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!