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I am studying beam-splitters in 2nd quantization and here is what I got so far

$$ \hat{BS}(\theta) = e^{i\theta(\hat{a}^{\dagger}_1 \otimes \hat{a}_2 + \hat{a}^{\dagger}_2\otimes \hat{a}_1)} $$

where $\hat{a}_1$ and $\hat{a}_2$ are the annihilation operators for the two outgoing paths of the beam-splitter. This seems to work if we ignore the polarization of photons.

My questions are:

  1. How does one include the polarization in this picture?
  2. Does the polarization of a photon introduce a phase shift somewhere?
  3. What if we try to write the operator of a polarizing beam-splitter?

Thank you!

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A beam splitter is essentially an interaction between two quantized modes with Hamiltonian (leaving out the hats because everything is quantized) $$H_{\mathrm{int}}\propto a_1^\dagger a_2+a_2^\dagger a_1.$$ This type of interaction leads to linear evolution among the creation or annihilation operators: $$\text{i}\frac{d a_1}{dt}=[a_1,H_{\mathrm{int}}]\propto a_2,\quad \text{i}\frac{d a_2}{dt}=[a_2,H_{\mathrm{int}}]\propto a_1.$$ Now, none of this has actually specified what the two modes $1$ and $2$ represent physically, so the same treatment can be done with any two modes that interact "linearly."

The answer to the overall question is: Write the creation and annihilation operators that involve the polarization degree of freedom. A waveplate operating on a single path, for example, is like a beam splitter, so it could be written with $$H_{\text{waveplate}}\propto a_H^\dagger a_V+a_V^\dagger a_H,$$ where $a_H$ annihilates a horizontally polarized photon and likewise $a_V$ for vertical polarization. A polarizing beam splitter has to act on both the path and the polarization, so you could write it with an interaction like $$H_{\text{PBS}}\propto a_{1,H}^\dagger a_{2,V}+a_{2,V}^\dagger a_{1,H},$$ where now we specify the polarizations and paths that are annihilated by the operators. In fact, these operators also need to encode the frequency of photon that they are annihilating, as well as all other modal properties of the field, but they can often be assumed to not change at this stage of the interaction and so we drop their label to conserve space.

If you want to include any phase shifts, you can write an interaction of the form $$H_{\mathrm{int}}\propto \xi a_1^\dagger a_2+\xi^* a_2^\dagger a_1$$ for any complex constant $\xi$. This also allows you to incorporate examples in which the beam gets split into other pairs of polarizations, like diagonal and antidiagonal, instead of $H$ and $V$.

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