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I was reading Horne, Shimony, Zeilinger "Two-Particle Interferometry" from 1989, an article showcasing how momentum-superpositioned particle pairs from a down-conversion crystal can lead to interference effects between detector pairs.

However I got hung up on a statement they make in the article which is not really used in their results. See the figure:

enter image description here

The beam-paths from the source S are A, B, C and D, and the superposition is AC+DB. So in both states we have a particle going to the beamsplitter H1 and one going to H2, so we always get a click in both the U1/L1 detector pair and the U2/L2 detector pair.

By varying the phase-shifters, various interference effects between the coincident detections at U1/U2, U1/L2, L1/U2, L1/L2 can be observed.

However, the article also states that the count rate of each of the 4 detectors is always going to be constant regardless of the phase-shifters:

enter image description here

I don't get this, if you look at only the H1 section for example, it looks just like the detector-end of a normal Michelson interferometer, in which you could adjust the phases of the incoming superposed photons to adjust the ratio of detections at U1 and L1 (for example you could completely cancel the output to one of the detectors by full destructive interference).

The addition of the down-conversion partner that goes into BC (and ends up at the H2 beamsplitter) shouldn't change this property at H1, or am I missing something completely here? :)

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So I think I figured this out - the interference at O1 (O2) is absent because the superpositioned photons A+D (B+C) as emitted from the source in this case are not phase-coherent (as would have been the case in my counterexample regarding a typical interferometer setup where a single photon is split by a beamsplitter, in which case the two legs are phase-coherent).

I didn't know this about SPDC crystals, I guess I assumed their outputs were coherent but thinking about it now it makes sense that they aren't, as the different output photon pairs could have originated from random points inside the crystal.

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