DC and AC Josephson Effect in a Superconductor-Luttinger Liquid-Superconductor System

TL;DR

This paper investigates how electron-electron interactions in a one-dimensional Luttinger liquid affect both DC and AC Josephson currents in superconductor-Luttinger liquid-superconductor systems, revealing algebraic and exponential decay behaviors, parity effects, and interaction-induced oscillations.

Contribution

It provides a comprehensive analysis of the impact of Coulomb interactions on Josephson effects in 1D systems, including new insights into decay behaviors, parity effects, and AC oscillations.

Findings

Critical current decays algebraically with distance at zero temperature.

Finite temperature induces a crossover from algebraic to exponential decay of the critical current.

Electron-electron interactions cause oscillations in the AC Josephson current amplitude.

Abstract

We calculate both the DC and the AC Josephson current through a one-dimensional system of interacting electrons, connected to two superconductors by tunnel junctions. We treat the (repulsive) Coulomb interaction in the framework of the one-channel, spin-$1/2$ Luttinger model. The Josephson current is obtained for two geometries of experimental relevance: a quantum wire and a ring. At zero temperature, the critical current is found to decay algebraically with increasing distance $d$ between the junctions. The decay is characterized by an exponent which depends on the strength of the interaction. At finite temperatures $T$, lower than the superconducting transition temperature $T_c$, there is a crossover from algebraic to exponential decay of the critical current as a function of $d$, at a distance of the order of $\hbar v_F/k_B T$. Moreover, the dependence of critical current on temperature shows non-monotonic behavior. If the Luttinger liquid is confined to a ring of circumference $L$, coupled capacitively to a gate voltage and threaded by a magnetic flux, the Josephson current shows remarkable parity effects under the variation of these parameters. For some values of the gate voltage and applied flux, the ring acts as a $\pi$-junction. These features are robust against thermal fluctuations up to temperatures on the order of $\hbar v_F/k_B L$. For the wire-geometry, we have also studied the AC-Josephson effect. The amplitude and the phase of the time-dependent Josephson current are affected by electron-electron interactions. Specifically, the amplitude shows pronounced oscillations as a function of the bias voltage due to the difference between the velocities of spin and charge excitations in the Luttinger liquid. Therefore, the AC Josephson effect can be used as a tool for the observation of