considering a simple model (spin in a magnetic field and coupled to a reservoir), we discuss the convergence behaviour and the positivity of these approximations. It is found that a wellknown positivity preserving Markov generator, introduced in the rigorous study of the weak coupling limit, does not necessarily represent a good approximation for physically relevant times, especially if the system is nearly degenerate.

Simplified virial expansions in the canonical ensemble. 
D. Loss, H. Schoeller and A. Thellung. Physica A155 (1989) 373384.
For a classical monatomic real gas the wellknown virial expansion of the equation of state and the reduced distribution function are rederived within the canonical ensemble formalism. In particular, two simplified methods are presented which lead directly to the desired result without making use of fugacity or perturbation expansions. The first method generalizes an argument given by Brout. This generalized argument is also used in the second method, which is based on cluster expansions and which is directly related to van Kampen's approach.

Quantumstatistical kinetic equations. 
D. Loss and H. Schoeller. J. Stat. Phys. 56 (1989) 175201.
Considering a homogeneous normal quantum fluid consisting of identical interacting fermions or bosons, we derive an exact quantumstatistical generalized kinetic equation with a collision operator given as explicit cluster series where exchange effects are included through renormalized Liouville operators. This new result is obtained by applying a recently developed superoperator formalism (Liouville operators, cluster expansions, symmetrized projectors,Pqrule, etc.) to nonequilibrium systems described by a density operator(t) which obeys the von Neumann equation. By means of this formalism a factorization theorem is proven (being essential for obtaining closed equations), and partial resummations (leading to renormalized quantities) are performed. As an illustrative application, the quantumstatistical versions (including exchange effects due to FermiDirac or BoseEinstein statistics) of the homogeneous Boltzmann (binary collisions) and ChohUhlenbeck (triple collisions) equations are derived.

A new quantumstatistical evaluation method for time correlation functions. 
D. Loss and H. Schoeller. J. Stat. Phys. 54 (1989) 765795.
Considering a system ofN identical interacting particles, which obey FermiDirac or BoseEinstein statistics, we derive new formulas for correlation functions of the type C(t)=⟨ΣNi=1Ai(t)ΣNj=1Bj⟩ (whereB j is diagonal in the freeparticle states) in the thermodynamic limit. Thereby we apply and extend a superoperator formalism, recently developed for the derivation of longtime tails in semiclassical systems. As an illustrative application, the Boltzmann equation value of the timeintegrated correlation functionC(t) is derived in a straightforward manner. Due to exchange effects, the obtained tmatrix and the resulting scattering cross section, which occurs in the Boltzmann collision operator, are now functionals of the FermiDirac or BoseEinstein distribution.

A new microscopic evaluation method for correlation functions: long time tails. 
D. Loss and H. Schoeller. Physica A150 (1988) 199243.
Considering a moderately dense fluid, whose N particles interact with a general shortrange repulsive potential, we show that the momentum autocorrelation function (associated with selfdiffusion processes) has a long time tail of the form (d = 2, 3: number of dimensions). Thereby new powerful concepts, in particular the Pqrule and the Pqsingularity (the latter being a generalization of van Hove's diagonal singularity) are introduced in order to select the most divergent terms of the density expansion of the correlation function in the thermodynamic and long time limit. It turns out that this method is more suitable for removing a special class of divergences than the Zwanzig inversion method applied by previous workers in this field.

The effect of dissipation on phase periodicity and the quantum dynamics of Josephson junctions. 
D. Loss and K. Mullen. Phys. Rev. A43 (1991) 21292138.
We consider systems described by compact, periodic variables, such as Josephson junctions and small normal metal rings. We examine how they can be coupled to a heat bath, via either momentum or position coupling, and how the two descriptions can be related by a unitary transformation. We show how it is critical to transform not only the Hamiltonian, but also the initial conditions. We then demonstrate that for certain types of initial conditions, paths of different winding number can interfere. Still other, ostensibly reasonable initial conditions, have no such interference. We conclude with a discussion of appropriate models to describe periodic systems.

