  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.
