zero-temperature statistics $$\begin{aligned}
\big\langle\hat{\xi}_k(t)\hat{\xi}_{k'}(t')\big\rangle &=& \big\langle\hat{\xi}_k^\dagger(t)\hat{\xi}_{k'}(t')\big\rangle = 0,\\
 \big\langle \hat{\xi}_k(t)\hat{\xi}_{k'}^\dagger(t')\big\rangle &=& \gamma \delta_{k,k'} \delta(t-t').\end{aligned}$$ As usual in Bogoliubov-like approaches, the interaction terms in (\[eq:q\_fluct\]) are ordered in increasing number of fluctuation operators. When the number of photons in the condensate $|\psi_0|^2$ is much larger than the number of fluctuations, one expects the effect of higher-order terms to be negligible [@castin_dum].

The first-order correction to the mean-field, summarized on the first line of (\[eq:q\_fluct\]), incorporates processes where two condensate particles collide and produce a pair of excitations with counter-propagating wavevectors $k$ and $-k$ and viceversa. In a Hamiltonian formalism, this corresponds to only retaining quadratic terms in the fluctuation operators $\hat{\phi}_k$. Restricting to these terms in (\[eq:q\_fluct\]) and dropping the ones on the second and third line, which contain terms with more than one fluctuation operator, results in a set of linear equations for the fluctuation fields $\hat{\phi}_k$. As done in [@Iacopo_spec], this set of linear stochastic equations is solved by means of a Bogoliubov transform to new operators $$\hat{\phi}_k = u_k\hat{\chi}_k + v_k\hat{\chi}^\dagger_{-k}\label{eq:bogo_transf}$$ that diagonalize the equations of motion $$\label{eq:bog_dyn}
i(\partial_t + \gamma/2) \hat{\chi}_k = \omega_k \hat{\chi}_k + u_k \hat{\xi}_k - v_k \hat{\xi}^\dagger_{-k}.$$ Here we have defined the quasiparticle energies 