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A delayed choice experiment with a paradox. What did I do wrong?

  1. Feb 19, 2005 #1

    Fredrik

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    Some time ago I read some stuff about delayed choice and quantum eraser experiments, but rather than reading the articles thorouhgly, I just glanced at them quickly, picked up a few ideas, and then tried to design a gedanken experiment of my own. Now I seem to have run into a bit of a problem. My gedanken experiment seems to either contain a paradox or violate the principles of quantum mechanics. I'm sure I'm making some sort of mistake, but so far I haven't been able to find it. Perhaps someone here can can help me figure this out.

    Look at the attached drawing. A "down converter" is a crystal that that emits two photons, each with half the energy of the original, when it's hit with just one. According to what I read, such crystals actually exist. I imagine that it would be difficult, if not practically impossible to build a down converter that always emits the two photons in the directions I've drawn in the picture, but I don't think it's impossible in principle.

    Imagine that we have carefully adjusted the lengths of the paths in this experiment so that the lower right detector never clicks, because of destructive interference. All the photons that reach the half-silvered mirror below the detectors will go to the left detector.

    If we imagine the same setup, but without the half-silvered mirror at the upper right, there wouldn't be any interference at the lower detectors. They would both detect the same number of photons. This is because the upper detectors would give us "which-way information" about the original photon. For example, the detector at the top of the picture will only click if the original photon was reflected at the first half-silvered mirror.

    The purpose of the half-silvered mirror at the upper right (and the mirrors at the upper left and lower right), is to eliminate the possibility to obtain which-way information that would destroy the interference. There's no way, even in principle, to determine if a photon that's been detected by one of the detectors at the upper right went through the last half-silvered mirror or was reflected by it.

    Let's move on to the paradox. Suppose that we sometimes remove the half-silvered mirror at the upper right, after one of the lower detectors has clicked. This will make it possible to determine which way the original photon went at the first half-silvered mirror, and this should destroy the interference at the lower two detectors. If the interference really is destroyed, then the lower right detector should detect a photon 50% of the times we do this (instead of 0%).

    This is retro-causation. The effect precedes the cause. The detector will click because someone at a later time removed a half-silvered mirror. If that isn't strange enough, suppose that we immediately after the lower right detector has detected a photon decide not to remove the half-silvered mirror. Then we have a paradox. If we never remove the half-silvered mirror, that detector should never click.

    Maybe it's just impossible to destroy the interference by removing the half-silvered mirror after one of the lower detectors has already clicked. But in that case, we can obtain which-way information about the original photon and still get interference. Doesn't this violate the principles of quantum mechanics?
     

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    Last edited: Feb 19, 2005
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  3. Feb 20, 2005 #2

    Fredrik

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    I had to shrink the image to upload it. That made it rather unpleasant to look at. If you'd like to see a version of that picture that's a bit easier on the eyes, you can use http://web.comhem.se/~u87325397/weird_experiment.bmp [Broken].
     
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  4. Feb 20, 2005 #3
    I spent a little time today trying to get my head around this. I need to spend some more. :smile:
     
  5. Feb 21, 2005 #4
    Hello Fredrik,

    I tried to follow your gedanken-experiment. It would be nice if you could give
    a link to the article you read, so we'd have background knowledge and understand your idea better.


    I don't agree with this part. Even if you remove the H-mirror (right top)
    and know which way the photon took at the first H-mirror, then, in my opinion, you would still get interference at the middle detectors.
    For example:
    Suppose we have no H-mirror (right top) and detect a photon at the right top
    detector. This means: We know the photon was NOT reflected at the first H-mirror, but it went through it.
    So, the photon goes straight through the first H-mirror, is downconverted,
    and then is incident on the H-mirror in the center. Because this mirror in the center is half-silvered, you don't know which way it takes (in the center) and hence still get interference.

    I think what you are considering has something to do with two-photon-interference, where you look if there's COINCIDENCE between the detectors in the center and on the top. Polarization plays a role there.
    http://www.bu.edu/qil/pdf/PRL-09-02-96.pdf [Broken]
     
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  6. Feb 21, 2005 #5

    Fredrik

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    Unfortunately I don't remember where I got most of the ideas. I know it started with an article by a guy named Ulrich Mohrhoff that I read about a year ago... I will do some googling when i get home today and see if I can find the articles I've read about similar experiments.

    I wouldn't call that interference. The interference I'm talking about will cause the left detector to click 100% of the time and the right detector to click 0% of the time (assuming that the path lengths have been carefully adjusted to make sure that happens). When things happen the way you describe, the left detector will click 50% of the time and the right 50% of the time. This happens because there's no interference between two different ways that a photon might reach the right detector (or the left detector).

