I am trying to understand Wigner functions of Gaussian states, specifically, that of a 2-mode squeezed state. I was wondering if anyone can explain what I am seeing in the context of what these plots physically represent in regards to quadrature measurements.
The Wigner function for an $n$ mode Gaussian state is given by:
$$W(\mathbf{X}) = \frac{\exp\left( -\frac{1}{2}\left( \mathbf{X} - \left<\hat{\mathbf{R}}\right> \right)^T\sigma^{-1}\left( \mathbf{X}-\left<\hat{\mathbf{R}}\right> \right) \right)}{\pi^n\sqrt{\det[\sigma]}}$$
where $\mathbf{X} = (x_1,y_1,\dots,x_n,y_n)^T$, $\hat{\mathbf{R}} = (\hat{x}_1,\hat{p}_1,\dots,\hat{x}_n,\hat{p}_n)^T$, and $\sigma$ is the $2n \times 2n$ covariance matrix.
For my problem in particular, I am looking at a zero mean Gaussian state, therefore $\left<\hat{\mathbf{R}}\right> = 0$, and I get:
$$W(\mathbf{X}) = \frac{\exp\left( -\frac{1}{2} \mathbf{X}^T\sigma^{-1} \mathbf{X} \right)}{\pi^n\sqrt{\det[\sigma]}}$$
For my particular problem, the covariance matrix takes the following form:
$$\sigma = \left(\begin{matrix} V_{11} & 0 & 0 & V_{14}\\ 0 & V_{22} & V_{23} & 0\\ 0 & V_{32} & V_{33} & 0\\ V_{41} & 0 & 0 & V_{44}\\ \end{matrix}\right)$$
This matrix is a function of a couple of different parameters, but the end result is a constant for each entry. The details are unimportant for this question so I will just write them as constants.
The inverse will then have the form: $$\sigma^{-1} = \left(\begin{matrix} S_{11} & 0 & 0 & S_{14}\\ 0 & S_{22} & S_{23} & 0\\ 0 & S_{32} & S_{33} & 0\\ S_{41} & 0 & 0 & S_{44}\\ \end{matrix}\right)$$
Okay so now, I just need to expand things out. For my case $n=2$ and I get:
$$\mathbf{X}^T\sigma^{-1}\mathbf{X} = \left(\begin{matrix} x_1 & y_1 & x_2 & y_2 \end{matrix}\right) \left(\begin{matrix} S_{11} & 0 & 0 & S_{14}\\ 0 & S_{22} & S_{23} & 0\\ 0 & S_{32} & S_{33} & 0\\ S_{41} & 0 & 0 & S_{44}\\ \end{matrix}\right) \left(\begin{matrix} x_1\\ y_1\\ x_2\\ y_2 \end{matrix}\right) $$ $$ \mathbf{X}^T\sigma^{-1}\mathbf{X} = \left(\begin{matrix} x_1 S_{11}+y_2 S_{41}\\ y_1 S_{22}+x_2 S_{32}\\ y_1 S_{23}+x_2 S_{33}\\ x_1 S_{14}+y_2 S_{44} \end{matrix}\right)^T \left(\begin{matrix} x_1\\ y_1\\ x_2\\ y_2 \end{matrix}\right) $$ $$ \mathbf{X}^T\sigma^{-1}\mathbf{X} = x_1^2 S_{11}+y_2x_1 S_{41} + y_1^2 S_{22}+x_2y_1 S_{32} + y_1x_2 S_{23}+x_2^2 S_{33} + x_1y_2 S_{14}+y_2^2 S_{44} $$
So the goal is to plot the Wigner function, but I am confused because of the mixed variables. And perhaps it is something fundamental I am confused about. But because there are 4 variables, to plot a 2D color plot, I need to choose 2 variables. So for example, I choose $(x_1,y_1) = (0.3,0.3)$, then I get the 2D color plot shown below:
Which looks good to me, it shows squeezing in one of the quadratures. But is it possible to plot the distribution of both modes? It seems like it would be a 4D surface, but I am wondering if anyone has a good way to visualize both mode distributions.
Also, from the literature I have read, I thought the squeezing was supposed to occur for different linear combinations of the quadratures, but in the above plot, it clearly shows that $y_2$ only is squeezed. If I instead choose the same single values for $(x_2,y_2)$, then I get squeezing in the other quadrature.
I know these represent quasi-probability distributions, but what do these results mean in the context of reality? Namely, quadrature measurements. Are $(x_1,y_1,x_2,y_2)$ electric field measurements with units of V/m?