The Physics of Frustration: "Quantum Simulation of Frustrated Classical Magnetism in Triangular Optical Lattices"

One of the benefits of having joined AAAS in order to get a reduced registration fee at their meeting is that I now have online access to Science at home. Including the Science Express advance online papers, which I don't usually get on campus. Which means that I get the chance to talk about the few cool physics things they post when they first become available, without having to beg for a PDF on Twitter. This week's advance online publication list includes a good example of the sort of cool ultra-cold atom physics that I talked about at and after DAMOP, so let's take a look at this paper in the usual Q&A format:

So, the title talks about frustrated magnetism. Have they started publishing papers about experiments that just won't work no matter what you try? No, the use of "frustrated" there is a term of art. "Frustrated magnetism" is the term for a situation where there's no way for a system of spins to get into the state that they really want to be in.

OK, what? Well, every system in physics always "wants" to have the lowest possible energy, like a dog who always wants to be napping in a comfortable spot. If you're clever, though, you can set up a system where there's no easy way for all of the particles in a large collection of things to arrange themselves so as to satisfy their individual desires.

The classic example of this is a bunch of spins on a triangular lattice, like the arrangement in this picture:

The spins at the corners of the triangle act like little magnets, and they're happiest when each spin is pointing in the opposite direction from its nearest neighbors. On a triangular lattice, though, there's no way to do that with simple up-and-down spins-- if spin 1 is up, and spin 2 is down, then there's no good state for spin 3. It wants to be down, because it's next to spin 1, but it also wants to be up, because it's next to spin 2.

Because this system can't find a simple arrangement that makes all the spins happy, it's called a "frustrated" system. This turns out to be an important problem in statistical and condensed matter physics, because it's really simple to make systems like this, but figuring out exactly what they're going to do is a tricky problem with a lot of rich physics in it.

So, what do they do? Well, there are a lot of different ways for this to work itself out, depending on the details of the lattice. They show the major options in this figure:

As you can see, with the exception of the "ferromagnetic" and "rhombic" states, these all involve spins pointing at odd angles, meaning they're combinations of up and down. Which of these arrangements the system settles into depends on the details of how the lattice sites talk to one another. If you minimize the communication between sites, they don't mind being all pointing in the same direction, the ferromagnetic state. If you let sites talk to each other along the vertical-ish directions, but not horizontally, you can get alternating rows, which is the rhombic state. If you let all of them talk, you get more complicated arrangements; exactly which one depends on how strongly the different directions are coupled to each other.

So, I get the part with the spin arrows on the lattice, but what are those fuzzy little pictures next to them? Those are experimental data, showing that they can, in fact, make all those different arrangements by adjusting their experimental parameters. As you can see, each configuration produces a unique experimental pattern, so they can try different combinations of parameters, and map out exactly what it takes to make each of these arrangements.

The fact that they're able to make all these different arrangements with a single experimental system is a big deal. It lets them generate all kinds of data that can be compared to the simulations that theorists build, and do it very cleanly, without needing to fabricate new samples all the time.

OK, but what are those pictures? Are they showing the individual spins? No, because this is a quantum simulation of the system. That is, the spins they're working with aren't really spins, but collections of ultracold atoms.

How does that make any sense at all? Well, they take a BEC containing a bunch of ultra-cold atoms (of rubidium, though they only say that in the supplementary material), and use three laser beams to chop it up into a bunch of long "tubes" of atoms arranged in a triangular pattern. Each of the "tubes" in the central part of their system contains a few hundred Bose condensed atoms, which have a limited ability to "talk" to atoms in the neighboring sites, by tunneling through the light field between tubes.

They can control the coupling between tubes by "shaking" the tubes along one direction or another by modulating the laser beam. Depending on the amount of shaking they apply, they can generate any tunneling probability they want, which lets them map out the possibilities for the configurations of the system.

When you write down the mathematical equations governing the behavior of this system, you find that there's a collective property of the atoms in each tube, related to the quantum-mechanical phase of the wavefunction in that tube, that behaves exactly like the spins in the frustrated magnetism system. This is a very common way of thinking about these sorts of systems, because lots of physicists learned about this kind of system in the spin context, so the language carries over very easily.

Yeah, but isn't that kind of dodgy? I mean, if they're not really spins, how does this tell you anything about spins? Well, mathematically, it's absolutely identical, so anything that happens in this system should also happen in the spin system. Granted, in an ideal world, you would want to do this with individual spins, but that's really, really hard. This collective atomic state gives you a way to explore the same physics, and provides a really easy readout.

How is this easy to read out? Well, through those pictures. The collective property that plays the role of the spin here is related to the phase of the wavefunction at each site. That phase is a critical factor in determining what will happen to the clouds of atoms when you let them go from the trap, and they expand and interfere with each other-- because, of course, these are quantum-mechanical particles, and have to behave like waves. If you change the phase of the wavefunction at different sites, that moves the peaks of the interference pattern produced from all those overlapping clouds of atoms around, and that, in turn, gives you all those different patterns for the different spin configurations.

So, to get those pictures, they load a BEC into the lattice, shake it around in the appropriate way for a few milliseconds, then turn off all the lasers, let the atoms expand, and then take a picture of where the atoms are. The quantum interference of the different phases gives you a clear way to distinguish the state of the spins, and the shaking trick gives you total control of the amount of communication between different spins.

OK, that's pretty clever. It's why this sort of research is such a big deal in both condensed matter and atomic physics.

So, what kind of things do they find? Well, for the moment, this is a proof-of-principle kind of thing. So they compare their interference patterns to theoretical predictions for various values, and show that they match up well with theory. They can also demonstrate "spontaneous symmetry breaking" in this system: they set up the conditions so that it's right on the boundary between two different configurations, and look at what they see. They find that, in general, the system tends to spontaneously fall into one configuration or the other, even though there's no reason to prefer one or the other in terms of the energy. This kind of phenomenon turns up all over the place, from condensed matter physics to particle physics to cosmology, so it's a neat thing to be able to measure so cleanly and directly.

This method can also potentially be used to look at the response of the system to changes in the parameters-- what happens when you set it up in one configuration, then suddenly change the conditions in a way that ought to favor a different arrangement? You can potentially map this out stroboscopically, by making the change, holding the system in the new state for some time, then taking a picture. Repeating this over and over for lots of different hold time will let you watch the evolution from one state to another, which is really cool (and damnably difficult to calculate theoretically, so having real data to look at will be a big bonus).

Thanks for that explanation. It's still kind of weird, but it makes more sense now. So, would you say that you're less frustrated as a result of reading this?

Don't make me hurt you. Sorry.

Struck, J., Olschlager, C., Le Targat, R., Soltan-Panahi, P., Eckardt, A., Lewenstein, M., Windpassinger, P., & Sengstock, K. (2011). Quantum Simulation of Frustrated Classical Magnetism in Triangular Optical Lattices Science DOI: 10.1126/science.1207239