Lab Visit Report: Unusual Lattices

In one of his March Meeting posts, Doug Natelson writes about laser cooling experiments that explore condensed matter phenomena:

While the ultracold gases provide an exquisitely clean, tunable environment for studying some physics problems, it's increasingly clear to me that they also have some significant restrictions; for example, while optical lattices enable simulations of some model potentials from solid state physics, there doesn't seem to be any nice way to model phonons or the rich variety of real-life crystal structures that can provide so much rich phenomenology.

I would dissent from this mildly. The use of optical lattice systems to try to seriously go after condensed matter/ solid state problems is a relatively recent development, and I don't think we've seen the full range of what's possible. I'm not sure how one would model phonons (quantized vibrations of the crystal lattice), but there are some experiments going on these days that significantly expand the range of lattice structures being explored. I visited two of them when I was in DC a little while ago.

The one that's already produced solid results is a lab at NIST headed by Trey Porto, where my friend Nathan is a post-doc. The other is a lab that Steve Rolston is building at Maryland.

But first, a little background.

The essential idea of an optical lattice is that you can use light to mimic the structure of atoms in a solid. The interaction between near-resonant light and atoms causes a shift in the energy levels of the atoms, and if you combine that with an interference pattern made by overlapping two beams of light, you can create an area of space in which the energy of an atom varies in a periodic way. Wikipedia only offers a stub of an article, but they do have the same picture of a lattice potential that's been floating around ever since I first got to NIST, showing a surface representing the energy of the atoms, with little balls floating in space above it.

As anyone who has taken high-school chemistry knows, atoms and molecules always try to minimize their energy, so the atoms tend to occupy places in the interference pattern where the energy is lowest. This may be where the light is brightest, or where the light is absent, depending on how the lasers are tuned. If you load a large number of atoms into a lattice, you can arrange things so you have one atom at each low-energy site ("potential wells" in the common jargon), and then the system is a pretty good analogue of a condensed matter system: the optical lattice wells represent the ions in a solid crystal, a periodic array of fixed locations where electrons want to collect, while the atoms in the optical lattice represent the electrons in the solid, which are free to move around between wells, provided their energy permits it.

This is an idealized analogue of solid-state physics, of course, but it has some advantages over the real thing. For one thing, the optical lattice is nearly perfect-- there are no impurities to disrupt the structure, and if you load from a dense enough sample, you can arrange things so that there aren't vacant sites in the lattice. Another important difference has to do with the interactions between atoms-- with real solid-state systems, you're stuck with electrons that have a fixed charge, and thus a (more or less) fixed repulsive interaction between them. In an optical lattice, though, the interaction between atoms is variable. It tends to be much weaker than the interaction between electrons, but it can be varied from repulsive to attractive, depending on the internal state of the atoms. You can even modify it on the fly, by applying light pulses or magnetic fields to the system.

These features make optical lattices a wonderful toy system for exploring phenomena that condensed matter theorists have been playing with for years. There are a huge number of optical lattice experiments going on that model the transition between a superconductor and a normal conductor, for example.

One weakness of the system, though, is, as Doug says, that the optical lattices need to be created by the interference of light, which limits the range of structures that people can make. The vast majority of optical lattice experiments have been done using simple cubic lattice structures, because those are very easy to make and maintain.

They're not the only thing possible, though, and that's what the Porto lab has been working on. They've set up a system that allows them to create more complicated structures in two dimensions, as described in this long preprint (I'm sure it's been published, but I'm too lazy to look for it). Their particular interest is in using the system for quantum information, so they focussed on making a lattice of double-well potentials, and demonstrating controlled interactions between atoms in those wells.

The basic idea behind their system is that by interfering four laser beams in a plane, they can make two different square arrays of potential wells, with different spacings between wells, and rotated with respect to one another. They can control which of these lattices they make by rotating the polarization of the lasers, and they can combine the two. Adding in the ability to control the relative phases of the beams lets them mix the two cases together to make a square array of double wells, holding two atoms in different internal states very close to one another.

So, it is possible to develop lattices with structures beyond a simple cubic array of single atoms. You can also do more exotic things than double-wells-- their most recent experiment (coming soon to a Physical Review Letters near you) adds radio-frequency light to mix different sublevels together, and lets them make a lattice of ring-shaped potential wells.

Now, I don't know that there's any burning interest in condensed matter circles for a model system in which "electrons" are trapped in ring-shaped wells (but, seriously, how cool is that?), but I think it demonstrates that optical lattice systems are more versatile than it may first appear. It's not easy to make more complicated structures, mostly because it's difficult to work out how to generate the necessary interference patterns, but it can be done, and as the field progresses, people are starting to do it.

The other experiment I wanted to mention is a project that Steve Rolston and his group are working on, to address something I said above that may have annoyed condensed matter people. I cited the lack of defects as a nice feature of optical lattice systems, but I've said that before only to have condensed matter physicists huffily respond that lattice defects are the really interesting part of condensed matter physics (it's usually experimentalists who respond this way-- theorists are perfectly happy to see the defects go away). If that's true, is there a way to look at systems where the ordering isn't so perfect?

The plan for the system they're working on at Maryland is to explore a somewhat extreme version of this. It's not easy to introduce local defects in a lattice, but large-scale disorder is something that can be explored, for example by adding together multiple lattices with incommensurate periods. For example, if you add together three sets of lattice beams with periods of 0.7071, 1.0, and 1.3 (chosen more or less at random), you get a structure that looks like this:


You've still got wells that will potentially trap atoms, but that's not a simple, boring crystal structure. This kind of system will allow them to look at the effects of disordered structures while still retaining the advantages of optical lattices with regard to interactions between particles.

So, I think there's still a lot to explore in optical lattice physics. The experiments to date have focussed on simple structures and simple interactions, because those are easier to deal with experimmentally. As the simple experiments get wrapped up, though, you can see the field moving toward more and more complicated structures, with more of the "rich phenomenology" of real systems.

Of course, I've done lattice work in the past, so I may be just a tiny bit biased...


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The complete-layman question: isn't there any way to numerically take a given fringe pattern and work backwards to the (complicated) laser system needed to produce it?

The complete-layman question: isn't there any way to numerically take a given fringe pattern and work backwards to the (complicated) laser system needed to produce it?

It's a hard problem, especially because you need to come up with something that's experimentally feasible. The double-well lattice in the Porto lab, for example, is pretty challenging to do, because it requires extremely good control over the polarization and phase of the lattice laser, and that's only a single beam on a folded path.

Thanks for the linkage. So, dumb question after I wrote my post.... In principle, could one define a holographic plate to form pretty much arbitrary lattices? Perhaps one problem with this idea is the relatively significant laser powers needed to do the trapping. My original post was prompted partly by a session that I went to at the meeting talking about strong correlation effects in a variety of materials, and some relatively simple systems (e.g., Fe3O4) have very complicated lattices. I think you'd need some holographic approach to get the details right.

As for disorder, one of Randy Hulet's former postdocs has a paper coming out sometime soon about looking at localization in a disordered 1d potential using cold atoms. Neat stuff. Individual defects in lattices would be tough, though.