![\"i-66f0286010718d4ed1413f1d870e5d14-sm_fig4-2.jpg\"](\"http:\/\/scienceblogs.com\/principles\/wp-content\/blogs.dir\/467\/files\/2012\/04\/i-66f0286010718d4ed1413f1d870e5d14-sm_fig4-2.jpg\")
<\/p>\nI seem to have been sucked into a universe in which I’m talking about the Many-Worlds Interpretation all the time, and Neil B keeps dropping subtle hints, so let me return to the whole question of decoherence and Many-Worlds. The following explanation is a recap of the argument of Chapter 4 of the book-in-progress, which will cover the same ground, with cute dog dialogue added.<\/p>\n
The central question here is what sorts of things count as producing a “new universe” in Many-Worlds. The scare quotes are because I’ve come around to the opinion that the whole “parallel universe” language does more harm than good for giving people an idea of what’s really going on. Hopefully, I’ll make it clear why as we go on.<\/p>\n
Anyway, to be concrete about it, let’s consider a really simple quantum system that may or may not involve the creation of a “new universe”: we have a single photon, hitting a 50-50 beamsplitter. There are two ways to talk about the photon after the beamsplitter: as a wavefunction with two components corresponding to the two different possible paths, or as a particle that has taken one of the two paths. Quantum mechanics tells us that this is properly described by a wavefunction in a superposition of the two paths, but everyday experience tells us that we only ever detect the particle on one path. Somehow, the quantum superposition has to evolve into the classical mixture. How does that happen?<\/p>\n
We can’t give a sensible answer to this question without having some way to distinguish between the two possibilities. So the first question we have to ask is, how do we know which situation we have?<\/p>\n
<\/p>\n
Well, the signature of quantum behavior is interference, so the only way to really tell which case we’ve got is to do an interference experiment, bringing the two paths back together. There are lots of ways of doing that, but let’s think about a Mach-Zehnder Interferometer:<\/p>\n
<\/p>\n
We take a couple of mirrors and steer the two possible photon paths back together, and use a second beamsplitter to combine those two paths on two detectors. If we’re dealing with waves, this should result in some interference that will depend on the relative lengths of the two paths. The probability of finding the photon at Detector 1 will range from 0% to 100%, and the probability of finding the photon at Detector 2 will range from 100% to 0%, in a complementary manner.<\/p>\n
If we’re dealing with a particle-like photon that definitely took one of the two paths, on the other hand, the probability of finding it at Detector 1 is 50%, and the probability of finding it at Detector 2 is 50%, no matter what you do with the path lengths. In this case, there’s no interference– there’s a 50% chance of the photon taking each of the two paths, and then a 50% chance of being directed to each of the two detectors.<\/p>\n
So, if we want to know whether we’re dealing with wavefunctions or definite particle trajectories, we need to do an interference experiment. But we’re talking about single photons, here– how do you get an interference pattern out of a single photon? A single photon will produce a single “click,” which is the canonical term for a detection event, even though nobody really uses detectors that make clicking noises any more. The photon will be detected at one detector or the other, and that’s all the information you get.<\/p>\n
The only way to detect interference of a single photon is to repeat the experiment many times, each time sending only one photon in. You record which detector “clicked” for each photon, and slowly build up a measurement of the probability of finding it at each detector. You can also vary the relative path length, repeating the experiment many times at various different lengths, and in this way you’ll trace out the probability distribution as a function of mirror position. If you’re dealing with wavefunctions, you’ll see an interference pattern ranging between 0% and 100% probability for each detector, and if you’re dealing with particles, you’ll find a constant 50% for each detector.<\/p>\n
So what do you find if you do this? Well, if you’ve set your interferometer up properly, with short path lengths and stable mirror mounts and all that technical stuff, you should see an interference pattern. So, a beamsplitter gives us a wavefunction in a superposition of two states.<\/p>\n
How does this turn into two photons that each take a single path, though? To see that, let’s think about making our interferometer huge:<\/p>\n
![\"i-c61ecb3e3b274fbd863021d5f7511b94-sm_fig4-3.jpg\"](\"http:\/\/scienceblogs.com\/principles\/wp-content\/blogs.dir\/467\/files\/2012\/04\/i-c61ecb3e3b274fbd863021d5f7511b94-sm_fig4-3.jpg\")
<\/p>\n
The wavy lines indicate a really long distance, say, a hundred kilometers, passing through air the whole way. What do we see then?<\/p>\n
Well, if we’re talking about a long distance in a turbid medium, there’s going to be a phase shift. If you think in terms of waves, there are going to be interactions along the way that slow down or speed up the waves on one path or the other. This will cause a shift in the interference pattern, depending on exactly what happened along the way. Those shifts are really tiny, but they add up. If you’re talking about a short interferometer in a controlled laboratory setting, there won’t be enough of a shift to do much, but if you’re talking about a really long interferometer, passing through many kilometers of atmosphere, it’ll build up to something pretty significant.<\/p>\n
That phase shift changes the interference pattern. If the probability of finding the photon at Detector 1 is 100% with no interactions, it could be, say, 25% with the right sort of interactions. Or 50%. Or 75%. Or 0%. The exact probability depends on exactly what happened to the piece of the wavefunction that traveled on each path.<\/p>\n
