Anomalous Andreev spectrum and transport in non-Hermitian Josephson junctions

We propose a phase-biased non-Hermitian Josephson junction (NHJJ) composed of two superconductors mediated by a short non-Hermitian link. Such a NHJJ is described by an effective non-Hermitian Hamiltonian derived based on the Lindblad formalism in the weak coupling regime. By solving the Bogoliubov--de Gennes equation, we find that its Andreev spectrum as a function of phase difference exhibits Josephson gaps, i.e., finite phase windows with no Andreev (quasi)bound states. The complex Andreev spectrum and the presence of Josephson gaps constitute particular spectral features of the NHJJ. Moreover, we propose complex supercurrents arising from inelastic Cooper pair tunneling to characterize the anomalous transport in the NHJJ. Additional numerical simulations complement our analytical predictions. We demonstrate that the Josephson effect is strongly affected by non-Hermitian physics.