= Out-mode number fluctuations in the in-vacuum
{title2=$\langle N_s^2\rangle=2n_s^2+n_s+|c_s|^2$}
If $b_s=\sum_j(A_{sj}a_j+B_{sj}a_j^\dagger)$ and the state is the <in-vacuum>, put $n_s=\sum_j|B_{sj}|^2$ and $c_s=\sum_j A_{sj}B_{sj}$. The <canonical commutation relations> give $N_s^2=b_s^{\dagger2}b_s^2+N_s$. Acting with $b_s^2$ on the vacuum produces $c_s|0\rangle$ and a two-particle vector whose squared norm is $2n_s^2$, because its two possible vacuum contractions have equal contributions. This proves the displayed second moment and $\operatorname{Var}(N_s)=n_s(n_s+1)+|c_s|^2$. A thermal-mode formula requires $c_s=0$; a general <squeezed vacuum state> need not satisfy that restriction.
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