Exploring Photon Trajectories in Double Slit Experiments

[DevilsAvocado]

... To avoid confusion, one can measure both for a classical electromagnetic field, which corresponds to the case when many photons travel through the slits at once. This corresponds to a simultaneous measurement of a classical position and momentum of a piece of "fluid" made up of many particles, which can be done. But as I already explained above, this is not what the goal is. It's "too classical" to be really interesting.

Thanks Demystifier, this seems very logical.

To a layman, it looks a little bit 'strange' to refer to classical Maxwell's field/wave equations published in 1862, when the modern use of "Quanta" was first introduced in 1900/1905 by Planck and Einstein ("Lichtquanta")...

I know there’s some on PF who tries to forbid me to use any form of "Dualism", and at the same time – they can’t explain how any "Monism" could provide a solution for the Double-slit experiment.

I say you need to have something "extended and continuous in space" to pass two slits simultaneously and to create interference with itself, and finally something "localized in space" to generate a "localized measurement".

Whether all this is should be called a pilot wave, wavefunction, particle, photon, discrete values, quanta, localized packets, beables or whatever – is still to find out for the scientists.

The Double-slit experiment won’t work in any classical 'monistic' description.

If one is interested in the interference pattern – one could just drop two stones in the water.


P.S.
A completely different question: Could the http://en.wikipedia.org/wiki/Aharonov-Bohm_effect" is be used in any kind of "weak measurement"?

Schematic of double-slit experiment in which Aharonov–Bohm effect can be
observed: electrons pass through two slits, interfering at an observation
screen, with the interference pattern shifted when a magnetic field B is
turned on in the cylindrical solenoid.

 

[Demystifier]

A completely different question: Could the http://en.wikipedia.org/wiki/Aharonov-Bohm_effect" is be used in any kind of "weak measurement"?

Schematic of double-slit experiment in which Aharonov–Bohm effect can be
observed: electrons pass through two slits, interfering at an observation
screen, with the interference pattern shifted when a magnetic field B is
turned on in the cylindrical solenoid.

One could use weak measurements to measure particle trajectories in region in which EM field (but not EM potential) is vanishing.

 

[Ken G]

Yes, that makes me happy for two reasons. First, because now we have something concrete to talk about. Second, because now I see that you only pretended that you didn't see what I meant by "equations".

That is complete hooey, by the way.

Poynting vector is indeed a natural way to associate trajectories with solutions of Maxwell equations. But can such trajectories be measured? Or more precisely, can the Poynting vector be measured by "ordinary" (i.e. not weak) measurements?

And here's my point all along: who cares? What I have repeated over and over is a stress on the information content of the key figure that everyone as touting as evidence of single-photon trajectories! Read this as many times as it takes: if I can make that figure with Poynting fluxes, then the figure contains no non-classical information no matter how it is actually constructed.

It's "too classical" to be really interesting.

Exactly, but you are missing the importance of this statement, because it applies to their "average trajectory" diagram, regardless of how they made it (if indeed it is qualitatively or quantitatively the same as the Poynting flux streamlines, an issue which was not raised in the paper and no one else seems to even recognize its significance). The whole point of a classical limit is a co-adding of quantum information until the uniquely quantum information isn't there any more, and it is perfectly obvious that this is what has happened here, if the figure they make is the same as a Poynting flux diagram. You still don't get it.

 

[DevilsAvocado]

One could use weak measurements to measure particle trajectories in region in which EM field (but not EM potential) is vanishing.

Sorry for not understanding 100% , but I guess you are saying we can measure that one electron did pass, but not any 'hint' thru which slit, right?

Seems compatible with this:

http://arxiv.org/abs/0807.1881

Time-resolved detection of single-electron interference

S. Gustavsson, R. Leturcq, M. Studer, T. Ihn, K. Ensslin, D. C. Driscoll, A. C. Gossard

Journal reference: Nano Lett. 8, 2547 (2008)

Abstract: We demonstrate real-time detection of self-interfering electrons in a double quantum dot embedded in an Aharonov-Bohm interferometer, with visibility approaching unity. We use a quantum point contact as a charge detector to perform time-resolved measurements of single-electron tunneling. With increased bias voltage, the quantum point contact exerts a back-action on the interferometer leading to decoherence. We attribute this to emission of radiation from the quantum point contact, which drives non-coherent electronic transitions in the quantum dots.


https://www.youtube.com/watch?v=sT6OyzJ8Oqw

The movie shows in real-time the gradual build-up of an interference pattern as we count electrons passing through an Aharonov-Bohm-ring. Since only one electron can pass through the ring at a time, the experiment shows that each electron is interfering with itself.

 

[Demystifier]

Sorry for not understanding 100% , but I guess you are saying we can measure that one electron did pass, but not any 'hint' thru which slit, right?

If we ignore information gained by weak measurements, then yes.

 

[Demystifier]

And here's my point all along: who cares?

Experimentalists do.

if I can make that figure with Poynting fluxes, then the figure contains no non-classical information no matter how it is actually constructed.

I agree. But my point is the following: It does not significantly contribute to a demystification of QM, as long as it cannot be generalized to the case of entangled particles. One reason why weak measurements and Bohmian mechanics are more cool than your classical-wave picture is the fact that they can be applied to entangled particles as well.

if indeed it is qualitatively or quantitatively the same as the Poynting flux streamlines, an issue which was not raised in the paper and no one else seems to even recognize its significance

I'm sure that many people would recognize it's significance if someone could generalize it to entangled particles as well. I would be the first. But if it works in the absence of entanglement only, then the significance of it is rather low.

 

[DevilsAvocado]

If we ignore information gained by weak measurements, then yes.

Okay, thanks.

