Experiment distinguishes Standard QM vs Bohm: Asymmetric beam splitter

[Ben-tanglement]

TL;DR Summary

The Bohm interpretation of a Mach–Zehnder interferometer in "particle mode" (1 beam splitter) swaps paths taken to D1 & D2, relative to the standard QM interpretation. An asymmetric beam splitter predicts different results for the 2 interpretations

Background
I was reading a paper, Delayed Choice Experiments and the Bohm Approach by Basil Hiley and Robert Callaghan. The Wheeler's Delayed Choice experiment was explained in a way that was very easy to understand. An interesting point in this paper is that when a Mach–Zehnder interferometer is in "particle mode" with 1 beam splitter, the Bohm interpretation says that the paths of the photons are swapped compared to the SQM (Standard Quantum Mechanics) interpretation.

See Figure 2 in the linked paper. The particle, a photon in this case, enters the apparatus from the lower left, and hits the only beam splitter, BS1, and is either reflected towards mirror M1 along Channel 1, or transmitted towards mirror M2 along Channel 2.

Hiley & Callaghan, section 3.1, Interferometer with BS2 removed:

Let us begin by first quickly recalling the SQM treatment of the delayed choice experiment. When BS2 is removed...If BS1 is a 50/50 beam splitter, then each particle entering the interferometer will have a 50% chance of firing one of the detectors. This means that the device acts as a particle detector, because the particle will either take path 1, BS1M1D1, trigging the detector D1. Or it will travel down path 2, BS1M2D2, triggering detector D2.

The above description of the paths is the same as described in the Wikipedia on Wheeler's Delayed Choice Experiment. See the figure in the "Simple interferometer" section in the "open" position with one beam splitter. Using the terminology in Figure 2 of Hiley & Callaghan: If the photon is detected in D1 then the photon is said to have gone down channel 1 with mirror M1. If the photon is detected in D2 then the photon is said to have gone down channel 2 with mirror M2.

Now let us turn to consider how the BI [Bohm interpretation] analyses this experiment. Here we must construct an ensemble of trajectories, each individual trajectory corresponding to the possible initial values of position of the particle within the incident wave packet. One set of trajectories will follow the upper arm of the apparatus, while the others follow the lower arm.

In the Bohm interpretation, the photon goes down either Channel 1 or Channel 2 (no superpositions), but the quantum potential goes equally down both channels. The region I2 (Figure 2), is of particular interest to this analysis. The quantum potential (pilot wave ripples?) traveling down both channels will interfere with each other. See Figure 3 for the Bohm trajectories within region I2, and Figure 5 for the overall Bohm trajectories.

Here the wave packets from each channel overlap and there will be a region of interference because the two wave packets are coherent...The particles following the trajectories then ‘bounce off’ this potential as shown in figure 3 so that the particles in channel 1 end up triggering D2, while the trajectories in channel 2 end up triggering D1.

The bold in the paragraph above is my emphasis. The conclusion is that the paths taken by photons are swapped in the Bohm interpretation compared to Standard Quantum Mechanics.

Experiment
Could there be any way to alter the Mach–Zehnder interferometer to distinguish between the two interpretations? Perhaps if the one beam splitter was non-symmetrical, let's say reflecting 52% of the time and transmitting 48% of the time (52-48) rather than 50-50, differences may emerge for the results predicted by the two interpretations.

Predictions are illustrated here with a Mach–Zehnder interferometer in "particle mode" with a non-symmetrical beam splitter:

This is a modified version of Figure 2 from the paper.
Blue highlights my modification to the interferometer.
Green highlights the predictions of the Standard Quantum Mechanics interpretation.
Purple highlights the predictions of the Bohm interpretation.

In the Standard Quantum Mechanics interpretation, the 52% of photons reflected at BS1 should reflect off M1 and arrive at detector D1. The 48% of photons transmitted at BS1 should reflect off M2 and arrive at detector D2.

In the Bohm interpretation, the 52% of photons reflected at BS1, traveling with a stronger quantum potential, should reflect off M1, enter region I2, 'bounce off' the weaker quantum potential arriving from the lower path, then head towards D2 at an angle bent slightly towards D1. The 48% of photons transmitted at BS1, traveling with a weaker quantum potential, should reflect off M2, enter region I2, 'bounce off' the stronger quantum potential arriving from the upper path, then head towards D1 at an angle bent slightly towards M2.

If one could gradually increase the reflectivity of BS1 from 50% to 100%, the number of photons in Channel 1 would gradually increase from 50% to 100%, and would exit region I2 at an angle that initially points at D2 but gradually shifts towards pointing at D1. The number of photons in Channel 2 would gradually decrease from 50% to 0%, and would exit region I2 at an angle that initially points at D1 but gradually shits towards pointing back to M2.

 

[pines-demon]

The bold in the paragraph above is my emphasis. The conclusion is that the paths taken by photons are swapped in the Bohm interpretation compared to Standard Quantum Mechanics.

What is exactly swapped again? In standard QM the wave function takes both paths?

