The whirlpool that comes in one size

The rotating-bucket experiment is the oldest question in the subject. Newton used it to argue about absolute space; a first-year laboratory uses it to demonstrate a paraboloid; and it is the experiment that shows most sharply what a superfluid is not.

An ordinary liquid does something entirely different, and it is worth having the contrast in full. Given long enough, viscosity drags it into rotation with its container: its velocity field becomes $\boldsymbol{\Omega} \times \mathbf{r}$, its surface becomes a paraboloid of height $\Omega^2 R^2/2g$, and every part of it turns at the same rate. The circulation round any loop is then $2\Omega$ times the area enclosed — a continuous function of how fast the bucket is driven, taking any value at all.

The paraboloid itself is ordinary hydrostatics: a pressure that knows only depth in a frame turning with the bucket, with the centrifugal term added, and the surface a spin decides is where that balance is worked through.

A superfluid cannot do that, and the reason is not that it has no viscosity. It is that its velocity field is not free to be anything.

A velocity that is the gradient of a phase #

The superfluid part of liquid helium below 2.17 kelvin is described by a single complex function — an amplitude and a phase, one phase for the whole of the liquid. The velocity is the gradient of that phase, times $\hbar/m$.

Where that single phase comes from is the transition itself: below it a macroscopic number of atoms occupy one state, and a macroscopic number of atoms in one state have one phase between them. That is what turns everything below into a statement about a bucketful rather than about an atom, and it is the only place in the argument where the number of particles enters.

Two consequences follow at once, and everything in this essay is one of them.

The curl is zero wherever the fluid is. The curl of a gradient is zero identically, so the superfluid cannot rotate as a body anywhere, however hard the bucket is spun.

And the circulation is quantised. Take any closed loop lying entirely in the fluid. Going once round it, the phase must return to its own value — a wavefunction has one value at each point — so it may change by a multiple of $2\pi$ and nothing else. The circulation, being $\hbar/m$ times the phase change, is therefore

$$\oint \mathbf{v}\cdot d\boldsymbol{\ell} = n\,\frac{h}{m} = n \times 9.97\times10^{-8}\ \mathrm{m^2\,s^{-1}}.$$

The argument is one this collection has met before. Only some notes fit on a string, because the wave has to close on itself — and a phase going round a loop in a superfluid has to close on itself for exactly the same reason, so the circulation comes in whole numbers of $h/m$ and in nothing between. What is unusual is not the argument but the size of the object it is applied to: a bucket, not an atom.

It is worth being careful about what has and has not been assumed there. The loop was required to lie entirely in the fluid, and that is the whole of the subtlety: if the loop can be shrunk to a point without ever leaving the fluid, the phase change must shrink continuously with it, and a multiple of 2π that varies continuously is zero. So a simply connected bucketful can carry no circulation at all. A loop that cannot be shrunk — one that encircles a line along which the fluid is absent — may carry any integer, and the fluid’s way of allowing itself a circulation is therefore to punch a hole in itself.

Nothing about helium enters that number except the mass of one atom. Planck’s constant divided by the mass of a helium-4 atom is $9.97 \times 10^{-8}$ square metres per second, and it is the whole answer.

The staircase #

Putting the two together gives a prediction that could hardly be more different from the classical one.

The flat part is the observation the subject was founded on. Hess and Fairbank cooled helium through the transition while the container was already rotating slowly, and found that the liquid stopped: it ended at rest in the laboratory while its container turned. That is the opposite of what an ordinary liquid does and the opposite of what a metastable state would do, because the liquid reached that condition by cooling into it rather than by being spun down.

How an ordinary liquid gets its rotation is by diffusion: momentum travels sideways through the fluid by viscosity, layer by layer, until every layer is going at the rate the boundary imposes. Take the viscosity away and there is no mechanism for it. Take it away and derive the velocity from a phase, and there is not even a permitted final state to diffuse toward.

