Real-time diagrammatic approach to transport through interacting quantum dots with normal and superconducting leads

We present a real-time diagrammatic theory for transport through interacting quantum dots tunnel coupled to normal and superconducting leads. Our formulation describes both the equilibrium and nonequilibrium superconducting proximity effects in a quantum dot. We study a three-terminal transistor geometry, consisting of a single-level quantum dot tunnel coupled to two phase-biased superconducting leads and one voltage-biased normal lead. 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 pi states as a function of gate and bias voltages. For the limit of large superconducting gaps in the leads, we describe the formation of Andreev bound states within an exact resummation of all orders in the tunnel coupling to the superconducting leads, and we discuss their signature in the nonequilibrium Josephson and Andreev currents and the quantum-dot charge.