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I have a question regarding the action of the $50:50$ beam splitter. According to Peter Knight's book, the action is $$B=\frac{1}{\sqrt{2}}\begin{pmatrix}1 & i\\ i & 1 \end{pmatrix}.$$

However, according to this paper on "Two-photon interference: the Hong-Ou-Mandel effect", the action is $$B=\frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1\\ 1 & -1 \end{pmatrix}.$$

Which of these is actually the action of the beam splitter?

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The two matrices both describe a beam-splitter, and they are related by phase shifts in the input/output modes. Specifically:

$$ \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 0\\ 0 & i \end{bmatrix} \begin{bmatrix} 1 & 1\\ 1 & -1 \end{bmatrix} \begin{bmatrix} 1 & 0\\ 0 & i \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & i\\ i & 1 \end{bmatrix} $$

Here the matrices

$$ \begin{bmatrix} 1 & 0\\ 0 & i \end{bmatrix} $$

represent phase shifts, the right one in an input mode, and the left one in an output mode.

The description of the BS using the complex matrix is sometimes preferred, because the interpretation of this matrix is that a photon receives a $\pi/2$ phase shift on reflection, independently of which port it enters in. However, operationally the two descriptions are equivalent. The reason for this is that there does not exist an absolute reference with which to measure the phase of the beam-splitter.

In practice, the way you would try to measure it is by constructing an interferometer, and observing the interference of two light fields on the beam-splitter. However, in an inteferometer the phase between the two fields depends on the path length difference of the interferometer arms, and this is a free parameter. You can never distinguish between the interferometer phase and the beam-splitter phase. This is true even for common-path interferometers, because in those you use the same beam-slitter twice, which is equivalent to squaring the beam-splitter matrix. Since both descriptions use Hermitian and unitary matrices, their squares are the identity matrix and you can't resolve the phase.

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