  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 generalise previous theoretical treatments of such
systems to analyse cases where two or more different photon generation
processes are resonant simultaneously. We also explore in detail a specific
case where 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.
