One of the key concepts in physics is that of a phase transition. Ice melting to form water is one example; another is the transition between magnetic and non-magnetic forms of iron. The underlying physics of these transitions is a story about correlations. Understanding a phase transition and, indeed, a phase of matter, is all about understanding the growth of correlations.

You would think that one of the cleanest and best understood physical systems wouldn't have a lot to offer physicists in terms of understanding correlations that develop through a phase transition. However, physicists got a bit of a surprise when they looked at particular correlations that arise as a dilute gas is cooled down until it forms a Bose Einstein condensate (BEC).

May I have the pleasure of correlated motion with you?

I think most people have a pretty good intuitive understanding of correlations, but let's begin with that anyway. Imagine that we have two swings that we set moving at exactly the same time. If we measure the position of one swing, then we also know the position of the other swing—even though they are not linked by anything more than the fact that we deliberately set them in motion that way. These swings are perfectly correlated.

Likewise, I can set the swings in motion such that when one is moving to the left, the other is moving to the right. But because they maintain this relationship, measuring the position of one swing still tells me the position of the other. These swings are also perfectly correlated, even though they are doing different things. There is a fixed relationship between the positions of the two swings.

However, things are seldom perfect. Usually, we would set the swings in motion, and things like ground vibrations and air currents will minutely change their motion. Early in their motion, they are perfectly correlated; later on, they are less perfectly correlated. The precision with which we can infer knowledge about one swing from the other decays with time.

So correlations provide a really useful measuring stick for telling us about events that set things—think particles, light fields, swings, genomes—in motion. More importantly, correlations also tell us how strongly the surrounding environment is modifying the trajectory set in motion by the original event.

Bose Einstein condensate, and really cold stuff: what's the difference? In quantum mechanics, every particle has wave-like properties and every wave has particle-like properties. Waves can interfere with each other, so why can't we (or any other large collection of particles) interfere with each other? Usually, the extent of the wave-like properties, called the wave packet, depends on the temperature and mass. The heavier the particle and faster its motion, the shorter the extent of the wave packet. In everyday life, the wave packet doesn't extend beyond the boundary of the particle. Once we cool atoms down to near absolute zero, their wave packets extend out well beyond the average reach of the atom's surrounding electron cloud. Now, when these atoms encounter each other, they do act like waves and interfere. But interference can be destructive or constructive, so what does this mean in terms of particles? Well, it turns out that some groups of particles destructively interfere with each other. What this means is that when their wave functions overlap, they are repelled by each other. But this repulsion doesn't necessarily mean a physical push—instead, it alters the quantum state of the particles. To keep from being pushed apart, every atom must be in a slightly different state. Particles that can adopt these different states are called fermions, and you can think of them as stacking from low energy to high energy. Other particles constructively interfere with each other. The atoms are attracted to each other and have no need to adjust their internal state to accommodate their closeness. The result is that two atoms will end up with a common wave function. Furthermore, the greater the amplitude of the wave function, the further it extends and dominates surrounding particles. The result is that these particles join together to form a single giant conglomerate with a single wave function. These particles are called bosons, and when they enter a common state, they have formed a BEC.

There can't be any interesting correlations in a BEC, right?

Surprisingly enough, certain aspects of the correlations in BECs have remained largely unexplored. This is because, as I have described in the sidebar, every particle within the BEC is part of the same wave packet, so they must all share the same degree of correlation to each other. In other words, after the phase transition to a BEC, all correlations should be the same everywhere.

To check this, a group of researchers in France and Austria measured a particular type of correlation for BECs. What they focused on were density fluctuations within the BEC. To measure these, they trapped atoms in a cigar-shaped trap and cooled them down. After reaching the chosen temperature, the researchers would turn off the trap and let the gas expand. They would then measure the density of the expanding cloud as a function of position and starting temperature.

They followed this up by calculating the expected correlations for the different starting temperatures. For temperatures that were too high for a BEC to form, they used the simple kinetic theory of gases (think of it as constantly moving pool balls bouncing off each other). For BECs, they used the ideal BEC theory (think of it as pool balls that can magically move through each other and never collide with anything).

For gases that had an initial temperature that was too high for a BEC to form, they found that the ideal gas theory perfectly described the correlations in the gas cloud. That is, at the very center, where the atoms can't move very freely, there are some correlations, but they drop off very sharply towards the edges.

The BEC, on the other hand, provided some surprises. It turns out that the simple BEC theory fails quite badly. This is because the trapped BEC still contains a small fraction of atoms that aren't part of the condensate. These atoms can scatter other atoms into and out of the condensate, redistributing it. That accounts for part of the theory's failure but, by itself, it's not enough to explain all the differences between its predictions and the measured correlations.

The other key feature that modifies the density correlations is the atom trap itself. The trap is basically a cigar-shaped area where an external force pushes atoms back toward the center. The further away from the center you are, the greater the force pushing you back to it. Since the atoms don't enter the trap motionless, they oscillate back and forth around the center just like the swings discussed above. In this case, however, the oscillations create pressure or density waves.

Along the long axis of the cigar, the force pushing inward grows more slowly with distance from the center, so the oscillations in this direction are relatively slow and involve less energy. On the short axis, the force grows more quickly and the oscillations involve more energy.

This creates density fluctuations along the long axis of the trap, but the temperature is typically too low to provide the energy needed to create density fluctuations along the short axis of the trap. As the temperature increases, there's enough energy around to excite more of the trap's oscillatory modes. As a result, the density fluctuations provide an accurate measure of the temperature of the BEC.

That in itself may be quite useful from an experimental perspective, but I must admit that I was quite surprised by the disagreement between theory and experiment. Part of the problem is that it is not clear from the article exactly what the theory they used entails. If it is purely the theory of BEC as a group of particles that do not interact with each other, then, yes, I can quite believe that theory and experiment don't agree. However, that is not the most sophisticated approach to BEC theory.

Interactions between BECs and a complement of atoms that are not in the condensed phase have been studied using a variety of techniques, but these are usually quite hard to get results out of—the equations that describe these models must be solved using a computer. The only conclusion this suggests to me is that this is a call to theorists to stop slagging off the mathematicians across the hall and get back to work.

Nature Physics, 2012, DOI: 10.1038/nphys2212