Josephson effect in a junction coupled to an electron reservoir

TL;DR

This paper extends the scattering theory of the Josephson effect to include coupling with an electron reservoir, revealing how quasiparticle escape influences supercurrent and connecting to non-Hermitian Hamiltonian models.

Contribution

It introduces a model incorporating quasiparticle escape in Josephson junctions and compares it with non-Hermitian Hamiltonian approaches.

Findings

Supercurrent is reduced by a factor involving the arctangent function.

The model links quasiparticle escape rate to supercurrent suppression.

Comparison with non-Hermitian models highlights different generalizations of the Josephson effect.

Abstract

We extend the scattering theory of the Josephson effect to include a coupling of the Josephson junction to a gapless electron reservoir in the normal state. By opening up the system with a quasiparticle escape rate $1/\tau$, the supercurrent carried at zero temperature by an Andreev level at energy $\varepsilon_{\rm A}$ is reduced by a factor $(2/\pi)\arctan(2\varepsilon_{\rm A}\tau/\hbar)$ . We make contact with recent work on "non-Hermitian Josephson junctions", by comparing this result to different proposed generalizations of the Josephson effect to non-Hermitian Hamiltonians.