Many Worlds, Many Headaches

So, I've put myself into a position where I need to spend a substantial amount of time thinking about weird foundational issues in quantum mechanics. This has revealed to me just why it is that not that many people spend a substantial amount of time thinking about weird foundational issues in quantum mechanics.

i-d92a65195a7c05474aa7445a78a6f0fc-machzehnder.gif Let's consider a Mach-Zehnder interferometer, of the type shown in the figure at left (click the figure to see the original source). A photon (or an electron, or an atom, or any quantum particle) enters the interferometer at the lower left, and is split onto two paths by a 50-50 beamsplitter-- half the light is reflected up, half is passed straight through. Those two paths separate, then are brought back together on a second beamsplitter, with half of each beam being transmitted, and half reflected, so that light from each path falls on each of the two detectors at the upper right. Those two beams interfere with each other, and determine the outcome of the experiment.

Now, if you do this with a single photon, you will get a single click, either from detector 1 or detector 2. If you repeat the experiment many times, you'll build up some number of counts in each of the detectors, and if you shift one of the mirrors slightly (thus making one path longer than the other), you'll find an interference pattern in the fraction of the total counts on each detector. At some positions of the mirror, you'll find absolutely no counts on detector 1, and in other positions you will never see a count on detector 2.

So far, so good. Now, we can start to make this weird by replacing the second beamsplitter...

Let's say we replace the second beamsplitter with a variable beamsplitter that can be set to either the 50-50 configuration of the original set-up, or a 100-0 configuration, where it transmits absolutely everything (this is relatively easy to do, using polarization optics, but the details aren't important). In the 100-0 configuration, the only light that reaches detector 1 is light that was reflected up at the first beamsplitter, and the only light that reaches detector 2 is light that was transmitted at the first beamsplitter. In this configuration, it doesn't matter where you position the mirrors, because there's nothing for the light to interfere with-- no matter what you do, you get half of your counts on detector 1, and half on detector 2.

The usual description of this is in terms of particle and wave behavior. In the first case, with the 50-50 beamsplitter, we have light behaving like a wave, with some of the light going on each path, and recombining to interfere with itself. Which detector records the count depends on both possible paths to the detector. In the second case, with the 100-0 beamsplitter, we have light behaving like a particle, with the photons choosing a definite path at the first beamsplitter, and staying on that path. Which detector records the count depends only on what happens at the first beamsplitter.

Now, we can make this even stranger, by allowing the switch between configurations to be very quick (again, this is relatively easy to do experimentally). And then we set it up so that we determine whether we have a 50-50 or 100-0 beamsplitter only after the photon has passed the first beamsplitter. This isn't trivial, but it's been done by a few different groups, most recently the Aspect group in France. What you find in this case is remarkable-- if you repeat the experiment many times, and keep track of the configuration, you find that when the beamsplitter was 50-50, the counts add up to give you an interference pattern, and when it was 100-0, you don't see any interference. Even though the configuration isn't set until after the photon has passed the first beamsplitter, somehow, it still turns out to have gone both ways when the second beamsplitter is 50-50, and only one way when the second beamsplitter is 100-0.

Spoooooky.

Now, here's the part that's bugging me: How do you think about this in the Many-Worlds Interpretation? The Copenhagen-ish version of the experiment is fairly simple-- you have a wavefunction that describes the whole apparatus and both light paths, and when you detect a photon, it collapses into one of the possible states. It's weirdly retroactive, but the collapse happens at a definite time, the moment when the detector records a count.

In the Many-Worlds picture, you ought to end up with four different universes, corresponding to the four possible paths the particle could take: reflected up, to detector 1; transmitted down to detector 2; both ways, ending up in detector 1; and both ways, ending up in detector 2. But when does the split happen? Does it split into four parts when passing the first beamsplitter? Two at the first beamsplitter, and two again at the second? Three at the first (one for each definite path, and one "both ways") then one of those splitting at the detectors? All four only when the light reaches the detectors?

I sort of lean toward a decoherence-based interpretation of the interpretation (if you follow), so I think that the right way to look at it is that you get all four after the light passes the first beamsplitter, and then those different paths experience decoherence as the photons move through the interferometer. They end up being completely separate by the time the photon hits one of the detectors, and you find yourself in one of the four through whatever mystical mechanism it is that usually causes you to see only one of the possible paths.

But then, that doesn't really seem to account for the delayed choice of what configuration to put the second beamsplitter in. So maybe you really want the 3-way split at the first beamsplitter, and then one of the three splitting at the second. Or maybe the two 2-way splits.

