We study the production of photons in a model of three bosonic atomic modes non-linearly coupled to a cavity mode. In absence of external driving and dissipation, the energy levels at different photon numbers assemble into the steps of an energy staircase which can be employed as guidance for preparing multi-photon states. We consider adiabatic photon production, driving the system through a sequence of Landau-Zener transitions in the presence of external coherent light pumping. We also analyse the non-equilibrium dynamics of the system resulting from the competition of the sudden switch of coherent photon pumping and cavity photon losses, and we find that the system approaches a plateau with a given number of photons, which becomes metastable upon increasing the rate of photon pumping. We discuss the sensitivity of the time scales for the onset of this metastable behaviour to system parameters and predict the value of photons attained, solving the driven-dissipative dynamics including three-body correlations between light and matter degrees of freedom.
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