Inverse spin Hall effect in superconductor/normal-metal/superconductor Josephson junctions

We consider dc supercurrents in SNS junctions. Spin-orbit coupling in combination with Zeeman fields can induce an effective vector potential in the normal conductor. As a consequence, an out-of-plane spin density varying along the transverse direction causes a longitudinal phase difference between the superconducting terminals. The resulting equilibrium phase-coherent supercurrent is analog to the nonequilibrium inverse spin Hall effect in normal conductors. We explicitly compute the effect for the Rashba spin-orbit coupling in a disordered two-dimensional electron gas with an inhomogeneous perpendicular Zeeman field.