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. We find that its Andreev spectrum as a function of phase difference exhibits Josephson gaps, i.e. finite phase windows, in which Andreev quasi-bound states do not exist. Remarkably, exceptional points emerge in the complex Andreev spectrum, constituting a particular spectral feature of NHJJs. Moreover, we propose complex supercurrents due to inelastic Cooper pair tunneling. At the exceptional points, the corresponding phase-biased supercurrent divergences. Our results illustrate that the Josephson effect is strongly affected by non-Hermitian physics.