Second virial coefficient of an interacting anyon gas. 
D. Loss and Y. Fu. Phys. Rev. Lett. 67(1991) 294297
We consider the statistical mechanics of an interacting anyon gas with a repulsive pair potential of the form g/r2. The second virial coefficient is calculated and shown to be a smooth function of the statistical parameter α for g>0, without the cusps previously found at the Bose point in a noninteracting anyon gas. As g→0 the cusps remain absent but a different type of singularity develops at the Bose point, showing that the order of limits is crucial.

Macroscopic quantum tunneling in antiferromagnetic ferritin particles. 
D. Awschalom, J. Smyth, G. Grinstein, D. DiVincenzo, and D. Loss. Phys. Rev. Lett. 68 (1992) 30923095. Erratum: 71 (1993) 4279.
We report lowtemperature measurements of the frequencydependent magnetic noise and magnetic susceptibility of nanometerscale antiferromagnetic horsespleen ferritin particles, using an integrated dc SQUID microsusceptometer. A sharply defined resonance near 1 MHz develops below T∼0.2 K. The behavior of this resonance as a function of temperature, applied magnetic field, and particle concentration indicates that it results from macroscopic quantum tunneling of the Néel vector of the antiferromagnets.

Resonant phenomena in compact and extended systems. 
K. Mullen, D. Loss, and H. T. C. Stoof. Phys. Rev. B 47 (1993) 26892706.
We consider the dynamics of Josephson junctions in two formulations: one where the phase is defined on a compact interval [0,2π], and a second where it is defined on an extended interval (∞,∞). We find that in general the two approaches are not equivalent: they have different sets of allowable initial conditions, and similar initial conditions can produce different values for observables. However, they do have identical predictions for the appearance of resonances. These resonances can be understood in a framework of ‘‘resonant Zener tunneling,’’ which encompasses resonant tunneling in superlattices, tunneling between WannierStark states in crystals, and LandauZener transitions in small metal rings. This framework also reveals a type of resonance in some quantum systems that are strongly driven (i.e., in the sudden limit), wherein the system has a large amplitude to transfer out of the diabatic state.

Quantum tunneling and dissipation in nanometerscale magnets. 
D. Loss, D. DiVincenzo, G. Grinstein, D. Awschalom, and J. Smyth. Physica B 189 (1993) 189203.
We summarize recent lowtemperature noise and AC magnetic susceptibility measurements on the nanometerscale magnetic protein horsespleen ferritin. The experiments show a narrow resonance peak at about 106 Hz, which is discussed in the framework of recentlydeveloped theories of macroscopic quantum coherence and tunneling in antiferromagnets; theory and experiment are argued to agree qualitatively, though quantitative discrepancies remain. We also review the recent analysis of a rather general spinparity effect for tunneling in magnetic systems: systems with appropriate symmetry may exhibit quantum tunneling for integer spin, but not for halfoddinteger spin, where destructive interference between different tunneling paths suppresses the tunneling. Finally, we study the effect on this spinparity phenomenon caused by dissipation, i.e. coupling to an environment consisting of a bath of harmonic oscillators. Using the realtime, FeynmanVernon path integral formalism, we find models where an arbitrarily small amount of ohmic dissipation completely destroys the spinparity effect (i.e., produces as such tunneling for halfoddinteger spins as for integer spins), and others where the effect appears to disappear gradually with increasing dissipation. Suprisingly, however, there is a sense in which the spinparity effect is preserved in both types of models: a CalderiaLeggett type of analysis shows that neither experiences any tunnel splitting of its ground state. We present simple arguments for how this intriguing paradox might be resolved.

Edgestate transport and conductance fluctuations in the metallic phase of the quantum Hall regime. 
D. Maslov and D. Loss. Phys.Rev.Lett. 71 (1993) 42224225.
We study mesoscopic conductance fluctuations in the metallic phase of the integer quantum Hall effect. We derive effective boundary conditions for the diffuson propagator which incorporate the edgestate transport along the sample boundaries. It is shown that the presence of edge states leads to a resonance structure in the conductance fluctuations which occurs as the Fermi level moves across the band of extended states. The result of this analytic treatment is consistent with that of our numerical simulations and can be tested experimentally.

