Quantum Mechanics Is Square: “Ruling Out Multi-Order Interference in Quantum Mechanics”

This week’s big story in physics is this Science paper by a group out of Austria Canada (edited to fix my misreading of the author affiliations), on a triple-slit interference effect. This has drawn both the usual news stories and also some complaining about badly-worded news stories. So, what’s the deal?

What did they do in this paper? The paper reports on an experiment in which they looked at the interference of light sent through a set of three small slits, and verified that the resulting pattern agrees with the predictions of the Born rule for quantum probabilities.

What does Matt Damon have to do with quantum physics? Born, not Bourne. Specifically, Max Born, a German physicist who worked out the connection between quantum wavefunctions and observable probabilities.

Okay, what? Allow me to explain. Actually, there’s too much to explain, so let me sum up.

Back in the early days of the 20th Century, Louis de Broglie proposed that electrons (and all other material particles) behave like waves, and not long after that, Erwin Schrödinger found an equation, now named after him, that describes a “wavefunction,” a mathematical object that is somehow associated with the state of the electron. The Schrödinger equation does a great job of determining the allowed states of the electron in a hydrogen atom, completely explaining the Bohr model, but there’s a slight problem: it’s not immediately obvious what the waveunction is.

Wait, they had an equation, but they didn’t know what it was for? Pretty much. The equation does a great job of giving you the allowed states of hydrogen and other atoms, but there’s nothing in it that tells you the physical meaning of the wavefunction– exactly what is waving, and what that means. Also, the wavefunctions necessarily contain imaginary numbers and complex exponentials, which is weird because all the physical quantities we measure are real nubers, pretty much by definition.

So, what did Born do? Born figured out that the way to deal with the wavefunction is to say that the square of the wavefunction (more or less) gives you the probability of finding the electron at a given position. This takes care of the imaginary numbers, because two complex numbers multiplied together in the right way will always give you a real number. Using Born’s rule, physicists could begin to discuss not only the energy of the electron states, but the physical distribution of the electron itself.

OK, but isn’t that ancient history? Yes and no. Born’s rule was established in the late 1920’s, but it’s always been a little arbitrary. That is, there’s no real solid physical reason why the connection between wavefunctions and probability should be through the square of the wavefunction, other than the fact that it happens to work out very nicely. It’s conceivable that there could be some other sort of rule, that would also get rid of the imaginary numbers, but give slightly different results.

People have worked very hard over the years on finding ways to derive the Born rule without additional assumptions– I heard Wojchiech Zurek give a talk on this several DAMOPs ago– but it’s a really hard problem, and there are lots of subtleties to it.

So, this paper is about testing that? Right. One o the most important results of the Born rule has to do with interference: if you send a quantum particle at a barrier containing two slits, the wavefunction on the far side contains two terms, one corresponding to the particle passing through each of the slits. When you square this to get the probability distribution, you end up with three terms (well, four, but you can usually lump two of them together)– two terms corresponding to the probability distribution for going through each individual slit, and a third “interference” term arising from the interaction between the wavefunction describing the particle passing through one slit, and the wavefunction describing the particle passing through the other.

This interference term is what’s responsible for the alternating pattern of bright and dark stripes that you see when you send quantum particles through a double-slit. It’s an unambiguous signature of wave behavior, and the key confirmation of de Broglie’s wave model of the electron.

But people have already done that, right? Right. What this experiment does is to look at the situation for three slits, rather than two.

How does that change things? The Born rule still gives you an interference pattern. In terms of the wavefunction, on the far side of the slits, you have a wavefunction with three terms, one for each slit. When you square that, you end up with six terms: three corresponding the the individual probabilities for the slits by themselves, and three corresponding to interference between each of the three possible pairs of slits (left-middle, left-right, and middle-right). Since you’re just dealing with the square of the wavefunction, according to the Born rule, there’s nothing else possible.

So how does this help? Well, if you used a different rule than the ordinary Born rule, you might expect an extra term to show up, one involving a three-way interference between waves from all three slits at once. For example, if you said that the interference involved the wavefunction to the fourth power, you could end up with a situation where you got rid of all the imaginary numbers, but the probability distribution would look a little different.

OK, but how can you tell the difference? I mean, as long as they’re all real numbers, what can you do? The thing that lets you test whether you’re dealing with the Born rule or something else is the fact that you can describe the Born prediction in terms of the individual probability distributions for each of the possible combinations of one or two slits. That is, the Born prediction is equal to the sum of the probability distribution for going through the left slit only, the probability distribution for going through the middle slit only, the probability for going through the right slit only, and terms related to the probability distributions for each of the possible pairs (left-middle, left-right, and right-middle).

Those are all things you can measure. If you send a quantum particle through three slits at once and record the probability distribution on the far side of the slits, and then record the distributions for all possible one- or two-slit combinations, you should be able to add the six individual distributions together to get the distribution for all three.

So, if you subtract the six individual distributions from the three-slit distribution, there should be nothing left? Exactly. And that’s exactly what they did in this paper. They made a large mask containing all the various patterns of slits they needed, and put it in front of a light source, then recorded the pattern on the far side for all the possible slit patterns. Then they subtracted the individual distributions from the three-slit distribution, and looked to see if there was anything left over. Any remaining pattern in the probability of finding light at different positions on the far side of the mask would have to be due to some non-Born contribution to the probability.

And they didn’t see anything? Nope. They did it three different ways: using a bright laser, so they could just take a direct picture of the intensity pattern; using an attenuated laser, and counting the individual photons as they arrived; and using a source that sent one photon at a time through the slits. In all three cases, the subtraction left nothing behind. Born’s rule worked perfectly for all of them, to within about 1% of their figure of merit.

So, that’s good news, then? If you’re the ghost of Max Born, or the author of an introductory quantum book, yes. This was disappointing news for some theorists, though, as there are a number of ways to approach problems like coming up with a theory of quantum gravity that would require some modification of the Born rule. This experiment rules those out, or at least it rules out any large modification of the Born rule. It’s still possible that there’s some really tiny contribution of other stuff to the probability distribution that was hidden in the noise in their measurements, but tiny modifications are harder to explain than big ones, which means more work for theorists.

And that’s a bad thing? If you’re a theorist, yeah. I’m an experimentalist, so I don’t mind that much. I can just revel in the ingenuity of the experimentalists who did this work.

What’s next, then? The news stories say they’re planning a more precise test, using beamsplitters rather than slits, which will give them a cleaner signal. And, of course, the theorists go back to working on ways to make the Born rule modifications of their theories small enough that they would’ve escaped detection here.

So, that’s the latest news from the world of quantum weirdness. Any more questions?

Just one last thing: Admit it, “The Bourne Rule” would be a great name for a Bourne Identity/ Good Will Hunting crossover story? Personally, I prefer The Bourne Approximation , but that’s a different area of physics.

Sinha, U., Couteau, C., Jennewein, T., Laflamme, R., & Weihs, G. (2010). Ruling Out Multi-Order Interference in Quantum Mechanics Science, 329 (5990), 418-421 DOI: 10.1126/science.1190545