In the previous installments, I talked about identical particles and symmetry, and what that means for fermions. Given that there’s only one other type of particle in the world, that sort of means that I need to explain what symmetry means in the case of bosons.

When I explain this to the first-year seminar, I generally do this by anthropomorphizing the particles somewhat, to describe fermions as “antisocial,” and bosons as “gregarious.” Not only does this give me the chance to use “anthropomorphize” and “gregarious” in class, thus confusing the hell out of a bunch of frosh, it’s actually fairly accurate. Pauli exclusion means that no two fermions can ever occupy the same state, while bosons are actually happiest when they’re all in the same state.

The canonical example of this is a laser. Photons are bosons, and you define the state of a photon with four numbers: energy, polarization, phase, and direction. A laser is a large collection of photons of a single energy, with the same polarization, all in phase with one another and organized into a tight beam headed in one particular direction. In other words, a laser is a collection of a huge number of photons all occupying a single quantum state.

Of course, this feels sort of like cheating, as photons aren’t material particles. They’re readily created and destroyed, unlike electrons, which means that a laser isn’t quite parallel to Pauli exclusion. If you really want a clear comparison, you ought to compare the behavior of large groups of electrons to the behavior of large groups of composite bosons. Which is where Bose-Einstein Condensation (BEC) comes in.

(Continued after the cut.)

The idea of Bose-Einstein Condensation traces back to the 1920′s, when an Indian physicist named Satyendra Nath Bose sent a paper to Einstein suggesting a new way of thinking about the statistical properties of photons. Previous attempts to get the paper published had failed, but Einstein saw the merit in the work, and quickly wrote a paper of his own to accompany Bose’s, and got both articles accepted in Zeitschrift für Physik.

What Bose did was basically to quantify the effect of symmetrization for bosons, in terms of the probability of finding some number of bosons in a particular energy state. This might not seem like the sort of thing that would have far-reaching consequences– it’s basically a matter of counting particles and states, after all, which seems like trivial arithmetic– but it leads to a dramatic prediction for low-temperature systems of bosons. Below a certain critical temperature, the probability of finding all of the particles in the same state blows up.

This is a consequence of the symmetry requirement for bosons, which need to occupy a wavefunction that is symmetric– that is, when you switch the labels on any two particles, you get the same state that you started with. The simplest way to arrange this is just to put all of the particles in exactly the same state, which is sort of trivially symmetric. So, not only is it all right for two bosons to be in the same state, it’s actually the preferred state of the system.

At high temperature, this doesn’t really show up, because of thermal fluctuations. At high temperature, the particles are scattered over a large number of states, and are constantly gaining and losing energy in collisions, in a process that is more or less random. The particles still “prefer” to be in the same state as other bosons, but there’s so much thermal energy rattling around that they don’t stay there for very long. As the temperature decreases, though, the amount of available thermal energy decreases, and the tendency of bosons to collect in the same state eventually wins out over the tendency of thermal energy to move them around.

Another way to think about this is through considering the physical effects of cooling quantum particles. Quantum theory tells us that particles have wave-like properties, and that the wavelength of these particles is inversely proportional to the momentum– that is, the slower a given particle is moving, the longer the wavelength associated with it. Temperature is a measurement of the average kinetic energy of a sample of particles, so as you cool a gas of atoms you’re making the individual atoms move more slowly, which in turn increases their wavelength.

Now, in order for the atoms to all end up in the same state, you need to have some way for them to “know” that there are other atoms around for them to group up with. This requires the atoms to be in contact with one another, and the “size” of an atom is given by the wavelength. So, at high temperature, when the atoms have very short wavelengths, they really don’t overlap much. As the temperature decreases, though, the wavelengths get longer, and the atomic wavepackets start to overlap. Eventually, you reach a point where the wavelength is comparable to the spacing between atoms, at which point essentially all of the atoms are in contact with one another all the time. At this point, they “realize,” in our anthropomorphic picture, that they’re bosons surrounded by other bosons, and that they would be happiest all occupying a single state.

You can write the condition for BEC in terms of the average thermal wavelength of a sample of atoms, and the average spacing between atoms, which is related to the density. The transition occurs when the density is high enough and the temperature low enough that the atomic wavefunctions overlap, at which point the atoms will all collapse into a single state, referred to as a Bose-Einstein Condensate.

There’s sort of a tricky balance here, because you’re talking about very low temperatures here, the sort of temperatures at which everything other than helium is a solid, and helium is a liquid. You need to somehow finesse things so that the density is high enough for the atoms to form a BEC, but not so high that they all clump together into a lump of solid. Working at lower density requires going to an even lower temperature, though, so nobody managed to get BEC in an atomic vapor until after the advent of laser cooling, despite several decades of effort. (Dan Kleppner’s group at MIT did manage to get BEC in hydrogen vapor by cryogenic techniques (i.e., without laser cooling), but groups working with laser-cooled alkali vapors beat them to it.)

If it only happens at really exotic temperatures and pressure, why is BEC interesting? Well, it turns out to be the explanation for a couple of deeply strange things that happen at low temperatures. One is the phenomenon known as superfluidity, in which liquid helium below about 2.7K acquires some bizarre properties– it flows completely without friction, resists rotation, and transfers heat with incredible efficiency. The transition between normal fluid and superfluid states occurs exactly where you would expect the helium atoms in the liquid to undergo Bose-Einstein condensation, and the effects can be explained through having a sizable fraction of the atoms occupying a single quantum state.