Quantum interference effects in inhomogeneous magnetic fields. 
D. Loss, H. Schoeller, and P.M. Goldbart. Physica B 194196 (1994) 11451146.
An approach is developed to study quantum transport in disordered conductors in the presence of inhomogeneous magnetic fields, which have the effect of introducing Berry phases. The possibility is raised of isolating and probing quantum interference phenomena through their sensitivity to the inhomogeneity of applied magnetic fields. Using semiclassical and diagrammatic techniques, the ideas are illustrated with applications to three examples: anomalous weakfield magnetoconductance, conductance oscillations in mesoscopic multiplyconnected structures, and sampledependent mesoscopic conductance variations.

Bloch states of a Bloch wall. 
H. B. Braun and D. Loss. J. Appl. Phys. 76 (1994) 61776179.
Bloch walls in mesoscopic ferromagnets can tunnel between periodically arranged pinning sites leading to a Bloch band. The quantum spin phase gives rise to a spin parity dependent shift in the dispersion. Static external magnetic fields induce magnetization oscillations and provide a magnetic analogue of the Josephson effect.

Comment on Have resonance experiments seen macroscopic quantum coherence in magnetic particles? The case from power absorption. 
D. Awschalom, J. Smyth, G. Grinstein, D. DiVincenzo, and D. Loss. Phys. Rev. Lett. 71 (1993) 4276.
A Comment on the Letter by A. Garg Phys. Rev. Lett. 71, 4442 (1993).

Quantum interference effects in inhomogeneous magnetic fields. 
D. Loss, H. Schoeller, and P.M. Goldbart. Physica B 194196 (1994).
An approach is developed to study quantum transport in disordered conductors in the presence of inhomogeneous magnetic fields, which have the effect of introducing Berry phases. The possibility is raised of isolating and probing quantum interference phenomena through their sensitivity to the inhomogeneity of applied magnetic fields. Using semiclassical and diagrammatic techniques, the ideas are illustrated with applications to three examples: anomalous weakfield magnetoconductance, conductance oscillations in mesoscopic multiplyconnected structures, and sampledependent mesoscopic conductance variations.

Dissipation and quantum propagation of Bloch walls. 
H. B. Braun and D. Loss. Europhys. Lett. 31 (1995) 555560.
Bloch walls in mesoscopic ferromagnets are shown to tunnel coherently between periodically arranged pinning sites and to form a band similar to a Bloch electron in a crystal lattice. Eliminating the spin wave degrees of freedom we derive an effective action for the Bloch wall position. The dissipative term in this action involves a spectral function with a gap and therefore damping due to spin waves is negligible at low temperatures. It is shown that the Berry phase gives rise to a spinparitydependent shift in the dispersion. We show that this band structure can give rise to Bloch oscillations of the magnetization in response to a static external field, which provides a magnetic analog to the Josephson effect.

Quantization of Superflow Circulation and Magnetic Flux with a Tunable Offset. 
Y. LyandaGeller, P.M. Goldbart, and D. Loss. Phys. Rev. B 53 (1996) 1239512399.
Quantization of superflow circulation and of magnetic flux are considered for systems, such as superfluid A3 and unconventional superconductors, having nonscalar order parameters. The circulation is shown to be the anholonomy in the parallel transport of the order parameter. For multiplyconnected samples free of distributed vorticity, circulation and flux are predicted to be quantized, but generically to nonintegral values that are tunably offset from integers. This amounts to a version of AharonovBohm physics. Experimental settings for testing these issues are discussed.
© 1996 The American Physical Society.

