Path-choice experiment in the framework of the MWI

[timmdeeg]

In a Mach-Zehnder interferometer the light arrives independent on the path at one the two detectors because of complete constructive resp. destructive interference at the final beam splitter. Is it possible to modify the device such that dependent on the path either detector A or Detector B will record a photon but with unequal probability, say Detektor A with a probability of 0.1 and B of 0.9?

If yes, how would one write down the superposition of the two paths? I dont't trust the naive

0.1(path 1 + Detektor A) + 0.9(path 2 + Detektor B).

How should one think of the representation of the two possible states - photon arrives at A and photon arrives at B - and their probabilities in the Hilbert space?

And sorry, naively again, is the outcome that there are x A worlds and 9x B worlds? And is x limited?

 

[mfb]

You can adjust the phase to make that possible. You want a phase shift close to pi for the low probability path and correspondingly a phase shift close to 0 for the other.

And sorry, naively again, is the outcome that there are x A worlds and 9x B worlds? And is x limited?

In the many worlds interpretation, after measuring, there are two worlds with different amplitudes. In other interpretation there are no different worlds. Before measuring you still have a coherent state and all interpretations agree that you just have a different amplitude.

 

[DarMM]

In Modern takes in Many Worlds you have an infinite number of worlds. Within the subset where your measurement took place 90% have the B detector activated and 10% the A detector.

 

[timmdeeg]

Thank you both for answering.

In Modern takes in Many Worlds you have an infinite number of worlds. Within the subset where your measurement took place 90% have the B detector activated and 10% the A detector.

Ok. So this subset represents a finite number of worlds and also the expected probability regarding the activated detector.
What determines the number of worlds in the subset if one measurement is performed?

 

[DarMM]

Thank you both for answering.Ok. So this subset represents a finite number of worlds and also the expected probability regarding the activated detector.
What determines the number of worlds in the subset if one measurement is performed?

The subset is also infinite in size, it's just the subset where your measurement took place, i.e. several branching points in the macroworld's history do not lead to your measurement occurring. The number of world where you did perform your measurement is still infinite (technically uncountably infinite), but it's not the totality of worlds.

 

[timmdeeg]

The subset is also infinite in size,

But then the photon is measured in an infinite number of worlds in Detetcor A and in an infinite number of worlds in Detector B as well. So, the probability 9:1 seems lost, if I see that correctly.

 

[DarMM]

But then the photon is measured in an infinite number of worlds in Detetcor A and in an infinite number of worlds in Detector B as well. So, the probability 9:1 seems lost, if I see that correctly.

The 90% comes from the measure of worlds or world volume. If it helps imagine the worlds being indexed by a real valued parameter $a$. Worlds with $a \in [0,0.9]$ are those where the B detector clicked.

 

[timmdeeg]

The 90% comes from the measure of worlds or world volume. If it helps imagine the worlds being indexed by a real valued parameter $a$. Worlds with $a \in [0,0.9]$ are those where the B detector clicked.

Thanks, that clarifies my question.