how the non-equilibrium condition allows for on-shell Beliaev-Landau scattering in one-dimension. We next explain how the truncated Wigner approximation fails to describe these processes in Sec. \[sec:TWA\]. In Sec. \[sec:HOC\] we introduce the third-order correlation functions to incorporate Beliaev and Landau scattering and construct a hierarchy of correlation functions. In Sec. \[sec:exp\] we discuss the imprint of Beliaev-Landau scattering on measurable quantities and discuss the expected signal for realistic parameters inspired from state-of-the-art semiconductor devices. Conclusions are finally drawn in Sec. \[sec:conclusion\]. Appendix A reports additional numerical TWA data in the absence of open Beliaev-Landau channels. Appendices B and C summarize technical details on the hierarchy of correlations and on the different truncation schemes. Details on the calculation of the effect of disorder are given in Appendix D.

The Model {#sec:model}
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We consider a 1D coupled array of $L$ nonlinear, single-mode photon cavities under a coherent drive with frequency $\omega_L$. The resonator frequencies $\omega_c$ are assumed to be uniform throughout the chain. After a unitary transformation to remove the time-dependence of the drive, we obtain the driven Bose-Hubbard Hamiltonian (we set $\hbar = 1$ throughout the article) $$\begin{aligned}
\nonumber
\hat{H} &=& -J \sum_{\langle j,l\rangle} \left(\hat{a}^\dagger_j \hat{a}_{l} +\hat{a}^\dagger_l \hat{a}_{j}\right) - 