What is the interference pattern with two sequential double slit screens and one detector? An experimental setup with a particle emitter, a double slit screen (A) with a detector at one slit to monitor particle transit, a second double slit screen (B) without any detectors and a capture screen (C) behind in.
Based on experimentation with the emitter, screen A and screen C only; the pattern will be two vertical lines with some spread.
Experiment 1:
Will inserting screen B between A and C create the same pattern on screen C or will there be a multiple line interference pattern?
If this experiment is repeated (experiment 2) switching screens B and A so that the detector is at slit one on the second screen, what is the interference pattern?
As the particle distribution and pattern arriving at the second set of slits is predictable from other experiments, the outcome of experiment 1 should inform if the probability wave is reinflated after travelling through screen B or if it remains collapsed and in experiment 2, if the second level observation collapses the probability curve and removes interference.
I cannot find such an experiment and if you could point me at one that would be great.
 A: If I understand it correctly, what you describe is the creation of a mixed state. Mixed in that the screen A with its detector splits the particle beam into two beams that have classical probabilities $\rho(A_1)$ and $\rho(A_2)$, which is the probability for a particle to have passed through the first slit on A or the second, respectively. These probabilities can no longer interfere, because the detector on screen A created mutually exclusive alternatives for a particle trajectory. All that remains is then to do two more calculations:

*

*The intereference pattern of a particle emitted at slit $A_1$, passing through the holes on screen B to the detector at C, let's call its intensity $I_1(x)$, where $x$ is the coordinate on the screen C.

*The interference pattern of a particle emitted at slit $A_2$, passing through the holes on screen B to the detector at C, let's call its intensity $I_2(x)$.

The resulting intensity should be close to $I(x) \propto \rho(A_1) I_1(x) + \rho(A_2) I_2(x)$, the proportionality constant arising from the original intensity of the beam.
Why close to and not exactly that? Because the slits $A_1$ and $A_2$ have a finite width, so we can't say that the particles start from a point source after the measurement on A. This is the usual idealization in the double slit experiment. So there will be some more corrections to the above expression. Yet I think they will not change the qualitative picture, at least for slit widths being of the order of magnitude of the de Broglie length of the particle. Correct me if I'm mistaken there.
