Within the standard model, photons are point particles, i.e., with no spatial distribution.
On the other hand, classical electromagnetic modes have field distributions.
Suppose there are two different single mode nanophotonic waveguides. The two waveguides are gradually bent and become adjacent to each other to form a coupler.
Two classical electromagnetic waves propagating along the two waveguides can couple to each other. For two photons, can they show the two photon interference effect in this structure?
If they can, how to calculate the probability that exactly one photon is detected at each exit, taking into consideration that the two waveguide eigenmodes have different field distributions?
I notice that, in the second quantization formalism, the creation operator is $$\hat{A}^{\dagger}(\alpha)=\int_{0}^{\infty} \mathrm{d} \omega \alpha(\omega) \hat{a}^{\dagger}(\omega),$$ where $\alpha(\omega)$ is the frequency distribution function of the excitation. Does this mean that the classical electromagnetic field distribution doesn't matter for quantum optics and the only important property is the frequency?
Update: In this paper, the probability that exactly one photon is detected at each exit is calculated as $$P\left(1_{\mathrm{a}}, 1_{\mathrm{b}}\right)=|t|^{4}+|r|^{4}+\left[t^{2} r^{* 2}+r^{2} t^{* 2}\right] I,$$ where $t$ and $r$ are the complex transmission and reflection coefficients. These coefficients can be calculated using the coupled mode theory which certainly takes the (classical) electromagnetic field distribution into consideration. This looks like a feasible formalism.
However, this calculation should be equivalent to some second-quantization formalism that involves the spatial field distribution, because this calculates the two photon interference possibility which should be a purely quantum mechanical quantity.
I wonder if there has been any literature that thoroughly established something like "waveguide quantum optics" in the language of second quantization.
