bose-einstein condensation

Measuring Temperature by Counting Atoms: "Suppressed Fluctuations in Fermi Gases"

drorzel — Tue, 27 Jul 2010 08:25:48 +0000

Measuring Temperature by Counting Atoms: "Suppressed Fluctuations in Fermi Gases"

When one of the most recent issues of Physical Review Letters hit my inbox, I immediately flagged these two papers as something to write up for ResearchBlogging. This I looked at the accompanying viewpoint in Physics, and discovered that Chris Westbrook already did most of the work for me. And, as a bonus, you can get free PDF's of the two articles from the Physics link, in case you want to follow along at home.

Since I spent a little time thinking about these already, though, and because it connects to the question of electron spin that I talked about yesterday, I think it's still worth writing these up. So:

OK, what are these papers about? Essentially, they're both talking about a new method for measuring the temperature of extremely cold samples of fermionic atoms, by looking at fluctuations in the number of atoms at different points in the cloud.

So, they've just made a better thermometer? Is that all it takes to get into PRL these days? Actually, it's more complicated than you might think. We're talking about samples of a few million atoms, at temperatures a few billionths of a degree above absolute zero. This isn't a matter of sticking a tiny little mercury bulb or thermocouple contact into a sample of atoms. There aren't enough of them to measure with any macroscopic object, and bringing them into contact with anything would destroy the sample completely. To measure temperatures at this level, you need to go back to the fundamental definition of the temperature of a gas.

OK, temperature is a measure of the average kinetic energy of the particles in a gas. So, you just need to measure how fast the individual atoms are moving. Big deal. That's how it's traditionally been done, but when you start talking about the particular systems they're dealing with, the situation becomes much more complicated. The quantum statistics of the particles comes into play, and it turns out that when you're dealing with ultra-cold fermions, measuring the average energy is no longer very effective.

OK, what? Let me back up a bit. The definition of temperature in terms of average energy is a perfect description of the problem if you're talking about an ideal gas: hard-sphere particles that are distinguishable from one another rattling around in a box. when you put quantum mechanics into the mix, though, some weird things start to happen when you get to very, very low temperatures.

Right, like a Bozo Condensate. That's Bose-Eisntein Condensate (BEC), and that's one example of the sort of thing I'm talking about. BEC is a phenomenon that takes place when you get a large number of the right sort of atoms together at very low temperatures. Atoms with an even number of particles making them up (protons, neutrons, and electrons) behave as bosons, and "want" to be in the same state. When the temperature gets very low, all of the atoms will "condense" into a single state, generally the lowest-energy state available to them.

At that point, the average-energy method of temperature measurement gets a little dodgy. All of the atoms have the same energy, and it's the lowest energy they could possibly have.

So they're at absolute zero? No, because there are still some fluctuations in the sample-- some atoms will wander in and out of the condensate, occasionally taking on a higher energy. But it becomes damnably difficult to measure the actual temperature, because you're talking about looking at small fluctuations in the energy, rather than the energy itself. It's very tricky.

So that's what they're doing here? sort of. They're using fluctuations to measure the temperature, but the atoms they're dealing with aren't bosons, but fermions, which makes things much more complicated.

How so? Well, unlike bosons, fermions can't stand to be in the same state. They're subject to the Pauli exclusion principle, which says that no two fermions can occupy exactly the same state. So, when you get a lot of them together, you necessarily end up with them occupying higher energy states.

The basic physics involved is the same thing that gives you chemistry. Electrons are fermions, so you can never have two electrons in exactly the same state. When you start adding electrons to an atoms, then, the first one goes into the lowest energy state, the second goes into the same energy state with the opposite value of spin, but then that fills up the lowest energy state. The next electron in has to go into the second-lowest energy level, and so on. As you move up through the periodic table, the last electron added to each atom is in a higher and higher energy state, and the exact energy state of that last electron is what determines the chemical behavior of the different elements.