 

[DevilsAvocado]

Sorry for going slightly off-topic, but is there any pro out there who could refute this little 'experiment' of mine. I know it’s wrong and that it will not work – but I can’t find the flaw:

Personal Gedankenexperiment; Aharonov-Bohm-ring & Atomic clock in a Double-slit experiment

1. It’s possible to measure the time when a single electron passed the slits via the Aharonov-Bohm-ring.
   
   
2. It’s possible to measure the time when a single electron hits the detector.
   
   
3. We know that the speed of light is constant, thus it doesn’t matter if the electron doesn’t travel in straight lines – it must take a predestinated amount of time to go from the slits to the detector.
   
   
4. The length of the traveling distance will vary depending on where on the detector the electron hits – and from which slit the electron went thru.
   
   
5. If the traveling distance varies, so will the traveling time.

Amazing amateur conclusion – we can use the traveling time to deice which slit the electron passed thru!

Now, what’s wrong with this...??

 

[unusualname]

Sorry for going slightly off-topic, but is there any pro out there who could refute this little 'experiment' of mine. I know it’s wrong and that it will not work – but I can’t find the flaw:

Personal Gedankenexperiment; Aharonov-Bohm-ring & Atomic clock in a Double-slit experiment

1. It’s possible to measure the time when a single electron passed the slits via the Aharonov-Bohm-ring.
2. It’s possible to measure the time when a single electron hits the detector.
3. We know that the speed of light is constant, thus it doesn’t matter if the electron doesn’t travel in straight lines – it must take a predestinated amount of time to go from the slits to the detector.
4. The length of the traveling distance will vary depending on where on the detector the electron hits – and from which slit the electron went thru.
5. If the traveling distance varies, so will the traveling time.

Amazing amateur conclusion – we can use the traveling time to deice which slit the electron passed thru!

Now, what’s wrong with this...??

Well, electrons don't travel at the speed of light for a start :-)

But, overlooking that detail, I think you would find that the Uncertainty Principle wouldn't enable such accurate measurement of time without blurring the energy/momentum of the electrons such that no interference could take place.

In a paper by Prosser I linked to above he discusses the case for single photons, and admits the interference pattern would not be observed, but he was just interested in whether the photons from each slit travel to both sides of the pattern, interference or not.

 

[DevilsAvocado]

Well, electrons don't travel at the speed of light for a start :-)

Oh yeah! Ever heard of the LHC and electromagnetic acceleratation??

No no no, of course you are right... I don’t know what I was thinking on... too fast, too enthusiastic... ()

There’s always 'something' with this darned experiment, isn’t it??

Without knowing anything about it, I can almost guarantee you that it will be impossible to time photons thru the slits – without disturbing them to 'non-interference pattern'. And I’m pretty sure it’s the other way around with electrons, no problem with timing – but then you don’t have any support from Einstein’s c ...

This is nuts!

But, overlooking that detail, I think you would find that the Uncertainty Principle wouldn't enable such accurate measurement of time without blurring the energy/momentum of the electrons such that no interference could take place.

I can’t say for sure, but shouldn’t you be able to 'surpass' HUP by scaling up the whole experiment to a size where this doesn’t matter – ie if your resolution is 1 second/meter, you drive 1000 meters and that 'resolution' will be sufficient. I think...

But as you pointed out - it won’t work with electrons anyway...

Many thanks for the help!

 

[SpectraCat]

Oh yeah! Ever heard of the LHC and electromagnetic acceleratation??

No no no, of course you are right... I don’t know what I was thinking on... too fast, too enthusiastic... ()

There’s always 'something' with this darned experiment, isn’t it??

Without knowing anything about it, I can almost guarantee you that it will be impossible to time photons thru the slits – without disturbing them to 'non-interference pattern'. And I’m pretty sure it’s the other way around with electrons, no problem with timing – but then you don’t have any support from Einstein’s c ...

This is nuts!
I can’t say for sure, but shouldn’t you be able to 'surpass' HUP by scaling up the whole experiment to a size where this doesn’t matter – ie if your resolution is 1 second/meter, you drive 1000 meters and that 'resolution' will be sufficient. I think...

But as you pointed out - it won’t work with electrons anyway...

Many thanks for the help!

I am not so sure the timing is the issue. I wonder if the following might actually work. Take a metal-double slit and super-cool it. Then measure the inductance-current in the *entire* double slit apparatus from the electrons (ions would work too), passing through it, and define the peak of the inductance current as t=0. In principle, there should be no "which-path" information from such a measurement.

I wonder what would be observed at the detection screen? My guess is that you would still see interference, even though you have nominal timing information. This is because the induction will couple the momentum of the electrons in the experiment to those in the conduction band of the metal screen, so although you will have timing information, you won't be able to use that to determine which slit the electron went through.

On the other hand, if you are clever, it might be possible to get which path information from the current measurements. If you connect wires to either side of the super cooled double slit, and measure the current at both ends, there should be a timing difference depending on which slit the electron passes through .. i.e. the response pulse should arrive slightly earlier at whichever side corresponds to the slit the electron passed through. I know this timing-based detection works in principle .. it is used to image electron distributions in photo-ionization and photo-detachment experiments. What I don't know is whether the temporal resolution is sufficient to discriminate between the closely spaced slits in the double slit experiment.

Anyway, for the purposes of a gedanken experiment ... I think it *is* possible to discriminate the timing. Can anyone figure out why measuring the current in the timing-sensitive method I mentioned above would destroy the interference pattern?

 

[Ken G]

Experimentalists do.

The issue here is not the experiment itself, but the way the data was packaged and sold. I never had any issue with doing weak measurements, which is the experimentalist part, my issue was with the "average trajectory" concept, which is not an experimental issue, it is an information manipulation issue. I wouldn't even call it "data reduction", it's more like "data obfuscation."

I agree. But my point is the following: It does not significantly contribute to a demystification of QM, as long as it cannot be generalized to the case of entangled particles.

