We investigate the superconducting Anderson impurity model for interacting quantum dot Josephson junctions with spin-orbit coupling and a term accounting for tunnelling through higher-energy orbitals. These elements establish the conditions required for spin polarization in the absence of external magnetic field at finite superconducting phase bias. This Hamiltonian has been previously used to model the Andreev spin qubit, where quantum information is encoded in spinful odd-parity subgap states. Here we instead analyse the even-parity sector, i.e., the Andreev pair qubit based on Andreev bound states (ABS). The model is solved using the zero-bandwidth approximation and the numerical renormalization group, with further insight from variational calculations. Electron-electron interaction admixes single-occupancy Yu-Shiba-Rusinov (YSR) components into the ABS states, thereby strongly enhancing spin transitions in the presence of spin-orbit coupling. The ABS states can thus become sensitive to local magnetic field fluctuations, which has implications for decoherence in Andreev pair qubits. For strong interaction U U , especially in the cross-over region between the ABS and YSR regimes for U \sim 2\Delta U ∼ 2 Δ , charge, spin, and inductive transitions can all become strong, offering avenues for spin control and quantum transduction.
We investigate the superconducting Anderson impurity model for interacting quantum dot Josephson junctions with spin-orbit coupling and a term accounting for tunnelling through higher-energy orbitals. These elements establish the conditions required for spin polarization in the absence of external magnetic field at finite superconducting phase bias. This Hamiltonian has been previously used to model the Andreev spin qubit, where quantum information is encoded in spinful odd-parity subgap states. Here we instead analyse the even-parity sector, i.e., the Andreev pair qubit based on Andreev bound states (ABS). The model is solved using the zero-bandwidth approximation and the numerical renormalization group, with further insight from variational calculations. Electron-electron interaction admixes single-occupancy Yu-Shiba-Rusinov (YSR) components into the ABS states, thereby strongly enhancing spin transitions in the presence of spin-orbit coupling. The ABS states can thus become sensitive to local magnetic field fluctuations, which has implications for decoherence in Andreev pair qubits. For strong interaction U U , especially in the cross-over region between the ABS and YSR regimes for U \sim 2\Delta U ∼ 2 Δ , charge, spin, and inductive transitions can all become strong, offering avenues for spin control and quantum transduction.
We theoretically study the dephasing of an Andreev spin qubit (ASQ) due to electric and magnetic noise. Using a tight-binding model, we calculate the Andreev states formed in a Josephson junction where the link is a semiconductor with strong spin-orbit interaction. As a result of both the spin-orbit interaction and induced superconductivity, the local charge and spin of these states vary as a function of externally controllable parameters: the phase difference between the superconducting leads, an applied magnetic field, and filling of the underlying semiconductor. Concomitantly, coupling to fluctuations of the electric or magnetic environment will vary, which informs the rate of dephasing. We qualitatively predict the dependence of dephasing on the nature of the environment, magnetic field, phase difference between the junction, and filling of the semiconductor. Comparing the simulated electric- and magnetic-noise-induced dephasing rate to experiment suggests that the dominant source of noise is magnetic. Moreover, by appropriately tuning these external parameters, we find sweet spots at which we predict an enhancement in ASQ coherence times.
We propose a variant of Andreev spin qubits (ASQs) defined in planar Josephson junctions based on spin-orbit coupled two-dimensional electron gases (2DEGs) in a weak out-of-plane magnetic field. The magnetic field induces a linear phase gradient across the junction, generating Josephson vortices that can host low-energy Andreev bound states (ABSs). We show that, in certain parameter regimes, the combined effect of the phase gradient and spin-orbit coupling stabilizes an odd-fermion parity ground state, where a single Josephson vortex binds a spinful low-energy degree of freedom that is energetically separated from the other ABSs. This low-energy degree of freedom can be exploited to define a special type of ASQ, which we dub the vortex spin qubit (VSQ). We show that single-qubit gates for VSQs can be performed via flux driving, while readout can be achieved by adapting standard circuit quantum electrodynamics (cQED) techniques developed for conventional ASQs. We further outline how an entangling two-qubit gate can be performed using an ac current drive. We argue that VSQs offer prospects for a substantial reduction in device complexity and hardware overhead compared to conventional ASQ implementations, while preserving key advantages such as supercurrent-based readout, single-qubit gates, and long-range two-qubit gates.
We solve the problem of an interacting quantum dot embedded in a Josephson junction between two superconductors with finite charging energy described by the transmon (Cooper pair box) Hamiltonian. The approach is based on the flat-band approximation of the Richardson model, which reduces the Hilbert space to the point where exact diagonalization is possible while retaining all states that are necessary to describe the low-energy phenomena. The presented method accounts for the physics of the quantum dot, the Josephson effect, and the Coulomb repulsion (charging energy) at the same level. In particular, it captures the quantum fluctuations of the superconducting phase as well as the coupling between the superconducting phase and the quantum dot (spin) degrees of freedom. The method can be directly applied for modeling Andreev spin qubits embedded in transmon circuits in all parameter regimes, for describing time-dependent processes, and for the calculation of transition matrix elements for microwave-driven transmon, spin-flip, and mixed transitions that involve coupling to charge or current degrees of freedom.
Opportunités de manipulation directe d'un qubit de spin d'Andreev piloté en phase 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.
We present a unified study of charge and spin qubits encoded in a single interacting quantum dot tunnel-coupled to Rashba fermionic leads. The dot is described by an Anderson impurity Hamiltonian with either repulsive or attractive local interactions and a magnetic field of arbitrary orientation. In the repulsive regime, a spin qubit is encoded in the singly occupied spin sector, whereas in the attractive regime, close to the degeneracy between the empty and doubly occupied configurations, these two states define a charge-qubit subspace. We combine a weak-coupling Lindblad master equation in the Markovian secular limit with numerically controlled matrix-product-state simulations of the full dot-reservoir system. In equilibrium, we identify magnetic pair breaking in the attractive regime and a weak Rashba-induced magnetic anisotropy when the magnetic field is rotated relative to the spin-orbit axis. In the dynamical regime, relaxation and decoherence are controlled by the energetic gap between the logical manifold and nearby leakage states, and are further enhanced by Rashba-dependent tunneling. These findings show that even a minimal Anderson-impurity description captures key ingredients for coherent spin and charge qubits: the energetic structure of leakage states and spin-orbit-dependent dot-lead hybridization.
Abstract The quantum mechanics of the Josephson effect is the core ingredient for quantum technologies with superconducting circuits. A new avenue was recently opened in this field by predicting that the Josephson quantum mechanics in the odd parity sector, when a quasiparticle is trapped in an Andreev bound state, is fundamentally different from the conventional one in the even sector. The focus was then on a Josephson junction surrounded by an electromagnetic environment formed of a collection of bosonic modes, including the case of an ohmic environment. Here we consider the distinct case of a superconducting qubit made of a single Josephson junction whose environment reduces to a capacitance. We find a novel structure for the low-lying discrete states in the odd sector, which is altogether different from the one that appears in the even sector. Our study of the bound-state spectrum ranges from the Coulomb-dominated (Cooper pair box) to the Josephson-dominated (transmon) regime. Our prediction could be tested in forthcoming experiments with superconductor/semiconductor/superconductor junctions, which have been studied intensively in recent years, both using nanowires as well as two-dimensional electron gases.