Something similar happens when you add fermionic atoms to an atom trap. The first atom in goes into the lowest possible energy state, the next one into the second-lowest state, and so on. The difference is that we're generally dealing with thousands or millions of atoms, way more than the tens of electrons in the atoms of the periodic table, so we tend to talk about this as more of a continuous distribution, because it's too difficult to count all the individual atoms. The situation is described by something called the "Fermi-Dirac distribution function" (sometimes just "Fermi function," for compactness), and it looks like the graph at right, which I grabbed from this page.

The Fermi function gives you the probability of a given state containing an atom, and it's usually plotted as a function of energy. It's equal to 1 for all the states up to a certain energy, the Fermi energy, determined by the total number of atoms you're working with, indicating that all the states below that energy contain exactly one atom. Above the Fermi energy, the Fermi function is zero, indicating that none of those states contain any atoms at all.

So you're saying that the last atom in your low-temperature sample of fermions actually has a high energy? It's not high in an absolute sense, but it's much higher than it would be in a system of bosons, where every atom would have the lowest possible energy.

That must play hell with measuring the temperature. Exactly. Once you get down to a low enough temperature, the average energy of your sample stops really changing. The last atom in has an energy determined by the Fermi energy, and that doesn't change. You can't measure the temperature by measuring the individual energy any more, you need to look at fluctuations.

What do you mean? Well, if you look at the graph above, you'll see that there are a bunch of different lines on the plot, corresponding to the Fermi function for different values of the temperature. you'll notice that as you raise the temperature, the step at the Fermi energy where the function drops from 1 to 0 gets more rounded.

That rounding of the edge occurs because when you have a finite temperature, atoms in your sample can move up and down in energy by an amount related to the temperature. An atom just below the Fermi energy can move up to a state just above the Fermi energy, which reduces the probability of finding an atom occupying the just-under-the-Fermi-energy state, and increases the probability of finding an atom occupying the just-over-the-Fermi-energy state.

This temperature effect shows up in real space as a fluctuation in the number of atoms at a given position. The low-energy states in a trapped sample of atoms correspond to atoms that are near the center of the trap, while the high-energy states correspond to atoms that spend most of their time near the outer edge of the trap. If you take a picture of the trapped cloud of atoms, you get a cross-section of energies, and you can measure how many atoms there are at various energies by looking at how the number of atoms varies with position.

So, you would expect a bigger variation in the number of atoms out at the edge of the cloud, where the high-energy states are? Exactly. The low-energy states are always fully occupied, because there's nowhere for the atoms to go-- they could use their thermal energy to jump up a small amount, but the state they would jump into is also occupied, so they can't do that. Out on the edge, though, the thermal energy can be enough to take an atom past the Fermi energy into an unoccupied state. That means the number of atoms in those states can fluctuate depending on which exactly atoms have jumped up or down.

What you expect to see, then, is that the fluctuation in the number of atoms you see in the center of a trapped sample of fermions at very low temperatures should be much, much lower than you would expect from normal counting statistics, while at the outer edge of the cloud, you would expect to see more fluctuations. And that's exactly what these two papers see.

How do they measure the fluctuations, though? Pretty much the simplest way you can imagine. They take pictures of the cloud of trapped atoms, and divide those pictures up into pixels. Then they count the number of atoms they see at each pixel, and repeat the measurement 20-odd times. Averaging all the different atom counts together gives a good measurement of the average number of atoms in each part of the cloud, and they can get the fluctuations by looking at the standard deviation of all their individual measurements.

Isn't that awfully tricky? I mean, lots of things could happen to make the number of atoms in a given pixel change. That's right. In order for this measurement to succeed, they need to have absolutely everything locked down as well as possible-- laser intensity, atom loading, cooling efficiency, etc. It's a major technical challenge, but these papers are from two of the very best experimental groups in AMO physics (Wolfgang Ketterle's group at MIT, and Tilman Esslinger's group in Zurich. And that's why this gets into Physical Review Letters.

