This is the last of the five papers that were part of my Ph.D. thesis, and at ten journal pages in length, it’s the longest thing I wrote. It was also the longest-running experiment of any of the things I did, with the data being taken over a period of about three years, between and around other experiments. As usual for this series of posts, I can sum up the key result in one graph:
(No spiffy color figure this time, as the experiment never made it onto the old web page, and my original figures are three or four computers ago.)
What we found was that when we prepared samples of metastable xenon at very low temperatures, some isotopes would collide as usual, making lots of ions, while other isotopes at the same temperature and density would not collide at all. This is a pure quantum effect– it’s not chemistry, as the only difference between the samples is a few neutrons in the nucleus– and comes about because of the Pauli Exclusion Principle.
This may seem like an odd thing to say, because we’re not used to thinking about Pauli exclusion in collisions. The familiar form of Pauli exclusion is high-school chemistry, where you draw little diagrams of up and down arrows filling electron shells in atoms. You can have one up arrow and one down arrow per shell, and then you need to move up to the next energy.
This happens because electrons are fermions, and is a result of a more general principle that says that fermions have to be in antisymmetric states. This means you can never find two electrons occupying exactly the same energy level in an atom or solid.
This rule also applies to fermionic atoms. If you put together an atom such that the total number of particles making it up (protons, neutrons, and electrons) is odd, that atom will behave like a composite fermion, and you can never get two of them to occupy exactly the same state. If the total number of component particles is even, that atoms will behave like a composite boson, and can form a Bose-Eisntein condensate and all that fun stuff. Adding a single neutron to the nucleus of an atom will flip it from a boson to a fermion, or vice versa, so atoms with multiple isotopes can come is both varieties. Xenon has nine stable isotopes, seven bosons and two fermions.
How does this affect collisions? Well, in order to explain why cold fermions won’t collide, I need to first say a bit about how to think about cold collisions in quantum terms. Quantum mechanics tells us that even system in the universe has discrete allowed states, and that it can only ever be observed in one of these states. It may not be immediately obvious, though, what is quantized when we talk about two colliding atoms.
It turns out that the relevant quantity is angular momentum. When two atoms collide, you can think of them as if they were rotating around a common center for a very short time, and figure out the angular momentum associated with those collisions. This angular momentum takes on discrete values, integer multiples of Planck’s constant, and for historical reasons, they’re assigned letter names: collisions with angular momentum of 0 are “s-wave” collisions, collisions with angular momentum of 1 unit of Planck’s constant are “p-wave,” collisions with 2 units are “d-wave,” and so on. If you look at the angular momentum involved at different temperatures, you find a graph that looks like this:
This is a log-log plot, covering four orders of magnitude in temperature, and it shows the relative likelihood of colliding in various angular momentum states as a function of temperature. At a temperature of about 100 microkelvin (100 one-millionths of a degree above absolute zero), the probabilities of s, p, and d-wave collisions are roughly comparable, but if you go down to a temperature of 1 microkelvin, s-wave collisions are almost a hundred times more likely than p-wave, and about 10,000 times more likely than d-wave.
You can understand this if you think about the collision in a semi-classical way: angular momentum, classically, is given by the mass of one of the particles multiplied by its velocity, multiplied by the distance of closest approach between the two (we imagine one of the two colliding particles to be stationary for this). The mass is fixed, so the velocity and approach distance (or “impact parameter“) determine everything.
If you want one unit of angular momentum, you can get it by having a fast-moving particle collide at a small distance, or by having a slow-moving particle collide at a large distance. There’s a limit to how far out the particle can be, though, because if the distance of closest approach is too great, it’s like the collision never occurred– the atoms won’t get close enough to ionize.
The collision velocity is determined by the temperature (the temperature is the average kinetic energy of an atom in the sample), which restricts the angular momentum options. At relatively high temperature, you have enough fast-moving atoms to get collisions with one or two units of angular momentum, but as the temperature gets lower and lower, fast-moving atoms are rarer and rarer, and eventually, you reach a point where the atoms are moving so slowly that there’s no way to get even one unit of angular momentum– if you move the atoms far enough apart to get one unit worth of angular momentum for that velocity, they’re too far apart for an ionizing collision, and just miss each other. At low enough temperature, the only collisions you can have are s-wave collisions.
What does this have to do with bosons and fermions? Well, recall that one of the defining characteristics of bosons and fermions is that they can only be in states of certain symmetry. Bosons can only be in symmetric wavefunctions, and fermions can only be in anti-symmetric wavefunctions.