Yes, we certainly agree that this experiment does not significantly contribute to demystification of QM. You are criticizing what they did not do, and I'm criticizing what they actually did, and the claims they made about what they actually did.

One reason why weak measurements and Bohmian mechanics are more cool than your classical-wave picture is the fact that they can be applied to entangled particles as well.

Maybe yes, but that has nothing to do with this experiment. My point is that, in this experiment, a figure was produced that does not clearly contain any non-classical information, and that's the reason there is no quantum demystification here. Had they done something involving entanglement, perhaps there'd be some interesting Bohmian consequences, perhaps not-- we'd have to see what can actually be done there.

But if it works in the absence of entanglement only, then the significance of it is rather low.

If its output is the same as a purely classical calculation, then its significance is zero. Yet the authors claimed it was something different from the way they were taught quantum mechanics. I'm saying I have not seen a shred of evidence for any of those claims, regardless of whether entanglement issues would make it more interesting.

 

[DevilsAvocado]

I am not so sure the timing is the issue. I wonder if the following might actually work. Take a metal-double slit and super-cool it. Then measure the inductance-current in the *entire* double slit apparatus from the electrons (ions would work too), passing through it, and define the peak of the inductance current as t=0. In principle, there should be no "which-path" information from such a measurement.

Maybe it’s me misunderstanding, but when it comes to getting exact timing for electrons leaving the double slit, I’m 99% it is already doable by utilizing the http://en.wikipedia.org/wiki/Aharonov-Bohm_effect" .

There’s a Time-resolved detection of single-electron interference using an Aharonov-Bohm-ring in this paper http://arxiv.org/abs/0807.1881

https://www.youtube.com/watch?v=sT6OyzJ8Oqw
The movie shows in real-time the gradual build-up of an interference pattern as we count electrons passing through an Aharonov-Bohm-ring. Since only one electron can pass through the ring at a time, the experiment shows that each electron is interfering with itself.

I wonder what would be observed at the detection screen? My guess is that you would still see interference, even though you have nominal timing information.

My guess is that you will see a perfect interference pattern – that could even be 'adjusted' by the Aharonov–Bohm effect!

https://www.youtube.com/watch?v=OgDPK5MLVnE

On the other hand, if you are clever, it might be possible to get which path information from the current measurements. If you connect wires to either side of the super cooled double slit, and measure the current at both ends, there should be a timing difference depending on which slit the electron passes through .. i.e. the response pulse should arrive slightly earlier at whichever side corresponds to the slit the electron passed through. I know this timing-based detection works in principle .. it is used to image electron distributions in photo-ionization and photo-detachment experiments. What I don't know is whether the temporal resolution is sufficient to discriminate between the closely spaced slits in the double slit experiment.

Anyway, for the purposes of a gedanken experiment ... I think it *is* possible to discriminate the timing. Can anyone figure out why measuring the current in the timing-sensitive method I mentioned above would destroy the interference pattern?

As a layman, the only thing I can think of is the detection causing a current at the slits – if this is a 'regular' measurement, it will cause decoherence...

If not – it’s really interesting!

 

[Ken G]

What I don't know is whether the temporal resolution is sufficient to discriminate between the closely spaced slits in the double slit experiment.

The timing difference is of order the slit separation divided by c, call it d/c. To get interference, you need the wavelength, call it l, to be of order d or larger. Since the period of the wave is l/c, this means the period must be of order or greater than the timing you are trying to resolve. You cannot resolve the timing of a coherent wave at scales of its own period. Put differently, your setup is having a little fight between the need for the wave to have a sharp frequency, so you get the interference pattern (that's why lasers are often used), and the need to have timing resolution of order the wave period. You can't get the latter with a sharp frequency, that is a basic wave property.

 

[DevilsAvocado]

Ken G, sorry for asking this really dumb question, but I can’t get it out of my head: Electrons have mass, they don’t travel at c (unless you 'scare' them with a LHC). According to CI electrons doesn’t 'exist' until you measure them, and before that we use the wavefunction to calculate the probability for the electron to 'popup' in a specific location.

Question – What speed does the wavefunction for electrons have (according to CI)?

(no entangled stuff)

 

[SpectraCat]

The timing difference is of order the slit separation divided by c, call it d/c. To get interference, you need the wavelength, call it l, to be of order d or larger. Since the period of the wave is l/c, this means the period must be of order or greater than the timing you are trying to resolve. You cannot resolve the timing of a coherent wave at scales of its own period. Put differently, your setup is having a little fight between the need for the wave to have a sharp frequency, so you get the interference pattern (that's why lasers are often used), and the need to have timing resolution of order the wave period. You can't get the latter with a sharp frequency, that is a basic wave property.

Well, aside from the fact that electrons don't propagate at c (as DA already mentioned), I think you have hit on an important issue. What you are saying is that, if the double slit is of the appropriate size to cause diffraction, then the question of which slit the electron passes through is meaningless. Since the wavelength of the electron is on the same order as the width and separation of the slits, the spatial resolution is not sufficient to determine through which slit the electron passed.

Of course, that analysis assumes that the electron has a uniquely wavelike interaction with the double slit. I see no reason to assume that the other aspects of the experiment I suggested (i.e. supercooling the double slit and measuring the induced current from either side) would do anything to change that. So it seems like there would be no way to get which path info from this gedanken, and thus a normal interference should be observed.

However, I would like to ask the Bohmian's about this experiment. I guess in the Bohmian picture, the pilot wave will diffract off both slits, but each single electron will take a well-defined trajectory through one slit to the screen. Is that correct? Also, in the Bohmian picture, are properties like charge associated with both the pilot wave and the particle, or just one or the other? It seems like charge would be a particle-only property, since charge is usually (always?) associated with localized detection events, but perhaps that is just a poor assumption on my part.

 

[Ken G]

Question – What speed does the wavefunction for electrons have (according to CI)?

