Mach–Zehnder interferometer question, path detection

[msumm21]

In a Mach–Zehnder interferometer experiment, can you measure which path the beam took by measuring the change in momentum of the (fully silvered) mirrors (i.e. if path A was taken then the mirror on path A was "pushed" a bit, if path B was taken then the mirror on path B was "pushed" a bit)?

 

[San K]

In a Mach–Zehnder interferometer experiment, can you measure which path the beam took by measuring the change in momentum of the (fully silvered) mirrors (i.e. if path A was taken then the mirror on path A was "pushed" a bit, if path B was taken then the mirror on path B was "pushed" a bit)?

i don't know if the path can be found out via push on the mirrors...

however

there are a couple of other ways to find out which path (in mach-zehnder, double-slit, delayed erasure etc) the photon/beam took.

the interference does not happen when the path is "found out"/known

 

[msumm21]

The reason I ask is because every time the experiment is done it would seem like the which-path information would be encoded in the momentum of the mirrors, and theoretically you could always go back and look at the momentum of the mirrors to see which path was taken, but then doesn't QM say there is no interference if that is the case?

Somehow the beam always goes into the same detector (there is interference) so evidentially the momentum of the mirrors is hidden somehow? I think I am missing something basic here.

 

[Cthugha]

In order to really measure the change in momentum induced by the photon, the initial momentum uncertainty of the mirror should be small compared to the momentum of the photon. Otherwise the momentum transfer is not detectable in principle. In any typical realization of a Mach-Zehnder interferometer the mirror momentum uncertainty is too large to allow which-way measurements utiliziung the momentum transfer by the photon.

 

[msumm21]

Theoretically (maybe this is not possible in practice) if you very precisely knew the momentum of the mirrors before the experiment, would the interference then not occur? So if you know the momentum of the mirrors well then the beam could pass to either detector?

Thanks for the info Cthugha.

 

[Matterwave]

Right, if you prepared the mirrors in a VERY precise momentum state (and therefore a very unprecise position state via the Uncertainty principle), you would not see the interference, because the position of the mirrors would be out of wack!

 

[Demystifier]

In a Mach–Zehnder interferometer experiment, can you measure which path the beam took by measuring the change in momentum of the (fully silvered) mirrors (i.e. if path A was taken then the mirror on path A was "pushed" a bit, if path B was taken then the mirror on path B was "pushed" a bit)?

If you are able to detect a change in the mirror, then the quantum state of the mirror after the income of the photon must be significantly different from that before the income of the photon. In that case, these two states of the mirror are nearly orthogonal, i.e., their scalar product is close to zero. But that means that the mirror causes decoherence of the photon state, which means that interference is destroyed.

Conversely, if you can see interference, then it means that there is no decoherence, and consequently that you are not able to detect a change in the mirror.

This is discussed more quantitatively in the book
B. Schumacher, B. D. Westmoreland: Quantum Processes, Systems, and Information (2010)

 

[lugita15]

Demystifier, have the "weak measurements" you always talk about been done for the Mach–Zehnder interferometer experiment?