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?(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# I Amplitude moving through detectors

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