$\varphi_j$ correspond to expectation values of symmetrically ordered products of quantum operators. In particular, for the number operator we find $$\label{eq:TWA_dens}
\big\langle\varphi_j^\ast \varphi_j\big\rangle_W = \frac{1}{2}\Big( \langle \hat{a}_j^\dagger \hat{a}_j \rangle +  \langle \hat{a}_j \hat{a}_j^\dagger \rangle \Big) = \langle \hat{n}_j \rangle + \frac{1}{2}.$$

As a consequence, the quantum vacuum is represented by a finite occupation of $1/2$ for the classical field $\varphi_j$. As long as $n_j \gg 1/2$, one does not expect this to cause problems but, when performing a TWA simulation to estimate the effects of Beliaev-Landau scattering, one finds surprisingly large corrections to the occupation numbers of quantum fluctuations, as can be seen in Fig. \[fig:2nd\]a). Even worse is that the occupation numbers may become negative at certain values of the momentum, even though the used parameters are well inside the supposed region of validity of TWA. In Fig. \[fig:2nd\] we fixed $Un_0 = 10\gamma$ and show the results for two interaction constants $U=0.02\gamma$ and $U=0.1\gamma$, such that the mean-field predictions for the number of particles per site are $n_0 = 500$ and $n_0 = 100$, respectively. The results were obtained by averaging out over a total number of about $10^6$ samples, which were collected by integrating (\[eq:TWA\]) in time 