with small enough time step $\Delta t$, and then taking a statistically independent sample each $\tau_s = 5\gamma^{-1}$.

As expected, the magnitude of the correction to the Bogoliubov theory is proportional to the single-photon interaction constant $U$ (or, equivalently, to the inverse of $n_0$ at a given mean-field energy $Un_0$). For both values of $U$, the negative occupation of some high-$k$ modes is a clearly unphysical prediction of the TWA.

To better understand the physical origin of this breakdown, one needs to take a closer look at the nature of the underlying physical processes. Through Beliaev scattering, a quasiparticle at a high momentum $k$ decays into two quasiparticles with smaller momenta $q$ and $k-q$. Since the occupation decreases for larger $k$-modes, one expects the importance of this effect to be suppressed at higher momenta. However, within the TWA the quantum field is represented as a classical field for which the occupation of high-$k$ modes does not decay to $0$ but to $1/2$, which represents the quantum vacuum fluctuations (see Eq. (\[eq:TWA\_dens\])). This finite occupation of all modes, even the highest-$k$ ones, results in the possibility of a nonphysical decay of the quantum vacuum through spontaneous Beliaev processes. The final states 