Josephson Current and Proximity Effect in Luttinger Liquids

TL;DR

This paper develops a theoretical framework for understanding the Josephson effect in one-dimensional Luttinger liquids coupled to superconductors, highlighting how electron interactions influence the supercurrent and condensate decay.

Contribution

It introduces a bosonic representation of fermions in Luttinger liquids with superconductor contacts and compares boundary condition and bosonization methods for calculating Josephson currents.

Findings

Josephson current unaffected by interactions with perfect interfaces

Significant current renormalization with poor interfaces

Power-law decay of induced condensate depending on interactions

Abstract

A theory describing a one-dimensional Luttinger liquid in contact with superconductor is developed. Boundary conditions for the fermion fields describing Andreev reflection at the contacts are derived and used to construct a bosonic representation of the fermions. The Josephson current through a superconductor/Luttinger liquid/superconductor junction is considered for both perfectly and poorly transmitting interfaces. In the former case, the Josephson current at low temperatures is found to be essentially unaffected by electron-electron interactions. In the latter case, significant renormalization of the Josephson current occurs. The profile of the (induced) condensate wavefunction in a semi-infinite Luttinger liquid in contact with a superconductor is shown to decay as a power-law, the exponent depending on the sign and strength of the interactions. In the case of repulsive (attractive) interactions the decay is faster (slower) than in their absence. An equivalent method of calculating the Josephson current through a Luttinger liquid, which employs the bosonization of the system as a whole (i.e., superconductor, as well as Luttinger liquid) is developed and shown to give the results equivalent to those obtained via boundary conditions describing Andreev reflection.