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Mach-Zehnder interferometer

The Mach-Zehnder interferometer (FIG. 1) is constituted by two arms. It is said to be (un)balanced if the two arms have (un)equal lengths. The difference in lengths generates different interference effect (for example, for some differences in lengths, photons should be detected by only one of the detectors). However, if the difference in length is greater than the coherence length, interference disappears, and the detection becomes independent of the lengths. Where does this decoherence come from?

But the question does not end here... Consider now the Franson interferometer:

Franson Interferometer

It is constituted by two unbalanced Mach-Zehnder interferometers, and emits one photon to each of them. The arms of each unbalanced interferometers have difference in length greater than the coherence length so that no interference is expected. Indeed, if one counts all photons detected, the same amount of photons will be detected in each detector. However, if one counts only the photons arriving at the same time in the detectors, it is impossible to know by which the two photons traveled (both by the longer or the smaller arms). Thus, the photons are entangled. However, this effect is independent of the difference between the lengths of the arms of each interferometer. Why decoherence do not apply here?

But the interference effects can again disappear if the difference between the differences of the interferometers arms are greater than the coherence length. Why now decoherence does apply?

What are the microscopic, of quantum details of the mechanism of (de)coherence in this experiment?

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The types of interferometers are associated with two different types of interference. In the Mach-Zehnder interferometer, the interference is just pure classical interference. The decoherence in this case is caused by the variation in relative phase for the different frequencies within the bandwidth. If you want to think of this in terms of photons, then you can express the state of the photon as a superposition of all the different frequencies.

The Franson interferometer is associated with quantum interference. What it means is that it involves multiple particles (photons). In effect, you are observing the interference among terms in a superposition where each term consists of multiple photons. Obviously, the situation now becomes much more complex than for the classical interference case. So, one would need to perform a careful analysis to see how this interference works in terms of all the time scales in the system.

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