This is called the "group velocity" of the wave, it's essentially the speed that the "wave packet" moves if we watch the solution of the wavefunction evolve in time. By the correspondence principle, it has to work fine in classical physics.

 

[Ken G]

Well, aside from the fact that electrons don't propagate at c (as DA already mentioned), I think you have hit on an important issue.

Yes, I answered it for photons, but the electron wavefunction is not qualitatively any different. Wave mechanics is wave mechanics.

What you are saying is that, if the double slit is of the appropriate size to cause diffraction, then the question of which slit the electron passes through is meaningless. Since the wavelength of the electron is on the same order as the width and separation of the slits, the spatial resolution is not sufficient to determine through which slit the electron passed.

Actually, you can tell which slit the electron passed through, but not without disrupting its wavefunction. You generally want to tell which slit it went through while just being a "fly on the wall", not disrupting anything, but this is just what you cannot do. Even quantum erasure experiments involve a change in the wavefunction, during parametric down-conversion.

In the above, I was talking about the period of the undisturbed wave-- you can always time it with arbitrary precision by blasting it with a bright radiation field, but the energy needed will disrupt the coherence of the wavefunction and alter the results.

So it seems like there would be no way to get which path info from this gedanken, and thus a normal interference should be observed.

That's harder to tell. My point was that if you can do the timing, then you cannot get an interference pattern, but I can't tell if your apparatus will allow the timing or not, because I can't tell how much it will disrupt the coherence of the wavefunction. If I had to guess, I'd agree with what you're saying here-- you should still get the interference pattern.

I guess in the Bohmian picture, the pilot wave will diffract off both slits, but each single electron will take a well-defined trajectory through one slit to the screen. Is that correct?

Yes. I can tell you that even without being a true believer in the Bohm interpretation!

 

[DevilsAvocado]

This is called the "group velocity" of the wave, it's essentially the speed that the "wave packet" moves if we watch the solution of the wavefunction evolve in time. By the correspondence principle, it has to work fine in classical physics.

Yes, I answered it for photons, but the electron wavefunction is not qualitatively any different. Wave mechanics is wave mechanics.

Thanks for the answer.

I also looked it up myself, the wavefunction for electrons and photons differ (mass/massless) due to the time-evolution of the field operator on Hilbert space, which is governed by the Hamiltonian.

I.e.:
photon wavefunction = c
electron wavefunction < c (and the actual speed depends on the energy)

More info in this thread: https://www.physicsforums.com/showthread.php?t=152053"

 

[Ken G]

Yes, the dependence of speed in a vacuum on energy for a particle with mass is a lot like propagation through a dispersive medium. This is why the connection between frequency and wavelength (which is how you find the group velocity) is called a "dispersion relation." In effect, a massless particle propagates dispersionlessly through vacuum at c, but a particle with mass propagates through vacuum as if it was slowed by interactions between its mass and the vacuum (I'm not formalizing this into a theory, just describing a picture). The result is essentially a dispersion relation, which says (nonrelativistically) that the connection between energy and momentum (frequency and inverse-wavelength) is E = p²/2m (which is a lot like saying frequency scales with inverse-wavelength squared divided by m). The group velocity is given by taking the derivative of the frequency with respect to the inverse-wavelength, so here that velocity comes out p/m.

 

[yoron]

I agree with Ken. I find all 'light paths' diffuse myself :)
To me a 'photon' has a recoil and a annihilation. We assume a propagation based on the way we've seen the universe macroscopically for a very long time. The difference between a football coming at you and a 'photon' becomes even weirder if you consider that all information you get of that footballs path is by those same photons :)

To me it's simpler to describe in terms of what we can measure, than to build theories pressing forward ideas that doesn't have the same clarity they once seemed to have. Einstein and Lorentz both questions our definitions of motion macroscopically, and QM does it 'microscopically'.

 

[SpectraCat]

Yes, I answered it for photons, but the electron wavefunction is not qualitatively any different. Wave mechanics is wave mechanics.

Yes and no .. photons cannot be described by wavefunctions .. I'd say that is a qualitative difference.

Actually, you can tell which slit the electron passed through, but not without disrupting its wavefunction. You generally want to tell which slit it went through while just being a "fly on the wall", not disrupting anything, but this is just what you cannot do.

Of course .. that was the whole motivation for my gedanken in the first place .. I was trying to understand why the "time-lag" detection of the induced current would lead to localization of the wavefunction on one of the slits.

Even quantum erasure experiments involve a change in the wavefunction, during parametric down-conversion.

I am not sure what you are talking about there ... the only PDC in the quantum eraser experiments I am familiar with is when entangled photon pairs are created at the source. The "erasure" step typically involves a normal optical element, as far as I am aware. In the Walborn setup it's a polarizer, while in the DCQE of Kim and Scully, it was just a beamsplitter. Furthermore, in those experiments the "erasure" is only evident in the two-photon coincidence measurements ... the interference pattern is never restored in the single-photon measurements. This can be understood through an analysis of the phase-relationship between the entangled photons, as has been discussed in detail recently on a few parallel threads here on the DCQE:
https://www.physicsforums.com/showthread.php?t=503667
https://www.physicsforums.com/showthread.php?t=505115

In the above, I was talking about the period of the undisturbed wave-- you can always time it with arbitrary precision by blasting it with a bright radiation field, but the energy needed will disrupt the coherence of the wavefunction and alter the results.

Not sure how that would work for free-electrons ... what I mean is that, while I can certainly see how the light field would perturb the electrons, it is hard to see how individual electrons would change the light field in a detectable way. I suppose in the gedanken-limit of perfect detectors, you could see the small change in the light field, I am just not sure about the details of how that would work.

