$\omega_k$ along with the transformation functions $u_k$, $v_k$ as $$\begin{aligned}
\label{eq:bog_transform}
\omega_k &=& \sqrt{\epsilon_k(\epsilon_k +2Un_0)},\\
u_k,v_k &=& \frac{\sqrt{\epsilon_k + 2Un_0} \pm \sqrt{\epsilon_k}}{2\sqrt{\omega_k}}.\end{aligned}$$ In Fig. \[fig:bel\_decay\](a) the Bogoliubov spectrum (\[eq:bog\_transform\]) is shown for different values of the renormalized detuning $\Delta$ (\[eq:delta\]). In contrast to equilibrium systems, note that a spectral gap is generally present in the Bogoliubov dispersion and only closes for $\Delta \rightarrow 0^{(-)}$, i.e. when the drive is exactly on resonance with the interaction-shifted mode [@Iacopo_QFL].

Due to the noise operators in (\[eq:bog\_dyn\]), one has a finite occupation of Bogoliubov modes with non-trivial anomalous correlations in the stationary regime, $$\label{eq:bog_distr}
n^{(\chi)}_k =\langle \hat{\chi}_k^\dagger \hat{\chi}_k \rangle = v_k^2,\;\;\;c^{(\chi)}_k =\langle \hat{\chi}_k \hat{\chi}_k \rangle= \frac{u_k v_k \gamma}{\gamma + 2i\omega_k}$$ It is important to note that the occupation of the Bogoliubov modes here, in contrast with an equilibrium system, is not at all set by a finite-temperature Boltzmann-Gibbs distribution but by the interplay of interactions, hopping, driving and dissipation [@Iacopo_spec]. This different origin is apparent in the slow, power-law decay of the occupation of high momentum modes, much slower than the usual exponential $\exp{\big[-E(k)/k_BT\big]}$ of equilibrium systems.

Moving back to the original $\hat{\phi}_k$ operators, one can derive a closed system of linear differential 