Squeezed-vacuum photon-number moments ($⟨N^2⟩=3s^2+2s,\mathrm{Var}N=2s(s+1)$)

Here $s={\sinh }^{2}r$. The pair moment is $⟨N(N-1)⟩=3s^2+s$. These expressions follow from the single-mode Bogoliubov transformation or by differentiating the even-occupation probability generating function $G(t)=[\cosh r\sqrt{1-t^2{\tanh }^{2}r}{]}^{-1}$.