Squeezed state of light

I was searching for the wigner function for the squeezed state, but I'm not finding it. So, what is the wigner function for it?

• Did you try to compute it? Nov 30, 2019 at 4:27
• Schleich's textbook, pp 135, 136. But as @flippiefanus asks, rhetorically, you really should be competing it for breakfast... Nov 30, 2019 at 19:44
• @CosmasZachos: is the textbook freely available? For the uninitiated such computations can be quite daunting. Dec 1, 2019 at 4:18
• @flippiefanus Not sure... This is the item; still under copyright. There could be bootleg versions out there... U libraries carry it... Surely asymmetric 2-d Gaussians are straightforward... Dec 1, 2019 at 17:52

The Wigner function for the squeezing operator requires a tedious calculation. It takes on a similar form as the operator, which is given by $$\hat{S} = \exp\left[\tfrac{1}{2}\hat{a}^2\zeta^*-\tfrac{1}{2}(\hat{a}^{\dagger})^2\zeta\right] ,$$ but the squeezing parameter turns into a tanh-function. So the result is $$W(q,p) = {\cal N} \exp\left[\alpha^2\tau^*-(\alpha^*)^2\tau\right] ,$$ where \begin{align} \alpha = & \frac{1}{\sqrt{2}}(q+i p) , \\ \tau = & \exp(i\theta)\tanh\left(\tfrac{1}{2}|\zeta|\right) , \\ {\cal N} = & \frac{1}{\cosh\left(\tfrac{1}{2}|\zeta|\right)} , \end{align} for $$\zeta=|\zeta|\exp(i\theta)$$.
The squeezing operation is a unitary operation. Its Wigner function is in the form of a Gaussian. It turns out that such a unitary operation in terms of Wigner functions only transforms the arguments of the Wigner function that it operates on. In the case of a squeezing operation, this transformation is called a Bogoliubov transformation. So in the end, all you need to do to get the Wigner function for the squeezed vacuum is to perform the appropriate Bogoliubov transformation on the Wigner function of the vacuum. The Bogoliubov transformation is given by \begin{align} \alpha & \rightarrow C^*\alpha-S\alpha^* \\ \alpha^* & \rightarrow \alpha^*C-\alpha S^* , \end{align} where \begin{align} C & = \cosh(|\zeta|) \\ S & = -\exp(i\theta)\sinh(|\zeta|) . \end{align}