So this method works well? Yep. They get fluctuation distributions that match exactly with what you would expect from the Fermi distribution, and they can work backwards from the distribution to obtain the temperature, which agrees very well with their estimates by other methods. In principle at least, this can be used to measure accurately to lower temperatures than any other method.

That's pretty cool. Isn't the fact that they can directly observe these fluctuations at all pretty awesome by itself, though? Yes, yes it is. Quantum mechanics is cool and weird, and this gives you a direct look at one of the coolest and weirdest aspects of the whole thing.

Westbrook, C. (2010). Suppressed fluctuations in Fermi gases Physics, 3 DOI: 10.1103/Physics.3.59

MÃ¼ller, T., Zimmermann, B., Meineke, J., Brantut, J., Esslinger, T., & Moritz, H. (2010). Local Observation of Antibunching in a Trapped Fermi Gas Physical Review Letters, 105 (4) DOI: 10.1103/PhysRevLett.105.040401

Sanner, C., Su, E., Keshet, A., Gommers, R., Shin, Y., Huang, W., & Ketterle, W. (2010). Suppression of Density Fluctuations in a Quantum Degenerate Fermi Gas Physical Review Letters, 105 (4) DOI: 10.1103/PhysRevLett.105.040402

drorzel Tue, 07/27/2010 - 04:25

Watching Individual Atoms Make a Phase Transition

drorzel — Tue, 22 Jun 2010 10:02:38 +0000

Watching Individual Atoms Make a Phase Transition

A press release from Harvard caught my eye last week, announcing results from Markus Greiner's group that were, according to the release, published in Science. The press release seems to have gotten the date wrong, though-- the article didn't appear in Science last week. It is, however, available on the arxiv, so you get the ResearchBlogging for the free version a few days before you can pay an exorbitant amount to read it in the journal.

The title of the paper is "Probing the Superfluid to Mott Insulator Transition at the Single Atom Level," which is kind of a lot of jargon. The key image is this:

Those by themselves probably don't clear things up all that much, though, so let's unpack that a little in Q&A format.

What's this about? Greiner's group has an apparatus that can detect the positions of individual atoms from a BEC in an optical lattice, which they have used to watch what happens to the distribution of atoms when they take the system from a "superconducting" state to an "insulating" state.

So, these atoms are conducting electricity? No, the atoms are playing the role of electrons. They're placed in an optical lattice, which plays the role of the atoms making up a solid. The transition in question is a transition from a state in which atoms move freely from one site to another to a state in which atoms are fixed in place at definite sites.

So what's an "optical lattice," again? An optical lattice is a periodic pattern of light that acts to trap atoms at specific points-- either spots where the light intensity is a maximum, or spots where the light intensity is a minimum, depending on the arrangement. The set-up Greiner is using is a two-dimensional square array of lattice sites, produced by projecting a predetermined pattern onto a BEC using a lens system that extends into the vacuum system, mounted just nine microns (0.000009 m, about a tenth the thickness of a human hair) away from the atoms.

How do they take pictures of the atoms, then? They use the same lens system that projects the lattice pattern, as described in an earlier paper (Nature, arxiv). It's a nice bit of engineering, but nothing all that unusual, optics-wise. They just put a really big lens right up next to their atoms.

So what's the big deal with that picture, then? OK, the top row of pictures in that figure are the actual images from their optical system. The bright green squares are producing a lot of light; the dark squares are producing very little light.

Those squares, which are 680nm on a side, correspond to different sites in the lattice where atoms can be trapped. The only way to get a lot of light from one of these sites is to have an atom sitting their absorbing and emitting photons, so the second row of pictures shows the output of an atom-spotting algorithm run on their images, which uses a threshold to assign either 1 or 0 atoms to the site .

