Multimode squeezed vacuum ($|0⟩=C\exp (\frac{1}{2}{a}^{\prime }{}^{†}M{a}^{\prime }{}^{†})|0^{\prime }⟩$)

For an invertible finite-mode Bogoliubov transformation $a=A{a}^{\prime }+B{a}^{\prime }{}^{†}$, the symmetric squeezing matrix is $M=-A^{-1}B$. The canonical commutation relations make its singular values strictly less than one. The normalized state has $|C|=\det (I-M{M}^{†}{)}^{1/4}$. Infinitely many modes additionally require bosonic mode mixing implementability.

Pair creation in two bosonic modes gives the normalized state displayed. The two occupations coincide in each term, while either mode alone has a geometric number distribution. At small $r$, the leading nonvacuum term is a single pair. This optical construction does not require particle-antiparticle modes.

Let $b_k=\cosh {\theta }_k{a}_k+\sinh {\theta }_k{a}_{-k}^{†}$ for real even ${\theta }_k$, and let every $b_k$ annihilate a normalized vacuum state. Inverting the Bogoliubov transformation gives $a_k=\cosh {\theta }_k{b}_k-\sinh {\theta }_k{b}_{-k}^{†}$. Only $⟨b_k{b}_q^{†}⟩={\delta }_{kq}$ 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 $\mathrm{sech}\theta {\sum }_{n\ge 0}(-\tanh \theta {)}^n|n,n⟩$, fixing the sign convention directly.

Tracing out one mode leaves geometric occupation probabilities with $q=|\lambda {|}^2$. In an oscillator basis of frequency $\omega$, they are Boltzmann weights with $k_{B}T=\hslash \omega /\ln (1/q)$. The joint state can be pure while each reduced state is mixed.

For a pair of distinct bosonic particle and antiparticle modes, the normalized state obeying $(a+q{b}^{†})|\psi ⟩=(b+q{a}^{†})|\psi ⟩=0$ is $\sqrt{1-|q{|}^2}{\sum }_{n\ge 0}(-q{)}^n|n,n⟩$, for $|q|<1$. It contains equal occupations and is generated by a pair-creation exponential. Infinite products require a separate implementability or regulator check.

The pair count of a two-mode squeezed vacuum has $P(n)=(1-r)r^n$, where $r=|q{|}^2<1$. Its mean is $r/(1-r)$ and its variance is $r/(1-r{)}^2$. Equal mode occupations follow from pair creation, not from independent particle and antiparticle draws.