How to Teach Physics to Your Dog

by CHAD ORZEL

  The answer, according to Everett, is that we can’t separate the observer from the wavefunction. The observer is included with the rest of the system—the components are things like “treat in the left-hand box plus a dog who knows the treat is in the left-hand box”—and as a consequence, we only perceive our own small part of the overall wavefunction. This branching is the origin of the quantum randomness that Einstein and others found so troubling: the wavefunction always evolves in a smooth and continuous way, but we only experience one branch of the wave-function at a time, and which branch we see is a random choice. Other versions of ourselves exist in the other branches, experiencing different outcomes (for this reason, the interpretation is sometimes referred to as the “many-minds” interpretation).

  None of the myriad other branches have any detectable influence over the events in our branch, and our branch has no detectable influence over the events in any of the others. For all intents and purposes, those other branches are self-contained parallel universes, completely inaccessible from our universe. This is the origin of the name “many-worlds” for the theory: it’s as if the universe forks every time a measurement is made, and is constantly spawning new parallel universes with slightly different histories.

  WAVEFUNCTIONS FALL APART: DECOHERENCE

  These noninteracting branches present a serious but subtle problem for the many-worlds interpretation. Every other two-part wavefunction we’ve seen has led to some sort of interference phenomenon. So, why don’t we see interference around us all the time if there are all these extra branches to the wave-function? What is it that seals these “parallel universes” off from us?

  The answer is a process called decoherence, which prevents the different branches of the wavefunction from interacting with one another. Decoherence is the result of random, fluctuating interactions with a larger environment, which destroy the possibility of interference between different branches of the wave-function, and make the world we experience look classical. Decoherence doesn’t just occur in the many-worlds interpretation of quantum mechanics—it’s a real physical process, compatible with any interpretation*—but it’s particularly important in the modern view of many-worlds (which is sometimes called “decoherent histories” as a result—the interpretation has almost as many names as universes†).

  Decoherence is absolutely critical to the modern view of quantum mechanics and quantum interpretations. The most common semiclassical explanations of decoherence leave a lot to be desired, though, as they are inaccurate, and often somewhat circular. The real theory of decoherence is subtle and difficult to understand. As with the uncertainty principle (chapter 2, page 40), though, it’s worth some effort to unpack it, because it provides a much richer understanding of the way the universe works.

  To understand the idea of decoherence, let’s think about the concrete example of a simple interferometer, a device consisting of two beam splitters that split a beam of light in half and a couple of mirrors that bring it back together again.* Interferometers like this are extremely important in physics, not only for demonstrating quantum effects, but because they form the basis for the world’s most sensitive detectors of rotation, acceleration, and gravity. These allow the measurement of tiny forces in physics experiments, and also find application in submarine navigation.

  Light enters the interferometer when it strikes a beam splitter, which passes half of the light through without perturbing it and reflects the other half off at a 90° angle. These two beams separate from each other, and then are steered back together using two mirrors, and recombined on a second beam splitter. The second beam splitter is lined up so that the transmitted light from one beam and the reflected light from the other follow exactly the same path and interfere with each other before falling on one of two detectors.

  You might think that each of the two detectors would detect exactly half of the light, because each detector receives one-quarter of the original beam from each of the two paths (¼ + ¼ = ½). Each detector can actually see anything from no light at all up to the full intensity of the initial beam, though, because the waves that took different paths interfere with each other, like the waves in the double-slit experiment in chapter 1 (page 18).

  If the paths followed by the two beams have exactly the same length, the two light waves undergo the same number of oscillations en route to Detector 2, and interfere constructively, giving a bright spot—all of the light entering the interferometer hits that detector.* On the other hand, if one path is longer than the other by one half of the wavelength of the light, the light along that path undergoes an extra half-oscillation, and the two waves interfere destructively: the crests of the wave hitting Mirror 1 line up with the troughs of the wave hitting Mirror 2, and they cancel out, giving no light at Detector 2. If we increase the length difference to one full wavelength, the peaks align again, and we get another bright spot, and so on. Between those two extremes we get some intermediate amount of light. We can generate an interference pattern by repeating the experiment many times, and changing the length of one path slightly. This will give us a pattern of alternating light and dark spots on Detector 2.

  The interferometer described in the text. Light enters from the left, is split by a beam splitter into two different paths, and then recombined by the second beam splitter. If the beam hitting Mirror 1 travels exactly the same distance as the beam hitting Mirror 2, interference causes all the light to hit the upper detector (Detector 1), and none to hit the detector at right (Detector 2).

