 Solution

Under the Gaussian assumption, $g(\mathbf{r},\mathbf{r})=n(\mathbf{r})$, so the coincident normally ordered density correlation is $(1+ϵ)n(\mathbf{r}{)}^2$. For a homogeneous nonzero density $n$, the large-separation limit is $n^2$, givingThese coincident-point bunching factors describe enhanced bosonic coincidence and excluded same-component fermionic coincidence. More generally the density-normalized ratio $C_{12}/(n_1{n}_2)$ tends to one at large separation. The unnormalized ratio printed in the paper additionally needs matching nonzero local and remote densities; $g_{12}\to 0$ alone does not ensure this. For a bosonic Fock product state, the local normalized value instead is $2-{\sum }_{\alpha }{N}_{\alpha }(N_{\alpha }+1)|{\phi }_{\alpha }(\mathbf{r}){|}^4/n(\mathbf{r}{)}^2$, and even a large-separation factorization must be checked separately.