Why the threshold, and why it is not sharp in the obvious way #

A vortex costs energy: the fluid has to move, at $\hbar/mr$, over a large volume. It also carries angular momentum. In the frame rotating with the bucket, what a system minimises is the energy less $\Omega$ times the angular momentum, so as the bucket speeds up there comes a point at which the vortex is worth its price.

Two singly quantised vortices cost $2\ln(R/a)$ against the $4\ln(R/a)$ of one doubly quantised one, and carry the same angular momentum between them. So a doubly quantised vortex is unstable against splitting, and a rotating bucket makes an array of single lines rather than one big whirlpool.

How a bucketful fakes rigid rotation #

If the fluid cannot rotate and a spun bucket nevertheless ends up looking exactly as though it does, something has to reconcile the two, and Feynman supplied the count.

The density is fixed rather than chosen. Circulation round a large loop must be the number of vortices inside it times the quantum, and it must also be $2\Omega$ times the area if the average flow is to look rigid; equate the two and the areal density is $2\Omega m/h$. At two radians per second that is one line every 0.17 millimetres, which is a macroscopic spacing — and the array is triangular, for the same reason any set of mutually repelling parallel lines is.

All of that threshold arithmetic is carried out in the bucket’s own frame, and the frame is not incidental. The fictitious forces that appear in a turning frame are what make $E - \Omega L$ the right quantity to minimise rather than $E$ alone, and the sign of the answer depends on it: a vortex is not energetically favourable in the laboratory frame at any speed whatever. It is favourable in the frame the bucket defines, above a threshold that frame also defines.

How a line was actually seen #

For thirty years the vortices were a deduction. Onsager suggested the quantisation in a footnote in 1949; Feynman worked out the array and its density in 1955; and the rotating-bucket experiments through the 1950s measured circulations consistent with whole quanta without ever seeing a line.

The reason the deduction was so hard to check directly is worth stating, because it is a measurement problem rather than a physics one. The surface of a rotating superfluid holding an array of vortices is, to any optical measurement, the classical paraboloid: the coarse-grained flow is Ω×r, and the departures live on the scale of the vortex spacing and are far too small to see in a meniscus. The quantisation is not hidden by being small. It is hidden by being averaged.

What finally worked was to put something in the fluid that vortices could trap. Negative ions — electron bubbles about a nanometre and a half across — are drawn into vortex cores, because a core is a region of low density and the bubble costs less energy there. Pulling the trapped ions out along the rotation axis with an electric field and letting them strike a phosphor gives a photograph of where the lines were, and in 1979 Packard and Williams obtained exactly that: a handful of dots, in the right numbers for the rotation rate, arranged as the array says.

The same picture is now taken routinely in dilute gases of trapped atoms, where the condensate is a few micrometres across, the vortex cores are a substantial fraction of it, and the array can simply be photographed. The lattice that took thirty years to see in helium is an undergraduate image in a rubidium experiment, and the reason is not that the physics is different. It is that the ratio of the core size to the container size is a hundred thousand times more favourable.

The wire that heard one quantum #

The photograph of the array came in 1979 and the circulation itself was measured eighteen years earlier, by a method that did not require seeing anything.

Stretch a fine wire along the axis of a container of helium II and set it vibrating. It has two transverse modes, at right angles to each other, and in a still liquid they have the same frequency. Put a vortex line along the wire and the symmetry is broken: the circulating flow adds to the wire’s motion on one side and subtracts on the other, so the two modes split, and the splitting is proportional to the circulation trapped on the wire.

Measuring a frequency difference between two modes of a vibrating wire is a straightforward laboratory matter, and the answer came out quantised: the splitting took discrete values, whose spacing corresponded to a circulation of $h/m$ for a helium-4 atom. Vinen’s measurement in 1961 is the direct confirmation that the staircase is a staircase, made a decade and a half before anybody saw a vortex.

Two features of the method are worth noticing. It measures the circulation trapped by the wire, which is a loop that cannot be shrunk — the wire is in the way — so the topological argument at the top of this essay applies literally and the answer must be an integer. And it is a null-ish measurement: what is read is a small frequency difference rather than an absolute quantity, so the accuracy is the accuracy of a frequency comparison rather than of anything mechanical.

