The Perpetual Motion of Superfluids (script) (Patreon)

The weird rules of quantum mechanics lead to all sorts of bizarre phenomena on tiny scales— particles teleporting through walls or being in multiple places at once or simultaneously existing and not. Shame all this magical behavior doesn’t happen on scales large enough for us to see. Except that there is a way for us to see large-scale quantum weirdness, and that’s Bose-Einstein Condensates

Perpetual motion machines are all scams. But perpetual motion itself is possible. Sort of. If you stir a cup of tea you’ll create a vortex that slows down over time. But there is a type of fluid in which the vortex will never stop—at least theoretically. Even unstirred, it will climb the walls all by itself, and even leak through the microscopic channels in the otherwise watertight porcelain. The fluid in question is liquid helium cooled to near absolute zero it becomes an exotic state of matter known as a superfluid.

Superfluidity is the most characteristic behavior of the bizarre state of matter called the Bose-Einstein Condensate. This is a state of matter in which the weirdness of the quantum world invades our large-scale reality. To understand this, we do need a bit of background—and that background is our last episode where we talked about how the behavior of matter depends on how nature counts its particles—described in the field of statistical mechanics. Watching that episode either right now or right after this episode will really help your understanding of this phenomenon. We’ll also be referring to this episode, but that you can definitely watch afterwards.

First, let's recap just a little of what we learned in the last episode. We saw that we can understand a lot of the behavior of matter by counting the number of different ways in which particles can be divvied up into the available energy states. Particles will tend to shuffle energy between each other until they have the most probable energy distribution. The shape of that energy distribution determines the thermodynamics—which means, how things like temperature, pressure, volume etc. relate to each other.

We also learned that there are different types of particles - bosons and fermions - which shuffle around their energies differently. To remind you: no two fermions can occupy the same quantum state, which means they can’t share exact energies. On the other hand, any number of bosons can occupy the same state. In everyday life, this difference doesn’t matter. Everything else being equal, a cloud of bosons and a cloud of fermions behave the same.

But there are circumstances where the differences become very important. That includes the world of the very small—the quantum world. But also the world of the very cold. There the quantum behaviors of these different particle types are also clear as day.

In another previous episode we talked about the reason for the different behavior of bosons and fermions—we saw how quantum spin determines the symmetry of the symmetry wavefunction, and its this symmetry that determines whether two particles are allowed to have identical wavefunctions. However that episode was on the mathy side, and also focused on fermions. For now I want to try to give you a more intuitive view - and one that lets us understand the behavior of bosons, which, afterall, is what we need to understand superfluids.

In quantum mechanics, we describe particles as wavefunctions—mathematical objects that encode the probability that we will observe a certain position, velocity, whatever, if we were to try to measure it. Wavefunctions are well named, because they’re wavy. For simplicity, let’s just look at one component of a particle’s position wavefunction, representing its blurry quantum location in space, which might look like this wave-packet-a sine wave that tapers off on either side. The height of the wavefunction at any point represents the probability of the particle actually being in that location.

Now let’s think about two particles moving together. At a certain point their wavefunctions will overlap - and what happens then depends on the symmetry of the wavefunction. In this context, when you have a symmetric wavefunction, then when they’re in the same quantum state their peaks and valleys line up perfectly. A pair of symmetric wavefunctions can stack, doubling the height of the peaks and the depth of the valleys. In the case of an anti-symmetric wavefunction, if you try to put them in the same quantum state, the peaks of one will always valley line up with the valleys of the other, and vice versa. In that case, overlapping the wavefunctions causes the entire thing to cancel itself out, with the peaks of one negated by the valleys of the other.

The case of the symmetric wavefunction is straightforward enough to interpret. The fact that they can stack without canceling each other means that particles of this type—bosons—can be placed in the same state without breaking any physics.

Fermions, on the other hand, with their antisymmetric wavefunctions, appear to cancel each other out. If the wavefunction encodes their possible positions, then when they cancel out there’s zero probability of either particle being anywhere. This doesn’t mean that pairs of fermions destroy each other. That would break several conservation laws—from conservation of energy, charge, even quantum information. A wavefunction can’t just be deleted from the universe. So rather than telling us that fermionic wavefunctions cancel each other out when the overlap, this canceling effect actually tells us that fermions can’t ever overlap perfectly. They can’t ever occupy the same state. In fact if you try to push two of the same type of fermion into the same quantum state, they resist with a force so powerful that, for example, it can help a dead star resist collapsing into a black hole. But that’s another story.

This way of describing the difference between bosons and fermions is extremely heuristic—it’s heavy on analogy and shouldn’t be interpreted too literally. For one thing, the part of the wavefunction that’s symmetric or antisymmetric doesn’t look like a simple sine wave—it’s the part of the wavefunction associated with the rotational state, not the position state. It’s connected to particle spin. In fact, spin defines whether a wavefunction is symmetric or antisymmetric. All particles with values of quantum spin that are an integer—so 0, 1, 2, etc. are symmetric and are bosons. Those with half-integer spin—½, 3/2, etc—and antisymmetric fermions. For a more nuanced description of all of this, including a lot more on this spin stuff, check the episode I mentioned earlier.

But for now, let’s just take one critical fact from all of this: bosons, with their integer spins and symmetric wavefunctions can be stacked, fermions, with their half-integer spins and antisymmetric wavefunctions, cannot. And by the way, all of the elementary matter particles—everything that makes up physical stuff—are fermions. And the inability for them to perfectly overlap is why physical stuff is physical in the first place—for example, atoms have structure because electrons can’t all fall into the lowest atomic orbital together. On the other hand, photons and other bosons can overlap completely. This is what makes lasers possible—a laser beam consists of many, many photon wavefunctions that are perfectly overlapping, or in phase with each other.

