Andreev non-Hermitian Hamiltonian for open Josephson junctions from Green's functions

TL;DR

This paper develops a non-Hermitian Hamiltonian approach derived from Green's functions to analyze transport in open Josephson junctions, accurately capturing Andreev bound states under specific conditions.

Contribution

It introduces a novel effective non-Hermitian Hamiltonian for Andreev levels, derived from Green's functions, to simplify and accurately model open Josephson junctions.

Findings

The NH model accurately predicts the supercurrent and density of states when ABS are well-separated from the continuum.

The approach is scalable for large devices and effective when ABS are far from the superconducting gap.

Benchmarking shows good agreement with exact Green's function calculations in applicable regimes.

Abstract

We investigate the transport properties of open Josephson junctions (JJs) through a minimal effective non-Hermitian (NH) approach derived from the equilibrium Green's function (GF) formalism. Specifically, we consider a JJ with a quantum dot barrier coupled to a normal metal reservoir. The coupling introduces an imaginary self-energy term in the JJ Hamiltonian which can be naturally accounted for in the NH formalism. While most approaches to similar problems work with the full junction Hamiltonian we propose a scheme for deriving an effective NH Hamiltonian for the Andreev levels only, that we compute from the singular part of the barrier GF. To establish the range of applicability of this NH model we benchmark our results for both the dot density of states and the supercurrent against exact GF predictions in different transport regimes. We find that, as a rule of thumb, the Andreev NH description is accurate when the spectral overlap between the Andreev bound states (ABS) and the near-gap continuum states is negligible, i.e. when the ABS energies lie sufficiently far from the superconducting gap relative to their line-width. This method not only highlights the effective physics of the JJ but also offers a scalable framework for studying large-size devices.