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The plain old double slit experiment displays interference when we don't measure which slit the photon passed through, and no interference when it is measured. Let's turn our attention to the case with no detectors at the slits. If instead of a back screen, we replace that with a focusing lens with a screen further back such that the lens focuses the optical image of the slits sharply on the screen further back, then just by looking at which spot the photon ends up in, we can state with absolute certainty which slit the photon passed through.

Afshar introduced detector wires where the interference troughs should have been to detect any photon hitting those wires, but allowing photons to pass through unimpeded in between the narrow wires. The wires have some thickness, and so, a small fraction of the photon still hit them anyway, but the important point is this fraction is pretty low, and consistent with an interference pattern. This is an example of what is known in the literature as a nondestructive measurement.

Anyway, this interference pattern seems to suggest the photon went through both slits. The catch is, at the screen further back, we still detect two sharp spots. The small fraction of photon intercepted by the wires blurs the sharpness a little, but it's still mostly sharp. Now, it appears we also know which slit the photon passed through.

But would you say the presence of the wires causes us to lose information about which slit the photon passed through despite the fact that we still see two sharp spots at the back, and even after knowing which of the two spots it ended up in, we still have to say it went through both slits despite the fact only a small fraction of the photons get intercepted?

Can we make any definite counterfactual statements about which slit the photon passed through even after knowing which spot it ended up in? Or are such questions meaningless?

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There is no reason to suppose that the photon which is detected in the top spot in the Afshar experiment when through the bottom slit. The result of the experiment is that you see two spots, corresponding to the two slits, but the spots have been spread out and recombined, and the process does not necessarily keep the photons which went through the bottom (inasmuch as this concept makes sense) going to the top spot, and the phtons which went through the top going to the bottom spot. There is no conservation of momentum, because of the spatial symmetry breaking by both the slits and the wires.

So while the interference pattern contains the information of the slit shape it is not giving you which-way information about the photons, because such information does not exist in quantum mechanics to begin with. This is only arguing against thinking of the quantum wavefunction as a parametrization of ignorance about stuff, and the quantum wavefunction is obviously not a parametrization of ignorance about anything, because only probability calculus is a parametrization of ignorance about things.

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Consistent histories is the sledgehammer to smash away confusions in quantum mechanics. Watch it smash away your confusion.

Consider the following three sets of complete orthonormal projection operators. The first is for the which way slit information. The second is for the wire detections. The last is for the final spot location at the back screen.

Consider first the case without the wires. There is a consistent histories framework combining the final spot projection operators with the which way slit information. It satisfies the consistency conditions. Not only that, there is a perfect correlation between which slit the photon passed through and the final spot it ends up in.

Now add the wires. The presence of the wires changes everything even though they only absorb a small fraction of the photons. Suddenly, the framework combining the final spot projection operators and the which way slit projection operators fail to satisfy the consistency conditions! There is a consistent framework combining the wire projection operators with the final spot projection operators. There is also another consistent frame consisting only of the which way slit projection operators but not the wire or final spot projection operators. Both frameworks are mutually incompatible with no common refinement. There isn't even a consistent framework combining the which way projection operators with the wire projection operators. To rephrase in the language of the OP, knowing which spot the photon lands on does not allow us to ask which slit it passed through.

It is true the absorption of some of the photons by the wires smears out the final spots somewhat. A cleaner example would be a doubled Mach-Zehnder interferometer with two "squares" touching at a common corner in the middle. The wire should be along the path in the second square which is never reached by a photon passing through the first square.

Is there any criteria to select which framework to use? Yes. The consistency condition used earlier is medium consistency. There is a stronger form, strong consistency. Valid histories ought to leave records, or generalized records. Generalized records are essentially what you get when records scramble up but are still present in principle. In Afshar's experiment, no records are left as to which slit the photon passed through. That framework is less physical.

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Ron, I think you have not even read Afshar's paper on the simple crossed-beam version (http://arxiv.org/ftp/quant-ph/papers/0701/0701039.pdf). A picture of the experiment can be found here. The light coming out of each hole has an uncertainty that allows the photon to be retrodicted to it's corresponding hole when detected. Assume that there are no wires at the plane the two beams intersect. Do the simple QM unitary time evolution for the wavefnctions emerging from each hole and see what happens. At the end of the setup where the detectors are placed, the photon can only have come from the opposite hole. All the uncertainty principle does in the above setup is the expand each of the beams, but at the detector plane, where the two beams no longer overlap, detection of a photon neccesitates its origin to the hole from which that wavefucntion emerged. If one claims otherwise, the photon must have suddenly changed its momentum at the crossing plane violating the conservation law.

In Einstein's versions, all of the which-way measurements were being done at the holes, where movement of the hole from which the photon passed would lead to loss of interference pattern. In Afshar's nothing disturbs the interference pattern, or the which way information.

You are talking about ERP where interestingly enough Bohr himself agrees that conservation of linear momentum necessitates entanglement, but that's completely different setup from Afshar's.