And here’s the thing: that shift is also random. What you get depends on exactly what went on when you sent a particular photon in. A little gust of wind might result in a slightly higher air density, leading to a bigger phase shift. Another gust might lower the density, leading to a smaller phase shift. Every time you run the experiment, the shift will be slightly different.<\/p>\n
So what happens to your interference pattern? Well, it goes away. The first photon may have a 100% chance of turning up at Detector 1, but the second will have a 25% chance, the third a 73.2% chance, the fourth a 3.6% chance, and so on. As you repeat the experiment many times these all smear together, and you end up finding that half of the photons end up on Detector 1, and the other half end up on Detector 2. The interaction between the photon and the air destroys your interference pattern. This process gets the name “decoherence,” because it’s destroying the “coherence” between the two pieces of the wavefunction, which is the technical term for “the property that allows us to see an interference pattern.”<\/p>\n
All right, then. Interactions with the environment destroy the coherence, so that’s what takes us out of the wavefunction situation and into the particle situation. So, photons that have interacted with a big environment become like classical particles and don’t interfere any more, right?<\/p>\n
Wrong. The photons always behave like waves, and they always interfere. The exact probability of detecting a given photon at Detector 1 or Detector 2 depends on both paths. The only thing the interactions with the environment do is to obscure the interference by making it impossible to build up a pattern through repeated experiments. If you could somehow keep track of all the interactions, say by measuring the precise state and trajectory of every air molecule along each path, you could recover the pattern by post-selection: simply choose to count only those photons for which the environmental conditions gave the same phase shift. Add them together, and you’ll see an interference pattern just as you did with the short interferometer.<\/p>\n
The cumulative effect of the interactions is to make the photons look as if they were classical particles, taking one definite path or the other. They’re always quantum objects, though, and they always interfere. Decoherence just keeps you from seeing the pattern.<\/p>\n
This is why I say that the standard Many-Worlds language about “separate universes” is pernicious and misleading. What’s going on here is not really a photon splitting into two photons in “separate universes,” one taking each path, it’s a photon wavefunction that is in a superposition state with random phases between the two pieces.<\/p>\n
Why do we talk about decoherence as if it produced “separate universes?” It’s really a matter of mathematical convenience. If you really wanted to be perverse, and keep track of absolutely everything, the proper description is a really huge wavefunction including that includes pieces for both photon paths, and also pieces for all of the possible outcomes of all of the possible interactions for each piece of the photon wavefunction as it travels along the path. You’d run out of ink and paper pretty quickly if you tried to write all of that down.<\/p>\n
Since the end result is indistinguishable from a situation in which you have particles that took one of two definite paths, it’s much easier to think of it that way. And since those two paths no longer seem to exert any influence on one another– the probability is 50% for each detector, no matter what you do to the relative lengths– it’s as if those two possibilities exist in “separate universes,” with no communication between them.<\/p>\n
In reality, though, there are no separate universes. There’s a single wavefunction, in a superposition of many states, with the number of states involved increasing exponentially all the time. The sheer complexity of it prevents us from seeing the clean and obvious interference effects that are the signature of quantum behavior, but that’s really only a practical limitation.<\/p>\n
Questions of the form “At what point does such-and-so situation cause the creation of a new universe?” are thus really asking “At what point does such-and-so situation stop leading to detectable interference between branches of the wavefunction?” The answer is, pretty much, “Whenever the random phase shifts between those branches build up to the point where they’re large enough to obscure the interference.” Which is both kind of circular and highly dependent on the specifics of the situation in question, but it’s the best I can do.<\/p>\n","protected":false},"excerpt":{"rendered":"
I seem to have been sucked into a universe in which I’m talking about the Many-Worlds Interpretation all the time, and Neil B keeps dropping subtle hints, so let me return to the whole question of decoherence and Many-Worlds. The following explanation is a recap of the argument of Chapter 4 of the book-in-progress, which… Continue reading Many-Worlds and Decoherence: There Are No Other Universes<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"1","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7,11],"tags":[],"class_list":["post-3171","post","type-post","status-publish","format-standard","hentry","category-physics","category-science","entry"],"_links":{"self":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts\/3171","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/comments?post=3171"}],"version-history":[{"count":0,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts\/3171\/revisions"}],"wp:attachment":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/media?parent=3171"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/categories?post=3171"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/tags?post=3171"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}