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

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, which 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 linewidth. This method not only highlights the effective physics of the JJ but also offers a scalable framework for studying large-size devices.