But then, both of those just seem weird. So maybe the way to do it is to go for the full-on Heisenberg treatment, in which nothing exists until it's detected, and say that the universe splits into four at the instant of detection. But then, that doesn't really seem to fit with the "decohering branches of a single unitarily evolving wavefunction" thing, which is sort of the whole point of Many-Worlds in the first place, and, well...

Well, now I know why more people don't write books about this stuff.

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This is **so** not my field :) but I was just this morning wondering idly about how probability arises in a Many-Worlds interpretation. Yes, I know about the Born rule, but that still doesn't seem to explain _why/how_ probability emerges, just that it's calculable.

Any insight? Do you buy Deutsch's 'differential thickness of branches in a branching multiverse'? Please bear in mind that - see sentence 1 - this is _utterly_ nothing I have any expertise in :)).

Is there an arrow missing at the top right, or am I misreading something?

As with Ewan, this is far, far from my field, but I wonder sometimes about the usefulness of "real-world (intuitive)" interpretations of this kind of thing. After all, we have no real intuitive understanding of quantum phenomena because we don't really experience them. Trying to think of light as waves or particles or wave-like particles and so on is really making an inapt anaology. Light isn't wave/particle in our intuitive sense, it's its own thing. Trying to understand what really happens to a photon in this kind of case is, in a sense, making billiard balls out of them.

But, what do I know? Maybe some day something useful will come from trying to explain it. So, carry on.

Gyah. I had to write a series of papers basically explaining the physical justification of all this to a CS/Math prof who was doing research on the computational aspect when I was in grad school.

Stuff still makes my head hurt.

And, similar to Ewan, my response would be, "Go bug David Deutsch. He's got all the answers for the many-worlds interpretation. Just ask him. He'll tell you. Again and again and again."

By John Novak (not verified) on 26 Sep 2007 #permalink

Is there an arrow missing at the top right, or am I misreading something?

Looks ok to me. Where do you think there ought to be another arrow?

You may be right that the problem here comes from trying to think about the photons too concretely. You still need to have four possibilities-- (50:50, 100:0)(detector 1, detector 2)-- but it may be wrong to think about it in terms of paths for the light. Maybe all the action comes at the second beamsplitter and the detectors.

Still, it feels like something ought to happen at the first beamsplitter...

This is an example where the horrible naming of many-worlds is the problem. All many-worlds is is just ordinary quantum mechanics. (Everett called it the relative state interpretation.) Splitting worlds and the like is just words people use to describe the unitary evolution of the wavefunction. If you want, you can think of decoherence as the physical basis of worlds splitting. Then, worlds only split when a measurement is encoded in a macroscopic instrument.

I thought there should be a reflected beam from the top left mirror to the second beam splitter to detector 2. The diagram seems to show only transmission.

I don't think you're taking Feynman integrals seriously enough. When you change the first beamsplitter (even "after the photon has passed [it]"), then you're changing some of the paths that the photon is (or, if you prefer, could with some probability be) taking. Talking about a definite time the photon passes the first beamsplitter is assuming the photon takes just one path (and stays a photon, too, but let's not go there).

We have temporal causality built into our language. Alas.

On first glance, I couldn't make full sense of the Science article (I'm a materials scientist, not a physicist). At least not the full mechanics behind how they switched between the two configurations. I need to go reread your primer on polarized light...

I do have one question though: How are they sure that the photon truly went "through" the first interferometer before the QRNG decided which configuration to create? I guess they use classical physics, i.e., they assumed light was traveling at the speed of light, so they knew whenabouts it was supposed to pass through the interferometer. What if this approach is wrong and light actually propagates differently? At least on this small scale.

I mean, we only know where a photon is or has been by detecting it (CHAD: IS THIS RIGHT?), and the experiment does not detect the generated the photon until everything has been put into place.

By Harry Abernathy (not verified) on 26 Sep 2007 #permalink

I agree with comment #8. The spookiness is built into path integral formalism.

And Chad's observation that foundations aren't studied much is very true. Putting this problem in the form of the 2 slit experiment, you are not allowed to interpret the intermediate states in an experiment, but instead only the initial and final.

My own favorite version of QFT takes that to the next natural conclusion, that if the intermediate states cannot be interpreted as specific particles at specific points, neither should the initial and final states. That is, one should work entirely with virtual particles.

As it turns out, spinor states are insufficient for virtual particles, instead one must use density matrices, so this all gets back to the density matrix formalism.

That the Mach-Zehnder interferometer experiment works in the lab is even stranger than you have considered.

The photon interacts with the mirrors and the beamsplitters:

Any potential path of the photon has to reflect off one of the mirror and this momentum change in the photon is balanced by a momentum change in the mirror.

