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Consider a simple setup with two independent but coherent light beams of identical intensity that converge on a 50:50 half mirror. One beam has a phase shift to the other such that they interferer and only one beam leaves the mirror in direction C while destructive interference causes no light to leave in direction D.

This setup is simply meant to compare two beams for coherence and ensure that their intensities are equal, since otherwise a detector positioned at D will measure something. In the following however I will assume perfect coherence, so only the sensitivity to intensity deviations remains a factor.

Now what happens when the both beam intensities are tuned down until there are only individual photons? Does the behavior at some point fundamentally change from the classical case – apart from the obvious that detectors will only measure individual photons instead of a continuous beam? [EDIT: of course the two incoming photons need to be assumed to be perfectly synced and arrive at the same time at the mirror] This is a case of two photon interference but unlike the setup of the HOM-effect there is phase shift in the incoming beams. If my quick mental calculation applying a displacement operator before the mirror matrix to the initial state is correct, there should still be no photons detected at D while C should measure both each time.

Now the thing I am interested in, is whether this setup still remains sensitive to the exact intensities, for example what happens if the initial beams are reduced even further e.g. by using additional beam splitters with adjustable splitting ratios ahead of the interference mirror? Would it be still possible from the detection frequencies at C and D to calculate the intensity ratios? Given that the calculus is completely linear I would assume that an interference of two half-photons should behave just the same as in the full case.

To give some background, I am trying to understand one of the central axioms of quantum mechanics that gives me the greatest headaches. And unlike the axioms defining the time evolution of a quantum system I know no experiments that would motivate it nor have I found any other convincing explanation so far. It usually just falls down from heaven without a comment. So i am trying to finding my own explanation by understanding how the above setup behaves.

This setup is simply meant to compare two beams for coherence and ensure that their intensities are equal, since otherwise a detector positioned at D will measure something. In the following however I will assume perfect coherence, so only the sensitivity to intensity deviations remains a factor.

Now what happens when the both beam intensities are tuned down until there are only individual photons? Does the behavior at some point fundamentally change from the classical case – apart from the obvious that detectors will only measure individual photons instead of a continuous beam? [EDIT: of course the two incoming photons need to be assumed to be perfectly synced and arrive at the same time at the mirror] This is a case of two photon interference but unlike the setup of the HOM-effect there is phase shift in the incoming beams. If my quick mental calculation applying a displacement operator before the mirror matrix to the initial state is correct, there should still be no photons detected at D while C should measure both each time.

Now the thing I am interested in, is whether this setup still remains sensitive to the exact intensities, for example what happens if the initial beams are reduced even further e.g. by using additional beam splitters with adjustable splitting ratios ahead of the interference mirror? Would it be still possible from the detection frequencies at C and D to calculate the intensity ratios? Given that the calculus is completely linear I would assume that an interference of two half-photons should behave just the same as in the full case.

To give some background, I am trying to understand one of the central axioms of quantum mechanics that gives me the greatest headaches. And unlike the axioms defining the time evolution of a quantum system I know no experiments that would motivate it nor have I found any other convincing explanation so far. It usually just falls down from heaven without a comment. So i am trying to finding my own explanation by understanding how the above setup behaves.

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