    Note that after the first half-silvered mirror, every line drawn in the diagram represents a photon that's in a superposition of existence and non-existence. In the language that's often used in discussions about similar experiments, each line represents a "half" photon. The interference I'm talking about is interference between two "halves" that have the last part of their paths to the detector in common. If we know that the original photon went through the first half-silvered mirror, then there was never any "half" photons around that could interfere with each other.

    Thank you for your comments. I haven't had time to read that article yet. I will check it out later today.
     
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  7. Feb 21, 2005 #6

    Fredrik

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    I appreciate comments like this too. It's good to know that someone is at least reading my post and thinking about it. Some people might choose not to read it, because it's rather long, but they don't know what they're missing. I believe that anyone who is even slightly interested in quantum mechanics will find this experiment very interesting.
     
    Last edited: Feb 21, 2005
  8. Feb 21, 2005 #7
    You're absolutely right. I wrote some bs there. I don't know how I got to that idea. The situation with 100% to click and 0% is the same as in the Mach-Zehnder-Interferometer http://www.upscale.utoronto.ca/GeneralInterest/Harrison/MachZehnder/MachZehnder.html
    I have to admit that I've understood for the first time, why this 100% and 0% situation occurs, the site above really gives a good explanation.

    I have to think about your gedankenexperiment.
     
    Last edited: Feb 21, 2005
  9. Feb 21, 2005 #8
    Ok, I think I fully understood your paradox. My thought:
    If you don't remove the H-mirror in the right top, only detector 1
    in the center will click due to interference (like in the Mach-Zehnder-interferometer). I don't think that you can destroy
    the interference afterwards. I can't imagine how that would look like.
    So I agree with your comment:
    But on the other hand:

    That is a good question. I'd say let the Gedankenexperiment become real
    and test it in the lab.
    :tongue:
     
  10. Feb 21, 2005 #9
    I think for the detectors in the right top the probabilities
    should also be 100% and 0%, because the outer mirrors also
    represent a Mach-Zehnder interferometer.
    I just noticed that your experimental setup
    consists of two Mach-Zehnders, an inner and an outer interferometer.
     
  11. Feb 21, 2005 #10

    Fredrik

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    We can choose those percentages to be whatever we want them to be, simply by making the right path a little bit longer than the left path.
     
  12. Feb 21, 2005 #11
    It's an interesting addition to the experiments to have as Edgardo says, and I also noticed, essentially two interferometers in inner and outer loops. :smile:

    I've been drawing the experiment as a superposition of two alternative histories which interfere.

    After the first H mirror, one of my drawings is the history which follows the path of photon up and then the paths of the two photons created this way, while the other drawing is the history which follows path of the photon right and then the paths of the two photons created that way.

    I'm imagining the drawings as then interfering and seeing if that gives me some clues.

    I'd guess the photon goes around the inner loop by two paths in superposition has inteference effects with itself whatever else happens.

    What is happening to the photon in the outer loop is what I'm wondering now.
     
  13. Feb 21, 2005 #12

    reilly

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    Fredrik-- the quote below greatly puzzles me.

    If we imagine the same setup, but without the half-silvered mirror at the upper right, there wouldn't be any interference at the lower detectors. They would both detect the same number of photons. This is because the upper detectors would give us "which-way information" about the original photon. For example, the detector at the top of the picture will only click if the original photon was reflected at the first half-silvered mirror.

    My instinct is to say "higher detectors" should replace "lower detectors" in the first sentence, there's a typo in other words.

    But, perhaps not. In which case note that you have a 'sort-of" standard interferometer with the lower detectors. That is to say, interference will be alive and well at the lower detectors no matter the state of the upper apparatus. This is certainly true for classical E&M, and for QM as well. the Basic idea can be shown by looking at thr quantum fields at the measurement points. Both classical and quantum fields share the same spatial form. So at Lower Detector 1 (left), or LD1, have for 2 photons, with momentum k, a wave function (poor nomenclature)

    EXP (-k{x11 + x21}) where x11 and x21 are the photon coordinates at LD1, fully symmetric in the two photons. Interference is demonstrated by varying x11 + x21. If the sum is pi/2 there is full destructive interference, and so on. Two channels feeding a detector guarantees interference. After the fact causality? I don't think so.