Ditto for any path the photon took that reflects off a beamsplitter. Transmitting though a beamsplitter is actually a momentum exchange at each air/glass interface.

The consequence of this argument can be stated in different ways:
* All of these momentum changes ought to entangle the photon with the state of the equipment and prevent any quantum interference.
* By simply watching the two mirrors you ought to know which path each photon took.
* The photon never appears to take both paths.

But since the experiment works without considering this issue, it means that something interesting is happening. I claim that if the equipment were at absolute zero degrees (ground state) then the experiment ought to fail. So the finite temperature of the equipment is washing out any ability to measure the momentum exchanged with the photon along the photon's path.

By Chris Kuklewicz (not verified) on 26 Sep 2007 #permalink

Some people know just enough about modern physics to make bad jokes.

By Johan Larson (not verified) on 26 Sep 2007 #permalink

If you want, you can think of decoherence as the physical basis of worlds splitting. Then, worlds only split when a measurement is encoded in a macroscopic instrument.

What exactly qualifies something as "macroscopic" here?

What exactly qualifies something as "macroscopic" here?

Nothing exactly. As there are more and more degrees of freedom in the measurement apparatus, it becomes more and more difficult to disentangle the apparatus from the stuff being measured.

Yes, that sounds weird, but: if you imagine (whatever that means) that the first BS splits the beam into two waves, and just deal with "waves" until there's an actual "collapse" at the detector, it makes sense. AFAICT you don't really have to imagine anything about "which way the 'photon' went" at all. Remember those theoretical treatments and experiments (bunching, anti-bunching, etc.) in which light could be treated (in many contexts) as if a classical wave, but the absorber (in effect, atomic levels) was the quantizing factor? But OTOH, what if you had a double-sided full mirror at the second BS instead of a 0-100 model - then, the wave can go to either A or B channel, right? But despite the wave being in two parts, it will somehow all get together at one or the other detector (and even if say, one leg was x-pol and the other was y-pol, etc.) - that looks more strange to me. "God only knows" about other universes.
I spend a substantial amount of time thinking about weird foundational issues in quantum mechanics (I used to be in top Google hit for "quantum measurement paradox"), and do I have a deal for you. If you want something really strange and "after the fact," please consider my following proposal: In a MZ interferometer, insert a gray filter G into leg L2. Given a traditional stream of light, that alters the amplitudes delivered to the beam-splitter/recombiner R. With a 25% transmitting filter, the L2 amplitude is 0.5 of that in L1 (which itself is sqrt (0.5) of the original pre-split input.) Hence, with symmetrical R, we get an output mix rather than all A channel output. We can adjust R to a compensatory split so that output is again all A channel. Using individual photons, the statistics should be the same. FEL optical physicist Michelle Shinn of J-Lab agreed with me that's so, despite the weirdness of the photon's wave function in L2 being attenuated by the chance that it could have been absorbed, even if it wasn't (well, superposition of absorption and not-absorption in the dye molecules in the filter, etc, right?) Also, as G gets darker, this has to be the limiting factor approaching the results of an opaque stop in L2. But what happens if we can find out whether a photon has been absorbed in the filter?
Consider the opaque stop: the stop clearly "reallocates" the WF all into L1, in a manner akin to the Renninger negative result problem, even though no actual "measurement" is taken. But there, a photon will just never get through. However, G may or may not absorb a photon, something we can in principle check on (There are semi-transparent optical detectors, no? Just consider film for example.) Now, while G is still "deciding" (in a state of superposition) whether it will absorb or not, it makes sense to consider the L2 wave to be attenuated relative to L1. Maybe that's the normal time scale to allow interference in R before that happens. But, after a certain time, if we check G carefully to look for evidence of absorption, it should be settled: absorption or not. If it did, there's no paradox. But if we find "no absorption," why in the world should the L2 wave continue attenuated? The measurement result was "no" for G, so there is no longer "a chance" that the photon might end up there. The filter might as well have been clear glass, right? If so, then the interference at R would be different (it would follow normal equal-balance rules instead.)
The really weird thing is, that reallocation should take place as soon as the absorption/detection issue is settled. If so, we could manipulate the pattern of hits (with sequential photon shots) at the output by looking for evidence of absorption in the filter, which would start rearranging the WF as per Renninger etc. In principle, there's nothing to stop this from being a true FTL signal, since manipulating G (or perhaps the distance to R) causes noticeable effects (not distant signal correlations) at R. Sure, that's problematical, but you can't just blow off the supposed effect on the WF of the negative measurement in G, can you? Have fun.