Andreev reflection and Josephson currents in Luttinger liquids. 
D. L. Maslov, M. Stone, P.M. Goldbart, and D. Loss. Phys. Rev. B 53 (1996) 15481557.
A theory describing a onedimensional Luttinger liquid in contact with a superconductor is developed. Boundary conditions for the fermion fields describing Andreev reflection at the contacts are derived and used to construct a bosonic representation of the fermions. The Josephson current through a superconductor/Luttinger liquid/superconductor junction is considered for both perfectly and poorly transmitting interfaces. In the former case, the Josephson current at low temperatures is found to be essentially unaffected by electronelectron interactions. In the latter case, significant renormalization of the Josephson current occurs. The profile of the (induced) condensate wave function in a semiinfinite Luttinger liquid in contact with a superconductor is shown to decay as a power law, the exponent depending on the sign and strength of the interactions. In the case of repulsive (attractive) interactions the decay is faster (slower) than in their absence. An equivalent method of calculating the Josephson current through a Luttinger liquid, which employs the bosonization of the system as a whole (i.e., superconductor, as well as Luttinger liquid) is developed and shown to give the results equivalent to those obtained via boundary conditions describing Andreev reflection. © 1996 The American Physical Society.

Luttinger liquids and composite fermions in nanostructures: what is the nature of the edge states in the fractional quantum Hall regime? 
M. Geller, D. Loss, and G. Kirzcenow. Superlattices and Microstructures 21, 49 (1997)
We study the AharonovBohm conductance oscillations of a constriction with an antidot in the fractional quantum Hall regime using a recently proposed compositefermion Fermi liquid theory, and also using Wen's chiral Luttinger liquid theory extended to include mesoscopic effects. The predictions of the compositefermion Fermi liquid theory are very similar to standard Fermi liquid theory and are consistent with recent experiments. In our chiral Luttinger liquid theory, which is valid in an experimentally realizable ‘strongantidotcoupling’ regime for bulk filling factorsg= 1/q(qodd), the finite size of the antidot introduces a new temperature scaleT0≡hv/ΠkBL, wherevis the Fermi velocity andLis the circumference of the antidot edge state. Chiral Luttinger liquid theory predicts the lowtemperature (T

Relationship between susceptibility and spin stiffness of mesoscopic quantum antiferromagnets. 
A. Scott and D. Loss. Physica A 239, 47 (1997).
We investigate the quantum antiferromagnetic Heisenberg spin chain of integer spinS in the semiclassical limit (S≫1) for a nonuniform system of finitesize and temperature. At low temperatures and in the mesoscopic regime with the correlation length being larger than the characteristic system size, boundary conditions significantly affect the scaling behavior of the spin stiffnessϱs and the magnetic response. Using a coherent spinstate path integral formulation and allowing for a nonuniform time varying external magnetic field we determine the dynamic susceptibility tensorχαβ. We find that stiffness and susceptibility satisfy a remarkable relation in the mesoscopic regime, viz. ϱs/c2=χzz. We briefly discuss the experimental consequences of this relationship.

Quantum spin dynamics in mesoscopic magnets. 
A. Chiolero and D. Loss. Physica E 1 (1998) 292296.
Molecular magnets present a new class of systems that can exhibit macroscopic quantum coherence. We show that magnetic fields can induce controllable tunnel barriers for the Néel vector in antiferromagnets such as the ferric wheel. The tunneling dynamics manifests itself in e.g. NMR resonances, the static magnetization or the specific heat. Numerical simulations confirm and complement the analytical results based on an extended nonlinear sigma model.

Mesoscopic effects in the fractional quantum Hall effect. 
M.R. Geller and D. Loss. Physica E 1 (1998) 120123.
Edge states of the quantum Hall fluid provide an opportunity to study mesoscopic effects in a highly correlated electron system that is both experimentally accessible and theoretically tractable. In this paper we review recent work on the persistent current and Aharonov–Bohm magnetoconductance oscillations in the fractional quantum Hall effect regime.