    Regards,
    Reilly Atkinson
     
  14. Feb 22, 2005 #13

    Fredrik

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    Nope, that's not a typo. It's rather obvious that there will be no interference between "half" photons at the upper detectors without the half-silvered mirror at the upper right, but that's not the point. The point is that if that mirror is gone, the detector at the top of the picture can only click if the orignal photon was reflected by the half-silvered mirror near the bottom of the picture. If the original photon is reflected there, rather than going into a superposition of being reflected and passing through, then there can't be any interference at the lower detectors.


    I'm not against the idea of "after the fact causality", but in this case it seems to lead to a paradox, and I'm against paradoxes. :smile:
     
  15. Feb 23, 2005 #14
    Hello Fredrik,

    I've read about quantum erasers. And read these websites:
    http://www.lns.cornell.edu/spr/1999-03/msg0015208.html

    and especialy this one:
    http://grad.physics.sunysb.edu/~amarch/ [Broken]

    It mentions, that the which way information is erased AFTER it made click. (at least that's how I understood it) In your gedankenexperiment I now think that the interference is destroyed, that is you don't get any interference although you take away the mirror AFTER it made click (at the center-detectors). However, that's my first thought after having read about these quantum erasure experiments.

    I will let you know, if I get any new insight into this.

    Regards

    Edgardo

    P.S. Check out this program about quantum erasure.
    http://www.physik.uni-muenchen.de/didaktik/Computer/interfer/interfere.html
    Note that you can rotate the polarizers, which I at first didn't get.
     
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  16. Feb 23, 2005 #15
    Talking of erasing the past, in Roland Omnes's Understanding Quantum Theory, he mentions the ideal von Neumann experiment.

    A particle is emitted and can go down one of two channels. There is a detector in one channel. If, after a certain time, the particle is detected then you know it went along the channel with the detector and if the particle is not detected then you know it went along the channel without the detector.

    So far, so simple. :smile:

    Now here comes the interesting part as, if you arrange it so that the two channels curve and later join up into the same direction, then when the particle reaches the point at which the channels join up then the detector result will go from "detected" or "not detected" into a superposition and "don't know" Schrodinger Cat state even if the particle is now a huge distance away from the detector! :bugeye:

    It's EPR and Schrodinger's Cat in one simple, fun package. :biggrin:

    Decoherence keeps this lunacy from happening in the real world, thankfully. :wink:
     
  17. Feb 24, 2005 #16

    Fredrik

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    This doesn't seem correct to me. The detector should go into a superposition of "detection" and "non-detection" even if you don't join the channels.
     
  18. Feb 24, 2005 #17
    some thoughts on the thought experiment

    This thought experiment has captured my interest, and I have been thinking about it for some days now. I do not know much about "quantum erasure" and ((S)P)DCs, and so I have only been able to take the setup at face value on the basis of heuristic considerations.

    The following "rule" appears to remove all elements of inconsistency:

    Interference effects will occur in the signal only if the 'decision' to erase the "which-way" information occurs a spacetime event which lies in the causal past of the signal detection event.

    If one would like to argue that this "rule" does not apply, namely that interference effects in the signal exist even when

    (i) the 'decision' to erase and signal detection are spacelike separated

    or, much more drastically,

    (ii) the 'decision' to erase lies in the causal future of signal detection ,

    then, in case (i) we can arrange for faster-than-light signaling, and in case (ii) we can arrange (with some abuse of the term 'decision') for an action performed now to 'undo' an event which has already taken place (... whatever that could mean).

    I think that this "rule" and the ramifications of its violation for cases (i) and (ii) above apply quite generally to all erasure-type scenarios, and not just to the thought experiment at hand.
     
  19. Feb 24, 2005 #18
    I'll quote a couple of relevant parts:

    This shows the most problematic aspect of an ideal measurement: the data it yields are not obtained once and for all. Apparently lost interferences can be regenerated later in the measuring device by an action on a distant system (the particle). There is no possibility for considering experimental facts as being firmly established. One may see the result as a particularly vicious consequence of EPR correlations or express it by saying that Schrodinger's cat cannot be dead once and for all, because evidence for his survival can always be retrieved.

    Another part, explaining a diagram of the apparatus:

    A von Neumann ideal meaurement does not produce a factual datum. The particle shown in Figure 19.1 is brought along direction 3, whatever its initial direction was. When the two paths join together, the state of the measuring device becomes a pure state, with no memory of a previous mixed state where the result of a measurement could be read.

    I can write up the full thing if you want. It's just over a page long. I'd just need to learn to do equations on this board and attach a diagram.

