# On ducks, waves, and superconductivity

When a duck decides to go for a swim with her ducklings, she possibly has a lot on her mind, but the one thing she doesn’t have to worry about is her ducklings keeping up with her. If you’ve ever watched a family of ducks in a pond, you’ll know that the ducklings follow mama duck around in single file, perfectly orderly, never straying from her path, as if they’re being pulled along by an invisible string. As it turns out, this illusion is the result of ducky instincts shaped by the fine scalpel of evolution — and classical wave mechanics. A recent study that went onto win the IgNobel prize in 2022 showed that when the ducklings cruise at just the right positions behind mama duck, they can quite literally ride the waves generated in her wake. To be more precise,* *the waves generated by mama duck *destructively interfere* with their own. This means that the waves, instead of creating a drag, propel the ducklings forwards. What’s more, each duckling effectively passes this assist onto its siblings behind them, so that they all move in unison.

Given the ubiquity of wave phenomena in our world, it is perhaps not surprising that classical wave mechanics shows up in the most unexpected places. I will describe another such instance, in a rather different setting — this time, with consequences that are arguably even more spectacular.

Electrons inside materials lead rather bumpy lives. Being charged particles, they strongly repel each other, and Pauli exclusion means they don’t ever want to be in the same place at the same time. In addition, they also bump into all manner of other objects in the atomic lattice that makes up the material, some stationary, and some moving, such as defects, vibrations, and impurities, so that the experience of an electron moving through a solid is a bit like biking through a crowded street market; you put a lot of effort into getting rolling, only to be stopped by something or someone moments later, and you have to start over. However, in some solids, if you go to sufficiently low temperatures, something astonishing happens. The very electrons that strongly repelled each other now instead *attract* each other and pair up. These pairs seemingly telepathically conspire to create a ‘condensate’ — a state in which they’re all in sync and move together like a perfectly conducted symphony. Set up an electronic current and it will last forever. Apply a magnetic field and it will be expelled with perfection. This is of course, a transition into the superconducting phase, one of the most dramatic manifestations of interactions in matter.

At the heart of all of this is the unexpected mutual attraction between electrons that drives them to form pairs. If you’re familiar with condensed matter physics at all, you might know that these are called ‘Cooper pairs’, named after Leon Cooper, who along with John Bardeen and John Schrieffer formulated the influential Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity. The glue that binds these pairs together, atleast in conventional superconductors, are quantized vibrations of the crystal lattice, called ‘phonons’. Given that electrons and phonons are quantum mechanical objects, one might reasonably expect that the mechanism underlying Cooper pairing is uniquely quantum mechanical. You will indeed find that most textbooks will derive the mutual attraction by considering how phonons influence the electrons’ quantum mechanical wavefunction.

It’s surprising then, to learn that Cooper pairing can be described perfectly well by the same physics that describes ducklings, tennis balls, clouds, you, and me — good old classical mechanics.

There is a completely general result in classical mechanics, as follows: if you have two objects interacting with a springy medium of some sort, or to be precise, a harmonic oscillator, the oscillator (with some caveats) will induce an *effective attraction* between the objects. You can show this by calculating what is called the ‘response function’ of the harmonic oscillator. The response function of an object mathematically describes its response to an external disturbance.

Now, this is classical mechanics, and is ostensibly not applicable to the quantum harmonic oscillator. The remarkable thing though, is that the result also holds in the quantum case. This is because the quantum harmonic oscillator has the unexpected and unique property that its response function is *identical* to its classical counterpart.

It’s enlightening to unpack this statement a bit. First, it is far from obvious. This statement is different from the famous ‘correspondence principle’ first formulated by Niels Bohr, which says that in the limit of large orbits or large quantum numbers, quantum mechanical systems behave approximately like their classical versions. Contrary to this, what we have here is *exact*, and it is true in *all* cases. You can show this by directly computing the response function using standard techniques such as the Kubo formula. The more interesting problem though, is not proving that they are identical, but understanding *why* they are identical — what makes the harmonic oscillator so special?

The answer lies in the fact that the harmonic oscillator is *linear*. Recall high school physics: the restoring force of a harmonic oscillator is linearly proportional to its displacement. Linearity is a rather special quality — among other things, for linear functions, the average of the sum is equal to the sum of the average. There is a well-known result derived from the Schrodinger equation called ‘Ehrenfest theorem’, which relates the average values, or ‘expectation values’ of quantum mechanical operators to the classical equations of motion. As it turns out, the linearity I just described, along with Ehrenfest theorem, directly implies that the equations of motion of the quantum and classical harmonic oscillators are exactly identical! This particular classical-quantum equivalence is unique to the harmonic oscillator. For square well potentials, cubic potentials, or whatever other potential you may care to draw up, the quantum mechanical expectation values will behave in ways that deviate from classical mechanics.

All this to say — the phonon-mediated effective attraction between electrons that leads to the formation of Cooper pairs in superconductors arises from a mechanism that is apparently quite classical! This also illustrates, in the simplest way possible, that any system that can be approximated as a harmonic oscillator, such as lattice vibrations (phonons) in the case of conventional superconductors, but also magnetic vibrations (magnons), or even electromagnetic waves (photons), can potentially mediate an attraction between electrons that otherwise repel each other. This is quite well established in condensed matter physics, for example, magnetic vibrations have been posited to be the pairing glue in copper-based high-temperature superconductors. There are theoretical proposals that describe how photons could induce such an effective attraction, though this has not yet been achieved experimentally.

Separately, and more generally, in a world where the word ‘quantum’ is tacked onto everything, and everything associated with this word is thought to be mysterious and counterintuitive, it’s worth asking — how much insight can you gain with good old classical mechanics, and what implications does that have for the physical nature of these things?

*This post was inspired by a section in the chapter on BCS superconductivity in ‘Quantum Liquids’ by Anthony Leggett. It’s a challenging (certainly for me) and rewarding book to work through, I recommend it to anybody interested in condensed matter physics.*