Universality of shot noise in multiterminal diffusive conductors 
Eugene Sukhorukov and Daniel Loss Phys. Rev. Lett. 80, 4959 (1998); arXiv:condmat/9802050.
We prove the universality of shot noise in multiterminal diffusive conductors of arbitrary shape and dimension for purely elastic scattering as well as for hot electrons. Using a BoltzmannLangevin approach we reduce the calculation of shotnoise correlators to the solution of a diffusion equation. We show that shot noise in multiterminal conductors is a nonlocal quantity and that exchange effects can occur without quantum phase coherence even at zero electron temperature. Concrete numbers for shot noise are given that can be tested experimentally.

Spin relaxation in Mn12acetate. 
M. Leuenberger and D. Loss. Europhys. Lett. 46 (5), 692698 (1999); condmat/9810156.
We present a comprehensive theory of the magnetization relaxation in a Mn12acetate crystal based on thermally assisted spin tunneling induced by quartic anisotropy and weak transverse magnetic fields. The overall relaxation rate as a function of the magnetic field is calculated and shown to agree well with data including all resonance peaks. The Lorentzian shape of the resonances is also in good agreement with recent data. A generalized master equation including resonances is derived and solved exactly. It is shown that many transition paths with comparable weight exist that contribute to the relaxation process. Previously unknown spinphonon coupling constants are calculated explicitly.

Quantum Computers and Quantum Coherence. 
D.P. DiVincenzo and D. Loss. J. Mag. Magn. Matl. 200, 202 (1999). Invited review paper for special issue of J. Mag. Magn. Matl. ("Magnetism beyond 2000"); arXiv:condmat/9901137.
If the states of spins in solids can be created, manipulated, and measured at the singlequantum level, an entirely new form of information processing, quantum computing, will be possible. We first give an overview of quantum information processing, showing that the famous Shor speedup of integer factoring is just one of a host of important applications for qubits, including cryptography, counterfeit protection, channel capacity enhancement, distributed computing, and others. We review our proposed spinquantum dot architecture for a quantum computer, and we indicate a variety of first generation materials, optical, and electrical measurements which should be considered. We analyze the efficiency of a twodot device as a transmitter of quantum information via the propagation of qubit carriers (i.e. electrons) in a Fermi sea.

Electron and Nuclear Spin Dynamics in Antiferromagnetic Molecular Rings 
Florian Meier and Daniel Loss. Phys. Rev. Lett. 86, 5373 (2001) 
Spinentangled electrons in mesoscopic systems. 
G. Burkard, E.V. Sukhorukov, P. Recher, and D. Loss. Phys.Usp. 44, 126 (2001)
Entanglement acts as a fundamental resource for many applications in quantum communication. We propose and theoretically analyze methods for preparing and detecting entanglement between the spins of electrons in a mesoscopic environment. The entanglement production mechanism which we present is based on two quantum dots coupled to a superconductor from which paired electrons are injected via Andreev tunneling. The spincorrelated electrons can then hop from the quantum dots into normal leads. For detection we propose to measure the shot noise which is produced by the entangled electrons after they have passed a beam splitter. The enhancement of the noise by a factor of two turns out to be a unique signature for the spin singlet, a maximally entangled state. In a different setting, the entangled ground state in two tunnelcoupled quantum dots is detected via the Aharonov–Bohm oscillations in the cotunneling current.