It also caught the quantum jumping. A single vortex would occasionally leave the wire or a second would arrive, and the splitting would step to a new value and stay there — the staircase observed one step at a time, in real time, on an oscilloscope.

The glitch #

The largest object obeying this arithmetic is not in a laboratory, and it announces itself by suddenly speeding up.

A neutron star’s interior is a superfluid. It rotates — some of them hundreds of times a second — so it contains an array of quantised vortices, at a density fixed by the same $2\Omega m/h$ as a bucket, which for a fast pulsar is of order $10^{18}$ lines per square metre. The star’s outer crust is a solid lattice of nuclei, and it spins down slowly as the star radiates.

The superfluid does not have to spin down with it. Its rotation rate is set by the vortex density, and the vortices can be pinned — held in place by the nuclei of the inner crust, which are regions of low superfluid density and therefore favourable places for a core to sit. So the crust slows while the superfluid, holding its vortices where they are, does not.

A lag builds up, and the force on the pinned vortices grows with it. Eventually a large number unpin at once, move outward, and deliver their angular momentum to the crust — which speeds up, abruptly, by about a part in a million, over a time too short to resolve.

That is a pulsar glitch, and it has been observed hundreds of times in dozens of objects. The subsequent recovery — the crust relaxing back over days to years — measures how strongly the two components are coupled, and the size and frequency of the glitches measure how much superfluid is available to hold the lag.

What makes it worth putting in an essay about buckets is that the argument transfers without alteration. The quantum of circulation is $h$ over the mass of a neutron rather than of a helium atom; the pinning sites are nuclei rather than surface roughness; and everything else is the same physics, at a density of $10^{17}$ kilograms per cubic metre and a radius of ten kilometres.

The defects a cooling makes #

The remark about a cooling universe deserves its mechanism, because it is one of the few places a laboratory experiment speaks to cosmology.

When a system cools through a transition into an ordered phase, it has to choose a phase — a value of the phase angle, for a superfluid — and it chooses independently in regions that cannot communicate quickly enough. Where three or more such regions meet with incompatible choices, the phase cannot be made continuous, and what is left is a defect: a line about which the phase winds by a multiple of $2\pi$, which is a vortex.

So the number of vortices left behind by a cooling is set by how fast the cooling was, since a slower quench lets the choices be correlated over larger regions and leaves fewer defects. The prediction is a power law relating the defect density to the quench rate, with an exponent fixed by the transition’s critical behaviour rather than by the substance.

Kibble proposed it for the early universe, where the defects would be cosmic strings; Zurek pointed out that superfluid helium does the same thing in a beaker and can be tested. It has been — by heating a small region of helium-3 with a neutron capture and letting it cool back through the transition in microseconds, and by quenching a trapped atomic gas into a condensate and counting the vortices that appear.

That is an unusual kind of experiment. It does not test a theory of the early universe; it tests a mechanism that is claimed to operate at any transition of the right kind, in a system where the mechanism can be run repeatedly and the exponent measured. What the cosmology gets is a piece of physics that has been checked rather than assumed.

What the quantum measures #

Because the quantum of circulation is $h$ divided by the mass of whatever carries the phase, measuring it is a measurement of that mass.

In helium-4 it comes out as $h/m_4$, and the phase is carried by single atoms. In helium-3 the atoms are fermions and cannot condense singly; they pair, and the measured quantum is $h/2m_3$. In a superconductor the same argument applies to the electrons, and the quantity that is quantised is the magnetic flux through a ring rather than the circulation — at $h/2e$, with the two again saying that the object whose phase winds is a pair.

The flux through a loop of perfect conductor cannot change, and in a superconductor it is also quantised, in units of $h/2e$. That is the same argument in another currency: a single-valued phase round a loop, with the vector potential doing the work the velocity does here. The factor of two in the denominator is how pairing was confirmed, in 1961 — seven years before the same measurement confirmed it in helium-3.

