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When two optical fibers A and B are brought close enough that there can be quantum tunneling between the two, we can write the "beam splitter" Hamiltonian as

$$H = a^\dagger b + b^\dagger a$$

If, instead, the two fibers are spliced together, such that (travelling from left to right) photons in B continue in B, but photons in A merge into B, how do I write the Hamiltonian for this?

My guess is

$$H = a b^\dagger.$$

It's not Hermitian, but maybe it's correct. I would be interested in references to any material that discusses something like this in the context of quantum optics and second quantisation representation. Thanks!

Tom
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    Non-hermitian hamiltonians are never correct, unless you have losses (i.e. actual absorption) in your system. – Emilio Pisanty Aug 08 '19 at 14:52
  • Related: Phase added on reflection at a beam splitter? – Emilio Pisanty Aug 08 '19 at 14:53
  • Hmm, ok I read the links, but still have no real sense of how to construct my Hamiltonian. Any ideas? – Tom Aug 08 '19 at 15:10
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    Hint: start with the classical optics, build a hamiltonian formulation in terms of modes, and then quantize. There's nothing about your question that cannot be figured out first in the classical-optics arena and then brought back to the QM side, and then you can just do old-school classical optics instead of wondering about non-hermitian hamiltonians. – Emilio Pisanty Aug 08 '19 at 15:15
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    @Emilio Pisanty I know that you deal with this stuff, so I am curious to understand how a beam-splitter Hamiltonian may work. Up to now, I only read descriptions of the BS device in terms of unitary matrices accounting for transition processes. A Hamiltonian $H$ would instead imply a true temporal process: is there a sort of time evolution? (Yes obviously!) If Yes, how is the evolutor operator $e^{-itH}$ related with the standard BS description? – Valter Moretti Aug 09 '19 at 09:43
  • (I might add that I am collaborating with a group of experimental physicists dealing with integrated optics and we have just submitted a joint experimental paper about violation of CHSH inequality by photons with intraparticle entanglement... so my question is not matter of pure curiosity) – Valter Moretti Aug 09 '19 at 09:46
  • @ValterMoretti That's a pretty subtle area and well worth a dedicated thread. As far as "is there a sort of time evolution? (Yes obviously!)", don't discount the subtleties, particularly because to get to this two-mode Hilbert space you need to assume a narrowband quasi-monochromatic limit, which means that the spatial extent of each laser pulse is at least $\Delta x = c/\delta\omega$, for $\delta\omega$ the bandwidth of the mode. Depending on the configuration, $\Delta x$ might be many kilometers, so the time dependence is a tricky thing to deal with as the pulse passes the beam splitter. – Emilio Pisanty Aug 09 '19 at 15:45
  • I see, thank you very much. – Valter Moretti Aug 11 '19 at 08:33
  • Is my question textbook material where I can find this specific example explained somewhere? Or it's more complicated than that? – Tom Aug 12 '19 at 10:21

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