  The fraction of the light reaching Detector 2 depends only on the time it takes for the light to travel each path. We can imagine the light moving down the two arms as two identical dogs setting out around a single block in opposite directions. They agree in advance that the first dog to arrive at the opposite corner chooses the path for the rest of the walk. If the two dogs walk at the same speed, and the two paths are the same length, they will always arrive at the opposite corner at the same time. If one path is slightly longer than the other, the dog taking that path will always be late arriving, and they will always follow the same route afterward.

  “Wait, what’s quantum about this? You’re just talking about waves and dogs.”

  “You can describe the basic operation of the interferometer classically, but it also works perfectly well if you send in one photon at a time. That’s a system you can only explain with quantum physics.”

  “Don’t you need lots of photons to see the pattern?”

  “Sure, but you can repeat the experiment many times, and build up the pattern. You set it up so the two paths have equal length, and repeat it 1,000 times, and you’ll see 1,000 photons at one detector. Then you move one mirror a little bit, and repeat the experiment another 1,000 times, and see 700 photons, and so on. If you keep doing this over and over again, you’ll trace out the same pattern you see with a bright beam.”

  “Like the way I measured the wavefunction of a bunny in the backyard by marking its position every night when I went out to chase it?”

  “Technically, you measured a probability distribution, the square of a wavefunction, not the wavefunction itself, but yes, that’s the basic idea.”

  “The bunny is most likely to be underneath the bird feeder, you know.”

  “Yes, because it eats the spilled seed. Try to focus, please.”

  “Okay.”

  When we move to a quantum description of the interferometer, talking about it in terms of single photons and wavefunctions, we say that the photon wavefunction splits into two parts at the first beam splitter. In the popular view of many-worlds, you might be tempted to say that this is also when the universe splits in two, as that is when we first acquire a second branch of the wavefunction.

  This is tempting, but wrong—the wavefunction has two branches, but they are not “separate universes” at this point. We know this because when we bring them back together, at the second beam splitter, we see interference. The probability of a photon reaching our detector changes
as we change the length of one of the paths in the interferometer, indicating that the two separate paths influence each other. The photon goes both ways at the same time, and interferes with itself when we recombine the branches.

  We see this interference pattern because the two parts of the wavefunction have a property called coherence. “Coherence” is a slippery word, but when we say that two wavefunctions are “coherent,” we mean that they behave as if they came from a single source.* In the case of the interferometer, they did come from a single source, and each of the pieces experiences essentially the same interactions as it goes through the interferometer, so they remain coherent all the way through. The only factor determining whether the interference is in phase or out of phase is the length difference between the two paths.

  To turn the two branches of the wavefunction into two separate universes, we need to destroy that coherence. Without coherence, the two pieces of the wavefunction will not interfere to give a detectable pattern, and we will not be able to see any influence of one path on the other. The photon will look like a classical particle in two different universes, passing straight through the beam splitter in one universe, and reflecting off it in the other.

  The coherence between the two branches of the wavefunction is destroyed by interaction with a larger environment. To see how this works, let’s imagine making the interferometer very long, so that the beams have to pass through a lot of air between the two beam splitters:

  An interferometer with a very long distance between the beam splitters.

  In our dog-walking example, we can imagine this longer interferometer as sending our two identical dogs around Central Park in opposite directions. The longer path allows more time for distractions to build up—squirrels to chase, dropped food to eat, horse droppings to roll in. Now the dogs no longer move at the same speed—sometimes they speed up, and sometimes they slow down, depending on what they see. The order in which the two dogs arrive depends not only on the length of the path they follow, but on what they encountered along the way.

  The same thing happens with light. As light travels through the interferometer, it occasionally interacts with the atoms and molecules making up the air. Light passing along one arm might encounter a region with more molecules than average, and slow down a little, or it might encounter a region with very few molecules, and speed up a bit. These effects are very small, but they add up over the length of the path, just like the distractions encountered by our walking dogs. The interference pattern is determined not only by the length of the two paths, but also by interactions with the environment in the form of the air that the light passes through.

  Interactions with the environment, like distractions around Central Park, are essentially random, and fluctuating—that is, they’re different from one place to another, and one time to another. A dog won’t find dropped food on the same block every day, and a photon won’t interact with a molecule in the same part of the interferometer every time. Interactions shift the pattern, and the shift is different each time we run the experiment. The end result—whether we see light at Detector 2, or which dog wins the race—will be completely random, determined by factors outside our control.

  Thanks to these random interactions, the light waves along the two paths are no longer coherent, so we no longer see a clean interference pattern when we bring them back together. Instead we see a pattern that’s constantly changing, shifting position millions of times a second. The bright and dark spots blur together, wiping out the pattern. This same effect carries over when we do the experiment with single photons—the two pieces of the wavefunction are no longer coherent. When the photons undergo random, fluctuating interactions with a large environment, the interference pattern is destroyed.