(I just put this up on sci.optics, sci.physics, etc. I plan to put it up on my blog later today.)

I already got a reply at sci.optics,physics which is interesting, and I thought of something else of relevance: what if we intermittently place a full absorber in leg two, instead of a gray filter? The long-term photon statistics coming into each side of R (second beamsplitter) are the same, which should mean the same amplitude of wave functions. However, the results really must be different (from being a mixture instead of a coherent superposition?):

"Ben Rudiak-Gould" wrote in message news:fdgtq6$96i$1@gemini.csx.cam.ac.uk...
> Neil Bates wrote:
>> But if we find "no absorption," why in the world should the L2 wave
>> continue attenuated? The measurement result was "no" for G, so there is no
>> longer "a chance" that the photon might end up there. The filter might as
>> well have been clear glass, right? If so, then the interference at R would
>> be different (it would follow normal equal-balance rules instead.)
>
> I think this is the core of your question. There's nothing peculiarly
> quantum mechanical about this situation. Classically, your corpuscle goes on
> path 1 with 1/2 probability, is absorbed on path 2 with 1/4 probability, and
> is not absorbed on path 2 with 1/4 probability. Suppose we know it's not
> absorbed; then the correct way to update our probability estimate is to toss
> out the excluded possibility and renormalize the others. This tells us that
> the corpuscle is on path 1 with 2/3 probability and on path 2 with 1/3
> probability, which is the correct answer: over many repetitions of the

Well, I see where you're going with that, and it seems the orthodox view (based on net averages of hits being equivalent to the wave function.) Yet I still have a problem, which is why the wave function should always be equivalent to the long-term statistics. I still don't see why getting a "no absorption" in the gray filter would maintain the original wave proportions. Why not equal in both? Why does the wave carry the information that at other times, there are absorptions, when there really wasn't one? Yes, sometimes atoms in the filter absorb photons, and sometimes they don't. If they don't, it's hard to see what effect they would have at all.

Here's a brain teaser to make you rethink this problem: Suppose that instead of a gray filter using atoms or molecules to absorb, we intermittently slipped in an opaque shield 1/4 of the time a photon shot was scheduled. Then, there's no way the effect at the recombiner could represent the chance that sometimes the opaque shield is there! It's a macroscopic object, and so during the times it is "out," the interference at R must be normal (unless you are getting *really* weird on me...) When it is in, half the time it won't absorb anyway and there will be some hits in either A or B output. (BTW That's the same regardless of whether we actually measure absorption in the barrier or not.)

However, the average effect of that and of the gray filter (with constant passing amplitude of 0.5) are not the same. (Consider that with a true 25% gray filter, a balancing R setting of transmit 80% and reflect 20% would always send interfering beams out the lower A channel. However, with intermittent block, we get 0.75[0.5*sqrt(0.8) - 0.5*sqrt(0.2)] + 0.25*0.5*0.2 = 6.25% for channel B (the first term is the 75% of the time the block is out, and so there's destructive interference of unequal beams to Ch. B, and the second term is for the 25% is with the block there, when half the time a signal goes to R at all with 20% chance of reflection into B.)

So, why should a "filter" work so differently from having a barrier there half of the time, which produces the same long-term statistics at R? Well, being a microscopic absorption possibility exists in a state of superposition, I can imagine the photon wave function exhibiting some correspondence to that (but I still wonder what happens when non-absorption is certain.) The long-term photon statistics coming into each side of R (second beamsplitter) are the same, which in conventional theory should mean the same amplitude of wave functions. However, the results really must be different. The barrier just can't reflect the long-term statistics coming in to R through leg two as an actual wave amplitude - it must instead be a shifting amplitude, sometimes full and sometimes nothing. Here's where we see that the classical concept can't just be transferred over to the quantum amplitudes. I think this alone is interesting, even if we don't get weird results from measuring non-detection in a gray filter. Maybe this is the result of sending a mixture to R instead of a coherent superposition.

(PS: I'd rather say B2 for beamsplitter two instead of R, so I can refer to reflectivity more easily.)

> experiment, there will be twice as many unabsorbed particles on path 1 as on
> path 2. Quantum mechanically it's the same. If you discover (via a
> measurement) that the particle wasn't absorbed, the remaining possibilities
> keep their relative amplitudes.
>
> -- Ben

Maybe I didn't understand you right. Most of the time when I read stuff about quantum mechanics I see something wierd, but all I saw here is the fact that you don't know which configuration your second lens is in until after the experiment. Are you saying it's not in any configuration until after the experiment -- or just that you haven't observed its configuration yet.