    I'm still learning quantum theory, so maybe I'm messing up somehow. :smile:
     
  20. Feb 24, 2005 #19

    Fredrik

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    I'm glad you joined the discussion Eye_in_the_Sky. I've read some of your previous posts and I think they're all very good. This one is too. You provided me with a piece of the puzzle, and helped me understand a few things better than I did before.

    You said "in the causal past of" when most people would have said "at an earlier time than". I like that. The "causal past" is of course the set of spacetime events that are at an earlier time for all observers.

    However, I have a feeling that the rule you're suggesting is too simple. It seems to me that the rule implies that the "collapse" of the wavefunction is something that happens with a finite speed. In EPR type experiments, the collapse doesn't behave this way, and I think that's a good reason to expect that it won't in this one. I don't expect a phenomenon like "wavefunction collapse" to behave differently in different experiments. (I'm not sure why. I'm using a lot of intuition here).

    I have a few ideas of my own about how to explain this experiment...

    When we write down a path integral expression for the amplitude of a particle's propagation from event A to event B, the path integral is a sum over all possible paths through space-time (i.e. all possible world lines). It's not just a sum over all paths through space. That means that we in general must consider paths that correspond to faster-than-light (FTL) and slower-than-light (STL) propagation. The reason we often just talk about paths through space is that the contribution from FTL and STL paths is negligible in comparison with the contribution from paths that look like classical world lines. This is the case e.g. in a standard double-slit experiment.

    I suspect that in this experiment, and in others that appear to lead to paradoxes, a careful examination of the path integral expression would reveal that the contribution from FTL paths is large.

    The reason I like the FTL paths here is that they seem to provide a "mechanism" (at least on a mathematical level) that lets the "half" photons on the outer paths in the picture "probe" the region in front of them and "determine" if the half-silvered mirror is in place.

    When one of the lower detectors is about to click, the "half" photons on the outer path have already "determined" if the half-silvered mirror is in place, so the "half" photons on the inner path will "know" if they're allowed to go to the right detector or not.

    If my explanation is correct, we can rule out your case (ii), but definitely not case (i). We could send signals faster than light from the location of the half-silvered mirror at the upper right to a recipient near the lower detectors. This could be a serious problem for causality (or my explanation :smile:), but I don't think it has to be. I will have to think some more about this to be sure

    If my explanation is correct, the decision to erase the which-way information can be made "after" the signal detection event. However it can't be made at an arbitrarily late time. The decision event must be either in the causal past of the signal detection event, or spacelike separated from it.

    What do you think?
     
  21. Feb 25, 2005 #20

    Chronos

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    It's elegant, but wrong.
     
  22. Feb 25, 2005 #21

    Fredrik

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    That's certainly possible. The possibility of sending signals between events that are spacelike separated bothers me, and might be a good reason to dismiss this explanation.

    If you can say that it's wrong with such certainty, can you also tell me (and everyone else who follows this thread) what the correct solution is?
     
  23. Feb 25, 2005 #22
    Hello Fredrik,

    I have taken some time to surf the web and look at various quantum erasure experiments.

    In some experiments the topology of the setup requires that the erasure event take place in the causal past of the signal detection event. So, in these cases there is no potential problem.

    In other experiments, the setup topology allows for the erasure event to occur at spacelike separation from signal detection. So, here we might suspect the possibility of being able to use the apparatus for FTL signaling. But this is not so. Why? ... Because in these scenarios the interference at the signal can only be inferred when the associated data set is compared with another data set generated at (or after) the eraser. In other words the "nonlocality" is to be found only in the correlations – just like in Bell.

    Also, in these experimental setups (where erasure and signal detection are spacelike separated) it is possible, at least in principle, to arrange for the erasure events to not just be spacelike separated from the signal detections but even to lie in their causal future. However, no problem ensues here. (Even though the experimenter at the eraser can receive a message containing the signal detection results before the 'decision' (to erase or not to erase), that data cannot be used to create a paradox (since we still need the other data set ... which does not even exist yet).)


    So, now I see that the thought experiment which you have proposed is of a fundamentally different character from any of the typical erasure experiments. Hmm ...

    I think that before I attempt to understand better just how this thought experiment can have the purported results, I had better review the entire setup again asking myself, "What did I do wrong? ... when I thought it was right."


    P.S. I am not sure I understood your explanation in terms of the path integral. In particular, how is it that case (ii) is ruled out. If your explanation is valid, then why can't the outer "half" photons probe the future all the way up to the causal future of the signal detection event?
     
    Last edited: Feb 25, 2005
  24. Feb 25, 2005 #23

    Fredrik

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    That's why it's so interesting (or just plain wrong).