Experiment
Could there be any way to alter the Mach–Zehnder interferometer to distinguish between the two interpretations? Perhaps if the one beam splitter was non-symmetrical, let's say reflecting 52% of the time and transmitting 48% of the time (52-48) rather than 50-50, differences may emerge for the results predicted by the two interpretations.

There are some free cloud quantum computers that allow you to test this. However by experience and playing with beam splitters, the results coincide with standard QM. Somebody would have noticed by now. Maybe what we call an 51-49 beam splitter is swapped and it is actually a 49-51 beam splitter, but how do we tell?

 

[Ben-tanglement]

What is exactly swapped again? In standard QM the wave function takes both paths?

Look at the diagram in the Wikipedia link to the Wheeler's Delayed Choice experiment, scroll down to the section "Simple interferometer" and look at the diagram of the interferometer in the "open" position with one beam splitter. Look at the paths taken to the two detectors. As far as I know, this is the standard way of describing what happens with the configuration with 1 beam splitter, and the same thing is said by the authors of the paper.

Now look at Figure 5 of the Hiley & Callaghan paper. The paths taken are swapped. The point here is that these two descriptions of what is physically happening should be exploitable, and a modified experiment should distinguish between these two descriptions of the reality.

 

[pines-demon]

Look at the diagram in the Wikipedia link to the Wheeler's Delayed Choice experiment, scroll down to the section "Simple interferometer" and look at the diagram of the interferometer in the "open" position with one beam splitter. Look at the paths taken to the two detectors. As far as I know, this is the standard way of describing what happens with the configuration with 1 beam splitter, and the same thing is said by the authors of the paper.

Now look at Figure 5 of the Hiley & Callaghan paper. The paths taken are swapped. The point here is that these two descriptions of what is physically happening should be exploitable, and a modified experiment should distinguish between these two descriptions of the reality.

I think I understand what you mean now. Anyway still look at my second comment. (1) this has been tried many times with different beam splitters, I would not be surprised if beam splitters are calibrated using the "open" inteferometer (2) if there is a swap in our definition of the percentages given by the beam splitter, then we cannot tell (can we?).

 

[Ben-tanglement]

(1) this has been tried many times with different beam splitters, I would not be surprised if beam splitters are calibrated using the "open" inteferometer

I've tried my best to search for any actual experiments, and I can't find any. I think the setup I'm describing is so simple and "boring" that nobody is expecting anything to happen. All experiments with one 50:50 beam splitter would give the same results for both interpretations. I can't find any experiments where someone has a beam splitter that reflects a different amount than it transmits, and placed it in a Mach-Zehnder interferometer. It could be that beam splitters are first tested and calibrated, and anything not 50:50 is discarded as a defect. I don't know a lot about how beam splitters are made and characterized.

(2) if there is a swap in our definition of the percentages given by the beam splitter, then we cannot tell (can we?).

The reason I proposed using an asymmetric beam splitter is that it does allow us to tell the difference. When the beam splitter is not 50-50, one detector or the other will receive more photons. If detector 1 receives more photons, then Copenhagen is correct. If detector 2 receives more photons, the Bohm interpretation is correct.

 

[pines-demon]

I've tried my best to search for any actual experiments, and I can't find any. I think the setup I'm describing is so simple and "boring" that nobody is expecting anything to happen. All experiments with one 50:50 beam splitter would give the same results for both interpretations. I can't find any experiments where someone has a beam splitter that reflects a different amount than it transmits, and placed it in a Mach-Zehnder interferometer. It could be that beam splitters are first tested and calibrated, and anything not 50:50 is discarded as a defect. I don't know a lot about how beam splitters are made and characterized.

Sure but you have to know that beam splitters exists since at least the 19th century far before quantum mechanics. When you realize an experiment you have to test first that your beam splitter is doing what it is supposed to do before adding more components to the experiment. That makes it extremely unlikely that there is an anomalous result that has never been noticed in the last 200 years.

What is more likely is that both interpretations give you the same result. Check point (2) below:

The reason I proposed using an asymmetric beam splitter is that it does allow us to tell the difference. When the beam splitter is not 50-50, one detector or the other will receive more photons. If detector 1 receives more photons, then Copenhagen is correct. If detector 2 receives more photons, the Bohm interpretation is correct.

Here is the inverse experiment. Imagine that you use an unknown percentage beam splitter, and you get 30% of photons in channel 1 and 70% of photons in channel 2, how do you know if it is a 30:70 beam splitter or 70:30 beam splitter? With the two current interpretations you cannot know, we have just chosen the standard one but the results would be the same using Bohm, you would just need to inverse the details of the beam splitter.

 

[Ben-tanglement]

Imagine that you use an unknown percentage beam splitter, and you get 30% of photons in channel 1 and 70% of photons in channel 2, how do you know if it is a 30:70 beam splitter or 70:30 beam splitter? With the two current interpretations you cannot know, we have just chosen the standard one but the results would be the same using Bohm, you would just need to inverse the details of the beam splitter.

This is easy. Instead of mirrors M1 and M2, place the detectors D1 and D2 at those locations and evaluate the beam splitter performance. Assume that D1 replaces M1. The beam splitter that reflects 70% of the time will put 70% of the photons at D1. The beam splitter that transmits 70% of the time will put 70% of the photons at D2.

