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The Hong-Ou-Mandel (HOM) effect in quantum optics consists, in a nutshell, in sending two identical photons through different input ports of a beam-splitter, to see that they bunch together at one of the output ports.

When implemented experimentally, the HOM effect gives a characterization of the single photon states sent to the beam-splitter. For example, it can be related to the indistinguishability, and thus purity $\text{Tr}[\rho^2]$, of the states.

Would it be interesting to realize the HOM experiment with multi-photon states? For example sending two two-photon states $|2\rangle$ as input? What information would it give, anything more than the states' purity? Was it ever done?

I know it is difficult to prepare $|2\rangle$ or higher, but my question is more of general interest.

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This isn't a precise answer, but I think this is a relevant place to start. And it is fairly recent, so you may get some good references from it as well (there are 44).

Quantifying n-photon indistinguishability with a cyclic integrated interferometer (2022)

Their experiments cover mainly variations on n=4 and n=8 photon modes, and how to achieve indistinguishability with those. But they also discuss HOM for 2 photon modes. I assume from their paper that the HOM effect cannot directly be used to demonstrate indistinguishability when n>2, and is instead done on pair-wise subsets.

Good luck!

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