I am trying to figure out how to derive the wigner function of the squeezed vacuum state, \begin{align} W(\alpha,\alpha^*)& =(2/{\pi})\times{e^{-2|\alpha'|^{2}}}\\ \alpha'& =\alpha \cosh{r}-\alpha^{*}e^{i\theta}\sinh{r} \tag 1 \end{align} I came across the expression in arXiv:quant-ph/0612024. I have so far been able to get to the Husimi $Q$-function of the squeezed vacuum state, $$ Q(\beta,\beta^*)=(\pi \cosh r)^{-1}e^{-|\beta|^{2}-\tanh(r)/2(e^{i\theta}(\beta^*)^{2}+e^{-i\theta}(\beta)^{2})} \tag 2 $$ I am trying to get the characteristic antinormal function, $$ C_{A}(\lambda,\lambda^*)= \int Q(\beta,\beta^*)e^{\lambda\beta^{*}-\lambda^{*}\beta}d^{2}\beta \tag 3 $$ after which I intend to get the Characteristic Wigner function $$ C_{W}(\lambda,\lambda^*)=C_{A}e^{|\lambda|^{2}/2} \tag 4 $$ From which I can get to the Wigner function, $$ W(\alpha,\alpha^*)=\int C_{W}*e^{\lambda^*\alpha-\lambda\alpha^*}d^{2}\lambda \tag 5 $$ I am stuck at (3). I have tried expressing the complex variables as $$ \beta=\frac{q+ip}{\sqrt 2} \hspace{0.5cm} \lambda=\frac{x+iy}{\sqrt 2} \tag 6 $$ and transforming the integral into 2D integral in real coordinates to no avail. I am unable to get a result with which I can proceed further. Is there any other way to deal with 2D Fourier transforms of complex gaussian distributions of the form in $(3)$?
PS: I have just stared learning quantum mechanics this semester along with Quantum optics and have zero formal knowledge of group theory.
