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Superconductor-Quantum Dot-Superconductor Josephson Structures with Attractive Intradot Interaction

Yu‐Chun Hsu, Wu-Jing Chen, Chien‐Te Wu

Published May 4, 2020• 0 citations
OpenAlex

Abstract

We consider a quantum dot with intradot attractive interaction sandwiched between two superconducting leads. We show that the system possesses quantum phase transitions of fermion parity as long as an external magnetic field is present. Due to the superconducting proximity effects, the possible electronic states for the embedded quantum dot are spin-polarized states with odd occupation and BCS-like states with even occupation. In this work, we adopt a self-consistent theoretical method to extend our considerations beyond the superconducting atomic limit to numerically investigate implications of an attractive interaction in electronic structures. We discuss the difference between results obtained in and away from the atomic limit. We find a reentrant behavior in the energy phase diagram when $\Delta$ is an order of magnitude larger than the hybridization strength. We also consider the Josephson current phase relations and find a number of examples showing the $0-\pi$ phase transitions that may offer important switching effects.

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We propose and theoretically investigate superconducting spin qubits. Superconducting spin qubit consists of a single spin confined in a Josephson junction. We show that owing to spin-orbit interaction, superconducting phase difference across the junction can polarize this spin. We demonstrate that this enables single-qubit operations and more complicated quantum gates, where spins of different qubits interact via a mutual inductance of the superconducting loop where the junctions are embedded. Recent experimental realizations of Josephson junctions made of semiconductor quantum dots in contact with superconducting leads have shown that the number of electrons in the quantum dot can be tuned by a gate voltage. Superconducting spin qubit is realized when the number of electrons is odd. We discuss the qubit properties at phenomenological level. We present a microscopic theory that enables us to make accurate estimations of the qubit parameters by evaluating the spin-dependent Josephson energy in the framework of fourth-order perturbation theory.

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We consider a quantum dot with intradot attractive interaction sandwiched between two superconducting leads. We show that the system possesses quantum phase transitions of fermion parity as long as an external magnetic field is present. Due to the superconducting proximity effects, the possible electronic states for the embedded quantum dot are spin-polarized states with odd occupation and BCS-like states with even occupation. In this work, we adopt a self-consistent theoretical method to extend our considerations beyond the superconducting atomic limit to numerically investigate implications of an attractive interaction in electronic structures. We discuss the difference between results obtained in and away from the atomic limit. We find a reentrant behavior in the energy phase diagram when Δ\DeltaΔ is an order of magnitude larger than the hybridization strength. We also consider the Josephson current phase relations and find a number of examples showing the 0−π0-\pi0−π phase transitions that may offer important switching effects.
We theoretically study the superconducting proximity effect in a quantum dot coupled to two superconducting leads when the intradot interaction between electrons is made attractive. Because of the superconducting proximity effect, the electronic states for the embedded quantum dot are either spin-polarized states with an odd occupation number or BCS-like states with an even occupation number. We show that in the presence of an external magnetic field, the system can exhibit quantum phase transitions of fermion parity associated with the occupation number. In this paper, we adopt a self-consistent theoretical method to extend our considerations beyond the so-called superconducting atomic limit in which the superconducting gap for the leads is assumed to be the largest energy scale. This method enables us to numerically investigate the electronic structure of the dot as a result of the attractive interaction. For energy phase diagrams in the regime away from the atomic limit, we find a reentrant behavior where a BCS-like phase of the dot exists in an intermediate range of the hybridization strength between the quantum dot and the leads. We also consider Josephson current-phase relations and identify a number of examples showing 0-$\ensuremath{\pi}$ phase transitions that may offer important switching effects.
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 calculate the equilibrium Josephson current through a disordered interacting quantum dot described by a Sachdev-Ye-Kitaev model fully contacted by two BCS superconductors, such that all modes of the dot contribute to the coupling, which encodes hopping and spin-flip processes. We show that, at zero temperature and at the conformal limit, i.e. in the strong interacting limit, the Josephson current is suppressed by U U , the strength of the interaction, as ln(U)/U and becomes universal, namely it gets independent on the superconducting gap. At finite temperature, instead, it depends on the ratio between the gap and the temperature. A proximity effect exists but the self-energy corrections induced by the coupling with the superconducting leads seem subleading as compared to the interaction self-energy and the tunneling matrix for large number of particles. Finally we compare the results of the original four-fermion model with those obtained considering zero interaction, two-fermions and a generalized q q -fermion model.
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.
We analyze the ground state properties of an array of quantum dots connected in series between superconducting electrodes. This system is represented by a finite Hubbard chain coupled at both ends to BCS superconductors. The ground state is obtained using the Lanczos algorithm within a low energy theory in which the bulk superconductors are replaced by effective local pairing potentials. We study the conditions for the inversion of the sign of the Josephson coupling ($\ensuremath{\pi}$-junction behavior) as a function of the model parameters. Results are presented in the form of phase diagrams, which provide a direct overall view of the general trends as the size of the system is increased, exhibiting a strong even-odd effect. The analysis of the spin-spin correlation functions and local charges give further insight into the nature of the ground state and how it is transformed by the presence of superconductivity in the leads. Finally, we study the scaling of the Josephson current with the system size and relate these results with previous calculations of Josephson transport through a Luttinger liquid.
We propose and theoretically investigate superconducting spin qubits. Superconducting spin qubit consists of a single spin confined in a Josephson junction. We show that owing to spin-orbit interaction, superconducting phase difference across the junction can polarize this spin. We demonstrate that this enables single-qubit operations and more complicated quantum gates, where spins of different qubits interact via a mutual inductance of the superconducting loop where the junctions are embedded. Recent experimental realizations of Josephson junctions made of semiconductor quantum dots in contact with superconducting leads have shown that the number of electrons in the quantum dot can be tuned by a gate voltage. Superconducting spin qubit is realized when the number of electrons is odd. We discuss the qubit properties at phenomenological level. We present a microscopic theory that enables us to make accurate estimations of the qubit parameters by evaluating the spin-dependent Josephson energy in the framework of fourth-order perturbation theory.