    I see what you mean. If virtual photons "probe" spacetime, they should be able to probe all of it except the causal past. The only way I see to save my explanation is to postulate that only FTL virtual particles can probe spacetime. With that additional assumption, we can rule out (ii) but not (i).
     
  25. Feb 26, 2005 #24

    Fredrik

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    I regret my choice of words in the previous post (#23). An additional postulate is the last thing we need. If it makes sense to think of FTL paths (or FTL virtual photons) as probing the surrounding region of spacetime, it must be possible to prove this by evaluating the path integral explicitly (or by using the methods of perturbative quantum field theory).

    I still have strong doubts about my own explanation. If it is correct, it is possible to send a message consisting of a single bit over a spacelike interval. It will of course fail 50% of the time, but that doesn't seem to be enough to avoid the usual paradoxes associated with superluminal signals.
     
  26. Feb 27, 2005 #25
    [EDIT: There is some need to edit this post. All such changes will appear in "red" (or some other color).]

    [NOTE: A second round of editing now appears in "blue".]

    Hello Fredrik.

    About that "rule" (which I suggested back in post #19), concerning which you indicated:
    Yes, yes. I agree. When I wrote down that "rule", I had in mind (ever so vaguely) that there could be a way to 'play around' with the idea of 'a decision' and thereby somehow avoid the conclusion which you have pointed to. But that (vague) 'reasoning' of mine turns out to be quite fallacious. Moreover, at that time, I was not at all appreciating the severity of taking such a "rule" seriously.

    ... And even more delusional was my thought:
    ............................. :surprised

    That being said, we still have this thought experiment to deal with; and if our current understanding of it is correct, we are then – of course – FORCED to take the "rule" (or at least a slightly weaker version of it) SERIOUSLY!

    However, as I indicated in my last post, I was not yet ready to do that, opting instead for a more careful reconsideration of the setup.

    On the other hand, you yourself have elected to examine this very question, posing it as follows:

    How can one 'derive' such a "rule" (or its weaker version) in terms of the mathematical machinery of path-integrals?

    In the course of your deliberations you arrived at:
    The "rule" (even its weaker version (which allows for FTL signaling)) is not consistent with conventional Quantum Mechanics. Therefore, the only possible way to 'derive' it – if at all! – would be through the agency of NEW postulates.

    For me, such a prospect is more than reason enough to go back and look for some misconception in the original analysis of the thought experiment.

    Let us do that now.
    ________________

    The hint of just where our misconception lies is to found in the very words:
    Indeed, in every "quantum-eraser" scenario, the which-way information is "erased" by means of:

    (a) the removal of a "measuring instrument" from the setup ,

    or

    (b) the insertion of a "measuring instrument" into the setup .

    [EDIT: My attempt to generalize has failed. In all of the type-(a) scenarios which I have considered, the which-way information is not "erased". Rather, a which-way 'tagging' scheme has been "removed" from the setup.]

    [EDIT: In connection with case (b), the prescription that the inserted "element" be of the "measuring-instrument" genre is required only when:

    the so-called "erasing" action takes place at the site of signal "counterparts" which exist in an appropriate state of quantum entanglement with the signal.]


    By the term "measuring instrument" I mean some entity (or entities) whose 'action' on the quantum system is represented in the formalism by a PROJECTION operator.

    Now, take a look at our alleged "eraser". It is nothing but a half-silvered mirror. The 'action' of such an entity is represented by a UNITARY operator. That is to say, in our current scenario, this 'action' is strictly local affecting only the outer "half"-photons while leaving the inner ones completely undisturbed. In other words, by inserting the upper half-silvered mirror into the experimental arrangement, the which-way information is not destroyed, but merely (and this point I think requires further consideration) rendered 'inaccessible'.

    [The last sentence in the above paragraph fails to properly convey the required idea. I would like to rewrite that sentence as follows:

    In other words, by inserting the upper H-mirror into the experimental arrangement, we eliminate the which-way information for the outer "half"-photons ... but this has no effect on the inner correlated counterparts whatsoever because there is NO COLLAPSE induced by that H-mirror.

    As for trying to give an equivalent characterization of this point in terms of such concepts as "destroyed" or "rendered inaccessible", further consideration is required.]

    Consequently, it must be the case that the lower detectors always click with 50:50 likelihood, even WITH the upper half-silvered mirror in place.

    We therefore do not have a so-called "quantum-eraser" scenario here.
     
    Last edited: Feb 27, 2005
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