Single-mode squeeze operator ($S(\gamma )$)

For a real squeezing parameter $\gamma$, the single-mode squeeze operator isIts exponent is skew-adjoint, so $S(\gamma )$ is a unitary operator, and it implements the Bogoliubov transformation

The squeezed vacuum is $|\gamma ⟩=S(\gamma )|0⟩$. Its position and momentum variances are rescaled in opposite directions,so it remains an minimum-uncertainty state with $\Delta X\Delta P=\hslash /2$.

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.