What it costs #

The two-fluid picture is doing work throughout. Below the transition, helium behaves as a mixture of a superfluid component with no entropy and no viscosity and a normal component with both. Everything above concerns the superfluid part; the normal part can rotate with the bucket, does, and contributes its own paraboloid.

Helium-3 does none of this the same way. Its atoms are fermions, so no two of them may occupy the same state and they cannot condense singly; they pair first, and everything above then applies with the mass of a pair.

The core is outside the model. The 1/r profile has infinite energy at the origin, and the logarithm in every energy quoted above is cut off at a core radius taken as about an angstrom. What is inside is a matter for a Gross–Pitaevskii or a microscopic treatment, and the results here depend on it only logarithmically — which is the reason they can be quoted at all with a core radius nobody has measured to better than a factor of two.

Nothing above says how the fluid gets from one step of the staircase to the next. The equilibrium argument says which winding number is favourable at each rotation rate and is silent about the barrier between them. That barrier is large — a vortex has to enter from the boundary, or a loop has to nucleate and grow — and it is why a superfluid can be spun well past the threshold and remain irrotational, in a metastable state that lasts as long as anybody has been willing to watch.

Vortices interact. The array’s triangular arrangement, its oscillation modes, and the tangle that forms when the rotation is changed quickly are all consequences of vortices pushing each other about, and none of it is in the single-line calculation.

Where the model stops #

Everything above is in equilibrium. Spin a bucket up quickly and the vortices do not simply appear at their equilibrium density; they nucleate at the walls, at defects, and in numbers that depend on how fast the change was made — which is the same physics that puts topological defects into a cooling early universe, and the analogy is close enough to have been tested in helium on purpose.

Turbulence in a superfluid is a different object. A tangle of quantised lines, each carrying the same circulation, reconnecting when they cross, is not the continuum of eddies that classical turbulence is made of — and yet it produces the same energy spectrum over a range of scales, which is one of the more surprising results in the subject and is not explained here.

And the threshold is idealised. The 0.256 radians per second above assumes a smooth cylindrical container and a vortex on the axis. A real container’s roughness nucleates vortices below it, and measured critical velocities are almost always lower than the calculation, by amounts that depend on the surface rather than on the helium.

And the scale that permits any of it. Everything has a wavelength, and a superfluid exists when the thermal wavelength of its atoms has grown comparable to the spacing between them. Above the transition that wavelength is far smaller and the atoms are distinguishable objects; below it they are not, and a bucketful acquires one phase.

What the pictures cannot show #

The array figure draws each vortex as a dot with a ring round it, and the ring is a lie of scale by a factor of about a million: the core is an angstrom and the spacing is a fifth of a millimetre. Drawn honestly, the picture would be an empty disc with nothing visible in it at all.

Nor can the staircase figure show what is discrete. What is quantised is a winding number — an integer attached to a loop, which is a topological property of a map and not a quantity anything possesses. The staircase’s height is a circulation, which is measurable; its integer-ness is a fact about phases, which is not a thing in the fluid at all. That the fluid’s density falls to zero along a line is the closest a picture gets, and even that is a consequence rather than the thing itself.

Where this ladder goes next #

Two rungs stand on superfluidity. The first found a liquid whose viscosity is not small but zero, and traced it to a macroscopic number of atoms in one state. This one asks what such a liquid does when it is asked to turn, and finds that it cannot — and that what it does instead is quantised, at a value containing nothing but Planck’s constant and the mass of an atom.

The habit worth carrying away is about where a quantum number can live. A quantisation condition does not need a small system; it needs a closed path and a single-valued phase. The loop here is centimetres across, and the integer is a property of the loop rather than of anything inside it — which is why the same argument gives flux quantisation in a superconducting ring, the Aharonov–Bohm phase round a solenoid and the Berry phase of a slowly turned system, in objects of every size.

What is left on this ladder is the mechanism that limits the flow rather than the rotation. A superfluid pushed through a channel fast enough stops being one, at a critical velocity set by the cheapest excitation it can make — and Landau’s argument for what that velocity is, from the shape of the excitation spectrum alone, is one of the most economical pieces of reasoning in the subject.