from Ref. [@optical_lattice_griffin]. We anticipate at this point that the experimental possibility of shifting the limiting scattering momenta in a well-controlled manner provides a genuine signature of Beliaev-Landau scattering.

Before continuing with our analysis, we would like to draw attention to an important consideration. Given a closed quantum system (e.g. a gas of ultracold atoms or a superfluid liquid Helium sample), the presence of detailed balance will unavoidably restrict the effect of Beliaev-Landau scattering to driving the system back into its thermal state once it is kicked out of equilibrium. Therefore most works on this physics are related to phonon-decay experiments, where one studies how externally injected phonons are damped through scattering with the condensate (Beliaev) or with the thermal cloud (Landau) [@Pitaevskii_Stringari_BL; @Giorgini_BL] or how a thermal equilibrium is reached again after a sudden global quench [@Menegoz_BL].

Consequently, it is exactly the *absence* of detailed balance in a driven-dissipative context which motivates us to study the effects of spontaneous Beliaev-Landau processes in the steady-state regime of the cavity array. In this section we have illustrated that two crucial conditions for these scatterings to be possibly relevant are indeed satisfied: 1) there is finite occupation of Bogoliubov modes over the 