The two in the flux quantum

Zero resistance freezes the flux through a superconducting ring: any change would drive a current that never decays, so whatever was threading the ring when it went superconducting is threading it still. That is a classical argument and it is correct as far as it goes.

It does not go far enough. The ring cannot hold whatever flux happens to be there. It can only hold multiples of one particular amount, and the amount is small enough to be about what the Earth’s field puts through a square micrometre.

Where a condition on a wavefunction becomes a condition on a field #

A superconductor’s carriers are all in one quantum state, described by one wavefunction with one phase. Go once round a ring and the wavefunction has to come back to itself, so the phase has to change by a whole number of turns. That is the only input.

The phase gradient of a charged condensate is not just its momentum; a magnetic field adds a term proportional to the vector potential, because the momentum that appears in the wavefunction is the canonical one. Integrating the phase gradient round the ring therefore gives a whole number of turns on one side and, on the other, the current plus the enclosed flux.

Deep inside a thick superconducting ring the current is zero — the screening currents live in a surface layer a penetration depth thick — so a path taken well inside the material contributes nothing from the current, and what is left is

$$\Phi = n\,\frac{h}{q}, \qquad n = 0, \pm1, \pm2,\dots$$

with $q$ the charge of whatever is carrying the current. The flux is quantised, and the size of the quantum contains the carrier’s charge in its denominator.

That last point is what makes the whole thing an instrument. Measuring the quantum measures $q$, and does so without measuring any charge, any current or any number of particles.

The distinction between the two statements is easy to lose and is worth keeping. Freezing is about a rate of change being zero and is a consequence of the resistance being zero. Quantisation is about the set of allowed values being discrete and is a consequence of the wavefunction being single-valued. A perfect classical conductor would freeze flux and would admit any value of it whatever, and the two experiments that distinguish those cases are the Meissner effect, which shows the state is thermodynamic, and this one, which shows it is quantum.

The two #

For a single electron, $h/e$ is $4.1357\times10^{-15}$ webers. For a pair it is half that. For a triple it is a third.

The measurement gives $2.0678\times10^{-15}$.

The candidates are separated by factors of two and three, so no question of experimental accuracy arises: the experiment does not have to be good, it only has to be done. It was done in 1961, independently by two groups, and it settled that the carriers have charge $2e$.

The timing is what makes it striking. Bardeen, Cooper and Schrieffer had proposed in 1957 that the carriers are bound pairs of electrons, and the proposal was a considerable leap — two electrons repel, and the attraction that binds them is a subtle effect involving the lattice. Flux quantisation weighed the carrier and found two electrons’ worth, by an experiment about a magnetic field.

The same single-valuedness argument runs with no charge in it at all. In a rotating superfluid the circulation is quantised — one, two, three quanta of $h/m$ — and the derivation is the identical one with the vector potential absent. The neutral case gives quantised circulation and the charged case gives quantised flux, and the difference between the two conditions is exactly one term. Which is why the two in the denominator is a statement about the charge carrier and about nothing else.

The same argument in a neutral superfluid gives circulation in one size rather than flux, because with no charge the vector-potential term is absent and what is left is the velocity. The two results are the same condition with and without one term, which is why helium’s quantum contains the mass of a helium atom and a superconductor’s contains twice an electron’s charge.

There is a general lesson in the shape of the argument that is worth separating from superconductivity. A quantum condition normally applies to something small — an orbit, a level, a mode in a box. Here it applies to a hole in a piece of metal, of any size, because what is quantised is the number of turns a phase makes and a phase can be coherent over a metre as easily as over a nanometre. What makes a system macroscopically quantum is not its size but whether one wavefunction describes all of it, and a superconductor is the standing demonstration that the two are independent.

What the ring does when the flux does not fit #

Applying an arbitrary external field to a ring poses an obvious question: what happens when the applied flux is not a whole number of quanta?

The ring holds a whole number anyway and makes up the difference with a circulating current. That current costs kinetic energy, and the energy goes as the square of the mismatch — which is what each parabola in the figure is.

The state the ring is actually in is whichever parabola is lowest, so the winding number changes at the half-integers, where two of them cross. Everything measurable about the ring is therefore periodic in the applied flux, with a period of one quantum.

That periodicity is the observable, and it is a strange one. A steady applied magnetic field produces a response that goes up and down as the field is increased, returning to the same value every $2.07\times10^{-15}$ webers. Nothing in ordinary electromagnetism does that.

It is worth writing the quantum out in the units the effect is met in. Two microteslas through a hundred square micrometres is one quantum; so is a hundred microteslas through two square micrometres. Since the Earth’s field is fifty microteslas, a ring a few micrometres across sitting on a bench is threaded by a few quanta from the Earth alone — which is why every measurement of this kind is made inside a magnetic shield, and why the periodicity rather than the absolute value is what is measured.

The measurement, and why the shape matters #

The cleanest version was Little and Parks’s, and its subject is a temperature rather than a current.

Take a thin-walled cylinder — thin compared with the penetration depth, so the current cannot hide in a surface layer and the whole wall carries it. The circulating current required to make the flux come out whole now costs enough kinetic energy to deplete the condensate, and depleting the condensate lowers the temperature at which the material becomes superconducting.