That's harder to tell. My point was that if you can do the timing, then you cannot get an interference pattern, but I can't tell if your apparatus will allow the timing or not, because I can't tell how much it will disrupt the coherence of the wavefunction. If I had to guess, I'd agree with what you're saying here-- you should still get the interference pattern.

That was the entire point of my question .. the difference between the "single-wire" detection of the induced current, where you get only longitudinal timing to set the t=0 for the travel time to the screen, and the "double-wire lag-time" detection, where you also try to get information about the transverse components of the electron momentum, seems very small to me. That's why I think your comments about the transverse wavelength of the electron being on the same size scale as the slits is the key point for understanding why the proposed "lag-timing" can not work. In any case, I certainly agree that if it *did* work, then there would be no interference pattern. I just can't figure out any reason why the apparently minor change in the detection electronics would affect the wavefunction so drastically. That's why ...

Ok .. I am going to stop arguing in circles now .. I am getting dizzy.

 

[Ken G]

Yes and no .. photons cannot be described by wavefunctions .. I'd say that is a qualitative difference.

Photons get described by wavefunctions all the time. Look at any quantum mechanical scattering calculation, or even the derivation of the cross section of an atomic transition. There are some technical matters given that these are typically done without using any relativistic physics at all, yet they serve quite well to determine these cross sections. Quantum mechanics, and quantum electrodynamics, can be conceived at many different levels, some so abstract that there's never any such thing as an evolving wave function (even the Heisenberg formulation does away with wavefunctions as anything but an index on the different possible evolutions for the operators). I don't know what can be said in general about wavefunctions, but I don't know of any fundamentally different behaviors of photons or electrons in two-slit experiments, unless you explicitly try to take advantage of the electron mass or charge.

I am not sure what you are talking about there ... the only PDC in the quantum eraser experiments I am familiar with is when entangled photon pairs are created at the source.

Yes, that's when their wavefunctions are altered. They cannot be treated as two identical input wavefunctions, so the PDC leaves some kind of subtle imprint on them which is quite important for what ensues.

That was the entire point of my question .. the difference between the "single-wire" detection of the induced current, where you get only longitudinal timing to set the t=0 for the travel time to the screen, and the "double-wire lag-time" detection, where you also try to get information about the transverse components of the electron momentum, seems very small to me. That's why I think your comments about the transverse wavelength of the electron being on the same size scale as the slits is the key point for understanding why the proposed "lag-timing" can not work. In any case, I certainly agree that if it *did* work, then there would be no interference pattern.

Yeah, that's the key isue for the Bohmian consequences. I wish I knew more about the detailed consequences of the apparatus you are describing, because "the devil is in the details" with this kind of thing. Every time we think we have found a way to shortcut the rules, reality inserts some subtle element we didn't anticipate to make sure that we can't.

 

[SpectraCat]

Yes, that's when their wavefunctions are altered. They cannot be treated as two identical input wavefunctions, so the PDC leaves some kind of subtle imprint on them which is quite important for what ensues.

That still doesn't seem to make much sense ... there is no requirement that the photons be identical for the quantum eraser experiment ... they could even have different wavelengths and it would still work just fine. The "subtle imprint" that I guess you are referring to would be the polarization entanglement. That guarantees that the photons maintain a well-defined phase relationship, which allows the reconstruction of the interference pattern based on polarization-filtered coincidence counting. In any case, the situation is very different than for the double-slit experiments we have been discussing here, because there is never an interference pattern observed for the photons that pass through the double-slit.

 

[Ken G]

That still doesn't seem to make much sense ... there is no requirement that the photons be identical for the quantum eraser experiment ...

Hmm, I say the photons aren't identical and you say that doesn't make sense, the photons aren't identical. I can't follow that logic. All I'm saying is that the down-conversion leaves an imprint on the wavefunction which is very important to what ensues, pursuant to the overall issue that you can't get which-way information without messing up the interference pattern, even in Bohmian mechanics unless there's some technicality that has not yet been tested.

The "subtle imprint" that I guess you are referring to would be the polarization entanglement. That guarantees that the photons maintain a well-defined phase relationship, which allows the reconstruction of the interference pattern based on polarization-filtered coincidence counting.

It's more subtle than that. Something causes the signal photons to divide into two sets, which each produce an interference pattern that is slightly shifted from the other, such that when you add them up, you get no interference pattern.

In any case, the situation is very different than for the double-slit experiments we have been discussing here, because there is never an interference pattern observed for the photons that pass through the double-slit.

I completely agree, that's why I've said so little about entanglement in this thread. Others seem to want to keep bringing it up, but I agree it's a red herring in regard to Bohmian trajectories in a simple two-slit experiment.

 

[JesseM]

Commenting on SpectraCat's original post, isn't there still a meaningful sense in which we can talk about "which-path information" in the Bohmian interpretation of a quantum eraser or delayed choice quantum eraser experiment? While it's true that if you look at the position the "signal" photon was detected you can always tell which slit it went through just by looking at the position on the screen, what if the only information you have is the position the idler was detected (and the experimental setup it passed through to get to the detector, like whether there was a polarizer in front of the detector in the basic quantum eraser experimenter), in some cases this information alone would be enough to tell you which slit the signal photon went through while in others not, right? In the delayed choice quantum eraser paper, if you look at the D0/D1 coincidence pattern in Fig. 3 on p. 4, in the Bohmian interpretation would it true here (as with the normal double-slit interference pattern) that half the signal photons making this pattern went through the left slit while half went through the right? And for the D0/D3 coincidence pattern in Fig. 5, in the Bohmian interpretation would it be true that all the signal photons making this pattern went through the same slit? If so, that would be a sense in which detecting the idler at D1 means the which-path information has been entirely "erased" while detecting the idler at D3 means the which-path information has been entirely preserved.

