Squeeze operator

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The squeeze operator for a single mode is

$\hat{S}(z) = \exp \left ( {1 \over 2} (z^* \hat{a}^2 - z \hat{a}^{\dagger 2}) \right ) , \qquad z = r e^{i\theta}$

where the operators inside the exponential are the ladder operators. The squeeze operator is ubiquitous in quantum optics and can operate on any state. For example, when acting upon the vacuum, the squeezing operator produces the squeezed vacuum state, which is represented in the photon number basis as:

$|z\rangle= ({\rm sech}\,r)^{\frac{1}{2}} \sum^{\infty}_{n=0} \frac{[(2 n)!]^2}{n!} \left [-\frac{1}{2} \exp (i \theta) \tanh r \right ]^n |2n\rangle$ ,

which oscillates in photon number, i.e. p(n)=0 for all odd n.

The squeezing operator can also act on coherent states and produce squeezed coherent states. It is important to note that the squeezing operator does not commute with the displacement operator:

$\hat{S}(z) \hat{D}(\alpha) \neq \hat{D}(\alpha) \hat{S}(z)$