When you look at the collision process in all its gory quantum detail, you find that collisions with even angular momentum (0, 2, 4,… units of Planck’s constant) are symmetric, while collisions with odd angular momentum (1, 3, 5,… units) are anti-symmetric. That means that the only way for two fermions to undergo an s-wave collision is for their internal states to be different (one spin up, the other down, say), so the overall wavefunction can be anti-symmetric. If the two fermions are in the same internal state (“spin polarized” is the jargon term for this), they are absolutely forbidden to undergo s-wave collisions, which means that they cannot possibly collide at all if the sample is cold enough.
(In the semi-classical picture, you can think of it as requiring the two fermions to be in the same place at the same time (because they have to end up right on top of each other to have zero angular momentum), in which case Pauli exclusion comes into play.)
Bosons, on the other hand, can only collide in even states, never odd. So, at ultra-low temperatues, we expect a spin-polarized sample of bosons to collide happily (the overall collision rate for a sample of bosons is shown as the dashed line in the figure above), while a spin-polarized sample of fermions should not collide at all.
That’s what we set out to measure in the metastable xenon system, which is extremely well suited to this, having seven bosonic isotopes and two fermionic. We cooled and trapped samples of different isotopes, and measured the collision rate for each in spin-polarized and unpolarized states, and looked to see the suppression due to quantum statistics.
Of course, there were a lot of details involved. The first step was to establish that the isotopes had the same rate to begin with, so as to be sure that any change we saw was a quantum effect. We had previously measured the absolute rate for xenon-132 in the optical control experiment, so we used that as our base number, and compared several other isotopes (xenon-134, -136, -131, and -129) to the rate for xenon-132. The results are shown in figure 3 from the paper:
As you can see, there’s no significant difference. The next order of business was to compare polarized to unpolarized samples over a wide range of temperatures. We could’ve done this by simply preparing trapped samples at a wide range of temperatures, and measuring the rate for each, but that would’ve been unbelievably tedious, so instead, we used a clever trick to get the whole temperature range of interest in a single experiment.
The trick we used relies on the fact that the velocity that matters for the collision is not the absolute velocity of the atoms, but rather the relative velocity of a colliding pair. We don’t need to have every atom moving slowly, as long as atoms that are close together are moving slowly relative to one another. The way to achieve this is simplicity itself: you just load a bunch of atoms in the trap, and then let them go.
When you have a bunch of atoms in an atom trap, they have a wide range of possible velocities, determined by their temperature, but they’re all in more or less the same place. That means that the range of possible collision velocities spans the entire range of velocities allowed by the temperature.
When you let the atoms go, they fly apart with whatever velocity they had at the moment of release. Some time later, you will find that at any given point, the relative velocity of any colliding pair of atoms is lower than it was when they were in the trap. This is because atoms moving with very different velocities end up in very different places, and can’t possibly collide. Atoms with similar velocities will end up close together, but the relative velocity between them will be low– one may be moving at 10.0 cm/s, and the other at 10.1 cm/s, but the collision velocity will be the difference between the two, or 0.1 m/s.
For the collision experiment, then, we collected a sample of atoms, let them go, and then measured the collision rate as they flew apart. We could then reconstruct the effective temperature for collisions at any given time, using the initial size and temperature of the original cloud of atoms. We did this with spin-polarized samples (where all the atoms were in the same internal state), and with “unpolarized” samples (for technical reasons, we chose to use samples that were prepared with a particular distribution of internal states– if you really want to know, ask, and I’ll explain in comments), and compared the two. When we did this for several different isotopes, we got the following data:
This is a semi-log plot, showing the ratio of polarized to unpolarized collisions for five different isotopes over just about three orders of magnitude in effective temperature. You can see that at high temperatures, the ratio is close to 1, meaning that there’s no difference between polarized and unpolarized samples, as we expect when there are lots of angular momentum states available. At very low temperatures, there’s a big split between the bosonic isotopes whose collision rate goes up slightly, and the fermionic isotopes whose collision rate goes down dramatically, exactly as we expect.
You might be wondering how well we really understand the details of this. Well, the figure at the top of this post shows the best boson data and the best fermion data compared to curves derived from a very simple theoretical model, which includes the exact distribution of states that we put in, including the fact that our spin polarization is not necessarily perfect (for technical reasons that aren’t worth going into here). The data match the theory very nicely, assuming a polarization between 90 and 100% for the bosons, and a bit lower for the fermions.
So, what we see is that adding one neutron to each atom in an ultra-cold sample can switch off the collisions completely. Quantum mechanics is weird stuff.
C. Orzel, M. Walhout, U. Sterr, P. S. Julienne, S. L. Rolston (1999). Spin polarization and quantum-statistical effects in ultracold ionizing collisions Physical Review A, 59 (3), 1926-1935 DOI: 10.1103/PhysRevA.59.1926