Andreev non-Hermitian Hamiltonian for open Josephson devices

In closed Josephson junctions (JJs) the supercurrent, in certain regimes, can be fully determined by sub-gap Andreev bound states (ABS). This picture becomes more subtle in open JJs coupled to external reservoirs. We focus on a superconductor-quantum dot-superconductor JJ coupled to a normal-metal lead. The coupling induces ABS broadening, turning them into Andreev quasi-bound states (quasi-ABS) with complex energy, that may spectrally overlap with the continuum states. By analyzing the dot Green’s function, we separate Andreev and continuum contributions to both the density of states and the supercurrent, and derive an effective non-Hermitian Hamiltonian for the quasi-ABS. We show that the non-Hermitian description accurately captures the JJ spectral and transport properties only when the quasi-ABS overlap with the continuum is negligible, i.e., when they lie sufficiently far from the superconducting gap.