Quantenphysik hat grossen praktischen Nutzen. 
Christoph Bruder and Daniel Loss. Basler Zeitung, Nr. 242, 17. Oktober 1998.
Fuer Laien ein Buch mit sieben Siegeln, fuer die Wissenschaft ein weites fruchtbares Feld: Die Quantenphysik hat viele ueberraschende Einsichten in die Struktur der Materie gebracht und ist noch fuer manche Ueberraschung gut. Die beiden Basler Quantenforscher Christoph Bruder und Daniel Loss erklaeren, fuer welche Entdeckungen die diesjaehrigen Nobelpreise fuer Chemie und Physik in Stockholm verliehen wurden.
Jeden Oktober schauen Wissenschafter aus aller Welt gespannt nach Stockholm: In dieser Zeit wird die Entscheidung des Nobelkomitees bekanntgegeben. Der von Alfred Nobel gestiftete Preis ist die hoechste Auszeichnung in den Naturwissenschaften ueberhaupt, seine Traeger, die vorher oft nur in Fachkreisen bekannt waren, gelangen zu Weltruhm und koennen sich fuer einige Tage des Ansturms von Journalisten und Fernsehkameras nicht mehr erwehren. Die diesjaehrigen Preise fuer Physik und Chemie wurden beide verliehen fuer Fortschritte in der Quantenphysik, der grundlegendsten und erfolgreichsten aller physikalischen Theorien.
Der diesjaehrige Nobelpreis fuer Physik geht an drei amerikanische Wissenschafter: den aus Deutschland stammenden Horst Stoermer, 1949, von der Columbia University in New York, den 1939 in China geborenen Daniel Tsui von der Universitaet Princeton und Bob Laughlin, 1950, von der Universitaet Stanford. Sie wurden ausgezeichnet ''fuer die Entdeckung einer neuen Art von Quantenfluessigkeit mit gebrochenzahlig geladenen Anregungen".

Stability of the conventional fixed point of the nonlinear sigmamodel in three dimensions. 
T. Sun and D. Loss. Europhys. Lett. 34 (1996) 355359.
The stability of the conventional fixed point of the nonlinear sigmamodel in (2 + epsilon)dimensions has been studied by calculating the anomalous dimensions of leading order O(n  1) symmetric gradient operators. The full dimensions, i.e. the canonical dimensions plus the anomalous dimensions, of these operators at the fixed point are found to be negative and therefore the fixed point is stable against the perturbation of these operators. The results indicate that as far as the O(n) symmetrybreaking regime is concerned, the conventional treatment of this model is adequate.

The electrical conductivity for inhomogeneous electric fields by the Liouville operator method. 
D. Loss and A. Thellung. Physica 144A (1987) 1728.
A derivation is given of the electrical conductivity for a metal in a space and timedependent electric field. Thereby it is assumed that the electrons are scattered elastically by randomly distributed impurities. The derivation starts from the KuboNakajima formula and is based on a perturbation expansion with Liouville operators, where use is made of van Hove's diagonal singularity property of the scattering potential. The formulae obtained are compact and the method is simpler and more transparent than the perturbation formalism developed by van Hove. It is shown that the lowest order approximation corresponds to the Boltzmann equation for electrons in inhomogeneous electric fields.

Correction terms to the Lamba2tlimit of van Hove by the Liouville operator method. II. Evaluation of the Kubo formula for the electrical conductivity. 
D. Loss Physica 139A (1986) 526542.
As an illustrative application of the oneresolvent method with Liouville operators recently developed by the author, the electrical conductivity tensor of metals for spherically symmetric impurity centres is calculated including all terms of order λ−2, λ−1 and λ0 in the interaction strength λ between electron and impurities. For each term the dependence on the impurity concentration is determined. Electronelectron interactions are neglected.
It is shown that the very compact and explicit result in terms of Liouville operators can be brought into the form previously obtained for the same case by Janner who used an extension of the doubleresolvent formalism of van Hove.

Correction terms to the Lamba2tlimit of van Hove by the Liouville operator method. I. A general perturbation treatment. 
Daniel Loss Physica 139A (1986) 505525
A large isolated quantum manybody system described by a Hamiltonian of the form H = H0 + λV is considered. In view of the calculation of transport coefficients general expressions to arbitrary order in λ are derived for the asymptotic values of the time integrals of matrix elements of Heisenberg operators. Thereby use is made of the superoperator formalism, in particular of the Liouville operator, the resolvent of which is the startingpoint for a perturbation treatment to general order in λ. The leading part of order λ−2 in the obtained power series for diagonal operators (i.e. diagonal with respect to the eigenstates of the unperturbed Hamiltonian H0) corresponds to the asymptotic value of a time integral calculated in the λ2tlimit introduced by van Hove.
The development is based on a thorough discussion of the considered Hamiltonian, unperturbed part and perturbation. Thereby it turns out that van Hove's diagonal singularity is essential for the formalism to make sense in the thermodynamic limit.
