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

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