# What's the unitary matrix equivalent to a beam splitter?

1. Jun 24, 2015

### Strilanc

I've seen various different matrices used to represent beam splitters, and am wondering which is the "right" one. Alternatively, are there various kinds of beam splitters but everyone just ambiguously calls them the same thing?

The matrices I've seen are the square-root-of-not-with-extra-phase-factor matrix $A = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & i \\ i & 1 \end{bmatrix}$ here and here, and the hadamard matrix $H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$ here.

A major difference between the square root of NOT gate and the Hadamard gate is that $H^2 = I$ whereas $A^2 \propto X$. Also, if you have a 180-degree phase shift gate $Z$ then $HZH = X$ whereas $AZA = A^2 e^{i \pi Z/2}$.

Which operation should I be thinking of, when an article says "beam splitter"?

2. Jun 29, 2015

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Jun 29, 2015

### cosmik debris

I'm no expert but I think the Hadamard matrix is the beam splitter and the phase factor is due to the length of the arm in an interferometer type setup.

4. Jun 30, 2015

### Strilanc

I also asked this on the physics stackexchange, and the answer there was that it's ambiguous. Typically the splitting will be even (as opposed to a photon being split 90% one way and 10% the other), but the phases may depend on the context (and are easily adjusted by placing half wave plates on one of the paths).