Which-Path from Global Momentum Measurement in Double Slit

[wnvl2]

Consider the standard double-slit experiment with photons (or massive particles), where a screen containing two narrow slits is illuminated by a coherent source and an interference pattern is observed on a distant detection screen. Assume that the slit screen is not rigidly fixed but is instead allowed to recoil, so that momentum exchange between the particle and the screen is, in principle, possible.

Suppose that one performs a measurement of the total momentum of the entire slit screen at the moment a particle passes through it, without resolving any spatial information about the individual slits and without directly measuring the particle’s position. The question is whether such a global momentum measurement provides which-path information and, consequently, whether it affects the visibility of the interference pattern.

 

[QuarkyMeson]

This is basically what you're asking:

https://arxiv.org/pdf/cond-mat/0101157

A global momentum measurement of the screen only yields which-path information if the recoil states are distinguishable. When they are, interference visibility drops accordingly. This occurs because the uncertainty principle requires the slit positions to become indefinite, smearing out the fringes on the detection screen.

 

[wnvl2]

In the paper, the experimental setup is asymmetric in the sense that the which–path measurement effectively takes place in only one arm of the interferometer. My question is whether, in a more symmetric configuration—where the momentum transferred by the electron is absorbed by the detector or screen as a whole—the interference pattern would still disappear. In particular, if the entire screen receives a net momentum kick (for example to the left) and this momentum transfer is measured, can one then infer whether the particle passed through the left or the right slit, and does this necessarily imply the loss of interference?

 

[QuarkyMeson]

Like before, a global measurement of the screen’s total momentum gives which-path information only if the two recoil states are distinguishable. In a symmetric two-slit experiment where the screen merely receives a kick to the left, that kick is tied to the particle’s outgoing transverse momentum, not which-path. So measuring it does not let you infer the path and does not by itself eliminate interference.

That said, you could definitely try to engineer a symmetric two-slit experiment that did encode which-path information. There's a ton of papers out there with different ways to mark which-path, here is another paper I think that's closer to what you're looking for then: https://arxiv.org/pdf/2507.19801.