Junctions with Superconducting Leads

To which extent quantum tunneling and proximity effects impact the transport properties of a quantum dot coupled to superconducting leads? We shall discuss transport across voltage biased Josephson junctions combining the Liouville approach with a particle conserving formulation of superconductivity. On the one hand the gapped character of the superconducting spectrum hinders pure quasiparticle transport at low bias. On the other hand anomalous tunneling processes involving Cooper pairs lead to a Josephson supercurrent. The current-voltage characteristics strongly depend on the relative alignment of the quantum dot levels with respect to the chemical potentials of the superconductors.