[Physics] What operation does a beam splitter apply

opticsterminology

What is the unitary matrix equivalent to the operation of a beam splitter?

I'm asking because I've seen different matrices used and was wondering if the term is just ambiguous or if there's an agreed upon meaning.

The "quantum chesire cat" paper treats beam splitters as a square-root-of-not-with-extra-phase-factor
$$A = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & i \\ i & 1 \end{bmatrix} \, .$$

On the other hand, the "QuVis" visualization project says that the correct matrix is the Hadamard gate
$$H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \, .$$

The difference between these two matters. For example, $H^2 = I$ but $A^2 \propto X$. Also, $HZH = X$ but $AZA = A^2 e^{i \pi Z/2}$. Which operation should I be thinking of, when an article says "beam splitter"?

Best Answer

Both operations are equivalent, up to a local phase in the second mode. In particular, if you shift the second basis vector's phase by $i$, then you will turn $H$ into $A$. In a beam splitter this is perfectly natural, because the phases of the output modes are not particularly well defined, and you can always model the difference between the two operations as an extra phase plate on one of the output modes. In any case, the phase difference between the two modes is not an experimental observable unless you bring the two modes together and interfere them, in which case you will want to introduce a variable phase delay between them to control the interference. This extra controlled phase delay will eat up this static par difference.

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