Hybrid normal-superconducting systems comprising interacting quantum dots

Quantum dots tunnel-coupled to both normal and superconducting leads exhibit a very rich physics due to the presence of superconducting correlations, quantum fluctuations, strong electron-electron interaction, and non-equilibrium. In order to study these systems, we have developed a real-time diagrammatic expansion in the tunnel coupling to the leads, which describes both the equilibrium and non-equilibrium superconducting proximity effects in the quantum dot. In the limit of a large superconducting gap, all orders in the tunnel-coupling strength to the superconductors can be included within an exact resummation scheme. Corrections due to finite values of the gap are evaluated within a 1/Δ expansion. This theory is applied to a single-level quantum dot tunnel coupled to two phase-biased superconducting leads and one voltage-biased normal lead. The normal lead is used to drive the dot out of equilibrium. We compute both the Josephson current between the two superconductors and the Andreev current in the normal lead, and analyze their switching on and off as well as transitions between 0- and π-states as a function of gate and bias voltage.