I Heuristic for simple two-photon scattering problems

[Swamp Thing]

I'm afraid this question is going to be a bit hand-wavy, but I'm hoping for a way of thinking about two-photon scattering that would be helpful to a beginner but also give correct results.

The starting point is that, for two photons in SPDC and other similar cases, the phase difference is a uniform superpositiion from 0 to $2\pi$.

Next, we consider scattering through a system that has 2 inputs and N outputs. This could be as simple as a beam splitter or something more complex, but let's say we know the amplitudes and phases to go from any input to any output.

Now we excite the two inputs with two coherent waves and vary their phase difference $\Delta\phi$ from 0 to $2\pi$. For each phase difference value, we can find the power going to each output port. If we integrate that over $\Delta\phi$ from 0 to $2\pi$ we can get the expectation of the power, which is the expectation of the photon count at that port up to a constant.

My question is, isn't there a similar logic that says if we integrate "something" over 0 to $2\pi$ we can get the probability of seeing two photons at a certain port, and if we integrate something else, we can get the probability of seeing one photon "here" and another "there".

Hope that makes sense...

And if that is possible, then how about three photons into three inputs -- can we double-integrate something over the phase difference between input A and B, as well as input B and C, to get the probability of seeing 2 photons over here and one over there?

 

[Swamp Thing]

Now we excite the two inputs with two coherent waves and vary their phase difference Δϕ from 0 to 2π. For each phase difference value, we can find the power going to each output port. If we integrate that over Δϕ from 0 to 2π we can get the expectation of the power, which is the expectation of the photon count at that port up to a constant.

In the last sentence in the quote, I mean that the same expectation of photon count, that we work out using coherent states as a test case, would also apply to non-coherent photons (Fock state) sent into the inputs of the same scattering problem, because non-coherent pairs have a Δϕ that is a superposition over 0 to 2π. And in a similar way, I'd like to derive probabilities for other classes of events as well.