Slightly better known is the phenomenon of superconductivity, in which electrons in certain materials at low temperature will suddenly start to flow with no resistance. This is of considerable technological importance, as lots of things run on electricity, and the performance of most of these devices (basically, anything that isn’t intended as a heater) could be greatly improved by getting rid of the electrical resistance. Superconductors are used extensively to make very large magnets and certain types of high-quality electronics, and there are groups working on making superconducting elements for power transmission (including one company based in the Albany area).

Superconductivity is also related to BEC, though it’s more complicated than superfluidity, as electrons are fermions, and first have to be induced to pair up into composite bosons. The theory explaining this process was developed by John Bardeen, Leon Cooper, and Robert Schrieffer, and won them a Nobel Prize in 1972. It’s also beyond the scope of the current discussion, so just take my word for it: BEC is involved, and it’s very cool.

Both superfluidity and superconductivity occur in systems where the particle density is pretty high. This has its good points and its bad points. On the positive side, this means that the temperatures required aren’t too low, and can be reached by reasonable cryogenic techniques. On the negative side, though, it means that it’s hard to separate the condensed particles from the rest of the system– they’re inside a strongly interacting liquid (in the case of superfluid He) or inside a solid (in the case of superconducting materials), and it’s very difficult to treat these systems theoretically.

Nothing in the BEC theory requires interactions between the atoms, though, so it ought to be possible to make a condensate in the gas phase, using neutral atoms, and look directly at the physics of the condensate without any other interactions to complicate matters. Doing this requires working at ridiculously low temperatures, which requires not only laser cooling, but a clever trick beyond laser cooling to get to the transition. And I’ll explain that in a future post…

Comments

  1. #1 NL
    November 3, 2006

    Impressively obscure reference.

  2. #2 Chad Orzel
    November 3, 2006

    I haven’t actually read it, but my father is fond of the quote, and it stuck in my head for some reason.

    It’s far from the most obscure line I’ve used as a post title, but given the paucity of sources Google turned up, it may be the most obscure reference not based on an inside joke with friends from college.

  3. #3 Tommaso Dorigo
    November 4, 2006

    Hi Chad,

    very nice post! I will link it from my page.
    Keep up with the good work…

    Cheers,
    T.

  4. #4 James Graber
    November 4, 2006

    Very nice series of three posts.
    Now let me ask a few very naive questions:
    What’s the difference between a BEC and a crystal?
    Once a BEC forms why doesn’t it just collapse
    to a crystal or something even more dense
    like a black hole or a naked singularity?
    i.e. what is the repulsive force?
    What determines the density of the BEC?
    Can you call a crystal a Fermi-Dirac Condensate
    (with any justification)?
    If not what would a Fermi-Dirac condensate look like?
    If Fermi-Dirac condensates do not exist, why not?
    If there are good references where I can read about these things, I will be happy to look them up.
    TIA
    Jim Graber

  5. #5 Chad Orzel
    November 4, 2006

    What’s the difference between a BEC and a crystal?

    The degree of order.
    A crystal is, by definition, a regular lattice of particles in a particular arrangement. A BEC is still a disordered gas, albeit an extremely cold and dense one.

    Once a BEC forms why doesn’t it just collapse
    to a crystal or something even more dense
    like a black hole or a naked singularity?
    i.e. what is the repulsive force?

    The individual atoms continue to act like “hard spheres,” and collisions between them give rise to what is effectively a repulsive force. The overall density of a BEC is something like ten million times lower than that of air, so these collisions are relatively rare.

    There are atoms for which the effective interaction from collisions is attractive, and those do tend to collapse when the atoms number gets high enough. The energy involved isn’t really all that great, so the collapse doesn’t lead to anything all that dramatic, but some very interesting work has been done on this by Eric Cornell and Randy Hulet, among others.

    What determines the density of the BEC?

    Collisions between atoms, and the Heinsenberg Uncertainty Principle.

    The wavefunction that the atoms occupy has a non-zero size, because of the zero-point energy of whatever potential the atoms are trapped in. You can never know both the position and momentum to infinite precision, so no matter how cold you make the sample, the atoms will still be moving around a bit, which gives the condensate a certain spatial extent, and determines a non-zero volume.

    Beyond that, collisions between atoms tend to push the cloud out a bit farther, lowering the density even more.

    Can you call a crystal a Fermi-Dirac Condensate
    (with any justification)?
    If not what would a Fermi-Dirac condensate look like?
    If Fermi-Dirac condensates do not exist, why not?

    Fermions are an interesting case, and I’ll probably talk about them more at some point. They can’t form a direct analogue to a BEC, because no two fermions can occupy the same state. As a result, there’s a limit to how much you can lower the energy of a gas of fermions.

    It turns out, though, that if you go about it in just the right way, you can induce a sample of fermions to pair up and form composite bosons (weakly bound molecules, in some sense), and those can Bose condense. This is a huge area of current research, and allows people to test fifty-odd years of condensed matter theories for the first time.

    Probably the first person to get this working was Debbie Jin at JILA, and the Ketterle group at MIT has also made some important contributions. If you check their web sites, you might find some useful stuff.

    As I said, though, I’ll probably talk more about this in a future post.