Cold Atoms and Cooper Pairs

So, last week, I talked about how superconductors work, and I have in the past talked about the idea of making cold atoms look like electrons. And obvious question, then, whould be:

Do cold atoms systems allow us to learn anything about superconductivity? The answer here is, unfortunately, “Yes and no.”

That’s pretty weaselly, dude. Yeah, well, there’s nothing I can do about that.

There are a huge number of experiments out there using ultracold atom systems to look at Bose Einstein Condensation, which is related to superconductivity, and that transition has been studied in great detail. Those experiments are necessarily done with bosons, though, and electrons are fermions. If you want to look at the superconducting transition, you need atoms that reproduce the spin characteristics of electrons, and then some way to create Cooper pairs with those atoms.

At the moment, there’s no way to use cold atoms for a complete simulation of the physics of superconductivity, but you can use them to look at an intermediate area of the theory, that can’t be explored any other way.

What do you mean, “intermediate?” It’s a reference to the size of the atom pairs that undergo the transition. Prior to experiments with cold fermions, there were two limits of the theory that had been studied: Bose-Einstein condensation (BEC) of very small particles, and the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity, which deals with very large things.

Hang on, I thought that the things condensing in a superconductor were electrons, which are point particles. How can they be considered “very large?” The important entities in the BCS theory are not electrons, but pairs of electrons. The Cooper pairs that form in a typical superconductor are very large– hundreds of times larger than the spacing between atoms in the superconducting lattice. This leads to behavior that is very different than the behavior of atoms in a BEC, which can mostly be thought of as point particles.

So the “intermediate” range you’re talking about is somewhere between these? Exactly. You can use cold atoms to explore the “BEC-BCS crossover” regime of theory, and searching on that phrase will turn up a whole bunch of stuff, including Cindy Regal’s Ph.D. thesis, which has a lot of information about the early experiments in this regime.

OK, so how does this work? Well, for starters, you need a gas of atoms that happen to be fermions rather than bosons. This can be arranged by simply taking a different isotope of one of the atoms people use for BEC experiments– if you add one neutron to a bosonic atom, it becomes a fermion, though it does not change its chemical properties appreciably. There’s generally a small “isotope shift” in all of the energy levels, and the hyperfine structure may be more complicated, but those are technical problems that are overcome with a bit of lab work. Lithium-6 is a popular choice for these experiments; other people work with potassium.

Clever experimentalists can cool gases of fermions down to the degeneracy point, where the difference between fermions and bosons becomes apparent, as in the recent experiment on number fluctuations. That’s the starting point for BEC-BCS experiments.

OK, but how do you stick the atoms together? When you talked about superconductivity, you said it’s the atomic lattice that pairs the electrons, but there’s no lattice here. So what gives? Well, you can get the idea pretty easily if you just ask yourself what it means to pair up atoms. What do you call two atoms that are stuck together?

A diatomic molecule, which is a molecule that has– One atom too many, I know. That’s a joke for a different post.

The key realization here is that two atoms stuck together make a molecule, which means you need to turn to molecular physics for the answer.

OK, but you’ve just got a bunch of atoms, not molecules. How do you turn them into molecules? Well, any time two atoms collide, you can describe their interaction in terms of molecular states. It’s a transient molecule, to be sure, because two free atoms colliding have too much energy to make a bound molecule, but even if they don’t stick together, they’re temporarily a molecule. And during the short time that they are a molecule, there’s a chance that they can jump to a state that is bound.

How? Because quantum mechanics is magic. The specific form of magic we’re talking about is a thing called a “Feshbach resonance,” which is illustrated here (Figure 3.1 in the thesis linked above):

The picture on the left shows the potential energy as a function of spacing between two atoms. The free colliding atoms we start with have potential energy given by the dark blue curve (“open channel”), and thus they start out at large separations, come together and then separate. If one of the atoms was in a different internal state, though, their interaction energy would be determined by the light blue curve. In that case, a pair of atoms with enough energy to be just above the dark blue curve at large separation would be below the light blue curve (“closed channel”). Atoms in the light-blue state with that amount of energy can never be very far apart, meaning that they’re only allowed to exist as bound molecules, staying much closer together.

OK, so you just need to flip the state of one of the atoms when they get close together? You can do that, though it’s a little tricky. There’s an even easier way, though, which is to apply a magnetic field, and let quantum mechanics do the job for you.

If you just pick two atoms at random, the odds are very poor that a particular pair of colliding atoms would have exactly the right energy to line up with a bound state in the other channel. If it did, though, a pair of colliding atoms could tunnel into the bound state, and spend a while existing as a bound molecule before tunneling back out and separating. You can make this happen, though, by using the fact that the different channels shift by different amount sin a magnetic field. This is illustrated by the graph on the right, which shows the energy of the allowed states of the open and closed channels as a function of magnetic field. You can see that the open channel changes a lot faster than the closed channel, so there’s a point where the two states overlap.

So, you put on a magnetic field of the right strength, and the colliding atoms spontaneously become molecules? Temporary molecules, yes. This changes the interactions between the atoms, in a way that can make the net effect of collisions between atoms look like an attractive interaction between them. This is known as a “Feshbach resonance,” after the theoretical nuclear physicist Herman Feshbach.

A net attractive interaction is what you need to make Cooper pairs, right? Exactly. The external magnetic field plays the role of the lattice in BCS theory, producing an interaction between atoms that pairs them up to make composite bosons. These “Cooper pairs” (technically, I suppose, the term ought to be reserved for lattice-mediated electron pairing, but everybody uses the term for atom pairs as well) then undergo a transition like that for electron pairs in a BCS superconductor.

So what’s the catch? Well, there are a couple of things. For one thing, the physics of a Feshbach resonance is very different than that of the electron-lattice interaction, so it’s not clear that this sheds all that much light on the behavior of electrons. Also, the size here is in a funny intermediate regime– the Cooper pairs produced via a Feshbach resonance are big, but nowhere near as big as a Cooper pair in a BCS superconductor. This puts the atoms in kind of an intermediate regime– it’s more like BEC than a pure BCS system, but they’re too big to be treated as real point objects.

What happens if you use real molecules, instead of these big pairs? You can do that, too, by changing the strength of the magnetic field in the right way. If you set thing sup right, you can produce a lot of atoms in bound molecular states that are also pretty big. This is closer to BCS theory than a pure BEC system, but definitely deals with bound molecules rather than big pairs with an effective attractive interaction between them.

Using magnetic fields and Feshbach resonances, you can map out a range of sizes, that defines this “BEC-BCS crossover” regime. This is a range of parameters that theorists have played around with over the years, but that was inaccessible in condensed-matter systems until ultracold atoms came along.

So, it’s kind of a toy model? Pretty much. It’s a nice check that the theoretical machinery is able to handle both sides of this, and provides some reassurance that we really understand what’s going on in a BCS superconductor. It would’ve been awfully surprising to find that the theory didn’t work, though, and this isn’t going to be the key piece that lets us figure out how high-temperature superconductors really work.

So, the short version of the answer is “Yes and no.”

Yep. You can use cold atoms to make something that is related to the superconducting transition in a real superconductor, but as yet, nobody has developed a way to make a cold-atom analogue of a high-temperature superconductor. Such a thing might be possible, and would be a real game-changer, but I don’t know of any good proposals for a way to do it.