One thing I've wondered about this type of experiment is whether it would be possible to arrange the idler-detectors in different ways so that one could have have "partial" which-path info in some sense that could be quantified, and whether a detector which gave partial which-path information would then show a partial interference pattern (somewhere between fig. 3 and fig. 5), with the closeness to a perfect interference pattern depending on the "amount" of which-path information. Since "amounts" of information are usually defined in terms of probabilities of a given outcome via the information-theory concept of entropy, in most interpretations of QM it would be a bit problematic to talk of partial information since we can't really talk about the probability that the particle "actually" went through one slit or another, but in Bohmian mechanics you can, so maybe this would be a convenient way to quantify partial which-path information and see if it corresponded to partial interference.

 

[Ken G]

Commenting on SpectraCat's original post, isn't there still a meaningful sense in which we can talk about "which-path information" in the Bohmian interpretation of a quantum eraser or delayed choice quantum eraser experiment?

I would say the Bohmian approach always dictates which way any given photon goes, but it can't count as "information" if we cannot extract it. Are you defining "information" by what "the universe knows about itself", or by what we can know about it? I suspect the latter, as that is the usual meaning.

In the delayed choice quantum eraser paper, if you look at the D0/D1 coincidence pattern in Fig. 3 on p. 4, in the Bohmian interpretation would it true here (as with the normal double-slit interference pattern) that half the signal photons making this pattern went through the left slit while half went through the right?

In my understanding, yes.

And for the D0/D3 coincidence pattern in Fig. 5, in the Bohmian interpretation would it be true that all the signal photons making this pattern went through the same slit?

I think so, yes.

If so, that would be a sense in which detecting the idler at D1 means the which-path information has been entirely "erased" while detecting the idler at D3 means the which-path information has been entirely preserved.

I'm not sure what you are saying here, it seems to me that this statement is true in all the interpretations, it's just a fact of the apparatus.

One thing I've wondered about this type of experiment is whether it would be possible to arrange the idler-detectors in different ways so that one could have have "partial" which-path info in some sense that could be quantified, and whether a detector which gave partial which-path information would then show a partial interference pattern (somewhere between fig. 3 and fig. 5), with the closeness to a perfect interference pattern depending on the "amount" of which-path information.

I would expect so, yes. This is because "partial" information would have to look like a probability that you know and a probability that you don't, so if you like, a mixture of a mixed state and a superposition state. But that's just a mixed state of a non-orthogonal basis, so a density matrix with nondiagonal elements that can't be diagonalized, I believe.

Since "amounts" of information are usually defined in terms of probabilities of a given outcome via the information-theory concept of entropy, in most interpretations of QM it would be a bit problematic to talk of partial information since we can't really talk about the probability that the particle "actually" went through one slit or another, but in Bohmian mechanics you can, so maybe this would be a convenient way to quantify partial which-path information and see if it corresponded to partial interference.

I actually think regular quantum mechanics can handle the partial information concept, we'd just talk about the probability that it went through slit 1, the probability it went through slit 2, and the probability that we don't know what slit it went through. In a sense we already have that with the delayed choice experiment if we set up a D4 detector symmetrically to D3 (why isn't it in there?). Then we have a 25% chance of detection of the idler at each of D1-4, so that means we have a 50% chance of not knowing which slit the signal photon went through, a 25% chance we'll know it was slit A, and a 25% chance we'll know it was slit B. I think we have those probabilities in any interpretation.

 

[JesseM]

I would say the Bohmian approach always dictates which way any given photon goes, but it can't count as "information" if we cannot extract it. Are you defining "information" by what "the universe knows about itself", or by what we can know about it? I suspect the latter, as that is the usual meaning.

This seems like a false dichotomy, "information" normally involves the question of how the facts you do know narrow down the range of possibilities for the facts you don't know, even if it is practically or theoretically impossible to determine the single correct truth about the facts you don't know. Consider statistical mechanics, where the thermodynamic entropy of a given "macrostate" (defined by the value of thermodynamic variables like temperature and pressure) is proportional to the logarithm of the number of possible "microstates" (precise specification of the state of all the particles that make up the system) consistent with that macrostate. So, the facts you do know (macrostate) allow you to narrow down the range of possible values for the facts you don't know (microstate), and since there are fewer possible microstates associated with macrostates that have a lower value of (thermodynamic) entropy, we can say that you gain more "information" about the system's microstate if you find it in a low-entropy macrostate than if you find it in a high-entropy macrostate, even though it may be impossible in practice to ever "extract" the true fact about the precise microstate. So I'm suggesting something similar for "which path information" in a Bohmian interpretation, where what you do know (the detection of the idler at some detector) gives you information about something you don't know, but which is assumed to have an objective value (which slit the signal photon went through), and where you can calculate theoretically the set of different trajectories which are consistent with the fact you do know about the idler's detection.

If so, that would be a sense in which detecting the idler at D1 means the which-path information has been entirely "erased" while detecting the idler at D3 means the which-path information has been entirely preserved.

I'm not sure what you are saying here, it seems to me that this statement is true in all the interpretations, it's just a fact of the apparatus.

The reason I discussed this is that SpectraCat had trouble understanding how we could still talk about the which-path information being "erased" under the Bohmian interpretation, since in the Bohmian interpretation even when you get an interference pattern on the screen you can still tell which slit the signal photon went through based on whether it's on the upper or lower half of the screen. My point was that if you throw out the knowledge of where on the screen the signal photon was detected, and only have knowledge of where the idler was detected, then some possible idler detections will correspond to the which-path information being "erased" because they alone tell you nothing about which slit the signal photon went through.

I would expect so, yes. This is because "partial" information would have to look like a probability that you know and a probability that you don't, so if you like, a mixture of a mixed state and a superposition state.

