Eigenstates of a small Josephson junction coupled to a resonant cavity

We carry out a quantum-mechanical analysis of a small Josephson junction coupled to a single-mode resonant cavity. We find that the eigenstates of the combined junction-cavity system are strongly entangled only when the gate voltage applied at one of the superconducting islands is tuned to certain special values. One such value corresponds to the resonant absorption of a single photon by Cooper pairs in the junction. Another special value corresponds to a two-photon absorption process. Near the single-photon resonant absorption, the system is accurately described by a simplified model in which only the lowest two levels of die Josephson junction are retained in the Hamiltonian matrix. We noticed that this approximation does not work very well as the number of photons in the resonator increases. Our system shows also the phenomenon of "collapse and revival" under suitable initial conditions, and our full numerical solution agrees with the two level approximation result.