Here's an easy challenge for you to prove your point using QM: In Afshar's simple crossed-beam experiment, when the wires are not there at the crossing plane, if a photon is detected at Detector D1 can we say it came from hole 1? If not please explain how you arrive at a different conclusion that does not violate the law of conservation of linear momentum.

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Sockpuppetry and argumentation are not pretty... –  user2963 Dec 24 '11 at 22:49
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What the heck are you talking about? 3 different people? Paranoid much? –  Quantum Expert Dec 25 '11 at 1:03
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I just realized what you think I did after reading your comments again. This is ridiculous. I did not post this question or use any other aliases. What is wrong with you exactly? –  Quantum Expert Dec 25 '11 at 2:50
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I read it, I disagree with it. When a photon is interacting with a slit and a grating, it gets arbitrary momentum at both the slit and the grating (and the lens). There is no reason to suppose that the photons that make the picture of the top lens came from the top lens. This assumption has no argument behind it. –  Ron Maimon Dec 26 '11 at 2:26
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With both slits open, the photons avoid the wires ... if you cover up one of the slits, the photons don't avoid the wires. So the situation isn't as simple as saying that each photon goes through one of the slits, and that there is no interference between the paths going through left slit and the right slit. –  Peter Shor Jan 23 '13 at 14:11
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There is no problem other than your confusion. The wires form a diffraction grating and what diffraction gratings do is change the direction of photons passing through them. If diffraction gratings can change the direction to a number of discrete directions at various angles, it is no surprise no one can tell which slit the photon went through.

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That's the beauty of Afshar's clever little experiment, it forces us to reconsider incompleteness of quantum theory by reductio ad absurdum. If you deny interference, then you are ignoring the null measurement by the wires, and if you deny which-way information you are denying validity of the law of conservation of linear momentum! So the experiment forces us to look at this as indication of New Physics. The lazy alternative would be to say using human language to explain the situation is meaningless, which in itself puts the validity of all of human efforts including physical laws (and that very statement) in question.

I've heard Afshar is working on his next set of experiments to explain exactly what is going on, and until then we have to live with the fact that we still don't know what the humble photon or any other quantum particle is, a wave, a particle, or both. Serious food for thought, given we don't even know what the 98% of universe is made of ;)

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The conservation of linear momentum is consistent with non-one-wayness. This is Einstein's first false argument in the Bohr-Einstein debates, that if you have a particle diffracting through a slit with a certain momentum, and you see it later at a different angular position, entangled with its momentum, the momentum is not conserved. This is false, because the slit itself imparts momentum to the particle, and if the slit has a definite position, so it can diffract, it's momentum is uncertain. –  Ron Maimon Dec 24 '11 at 14:55
    
Ron, I think you have not even read Afshar's paper on the simple crossed-beam version (arxiv.org/ftp/quant-ph/papers/0701/0701039.pdf). A picture of the experiment can be found [here][1]. See the response above for more details. –  Quantum Expert Dec 24 '11 at 18:27
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I glanced at the paper, I understand his point, I disagree with it. I repeat: there is no reason to call the photons which make the bottom image "photons which came from the top slit", even though they would be were you to remove the mesh and close one slit. The presence of the mesh means that if you look at just the photons passing through the top slit, their wavefunction would not reproduce the bottom spot. The bottom spot is a combination of top and bottom slits which is obscured because of the null measurement. –  Ron Maimon Dec 26 '11 at 2:17
    
Now we are getting somewhere. Let's do this one step at a time, when no wires are present (in the version where there is not even a lens, just crossed beams) when both holes are open, does seeing a photon at D1 mean the photon came from hole 1? If not, please explain using QM, without violating conservation laws. –  Quantum Expert Dec 26 '11 at 16:05
    
@Quantum Expert: how can you know what momenta the photons have if you are measuring their positions? –  Peter Shor Jan 23 '13 at 16:08
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Since this came up again I want to once more point to this two slit experiment,with particle detectors at the slits.

two slit

electron build up over time.

A clear interference pattern, a quantum mechanical probability distribution, appears over time, even though the slit through which the electron went through is known. It shows that it is the method used to identify the slit that destroys the pattern, not the identification itself.

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I do not want to kick this up front page again so I am making a correction here. It seems that the photo above is from electrons built up in a usual two slit experiment. It shows the build up individually. It had been positioned in the paragraph in the link about detectors at the slit , and I had been misled that it was from those experiments.arstechnica.com/science/2012/05/… –  anna v Apr 11 '13 at 3:35
    
Quote from link "However, Ralf Menzel, Dirk Puhlmann, Axel Heuer, and Wolfgang P. Schleich entangled two photons and allowed one to pass through a barrier with two slits. The entanglement enabled them to determine which opening the photon went through, but a detector on the other side still picked up an interference pattern, demonstrating light's wave- and particle-like characteristics simultaneously." –  anna v Apr 11 '13 at 3:35
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