Tunneling through an Anderson impurity between superconductors

We consider an Anderson impurity (A) weakly connected to a superconducting electrode (S) on one side and a superconducting or a normal metal electrode (N) on the other side. A general path integral formalism is developed and the response of SAN and SAS junctions to a constant voltage bias V is elucidated, using a combination of the Keldysh technique (to handle non-equilibrium effects) and a dynamical mean field approximation (to handle repulsive Hubbard interactions). An interesting physics is exposed at sub-gap voltages ($eV < \Delta$ for SAN and $eV < 2 \Delta$ for SAS). For an SAN junction, Andreev reflection is strongly affected by Coulomb interaction. For superconductors with p-wave symmetry the junction conductance exhibits a remarkable peak at $eV < \Delta$, while for superconductors with s-wave symmetric pair potential the peak is shifted towards the gap edge $eV=\Delta$ and strongly suppressed if the Hubbard repulsive interaction increases.Electron transport in SAS junctions is determined by an interplay between multiple Andreev reflection (MAR) and Coulomb effects. For s-wave superconductors the usual peaks in the conductance that originate from MAR are shifted by interaction to larger values of V. They are also suppressed as the Hubbard interaction strength grows. For p-wave superconductors the sub-gap current is much larger and the I-V characteristics reveal a new feature, namely, a peak in the current resulting from a mid-gap bound state in the junction.