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?

  • 1
    $\begingroup$ Did you try to compute it? $\endgroup$ Nov 30, 2019 at 4:27
  • $\begingroup$ Schleich's textbook, pp 135, 136. But as @flippiefanus asks, rhetorically, you really should be competing it for breakfast... $\endgroup$ Nov 30, 2019 at 19:44
  • $\begingroup$ @CosmasZachos: is the textbook freely available? For the uninitiated such computations can be quite daunting. $\endgroup$ Dec 1, 2019 at 4:18
  • $\begingroup$ @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... $\endgroup$ Dec 1, 2019 at 17:52

1 Answer 1


Without giving the expression for the final answer, I sketch out the steps one needs to take to do the computation.

One can get Wigner functions for states as well as operators. The squeezed state that you are interested in seem to be the squeezed vacuum. So, if you have the Wigner function of the squeezing operator and that of the vacuum state you can compute the Wigner function of squeezed vacuum.

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 Wigner function for the vacuum is very simple. If you know the Wigner function for the coherent states, you can just set the parameter to zero and then you have it for the vacuum.

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}

Hopefully I have given you enough of the details now to perform the final calculation. There are some subtleties. See if you can figure it out. If not, ask again and I'll provide more details.


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