= Bosonic squeezed-vacuum contractions
{title2=$\langle a_k^\dagger a_q\rangle=\sinh^2\theta_k\,\delta_{kq},\quad\langle a_ka_q\rangle=-\sinh\theta_k\cosh\theta_k\,\delta_{k,-q}$}
Let $b_k=\cosh\theta_k\,a_k+\sinh\theta_k\,a_{-k}^\dagger$ for real even $\theta_k$, and let every $b_k$ annihilate a normalized <vacuum state>. Inverting the <Bogoliubov transformation> gives $a_k=\cosh\theta_k\,b_k-\sinh\theta_k\,b_{-k}^\dagger$. Only $\langle b_kb_q^\dagger\rangle=\delta_{kq}$ survives, proving the displayed <vacuum expectation values>. Taking the <Hermitian adjoint> gives the same real negative pair amplitude for two <creation operators>. The normal <occupation number> is quadratic at small $\theta_k$, whereas the pair amplitude is linear. For one opposite-<momentum> pair the <two-mode squeezed vacuum> is $\operatorname{sech}\theta\sum_{n\ge0}(-\tanh\theta)^n|n,n\rangle$, fixing the sign convention directly.
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