Hamiltonian approach to the ac Josephson effect in superconducting-normal hybrid systems

The ac Josephson effect in hybrid systems of a normal mesoscopic conductor coupled to two superconducting (S) leads is investigated theoretically. A general formula of the ac components of time-dependent current is derived, which is valid for arbitrary interactions in the normal region. We apply this formula to analyze a S-normal-$S$ system where the normal region is a noninteracting single-level quantum dot. We report the physical behavior of time-averaged nonequilibrium distribution of electrons in the quantum dot, the formation of Andreev resonance states, and ac components of the time-dependent current. The distribution is found to exhibit a population inversion; and all Andreev resonance states between the superconducting gap $\Delta$ carry the same amount of the current and in the same flow direction. The ac components of time-dependent current show strong oscillatory behavior in marked contrast to the subharmonic gap structure of the average current.