Josephson current through a Luttinger liquid

TL;DR

This paper investigates how electron interactions in a one-dimensional Luttinger liquid affect the Josephson current, revealing algebraic decay, parity effects, and flux-dependent behavior, highlighting qualitative differences from non-interacting systems.

Contribution

It provides a detailed analysis of the Josephson effect in a Luttinger liquid, including interaction-induced decay, parity effects, and flux dependence, which are new insights compared to previous non-interacting models.

Findings

Critical current decays algebraically with distance

Parity effects depend on electron number modulo 4

Josephson current varies with gate voltage and magnetic flux

Abstract

We study the Josephson effect through a one-dimensional system of interacting electrons, connected to two superconductors by tunnel junctions. The interactions are treated in the framework of the one-channel Luttinger model. At zero temperature, the Josephson critical current is found to decay algebraically with increasing distance between the junctions. The exponent is proportional to the strength of the Coulomb interaction. If the Luttinger liquid has a finite size, the Josephson current depends on the total number of electrons modulo 4. These parity effects are studied for the ring, coupled capacitively to a gate-voltage and threaded by a magnetic flux. The Josephson current changes continuously as a function of the gate voltage and {\em stepwise} as a function of the magnetic flux. The electron-electron interaction introduces {\em qualitatively} new features compared to the non-interacting case.