I have a question (my very first here) related to 50/50 beam splitters as used in the Mach-Zehnder interferometers (see for example the Wikipedia page).
Let's concentrate on the input beam splitter: A continuous light beam (the input) is split 50/50, one in the 90° direction and one in the forward direction of the input beam. Momentum/energy conservation teaches us that deflected part of the beam must exert a (tiny) force on the beam splitter. The forward part of the beam will not.
In the single photon (quantum) case, the beam-splitter should get a slight "kick" if the photon is deflected, while the beam splitter is left in its original state if the photon goes straight through.
If I got this right so far, we can now increase the energy (and momentum) of the photon and decrease the mass of the beam splitter (now coupled to a sensitive piezo transducer or something) so that a photon "kick" can be recorded if deflected. Now "which way" information is available and interference should be destroyed as I understand this.
Would interference be restored if the transducer is disconnected? If yes, how do the photon "know" whether the transducer is connected or not? Maybe this is related to how much the photon/beam splitter system can become entangled to the environment?
If the answer is "No", how weak must the photon kick be in order for the interferometer to work? We already know that it works for ordinary light (very tiny photon kicks).
In other words: Why doesn't the momentum exchange (or lack thereof) between the photon and the beam-splitter (and the trace this leaves in the environment) destroy the interference?
Alternatively, couls someone just point me to relevant literature which discusses the transfer of momentum between between photons and optical elements?