As you go from left to right in the figure, the pictures correspond to samples prepared in optical lattices of different "depths." The deeper the lattice site, the lower the probability of atoms moving from one site to another, so the leftmost column of images corresponds to a "superconducting" state in which atoms move about freely, while the rightmost column of images corresponds to an "insulating" state where the atoms cannot move.

So, basically, you have more atoms in the insulating state? It looks that way, but only because of a detail of their imaging system. When they take their pictures, any sites containing two atoms get emptied out very quickly, thanks to an optical enhancement of the collision rate which knocks both atoms out of the trap. So the bright spots don't indicate sites with atoms as opposed to sites without atoms, but sites with an odd number of atoms-- a bright spot is a site that ended up with a single atom in it after all the possible pairs collided and went away.

So, the patchy-looking pictures on the left in that figure don't show just occupied and unoccupied sites, but sites with odd or even numbers of atoms. Some of those dark spots are sites that originally contained two atoms, and some of the bright spots are sites that originally contained three atoms. The picture all the way on the right shows the same overall number of atoms, but they're distributed more uniformly-- every lattice site has a single atom in it.

This is the difference between a superconducting state and an insulating one. In the superconducting phase, atoms can move from place to place, and thus the atom number per lattice site can fluctuate, even when it's in the lowest possible energy state. In the insulating phase, the atoms can't move from one site to another, so the lowest-energy state of the system has exactly one atom per site. What you see in the pictures is exactly what you expect to see on the microscopic level during this sort of phase transition.

Yeah, but how do you know that they're really uniformly distributed, and you didn't just get really unlucky with the distribution of pairs? That's the point of the third row of pictures. These are images of the atom cloud some time after it was released from the lattice. The distinct bright spots in the leftmost picture are an interference pattern resulting from atoms expanding out from different lattice sites, which is a signature of the superconducting phase. When you move to the insulating phase, the reduction in the fluctuation in the number of atoms per site leads to a random phase shift for the wavefunction at various sites, wiping out the interference pattern and producing the indistinct blob on the right. This provides confirmation of the phase transition.

That's pretty cool. What else can they do? Well, if they bump up the number of atoms, they see pictures like the ones at right. The atoms are confined to a limited range in two dimensions, so as they increase the atom number, at some point, they start to pile up in the center, and you get a dark region in the center where each site has exactly two atoms in it, and a bright ring where each site has exactly one atom. Keep going, and you get a bright central region with three atoms per site, then a dark region with four, and so on.

Those shells are kind of blobby, aren't they? Yeah. In an ideal case, you would see nice concentric rings, but imperfections in their optical system lead to distortions of the lattice, and the irregular shape that you see. They can correct this distortion with an adaptive optical element, which is what you see in the bottom row of the figure at right.

OK, so they can see transitions from superconducting to insulating in a square lattice. This is going to lead to room-temperature superconductors and flying cars? Not any time soon. It does give a nice way of studying that phase transition, though, which is very exciting for people working on those sorts of models. The way they generate the lattice potential is also nice, because they could easily use it to make different patterns, not just a square array. And the fact that the atoms they're using (rubidium, God's atom) have an array of different spin states means that they could use this imaging system to study a variety of magnetic effects and interactions.

None of this is going to produce flying cars in the immediate future, but it is a really nice, very clean way to look at what's going on in the atomic analogue of a condensed matter system. Which is a clever trick, and some great science.

Waseem S. Bakr, Amy Peng, M. Eric Tai, Ruichao Ma, Jonathan Simon, Jonathon I. Gillen, Simon Foelling, Lode Pollet, & Markus Greiner (2010). Probing the Superfluid to Mott Insulator Transition at the Single Atom
Level Science arXiv: 1006.0754v1

Bakr, W., Gillen, J., Peng, A., FÃ¶lling, S., & Greiner, M. (2009). A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice Nature, 462 (7269), 74-77 DOI: 10.1038/nature08482

drorzel Tue, 06/22/2010 - 06:02