I've been trying to make a simple puzzle game based on quantum optical circuits. Problem is, I don't actually know the physics that well (this is a hobby project with the goal of increasing that understanding). In particular, I'm not sure how to deal with cycles or a photon arriving at a detector at different times.
For example, start with a Sagnac interferometer and flip the half-silvered mirror.
Presumably this creates a situation where the photon 'decays' out of the loop exponentially. I think the state of the system evolves like this:
[note: s = 1/sqrt(2), i = sqrt(-1)] 1 |PhotonLeavingSource> is |PhotonHeadingToDetector> + s |PhotonHeadingToBottomRight> is |PhotonDetected> + is |PhotonHeadingToTopRight> is |PhotonDetected> + -s |PhotonHeadingToTopLeft> is |PhotonDetected> + -is |PhotonHeadingToBottomLeft> is |PhotonDetected> + -i/2 |PhotonHeadingToDetector> + 1/2 |PhotonHeadingToBottomRight> i(s-1/2) |PhotonDetected> + 1/2 |PhotonHeadingToBottomRight> ... i(s-s^2-s^3-s^4-...) |PhotonDetected> = -i |PhotonDetected>
I'm pretty sure I've made a mistake in there. The final state has an amplitude of 1, but some of the intermediate state don't (they aren't unitary). Am I supposed to be normalizing as I go? Should the detected states be parametrized by arrival time and thereby not interfere with each other? Does that happen conditionally based upon the type of detector?
Any insights would be appreciated.