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Adiabatic polaron dynamics and Josephson effect in a superconducting molecular quantum dot

We study the Josephson current through a resonant level coupled to a vibration mode (local Holstein model) in the adiabatic limit of low oscillator frequency. A semiclassical theory is then appropriate and allows us to consider the oscillator dynamics within the Born-Oppenheimer approximation for arbitrary electron-vibration couplings. The resulting Fokker-Planck equation has been solved in the most relevant underdamped limit and yields the oscillator distribution function and the Josephson current. Remarkably, a transition from single-well to double-well behavior of the effective oscillator potential surface is possible and can be tuned by variation of the superconducting phase difference. The Josephson current is shown to be only weakly affected by the electron-vibration coupling due to strong phonon localization near the bottom of the potential surface.