Effects of electron-electron interaction and spin-orbit coupling on Andreev pair qubits in quantum dot Josephson junctions

TL;DR

This work analyzes the superconducting two-channel Anderson impurity model with spin-orbit coupling and background tunneling to study Andreev pair qubits formed from even-parity Andreev bound states. By combining zero-bandwidth approximations, numerical renormalization group calculations, and variational insights, it reveals that electron-electron interactions admix Yu–Shiba–Rusinov components into ABS, generating local moments and SOC-enabled spin polarization without external fields. The results show that in the ABS–YSR crossover region around $U\approx 2\Delta$, charge, spin, and inductive transitions become simultaneously strong and tunable, enabling spin control and potential quantum transduction, albeit with enhanced decoherence risks due to magnetic fluctuations. These findings have design implications for superconducting qubits based on ABS, highlighting how SOC, phase bias, and background tunneling can be leveraged to tailor qubit properties and coupling channels. The work also points to the importance of extending to multi-orbital junctions to capture realistic device behavior.

Abstract

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 $U$, especially in the cross-over region between the ABS and YSR regimes for $U \sim 2Δ$, charge, spin, and inductive transitions can all become strong, offering avenues for spin control and quantum transduction.

Figures (14)

• Figure 1: Schematic representation of the sub-gap states in the even-parity subsector. The lines represent the QD levels, the arrows represent Bogoliubov quasiparticles located either inside the junction (dark) or in the bath (light). The ABS components (left) differ in the number (zero or two) of trapped quasiparticles, the YSR components (right) differ in the wavefunction of the quasiparticle in the bath.
• Figure 2: Energy-phase relationships (dispersion curves) of the "doublet" (odd parity, the states relevant in the Andreev spin qubit) and "singlet" (even parity, the states relevant in the Andreev pair qubit) subgap states for the reference set of parameters. (ZBW results)
• Figure 3: Top: Eigenenergies of the normal-state ($\Delta=0$) single-electron part of the Hamiltonian for three values of the dimensionless SOC parameter $\lambda$ as a function of the background tunnelling parameter $\tau=t/V$. Bottom: Corresponding probabilities for electron occupation on the QD orbital for lowest (blue), middle (orange) and highest-energy state (green).
• Figure 4: Local-moment fraction, i.e., the probability of single-electron occupancy on the quantum dot, $P_1=\langle n-2n_\uparrow n_\downarrow \rangle$, in ground (G) and excited (E) subgap states; ZBW results. (a,b) Dependence on the electron--electron repulsion $U$ and the hybridization $V$. (c,d) Dependence on the displacement from half-filling, $\nu$, and the relative strength of background tunnelling through high-energy orbitals, $\tau$. (e,f) Dependence on the strength of spin-orbit coupling, $\lambda$, and the phase bias $\phi$. Reference parameter values (Sec. \ref{['secparam']}) are indicated by the blue markers.
• Figure 5: Magnetic properties of the subgap states as a function of the SOC strength $\lambda$ and the phase bias $\phi$; ZBW results. (a,b) Spin polarization along the SOC axis ($x$); note the difference in signs. (c,d) Strength of the SOC effective magnetic field. (e,f) Size of the YSR spin-triplet component quantified by $\langle \mathbf{S}_\mathrm{total}^2 \rangle$.