Spontaneous Beliaev-Landau scattering out of equilibrium

We investigate Beliaev-Landau scattering in a gas of interacting photons in a coherently driven array of nonlinear dissipative resonators, as described by the 1D driven-dissipative Bose-Hubbard model. Due to the absence of detailed balance in such an out-of-equilibrium setup, steady-state properties can be much more sensitive to the underlying microscopic dynamics. As the popular truncated Wigner approximation dramatically fails in capturing this physics, we present an alternative approach, based on a systematic expansion beyond the Bogoliubov approximation, which includes the third-order correlation functions in the dynamics. As experimentally accessible signatures of Beliaev-Landau processes, we report a small but nonnegligible correction to the Bogoliubov prediction for the steady-state momentum distribution showing a characteristic series of peaks and dips, as well as non-Gaussian features in the statistics of the cavity output field.