amplitude $\Omega$ is always implicitly determined by choosing a value for $n_0$.

Bogoliubov theory and Beliaev-Landau processes {#sec:beliaev}
==============================================

Bogoliubov dispersion and the non-equilibrium steady-state
----------------------------------------------------------

The case of a uniform drive field allows for a convenient parametrization of the full quantum field in terms of a homogeneous, coherent field $\psi_0$ and quantum fluctuations. Expanding the latter in their $\hat{\phi}_k$ spatial Fourier components, one can write $$\label{eq:ansatz}
\hat{a}_{j} = \psi_0 + \frac{1}{\sqrt{L}} \sum_k e^{ikj} \hat{\phi}_k$$ where the sum over $k$ is restricted to the interval $[-\pi,\pi]$ with a spacing equal to $2\pi/L$ in the case of a finite number $L$ cavities and periodic boundary conditions.

While the time evolution of the mean-field $\psi_0$ is governed by the classical equation (\[eq:MF\]), the dynamics of the quantum fluctuations is governed by a quantum Langevin equation [@gardiner_quantum_noise; @Iacopo_QFL] $$\begin{aligned}
\label{eq:q_fluct}
\nonumber
i\partial_t \hat{\phi}_k &=& (\epsilon_k + Un_0 - i\gamma/2)\hat{\phi}_k + U\psi_0^2 \hat{\phi}^\dagger_{-k} +  \hat{\xi}_k \\\nonumber
&&+ \frac{2U\psi_0}{\sqrt{L}} \sum_q \hat{\phi}^\dagger_q \hat{\phi}_{k+q} + \frac{U\psi_0^\ast}{\sqrt{L}} \sum_q \hat{\phi}_q \hat{\phi}_{k-q}\\
&&+ \frac{U}{L} \sum_{q,l} \hat{\phi}^\dagger_q \hat{\phi}_l \hat{\phi}_{k+q-l},\end{aligned}$$ where we have set $$\label{eq:epsilon}
\epsilon_k = -\delta + Un_0 - 2J\cos{k}.$$ The Markovian losses are responsible for quantum noise with Gaussian statistics, represented by the operators $\hat{\xi}_k$, that assume the following 