Particle conserving approach to ac-dc driven interacting quantum dots with superconducting leads

The combined action of a dc bias and a microwave drive on the transport characteristic of a superconductor-quantum dot-superconductor junction is investigated. To cope with time dependent nonequilibrium effects and interactions in the quantum dot, we develop a general formalism for the dynamics of the density operator based on a particle conserving approach to superconductivity. Without invoking a broken U(1) symmetry, we identify a dynamical phase connected to the coherent transfer of Cooper pairs across the junction. In the weak-coupling limit, we show that besides quasiparticle transport, proximity induced superconducting correlations manifest in anomalous pair tunneling involving the transfer of a Cooper pair. The resulting generalized master equation in presence of the microwave drive showcases the characteristic bichromatic response due to the combination of the ac Josephson effect and an ac voltage. Analytical expressions for all harmonics in the driving frequency of both the current and the reduced dot operator are given for arbitrary driving strength. For the net dc current, the resulting photon assisted processes give rise to rich current-voltage characteristics. In addition to photon assisted subgap transport, we find regions of total current inversion in the stability diagram. There, the junction acts as a pump with the net dc current flowing against the applied dc bias. The first harmonic of the current, being closely related to the nonlinear dynamic susceptibility of the junction, is discussed at finite applied dc bias