So what would it look like if we could turn fermions into bosons? Would it be possible to make an electron laser? Could we make atoms that collapse into super-dense states? Actually, we can turn fermions into bosons, and the result is the Bose-Einstein condensate, and the superfluid that I’ve been teasing at for the past several minutes.

Fermion wavefunctions have half-integer spin. The spin of multiple fermion wavefunctions that are somehow connected to each other have spins equal to the sum of those spins. For example, a proton is made of three quarks, two with +½ and one with -½, adding up so the proton is also a spin ½ fermion.

But what if we could connect just two spin-½ particles? That would give a spin of 1 which is an integer and should produce a boson.

Take the helium-4 atom. It has two protons, two neutrons, and two electrons - all spin-½ particles. In this case, 3 are spin +½ and 3 spin -½, which gives the overall atom a spin of 0. That’s an integer spin, therefore Helium-4 is a boson. Its component particles remain fermions and behave that way with respect to each other within the atom - they have different internal energy states. But the entire helium-4 atom acts like a boson with respect to its neighboring helium-4 atoms. That doesn’t mean they overlap in physical or position space—their internal fermions would resist that. But the atoms as a whole can occupy identical energy states..

We really only see the effect of this at extremely low temperatures. So let’s do an experiment and cool down some helium-4. At atmospheric pressure and room temperature the helium is a gas. At this point it wouldn’t matter what the gas was—the shape of this distribution would look the same and only depend on the temperature. As temperature drops, more and more particles move to lower energies. At around 5.2 K the gas condenses into a regular liquid. As we cool things down further, things start to look different. If these were a gas of fermions, the inability of particles to occupy the same energy levels would prevent all particles from falling to the lowest energy states. That has its own bizarre effects, especially at very high pressures, but we’re talking about bosons here.

For the boson Helium-4 atoms, there’s nothing stopping all particles entering the lowest energy state. When we hit a temperature of 2.17 K, that’s exactly what happens. Rather than having a distribution of energies, all particles have the same energy—the lowest possible energy. In fact they can’t have any other energy because at this low temperature, there isn’t enough energy to jump to higher states. But they also can’t lose energy - they already have the least possible energy.

In normal states of matter, when two atoms get close to each other they repel each other, exchanging energy. Typically one loses energy and one gains it. But when all the particles have the same energy then interactions that require the exchange of energy aren’t possible. For example, a fluid made of such particles loses all friction. Normally, when fluids flow, currents or streamlines within them that have different speeds within the fluid drag on each other—we call that viscous drag. That drag is caused by particles at the streamline boundaries colliding and exchanging energy, and it causes the flows to slow down. Such a fluid has viscosity—the higher the viscosity, the more drag, and the less easily the fluid flows. But the streamlines in very cold helium-4 don’t exchange energy with each other—they slide past each other without friction, and so the fluid has zero viscosity. It is, in fact, a superfluid.

And now we get back to stirring our cup of tea. Particle interactions with the walls and with each other cause the circular currents to slow down and stop. But if your tea is superfluid helium-4, then these interactions can’t happen. Stir the cup and the vortex should last forever. In reality, this would also require the walls of the cup to be perfectly smooth. According to this cool simulation, when superfluid flows around the microscopic bumps of a container wall it forms tiny perfectly-flowing vortices that do actually slow down the overall flow, but in a different way to a regular fluid.

By the way, saying that all particles in a superfluid have the same energy is a bit simplistic. Energy states can vary a bit across the fluid due to different speeds of its streams, different heights in the gravitational field, etc. But the point is that particles near each other are extremely close to having the same energy state and so are restricted in how they can interact.

There are a few other weird superfluid behaviors. All fluids will creep up the walls of their containers a little bit due to the mild attraction between the atoms of the fluid and container. But in a superfluid, the frictionlessness allows the fluid to literally flow up the wall, at least once that wall is coated with enough fluid to itself be frictionless. Then when the flow drops over the other side, it acts like a siphon, and can drain your container of your precious helium-4. Superfluids will also flow through microscopic fissures and pores in a container, causing it to appear to leak through apparently solid material.

This Houdini-like nature of helium-4 superfluid reminds me of quantum tunneling—how quantum objects can teleport short distances and through solid barriers. This is NOT quantum tunneling, but it IS a true quantum effect, visible on a macroscopic scale. All of the behaviors of superfluids are due to the quantized nature of the energy levels in the fluid.

Superfluids aren’t just a novelty found in physics labs. Super fluids are believed to exist inside neutrons stars, where fermionic neutrons team up to act like bosons, and form these whirlpools trom the surface of the star to its core that hold insane amounts of energy. They last for a long time because-superfluid-but when they finally do break they are believed to cause the neutron starquakes that we see in the glitches of pulsar signals.And superfluidity is basically what electrons are doing inside the otherwise solid crystal lattices of a superconductor, as we showed you in our episode on quasiparticles. In that case, electrons connect with each other over large distances to form Cooper pairs, so that these fermion pairs also act as bosons, leading to frictionless flow of the electrons. This manifests as superconductivity.

At a more fundamental level, the fermionic quarks can join in pairs to become bosons, which allows them to serve as strong-force-carrying mesons. The bosonic nature of the Higgs field allow its particle to form a slightly different kind of condensate—one which fills the entire universe, and which is responsible for giving elementary particles their mass. I feel like we’ve pointed you to earlier episodes even more than usual this episode. But, yeah, we have one on the Higgs stuff also. But by now I think we’ve all realised that, just like with bosons, there’s no limit to the number of overlapping references to past episodes of Space Time.