"Mixture of a mixed state and a superposition state" doesn't seem to make sense, wouldn't that just be another mixed state? Are you just thinking out loud or have you seen such a "mixture" defined in a mathematical sense somewhere? In any case I didn't mean anything like that by "partial" information, I just meant putting the detector at some position midway between a position where you completely lose the which-path information (the position of D1 on the last page of the DCQE paper for example) and a position where you completely preserve the which-path information (the position of D3 for example). In that case there might be a way to quantify a "partial" which-path information, and the amount of partial which-path information might correspond to the degree to which the coincidence count resembled the D0/D1 coincidence count (fig. 3) vs. the D0/D3 coincidence count (fig. 5). My suggestion was that the Bohmian interpretation might give a nice way to quantify "partial information" since for every idler detected there will be a definite truth about which path the signal photon took, so you can calculate theoretically the proportion of photons in the coincidence count that went through one slit vs. the other (then the idea would be that the closer they are to evenly distributed between slits the closer you are to zero which-path information, and the closer they are to all having come through one of the two slits the closer you are to 1 bit of which-path information)

I actually think regular quantum mechanics can handle the partial information concept, we'd just talk about the probability that it went through slit 1, the probability it went through slit 2, and the probability that we don't know what slit it went through.

How can regular quantum mechanics give quantitative values to "the probability it went through slit 1" vs. "the probability that we don't know"? I see below that you define what these would mean in the specific case of the setup of the delayed choice quantum eraser, but as I note below I don't see how this could be generalized to any arbitrary setup.

In a sense we already have that with the delayed choice experiment if we set up a D4 detector symmetrically to D3 (why isn't it in there?).

Well, note that the paper on the delayed choice quantum eraser that I referred to does have a D4 symmetric to D3 in the schematic Fig. 1, though not in Fig. 2, I guess they didn't think any interesting new information would come from it so they didn't bother.

Then we have a 25% chance of detection of the idler at each of D1-4, so that means we have a 50% chance of not knowing which slit the signal photon went through, a 25% chance we'll know it was slit A, and a 25% chance we'll know it was slit B. I think we have those probabilities in any interpretation.

OK, but this is only for the special case where each detector is positioned in such a way as to 100% erase or 100% preserve the which-path information (giving either a perfect interference pattern or a perfect non-interference pattern), it doesn't help at all with the question of quantifying "partial information" if you have a detector somewhere else where this isn't true, that was the case my question was meant to discuss, sorry if it wasn't clear.

 

[Ken G]

So I'm suggesting something similar for "which path information" in a Bohmian interpretation, where what you do know (the detection of the idler at some detector) gives you information about something you don't know, but which is assumed to have an objective value (which slit the signal photon went through), and where you can calculate theoretically the set of different trajectories which are consistent with the fact you do know about the idler's detection.

Here's my problem with the whole "Bohmian trajectory" concept. For a trajectory to represent new information, it has to be predictive. Otherwise a trajectory is just a semantic label for a bunch of information we already have. If I say "I know the photon went through the left slit, so I can predict it will hit the left detector", then we have information content in that trajectory concept. If we say "I know the photon hit the left detector, and I'm calling that (via Bohmian trajectories or any way you like) a photon that went through the left slit", then I call that trajectory concept a longwinded label for the same photon, containing no actual new information.

So much for the information content of the Bohmian approach, what about the determinism issue? After all, that's the main motivation. If I can imagine a definite trajectory, then I can connect the left and right sides of that trajectory, and say nothing happened in between that breaks the determinism. But do I really have a deterministic process? Not really, because nobody who isn't using a deterministic approach (say, CI) ever claimed that the two-slit apparatus broke what was otherwise a deterministic process-- they would have said the process was not deterministic from the get go. So plotting trajectories through the apparatus completely begs the issue of whether or not this is actually a deterministic process. If a Bohmian says "the photon followed this path because it hit the detector here", I ask "but what determined that the photon would follow that path in the first place?" There's a lack of new information there, the Bohmian trajectories are going to start out with some kind of ergodic assumption at the input boundary, which is a stochastic assumption in the first place. You cannot argue a process is deterministic because there's no break in determinism, you have to argue that it starts out determined and also there is no break in the determinism. No Bohmian calculations ever actually do the latter, only the former.

So I claim the Bohmian approach has limited meaning for two reasons:
1) Bohm trajectories are never predictive, so they are in effect just longer-winded semantic labels for whatever information has already been established by the apparatus, and
2) Bohm trajectories do not represent a deterministic process, they only represent a process which does not alter its deterministic or nondeterministic status in the course of the development of the envisaged trajectories. Since other interpretations hold that no part of the process is strictly deterministic, providing a way to maintain that status during propagation is of no significance.
So in all, what Bohm accomplishes is a way to let you continue believing the process is deterministic if you already wanted to believe that for other reasons, but it offers no evidence that the process actually is deterministic, nor does it help you gain information that is dynamically active.

The reason I discussed this is that SpectraCat had trouble understanding how we could still talk about the which-path information being "erased" under the Bohmian interpretation, since in the Bohmian interpretation even when you get an interference pattern on the screen you can still tell which slit the signal photon went through based on whether it's on the upper or lower half of the screen.

I see what you are saying about the meaning of "erasing" the information, but note that when information is erased, it is real information-- the dynamically active information, and it really is erased. The semantic label that the Bohmians attach to photons that hit the left side of the detector cannot be erased because it isn't really information at all, it's just a longer label for the same information and it presents nothing testable or dynamically active about those photons. A Bohmian trajectory could pass around Mars for all the difference it would make-- it's just a label attached to the photons.

"Mixture of a mixed state and a superposition state" doesn't seem to make sense, wouldn't that just be another mixed state?

Yes, but as I said above, I believe it would be a particularly unusual type of mixed state-- one with nondiagonal elements (if "superposition state" carries the implication of including more than one eigenfunction of the observable, with phase information not normally present in a mixed state, as would be appropriate for weak measurements).

