How does one apply multiple beamsplitters to a covariance matrix in the covariance matrix (2nd moments) formalism?

Question

I would like to propagate a two-mode squeezed vacuum state through a collection of beamsplitters. I'd like to represent this in the second moments, that is, the covariance formalism such as used in quantum information, so that I start with a covariance matrix, $\gamma$ and apply the many beamsplitter symplectic matrices, $S_1, S_2, S_3, ...$. If I apply $S_1$ first, $\gamma_1=S_1^T\gamma S_1$, this of course makes $\gamma_1$ a diagonal matrix, so that I think it would then be improper to apply $S_2$. What is the appropriate way to do this in order to get a physically correct final covariance matrix?

More specifically, suppose we have the following (in xp-order):

$$\gamma = \left(\begin{array}{cc} 2\mu+1 & 0 & 0 & 2\sqrt{\mu(\mu+1)} & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ -2\sqrt{\mu(\mu+1)} & 0 & 0 & 2\mu+1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1\\ \end{array}\right)^{⊕2}$$, where ⊕2 is to give the p-block for gamma Then, say we have two beamsplitter matrices: $$S_1 = \left(\begin{array}{cc} \sqrt{\eta} & \sqrt{1-\eta} & 0 & 0 & 0 & 0\\ -\sqrt{1-\eta} & \sqrt{\eta} & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & \sqrt{\eta} & 0 & \sqrt{1-\eta}\\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -\sqrt{1-\eta} & 0 & \sqrt{\eta}\\ \end{array}\right)^{⊕2}$$ $$S_2 = \left(\begin{array}{cc} \sqrt{t} & 0 & 0 & \sqrt{1-t} & 0 & 0\\ 0 & \sqrt{t} & 0 & 0 & \sqrt{1-t} & 0\\ 0 & 0 & \sqrt{t} & 0 & 0 & \sqrt{1-t}\\ -\sqrt{1-t} & 0 & 0 & \sqrt{t} & 0 & 0\\ 0 & -\sqrt{1-t} & 0 & 0 & \sqrt{t} & 0 \\ 0 & 0 & -\sqrt{1-t} & 0 & 0 & \sqrt{t}\\ \end{array}\right)^{⊕2}$$

So the question put more succinctly is: $\gamma_{final}=(S_2^T(S_1^T \gamma S_1) S_2)\\$ or $\gamma_{final}=(S_2^T S_1^T (\gamma) S_1 S_2)$ or something else?

Answer 1

Yes, $\gamma_\mathrm{final}$ is exactly what you specify: $$ \gamma_{\mathrm{final}} = S_2^TS_1^T\gamma S_1 S_2 = S^T \gamma S\ , $$ with $S=S_1S_2$.

This is evident since under $S$, $\gamma$ transforms as $\gamma\mapsto S^T\gamma S$. If you apply two such maps, you get their concatenation. Where you put your brackets does not matter, as matrix multiplication is associative.