Josephson photonics with simultaneous resonances

Inelastic Cooper pair tunneling across a voltage-biased Josephson junction in series with one or more microwave cavities can generate photons via resonant processes in which the energy lost by the Cooper pair matches that of the photon(s) produced. We generalize previous theoretical treatments of such systems to analyze cases where two or more different photon generation processes are resonant simultaneously. We also explore in detail a specific case where the generation of a single photon in one cavity mode is simultaneously resonant with the generation of two photons in a second mode. We find that the coexistence of the two resonances leads to effective couplings between the modes which in turn generate entanglement.