  This process of random interactions destroying quantum effects is known as decoherence, because it’s destroying the coherence between the different parts of the photon wavefunction. Decoherence prevents the various branches of the wave-function from affecting one another in any detectable way—while we would expect the different branches to come back together and produce interference patterns, we don’t see that. Thanks to decoherence, the interference between different branches is random, and thus can’t be detected.

  “So, now we have two different photons that don’t interfere?”

  “No, it’s more subtle than that. We still have only one photon, but it appears in two different branches of the wavefunction. We don’t see any interference because of decoherence.”

  “It’s one photon that doesn’t interfere with itself?”

  “It interferes with itself, but the interaction with the environment leads to random shifts that make the interference pattern different every time. We can’t build up a pattern through repeated measurements, because it’s always moving around. The pattern shifts millions of times a second, and you’re as likely to get a bright spot as a dark spot, smearing the whole thing out into—”

  “An in-between spot?”

  “Exactly.”

  “So, it’s like if I tried to measure the bunny wavefunction by marking its position when I went out to chase it, but sometimes I was chasing squirrels instead of bunnies?”

  “Well, the squirrels and bunnies both tend to be found under the bird feeder, so that wouldn’t make much difference. It’s more like trying to measure the bunny wavefunction by recording the position every night, while I keep moving the bird feeder to different spots in the yard.”

  “Oh. That would be mean. Don’t do that.”

  “I’m not going to. It’s just an analogy for the way that decoherence wipes out the interference pattern. Even though each individual photon interferes with itself, the two parts of the wavefunction aren’t coherent, so we can’t build up a pattern from repeated measurements.”

  “But if we don’t see a pattern, how do we know that there’s interference? What’s the difference between quantum particles that don’t produce an interference pattern and ordinary classical particles?”

  “That’s exactly the point: there is no difference. Because of the random interactions, the quantum effects are smeared out, and we’re left with something that doesn’t look like an interference pattern, even though there is interference happening all the time. You end up detecting a photon 50% of the time, exactly as you would if it were a classical particle with no wave properties.”

  “I don’t know. It’s kind of Zen, isn’t it? ‘What is the pattern of one photon interfering?’ ”

  “Hey, that’s pretty good.”

  “Thanks. I have Buddha nature, you know.”

  THE INFLUENCE OF THE ENVIRONMENT: DECOHERENCE AND MEASUREMENT

  You will sometimes see explanations of decoherence describing it as a measurement process, saying things like “when a photon interacts with an air molecule, that’s the same as measuring the position of the photon, which destroys the interference pattern.” This is almost exactly backward—decoherence isn’t a result of measuring the photon through interactions, it’s the result of not measuring the interaction between the photon and its environment.

  This is a subtle point, but it’s critical to the modern understanding of decoherence. As we send individual photons through the interferometer, each one interferes with itself, according to a wavefunction that shows some interference pattern. If we could send in thousands of photons, and get the same interaction each time, we could repeat the measurement over and over, and trace out an interference pattern like the first dashed curve shown in the figure on the next page.

  We can’t guarantee exactly the same interaction between the photon and the environment every time, though, any more than we can guarantee that a Central Park tourist will drop food in the same place every time our dogs go for a walk. As a result, the second photon sent in will interfere with itself according to a different wavefunction. If we could repeat this experiment thousands of times, we would trace out a pattern like the second dashed curve in the figure. The third photon has yet another different wavefunc
tion, which would trace out a pattern like the third curve, and so on.

  In the end, we get one photon drawn from each of these patterns, and thousands of others, each with peaks in different positions. The cumulative effect is to trace out a pattern that is the sum of many different interference patterns. This is shown as the solid line in the figure, which barely shows any interference at all.

  What does this have to do with measuring the environment? Well, when each of the photons we send in interacts with the air, it makes a small change in the state of the environment—an air molecule is moving a little bit faster, or a little bit slower, or the internal state of that molecule is changed in some way. Exactly what happens to the environment depends on exactly what sort of interaction went on, and that, in turn, determines what happens to the photon.

  The dashed lines show the interference patterns for photons with three different phases. The solid line represents the sum of several such patterns, showing that the interference pattern is almost completely wiped out.

  If we could keep track of everything that happened to the environment—the exact state of every air molecule in the two paths through the interferometer—we could use that information to work out what happened to the photon, and choose to look only at photons whose wavefunctions produce identical patterns. Only a tiny number of photons will have exactly the same result, but if you repeat the experiment often enough, you’ll find some—the 159th photon through might produce exactly the same pattern as the first, and then the 1022nd, and the 5,674th, and so on. If you look only at those photons, you’ll see them trace out an interference pattern, just as if there were no decoherence.