Nonequilibrium supercurrent through a quantum dot: Current harmonics and proximity effect due to a normal-metal lead

We study the out-of-equilibrium current through a quantum dot which is placed between two superconducting leads held at fixed voltage bias, considering both cases of the absence and the presence of an additional normal lead connected to the dot. Using the nonequilibrium Keldysh technique, we focus on the subgap bias region, where multiple Andreev reflections are responsible for charge transfer through the dot. Attention is put on the dc current and on the first harmonics of the supercurrent. Varying the position and/or the width of the dot level, we first investigate the crossover between a quantum-dot and quantum point-contact regimes in the absence of a normal lead. We then study the effect of the normal electrode connected to the dot, which is understood to lead to dephasing, or alternatively to induce reverse proximity effect. By increasing the dot coupling to the normal probe, we show the full crossover from zero dephasing to the incoherent case. We also compute the Josephson current in the presence of the normal lead and find it in excellent agreement with the values of the nonequlibrium current extrapolated at zero voltage