Josephson effect in superconductor/normal-dot/superconductor junctions driven out of equilibrium by quasiparticle injection

We study theoretically the large variations of supercurrent through a normal dot that are induced by small quasiparticle injection current from leads connected to dot. find decomposes into subgap contribution, which depends on voltages applied leads, as well contribution with opposite sign energies outside gap, is insensitive voltages. As gradually suppress critical voltage exists above gap dominates, leading reversal current-phase relation, namely transition so-called $\pi$-junction behavior. determine and analyze robustness effect respect temperature inelastic relaxation in