Trying to understand Wigner function of zero-mean two-mode squeezed Gaussian State

Question

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?

Answer 1

The mixed variables are a result of a non-diagonal $\sigma^{-1}$ matrix. If you diagonalize $\sigma^{-1}$ you will obtain new variables $X_1$ and $Y_2$ as a combo of $x_1$ and $y_2$, and another pair $Y_1$ and $X_2$ as a combo of $y_1$ and $x_2$, where the cross-terms will be gone. There may not be an obvious physical interpretation to those new variables, i.e. they might represent some obscure combination of mode operators.

In general there is no way of plotting the two pairs simultaneously. However, by going to $X_1,Y_1,X_2$ and $Y_2$, you can "see" the squeezing through the eigenvalues of $\sigma^{-1}$ and then "select out" appropriate illustrative slices (which need not have some coordinates at $0$) and get a sense of the higher-dimensional squeezing by presenting a sequence of slices like the way the mind reconstruct a 3D image from a stack of 2D images. As far as I know there's no canonical way of selecting the slices unless you have a Hamiltonian that provides insight into what might remain constant.

Finally, the use of quadrature measurements is really a constraint that stems from actual experimental techniques. It's possible to get squeezing in any direction in the $(x,p)$ plane and indeed one can get "number-phase" squeezing as in

taken from

Didier, Nicolas, et al. "Heisenberg-limited qubit read-out with two-mode squeezed light." Physical review letters 115.9 (2015): 093604

(arxiv version here)

There are some details also in

Collett M.J. (1993) Generating number-phase squeezed states. In: Ehlotzky F. (eds) Fundamentals of Quantum Optics III. Lecture Notes in Physics, vol 420. Springer, Berlin, Heidelberg

or in

Kitagawa, M., and Y. Yamamoto. "Number-phase minimum-uncertainty state with reduced number uncertainty in a Kerr nonlinear interferometer." Physical Review A 34.5 (1986): 3974.