Thesis presented April 09, 2024

Abstract:
The research presented in this thesis delves into a theoretical exploration of Andreev bound states (ABS) hosted in Josephson junctions with spin-orbit coupling (SOC). ABS are fermionic bound states that appear at the junction between two superconducting electrodes forming a Josephson junction. These bound states are at the core of the microscopic description of the Josephson effect and provide an explanation for the transfer of Cooper pairs from one superconducting electrode to the other. Crucially, when the superconducting phase difference between the two electrodes is non-zero, a supercurrent can flow in the junction, depending on the occupancy of the ABS. The presence of this supercurrent facilitates the coupling between a microwave resonator and a junction due to its sensibility to the electromagnetic field. This coupling then allows for the detection and manipulation of ABS. ​Furthermore, in the presence of SOC along with a superconducting phase bias, the spin degeneracy of ABS can be lifted. Therefore such a junction provides a unique opportunity to realize a special kind of spin qubit known as Andreev spin qubit (ASQ). The qubit operation can be performed via an AC modulation of an electrostatic gate or magnetic flux, thanks to the sensitivity of the ABS to the electric potential or the phase difference, respectively. So far, the coherent manipulation of such qubits has been realized in gate-driven experiments as well as in flux-driven experiments with the use of additional Andreev levels. However, the direct transition between the two ABS forming the ASQ has remained out of reach in flux-driven experiments. In this thesis, we investigate two distinct types of Josephson junctions with SOC with the aim of estimating the amplitude of the matrix elements of the current operator between two ABS forming an ASQ. These elements characterize the coupling strength between the qubit and an external flux drive, indicating which transitions between ABS are within reach. Our first project focuses on a superconductor-normal-superconductor junction where the normal region consists of a nanowire with Rashba SOC. As a minimal model, we consider a generic scatterer located along the nanowire. Unless this scatterer possesses additional spatial (mirror) symmetries, it is responsible for inducing spin-flip transmission probability. Using the scattering formalism, we derive the energy spectrum of ABS and assess how SOC influences the spin-splitting between opposite spin states. We then obtain analytical expressions for the matrix elements of the current operator. Notably, our study reveals that SOC allows one to have finite elements between opposite spin states. In a second project, the system we study consists of a superconductor-double quantum dots-superconductor junction. A quantum dot consists of a tiny region defined by electrostatic gates or impurities. Due to its small size, Coulomb repulsion can be significant. However, in the simplest case of a quantum dot with a single level, even in the presence of SOC in the coupling between the dot and the leads, spin degeneracy of ABS is still preserved. However, recent experiments showed that additional levels allow to lift the spin degeneracy. Here, we introduce a second quantum dot which provides an additional level and offers more control on each level. By using an effective model, we identify the minimal ingredients that are necessary to lift the spin degeneracy. Our results reveal that this splitting is achievable through spin-dependent couplings with the leads and is reduced by the Coulomb repulsion. Additionally, the presence of a finite coupling between the two dots, combined with SOC in the couplings with the leads, allows one to have finite matrix elements of the current operator between opposite spin states, which we obtain numerically. In short, we present in this thesis two platforms for the realization of phase-driven ASQ.

Keywords:
Superconductivity, Quantum computer, Quantum transport