When does the distribution change in interference/which way measurements compared to the time of the type change of the measurement? Does anyone know of any experiments that have looked at the temporal relationships of this?
Measurements that show the dual nature of the particle/wave are well known. The fundamental feature of which is that the output distribution as a result is measurably and characteristically different. There are several possibilities for particles/photons and if a measurement cannot determine which one is realised, an interference distribution is obtained. However, if we change the type of measurement and it is possible to determine which possibility is followed (which way measurement), the interference disappears. The result is a distribution that is consistent with the nature of the particle (e.g. double slit experiments, interferometric experiments and the other similar experiments).
The natural and important question arises in this way: relative to the event when the which way detector is switched on (i.e. the character of the measurement changes), when exactly is the first change in the output distribution observed? The issue is interesting and can be investigated if the which way detector that can be switched on is located spatially ahead of the screen or detector that measures the output.
Unfortunately, I am not aware of any experiments or studies that answer this question, which is why I am asking on this forum.
Essentially 3 cases are possible in my opinion:
A). Only those particles/photons that have not yet passed through the equipment at the point of the switch-on event will give the different distribution. Those that have already passed the ww-detector site at the time of the switch-on event are still involved in the formation of the interference pattern.
B) Any particles/photons that reach the final screen or detector within the light cone of the switch-on event will already be producing the altered distribution. But those outside the light cone still form the initial interference distribution. So we imagine an unknown effect propagating at speed c, starting from the location of the ww detector when it is switched on - and the first change in the output distribution is when enough time has elapsed for this effect to reach the output screen or detector.
C) No time delay or other time dependence is observed in the change of the distributions. So immediately after we turn on the ww detector, it changes immediately; the result is a transformation from interference to particle distribution.
The laws and known calculations of quantum mechanics are in favour of case C) as far as I know. However strange or fantastic this possibility may seem. After all, the calculations describing the measurement process do not include time as a dependent variable. Furthermore, the states of the which way detectors and the measured particles/photons are entangled; and at the end of the measurement, these superposed, entangled states are reduced to a specific eigenstate. And the reduction of entangled states is an instantaneous, time-less process, both theoretically and empirically proven.
But of course we are correct in our thinking if we maintain the possibility of cases A) and B) being realised by some as yet unknown factors. We can't dictate to Nature in advance how it should behave - actual experiments must decide which one will be realised (or perhaps some strange fourth possibility that cannot be formulated in terms of what we have seen so far).
Versions A and B are different. If they are particles, it is because they are moving in the apparatus at a speed less than c, so the light cone boundary will include some of them that have already passed the ww detector at the time of switch-on. And when it comes to photons, it is because the geometry of the experimental setups is usually such that the photons under test travel longer distances than the shortest distance (for example, in MZI they travel along the sides of a rectangle).
It is important that such an experiment is not done in a single way, but with a strong source. So many photons/particles entering the apparatus at the same time, which will themselves form the appropriate distributions in a continuous manner. Do not wait for them to enter the measurement and arrive at the end one after the other (as the question itself then becomes pointless). Also, the which way detector, whose switch-on causes a change in the type of measurement, should be spatially located at the front of the equipment. It should be located in front of the final screen or detector. Because if it is located after them (as in Wheeler's original delayed choice experiments), it is also not suitable for the time delay versus instantaneous change issue.
I will describe the specific situation for the well-known, relatively simple Mach-Zehnder interferometer setup to which the question applies:
Consider the MZI interferometer with two beam splitters. A ww detector can be inserted rapidly into the first half of the lower arm, which then blocks and absorbs the photons passing through it. The source is sufficiently strong so that a sufficient number of photons can be continuously and simultaneously received by the two output detectors.
Situation 1: The ww detector is outside the arm. Thus, due to phase shifts and constructive interference, all arrive in detector D1, and none arrive in D2 due to destructive interference, i.e.  it does not signal.
Situation 2: quickly insert the operating ww detector. Then the which way measurement is performed and the ones in the lower arm are absorbed. The photons of the upper arm split in two at the second BS and about 25% of the original intensity is already arriving at the D2 detector, so it starts to signal. The intensity entering in D1 is also reduced to about 25%.
So inserting a ww detector causes substantial and well measurable differences for the D1 and D2 detectors.
Our question is how long after the ww detector is inserted do these differences appear? The general versions A), B), C) formulated above need to be considered specifically for this. The figure shows the point. For briefness, I will highlight only one point:  
If we insert the ww detector at time T0, the boundary of the light cone of this event is a spherical surface of distance c*∆t away from the detector location in 3-dimensional space. Should we wait for this to reach the distance between the ww detector and the location of the detectors D2, D1? Or does it already happen before then that D2 is triggered and the intensity of the measured signal in D1 is reduced?
The delayed choice measurements that I know of do not answer this, because in those the change in measurement type is not achieved by an operation in the MZI arms, but already outside, locally after the second BS.
But perhaps there have been experiments that have investigated the question I have raised; or, if not intended to do so, have "incidentally" yielded the relevant data.
I am of course aware that if version C) is confirmed, it opens the door to the possibility of FTL communication. But I don't intend to discuss this now; it should first be found out how it happens. I also do not want to get into the complicated topic of quantum erasers and retrocausality. For the moment, I am looking for answers to the relatively simple arrangement and question above. Of course, it is easy to formulate the same for the case of double slit experiments. If anyone can show a meaningful measurement not for MZI but for double slit that deals with the question of time, I would also welcome a response.
 A: I cannot offer any reports of the experiment you are asking for, but I can motivate why such an experiment might be considered moot in view of the currently available delayed choice quantum eraser experiments.
First of all, QM predicts option A. Your option C would require the WW detector to be able to affect photons which have already passed that point in space. There is no reason one would expect this to happen. The QM prediction would follow an argument of the following form: before insertion of the WW detector, the state arriving at beam splitter 2 is something like $|\psi_1\rangle = (|1\rangle +|2\rangle)/\sqrt{2} $ where 1 and 2 indicate the different paths. At $t=t_0 + L/c$, where $L$ is the length from the WW detector to beamsplitter 2 and $t_0$ is the time of insertion of the WW detector, this will change to the state $|\psi_2\rangle = |x\rangle $, where $x$ is 1 or 2 with equal probability, and no interference will happen.
In the context of quantum eraser experiments, what they find is that for a given entangled photon pair, if one photon reveals its which-way information, the other photon will contribute to the particle pattern on the other detector, i.e. will not interfere with itself. This does not necessarily require communication between the photons, it is entirely predicted and explained by the entangled quantum state.
How can we use this information to answer your question? Given the results of other quantum eraser experiments, those photons which reveal their which-way information upon insertion of the detector will be correlated with zero interference at beam splitter two. We would thus expect that the interference will stop at a time $t=t_0 + L/c$.
