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arXiv:2402.02118 · Physical review. B./Physical review. B

Generalized transmon Hamiltonian for Andreev spin qubits

Luka Pavešič, Rok Žitko

Published Apr 26, 2024• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.109.155164

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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.
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 realize a hybrid superconductor-semiconductor transmon device in which the Josephson effect is controlled by a gate-defined quantum dot in an InAs-Al nanowire.Microwave spectroscopy of the transition spectrum of the transmon allows us to probe the ground-state parity of the quantum dot as a function of the gate voltages, the external magnetic flux, and the magnetic field applied parallel to the nanowire.The measured parity phase diagram is in agreement with that predicted by a single-impurity Anderson model with superconducting leads.Through continuous-time monitoring of the circuit, we furthermore resolve the quasiparticle dynamics of the quantum dot Josephson junction across the phase boundaries.Our results can facilitate the realization of semiconductor-based 0-π qubits and Andreev qubits.
We use a hybrid superconductor-semiconductor transmon device to perform spectroscopy of a quantum dot Josephson junction tuned to be in a spin-1/2 ground state with an unpaired quasiparticle. Because of spin-orbit coupling, we resolve two flux-sensitive branches in the transmon spectrum, depending on the spin of the quasiparticle. A finite magnetic field shifts the two branches in energy, favoring one spin state and resulting in the anomalous Josephson effect. We demonstrate the excitation of the direct spin-flip transition using all-electrical control. Manipulation and control of the spin-flip transition enable the future implementation of charging energy protected Andreev spin qubits.
Semiconductor-based Josephson junctions embedded within a Cooper-pair-box can host complex many-body states, such as interacting Andreev states and potentially other quasi-particles of topological origin. Here, we study the insights that could be revealed from a tomographic reconstruction of the Cooper-pair charge distribution of the junction prepared in its ground state. We posit that interacting and topological states can be identified from distinct signatures within the probability distribution of the charge states. Furthermore, the comprehensive dataset provides direct access to information theory metrics elucidating the entanglement between the charge sector of the superconductor and the microscopic degrees of freedom in the junction. We demonstrate how these metrics serve to further classify differences between the types of excitations in the junction.
In the field of nanoelectronics, one seeks to combine quantum mechanical effects with macroscopic entities such as currents in order to find new ways of storing and manipulating information. One possibility to accomplish this is to use the quantum correlations found in superconductors. Teaming superconductors with normal metals and magnetic systems, one can obtain novel correlation and fluctuation effects. \n\nIn the first work of this thesis, we examine how charge correlations in a metallic dot are modified due to the presence of a capacitively coupled superconducting island. It is possible to overcome the Coulomb repulsion between electrons in a hybrid double-dot system consisting of a metallic island coupled capacitively to a Cooper-pair box. The superconducting island S acts as a dynamic gate on the metallic island N and in the strong coupling regime the electric charge of the Cooper pairs in S and their fluctuations are able to overscreen the repulsion between electrons in N. This produces an attractive interaction between two additional electrons in the metallic island and results in a non-monotonous charging curve of the metallic island. \n\nThe second part of the thesis considers how the dynamics of a precessing molecular magnetic moment affects the Josephson current. The precession gives rise to a time-dependent tunnel potential which not only creates different tunneling probabilities for spin-up and spin-down quasiparticles, but also introduces a time-dependent spin-flip term. In particular, we study the effects of the spin-flip term alone on the Josephson current between two spin-singlet superconductors as a function of precession frequency and junction transparency. The system displays a nonequilibrium steady-state solution which depends on the precession frequency of the classical spin.\n\n\n
The case of a quantum dot connected to two superconducting leads is studied by using the narrow-band limit to describe the superconducting degrees of freedom. The model provides a simple theoretical framework, almost analytical, to analyze the interplay between the Kondo effect, superconductivity, and finite temperature. In the quantum dot Kondo regime, the model is completely characterized by the ratio $\ensuremath{\Delta}/J$, with $\ensuremath{\Delta}$ the superconducting gap and JJJ an effective antiferromagnetic exchange coupling between the dot and the leads. The model allows us to calculate, at any temperature TTT, the equilibrium Josephson current through the dot in a very straightforward way as a function of $\ensuremath{\Delta}/J$. The behavior of the current allows us to distinguish the four types of hybrid junctions: 0, ${0}^{\ensuremath{'}},{\ensuremath{\pi}}^{\ensuremath{'}}$, and $\ensuremath{\pi}$. The presence of the 0- and ${0}^{\ensuremath{'}}$-junction configurations are intrinsically linked to the Kondo effect in the quantum dot, while the $\ensuremath{\pi}$- and ${\ensuremath{\pi}}^{\ensuremath{'}}$-junction configurations are driven by the superconductivity in the leads. The Josephson critical current has a non-monotonic behavior with temperature, that may be used for the experimental characterization of the fundamental $0\ensuremath{-}\ensuremath{\pi}$ transition. The model allows us to obtain easily a phase diagram $\ensuremath{\Delta}/J$ vs temperature, from where we can obtain an overall picture on the stability of the different types of junctions. From the explicit analytical expressions for the ground-state, low-energy excitations, free energy, and Josephson current, it is easy to understand the physical nature of the main features of the critical current and the phase diagram. The results, obtained with a minimum of numerical effort, are in a good qualitative agreement with more demanding calculational approaches aimed to solve the full model.