Josephson effect in a junction coupled to an electron reservoir

We extend the scattering theory of the Josephson effect to include a coupling of the Josephson junction to a gapless electron reservoir in the normal state. By opening up the system with a quasiparticle escape rate 1/τ, the supercurrent carried at zero temperature by an Andreev level at energy εA is reduced by a factor (2/π)(2εAτ/) . We make contact with recent work on "non-Hermitian Josephson junctions", by comparing this result to different proposed generalizations of the Josephson effect to non-Hermitian Hamiltonians.