Josephson effect through a resonant level coupled to a single oscillator mode

Motivated by very recent experiments on superconducting transport through single-walled carbon nanotubes, we investigate the supercurrent through a one-level quantum dot connected to a single phonon mode. Calculations are done using non-equilibrium Green's function methods within a self-consistent Born approximation, i.e. assuming that the tunneling rate is much larger that the effective electron-phonon coupling. We calculate both the modified Andreev-bound state spectrum and the renormalization of the phonon density-of-states in situations when the phonon-mode is either in or out-off thermal equilibrium with the electron system. Based on our calculations we discuss possible Andreev-state spectroscopy.