 

[pines-demon]

This is easy. Instead of mirrors M1 and M2, place the detectors D1 and D2 at those locations and evaluate the beam splitter performance. Assume that D1 replaces M1. The beam splitter that reflects 70% of the time will put 70% of the photons at D1. The beam splitter that transmits 70% of the time will put 70% of the photons at D2.

Fair enough, but then what happens with the Bohmian trajectories in that case?

I will recommend you to find out what the Bohmian trajectories are for (1) the case where the mirrors are replaced by detectors (2) the case where the beam splitter is asymmetric as in your experiment.

Naive prediction: in (2) it allows for some mixing of trajectories to reach the detectors, allowing for the 30-70 ratio to be confirmed.

 

[PeterDonis]

Moderator's note: Thread moved to interpretations subforum.

 

[PeterDonis]

The conclusion is that the paths taken by photons are swapped in the Bohm interpretation compared to Standard Quantum Mechanics.

This doesn't make sense. The Bohmian interpretation, AFAIK, has been proven to be mathematically equivalent to standard QM. That means it must make the same predictions for all experimental results. The conclusion quoted above contradicts this, so I am extremely skeptical of it. I suspect they are making a mistake somewhere in their Bohmian analysis, introducing something that is not mathematically equivalent to standard QM and is therefore not part of the actual Bohmian interpretation.

 

[Demystifier]

In the Standard Quantum Mechanics interpretation, the 52% of photons reflected at BS1 should reflect off M1 and arrive at detector D1. The 48% of photons transmitted at BS1 should reflect off M2 and arrive at detector D2.

In the Bohm interpretation, the 52% of photons reflected at BS1, traveling with a stronger quantum potential, should reflect off M1, enter region I2, 'bounce off' the weaker quantum potential arriving from the lower path, then head towards D2 at an angle bent slightly towards D1. The 48% of photons transmitted at BS1, traveling with a weaker quantum potential, should reflect off M2, enter region I2, 'bounce off' the stronger quantum potential arriving from the upper path, then head towards D1 at an angle bent slightly towards M2.

Both arguments are wrong. Neither standard nor Bohmian QM work the way you described.

 

[Demystifier]

I've sketched what actually happens according to Bohmian mechanics. For simplicity I've taken the case of 66% - 33% beam splitter and presented 3 characteristic Bohmian trajectories. Both standard and Bohmian QM predict that 66% photons will end up in D1, and 33% photons will end up in D2.

 

[Ben-tanglement]

This doesn't make sense. The Bohmian interpretation, AFAIK, has been proven to be mathematically equivalent to standard QM. That means it must make the same predictions for all experimental results. The conclusion quoted above contradicts this, so I am extremely skeptical of it. I suspect they are making a mistake somewhere in their Bohmian analysis, introducing something that is not mathematically equivalent to standard QM and is therefore not part of the actual Bohmian interpretation.

Note that I am using the term "Bohm interpretation" as the authors do, to distinguish from "Bohmian mechanics" (see their footnote at the bottom of page 3). In the "Background" part of my post, I am only regurgitating what the authors wrote. The author Basil Hiley was a close colleague of Bohm and co-wrote books with Bohm on the pilot wave theory. It is his claim that the particles that end up at D1 and D2 took paths that are swapped compared to "Standard Quantum Mechanics" which I take to be Copenhagen. That's the part I quoted & bolded at the end of my Background section.

 

[Ben-tanglement]

I've sketched what actually happens according to Bohmian mechanics. For simplicity I've taken the case of 66% - 33% beam splitter and presented 3 characteristic Bohmian trajectories. Both standard and Bohmian QM predict that 66% photons will end up in D1, and 33% photons will end up in D2.

Thanks for the sketch and info. One of the things that is possibly confusing is "Bohmian mechanics" versus the "Bohm interpretation" that Bohm's colleague Basil Hiley presents in the paper that I referenced.

One thing that could help me out: Can you look at the particle trajectories they show in Figure 3? The way they show it, particles arriving from M1 and M2 end up making a 90 degree turn in region I2, to end up ad D2 and D1, respectively. My question is this: if the beam splitter was asymmetric, would all particle trajectories continue to make 90 degree turns, or would some particles now go perfectly straight through I2, or could there be trajectories that shoot off at an angle, neither straight nor 90 degrees?

 

[Demystifier]

My question is this: if the beam splitter was asymmetric, would all particle trajectories continue to make 90 degree turns, or would some particles now go perfectly straight through I2, or could there be trajectories that shoot off at an angle, neither straight nor 90 degrees?

As my drawing illustrates, some will go perfectly straight, but neither will go at an intermediate angle.

 

[Demystifier]

One of the things that is possibly confusing is "Bohmian mechanics" versus the "Bohm interpretation" that Bohm's colleague Basil Hiley presents in the paper that I referenced.

There is no important difference between the two notions. Bohm never referred to it as a kind of "mechanics", so Hiley, as his ex collaborator, does not like to call it "mechanics" either.