Amplitude moving through detectors

[Swamp Thing]

We have two thin parallel detectors, and a $|1\rangle$ photon state passes through them. Each detector has a 10% chance of catching the photon. How can I write the final state?

I'm thinking something like $\sqrt{0.1}|1_A\rangle |0_B\rangle |0\rangle+\sqrt{0.09}e^{-j\theta_1}|0_A\rangle |1_B\rangle |0\rangle + \sqrt{0.81}e^{-j\theta_2}|0_A\rangle |0_B\rangle a^{\dagger}a$

Is this correct? (If not, what is the correct way?)

About the phases $\theta_1$ and $\theta_2$ , would they be the same as if the medium was lossless? To keep it simple, let's say the speed of light in the lossy medium is the same as vacuum.

But even if it is technically OK, it means that I have set the coefficients by hand, based on what I expect to see. Can it be written so that one puts in only 0.1 and the rest comes out from "shut up and calculate"? I mean, this is a toy problem, but how do we "automate" more complex stuff?

 

[PeterDonis]

How can I write the final state?

You can't, because there isn't a single "final state". There are three possible "final states": detector #1 detects a photon, detector #2 detects a photon, or neither detector detects a photon. And each one only happens if the ones before it do not (e.g., if detector #1 detects a photon, there is no possibility of detector #2 detecting one). There is no way to write down a single "state" (i.e., wave function/linear combination of kets) that covers all of these possibilities (unless you want to include the detectors as quantum objects and add their states and their entanglement with the photon).

 

[Swamp Thing]

Thank you, I found that really useful.