Hamiltonian for an Optical Phase Shifter?

[Twigg]

Hey all,

I was reading Efficient Linear Optics Quantum Computation by Knill, Laflamme, and Milburn, when I came across their expression for the Hamiltonian for a phase shifter, given as $\textbf{n}^{(\ell)} = \textbf{a}^{(\ell)\dagger} \textbf{a}^{(\ell)}$, where $\ell$ indicates the mode. (They ignore polarization and work in the formalism of spinless bosons.) How does one get this?
I was able to get the beam splitter Hamiltonian by looking at the evolution operator (the unitary operator that takes an input photon in the x direction and spits out a superpostion of x and y photons) and subtracting out identity. But this gives a scalar Hamiltonian of $e^{i\phi} - 1$ when I think about following the same procedure for the phase shifter. Am I looking at this the wrong way? I'm completely new to quantum optics. I'm familiar with the quantization of the EM field, but that's about it.

Thanks!

 

[vanhees71]

Maybe this already helps:

https://en.wikipedia.org/wiki/Optical_phase_space

For a clearer definition of the coherent states $|\alpha \rangle$, see

https://en.wikipedia.org/wiki/Coherent_states