ContactDepartment of PhysicsUniversity of Basel Klingelbergstrasse 82 CH4056 Basel, Switzerland

As of 10/2017 I have relocated to the group of Prof. Liang Fu at the Massachusetts Institute of Technology (M.I.T.) as a postdoctoral research fellow and therefore this website will no longer be updated. My new website is www.constantinschrade.com,
Short CV
2013  2017  PhD studies in the Condensed Matter Theory & Quantum Computing group at the University of Basel, under the supervision of Prof. Daniel Loss 
2012  Masters Thesis at the Boston University Physics Department, Boston under the supervision of Prof. Claudio Chamon and Prof. Manfred Sigrist 
2008  2013  Undergraduate studies at ETH Zurich, Faculty of Physics 
Publications
Show all abstracts.1.  DIII Topological Superconductivity with Emergent TimeReversal Symmetry 
Christopher Reeg, Constantin Schrade, Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 96, 161407(R) (2017)
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.
 
2.  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)
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 swave 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.
 
3.  Detecting Topological Superconductivity with $\varphi_{0}$ Josephson Junctions 
Constantin Schrade, Silas Hoffman, and Daniel Loss. Phys. Rev. B 95, 195421 (2017)
The interplay of superconductivity, magnetic fields, and spinorbit interaction lies at the heart of topological superconductivity. Remarkably, the recent experimental discovery of $\varphi_{0}$ 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 φ0 Josephson junction can serve as a new qualitative indicator for topological superconductivity: Microscopically, we find that the phase shift in a junction of s−wave superconductors is due to the spinorbit induced mixing of singly occupied states on the qantum 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 $\varphi_{0}$ 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.
 
4.  Universal Quantum Computation with Hybrid SpinMajorana Qubits 
Silas Hoffman, Constantin Schrade, Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 94, 045316 (2016)
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.
 
5.  ProximityInduced $\pi$ Josephson Junctions in Topological Insulators and Kramers Pairs of Majorana Fermions 
Constantin Schrade, Alexander Zyuzin, Jelena Klinovaja, and Daniel Loss. Phys. Rev. Lett. 115, 237001 (2015)
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.
