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 $1\mathrm{/}\tau$, the supercurrent carried at zero temperature by an Andreev level at energy ${\epsilon }_{\mathrm{A}}$ is reduced by a factor $(2\mathrm{/}\pi )\arctan (2{\epsilon }_{\mathrm{A}}\tau \mathrm{/}\mathrm{\hslash })$ . 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.