Are you just thinking out loud or have you seen such a "mixture" defined in a mathematical sense somewhere? In any case I didn't mean anything like that by "partial" information, I just meant putting the detector at some position midway between a position where you completely lose the which-path information (the position of D1 on the last page of the DCQE paper for example) and a position where you completely preserve the which-path information (the position of D3 for example).

And I'm saying that's exactly the kind of partial information I'm talking about. The eigenvalues are which slit, and if you get partial information about that, to whatever extent you decohere which slit, you'll get a mixed state of the which-slit eigenfunctions, and to whatever extent you don't decohere those eigenstates, you'll still have the superposition of both slits. When you go to make the density matrix for that state, you'll get diagonal contributions that look like the which-slit information, and you'll get off-diagonal elements that reflect the coherences you did not decohere with your detector placements in the which-slit basis.

In that case there might be a way to quantify a "partial" which-path information,

The trace of the density matrix?

and the amount of partial which-path information might correspond to the degree to which the coincidence count resembled the D0/D1 coincidence count (fig. 3) vs. the D0/D3 coincidence count (fig. 5).

No question, I would say.

My suggestion was that the Bohmian interpretation might give a nice way to quantify "partial information" since for every idler detected there will be a definite truth about which path the signal photon took, so you can calculate theoretically the proportion of photons in the coincidence count that went through one slit vs. the other (then the idea would be that the closer they are to evenly distributed between slits the closer you are to zero which-path information, and the closer they are to all having come through one of the two slits the closer you are to 1 bit of which-path information)

And again I believe that would all just be another semantic relabeling of the exact same information contained in the density matrix of the joint wavefunction of both photons, subjecting it to the same criticism I leveled above for the one-photon situation. It gets complicated with the entanglement, but I'd happily bet that once again all we get from the Bohm approach is longwinded trajectory-sounding labels for the same information we get in CI. Bohm mechanics isn't mechanics at all, it's language, which is fine if you like that language, but we should not be fooled that there is any different information content.

OK, but this is only for the special case where each detector is positioned in such a way as to 100% erase or 100% preserve the which-path information (giving either a perfect interference pattern or a perfect non-interference pattern), it doesn't help at all with the question of quantifying "partial information" if you have a detector somewhere else where this isn't true, that was the case my question was meant to discuss, sorry if it wasn't clear.

I don't agree, that just happened to be an easy example I gave. If the detectors are set up to give partial information, like beamsplitters that split 75% - 25% or some such thing, then you can still calculate a probability of which slit the signal photon went through if the idler is detected in the non-erasing detectors, and the fraction of the time that the idler is detected in the partially erasing detectors, you have a problem you can use classical methods to determine. An erasing detection will look like a target-shaped interference pattern on the detector, and a non-erasing detection will look like a simple beam on the detector. Maxwell's equations could do it, or so it seems to me anyway.

 

[DevilsAvocado]

... In the delayed choice quantum eraser paper, if you look at the D0/D1 coincidence pattern in Fig. 3 on p. 4, in the Bohmian interpretation would it true here (as with the normal double-slit interference pattern) that half the signal photons making this pattern went through the left slit while half went through the right? And for the D0/D3 coincidence pattern in Fig. 5, in the Bohmian interpretation would it be true that all the signal photons making this pattern went through the same slit?

Ehh... maybe for me and any other layman out there, let’s recap and state exactly what we are talking about. This colored image is easier to follow:

• Single (laser pumped) photons from the double-slit are entangled in the BBO crystal.
  
  
• The five (black) detectors are labeled D₀, D₁, D₂, D₃, and D₄.
  
  
• The three (green) beam splitters (50% thru) are labeled BS_a, BS_b, and BS_c. The two (gray) mirrors (100% reflection) are labeled M_a and M_b.
  
  
• The upper 'red slit' going thru the BBO is called B, and photons from this slit are marked with red lines.
  
  
• The lower 'light blue slit' going thru the BBO is called A, and photons from this slit are marked with light blue lines.
  
  
• The "signal" photon (either from A or B) passes the (yellow) Lens to go to detector D₀.
  
  
• The "idler" photon (either from A or B) is deflected by a prism PS that sends it along divergent paths depending on whether it came from slit A or slit B.
  
  
• The detection time of the signal photon is 8ns earlier than that of the idler.

Therefore AFAIK, the signal photons are ALWAYS 50% A and B, no matter what coincidence pattern you are looking at. Or did I miss something (again)??

AFAIK, the 'magic' in the Delayed choice quantum eraser experiment is that there is NO way for the signal photons at detector D₀ to 'know' – in advance – what will happen with the idler photons at detectors D₁, D₂, D₃, and D₄ – whether there will be a 'definite path knowledge' or not ...??

That means that NO interference pattern could be observed in the corresponding subset of signal photons at detector D₀, IF the idler photon detection was made at detectors D₃ or D₄ – but detector D₀ doesn’t have this information at the time of detection...

That is weird.

I made a very unscientific 'investigation' and compared R₀₂ & R₀₃ in http://arxiv.org/abs/quant-ph/9903047" (Fig. 4 & 5) by putting them on top of each other and made one 50% transparent (+ inverted colors). Unfortunate, there seems to be a difference in the scale of the two pictures, I stretched width & height to get as good match as possible.

My thought was that I should be able to figure out which data (points) comes from the signal photons at detector D₀, to get a better chance to see what really happens here... (please don’t laugh )

Either I did something wrong, or I’m missing something fundamental about this 'data processing' – because there are only two (2) matching data points in R₀₂ & R₀₃ ...?? They are marked with red circles in the picture below:

One would expect it to be much more matching 'signal data points' from D₀... 50% or something like that...

What did I miss?? What is wrong?? And why did the authors not make it much easier to 'filter out' signal and idler in the diagrams of "joint detection" rate (by different symbols)??

That would have been extremely interesting!

Clues anyone...?