The depression goes as the square of the mismatch, so plotting the transition temperature against the applied flux gives a periodic curve with a cusp at each half-integer. The size is a few millikelvin, and the shape is what makes the interpretation unambiguous:

Periodic, in a steady field. No ordinary magnetoresistance does this. A smooth field effect gives a monotonic curve, and a curve that returns to its starting value every time the flux increases by a fixed amount is a statement about counting.

Cusped, not sinusoidal. The cusps come from the ring switching between winding numbers, and a smooth mechanism cannot produce them.

And the period fixes the quantum. Measuring the field and the cylinder’s area gives the flux per period directly, which is the whole measurement.

The quantum as an instrument #

A periodic response to a magnetic field, with a period fixed by constants of nature, is the beginning of the most sensitive measurement in physics.

Two paths recombining with a phase between them is an interferometer, and a SQUID is that device with the two paths through a superconductor and the phase supplied by the enclosed flux. One fringe corresponds to one quantum, which is what makes the device an instrument rather than a demonstration: counting fringes counts quanta, and a quantum is $2.07\times10^{-15}$ Wb.

Interrupt the ring with two weak links — thin barriers a Cooper pair can tunnel through — and the current the ring will carry depends on the phase difference across them, which depends on the enclosed flux. Sweeping the flux therefore sweeps the critical current through a full oscillation for every quantum, and the ring becomes an interferometer in which the two paths are the two arms and the phase is supplied by the field.

The consequence is a magnetometer whose calibration is a ratio of constants. A superconducting quantum interference device resolves a small fraction of a flux quantum, which for a loop of a few tens of micrometres is a field of the order of $10^{-15}$ teslas — some ten orders of magnitude below the Earth’s, and small enough to detect the field of an electric current in a nerve from outside a skull.

Two features are worth separating from the sensitivity. The device measures flux, not field, so its sensitivity to a field is traded against its loop area, and a large loop that catches more flux also catches more noise. And it is periodic, so a raw reading is ambiguous by a whole number of quanta; a working instrument therefore runs a feedback loop that holds the flux at one point on the fringe and records the correction, exactly as a tunnelling microscope holds its current and records the piezo voltage.

What is being counted #

There is a way of reading the winding number that makes the robustness of the effect less surprising.

What makes the winding number a well-defined thing to count is that the phase itself is not. Two vector potentials describing the same field differ by the gradient of anything, and the condensate’s phase shifts along with them — but the number of times that phase turns round a closed loop cannot change, because it is an integer and nothing continuous can move it. The quantity being counted is a topological one, which is why it comes out exact rather than approximate.

The number of times the phase turns going round the ring is an integer, and an integer cannot change gradually. To change it, the phase has to become undefined somewhere — which means the condensate has to be destroyed along some path across the ring, at least momentarily.

That is why a persistent current persists. It is not merely that the resistance is small; the state carrying it is labelled by a whole number, and getting to a neighbouring number requires an excursion over a barrier rather than a slow leak. Measured lifetimes of persistent currents exceed a hundred thousand years — a stability of the same kind, and for the same reason, as the one a knotted field has.

It also explains why the effect survives disorder, irregular ring shapes and impurities. None of those can change an integer. What can is a large enough current, which is why there is a critical current above which the winding number does slip — one quantum at a time, in discrete jumps, each accompanied by a voltage pulse.

The experiment as it was actually done #

The 1961 measurements are worth describing, because the difficulty was not the physics.

The experiments used cylinders a micrometre or two across, and the reason is arithmetic. A flux quantum divided by any larger area is a field too small to control: over a square millimetre it is $2\times10^{-9}$ T, well below the Earth’s field and below what shielding of the day could hold steady. Over a micrometre-scale cylinder it is a couple of millitesla, which a small coil produces and a laboratory can step through.

The flux quantum is $2\times10^{-15}$ webers, so getting one quantum through a ring requires a field of two millitesla through a square micrometre, or two microtesla through a square of ten micrometres — which is a twentieth of the Earth’s field, and therefore below the ambient. The cylinders had to be tiny, and they were made by evaporating tin onto a quartz fibre a micrometre and a half across.

Deaver and Fairbank measured the magnetic moment of the trapped current by vibrating the cylinder in a pickup coil and watching the induced signal step as the applied field was raised. Doll and Näbauer measured the same thing by hanging the cylinder from a torsion fibre and watching it twist in a transverse field.

Both saw steps rather than a smooth rise, both found the step size to be $h/2e$ rather than $h/e$, and both published within weeks of each other. Neither group had set out to test the pairing hypothesis; both were testing quantisation, and the factor of two was what they found.

The Little–Parks measurement, from the same year, is the version reproduced in teaching laboratories, because measuring a resistance against temperature is easier than measuring a magnetic moment on a fibre.

The same ring, without a condensate #

A period of one quantum in a magnetic field is not by itself proof of pairing, and the cleanest way to see that is to run the experiment on a ring that is not superconducting at all.

Cool a gold ring less than a micrometre across to a few millikelvin and measure its resistance against applied field. It oscillates, periodically, exactly as the superconducting cylinder’s transition temperature does — and the period is $h/e$, not $h/2e$. There is no condensate and no pairing; what is interfering is a single electron with itself, arriving at the far side of the ring by the two arms and picking up a phase difference proportional to the enclosed flux. The experiment was done by Webb and colleagues in 1985, and the requirement is that an electron keep its phase for the whole way round, which is why it needs a small ring and a very low temperature.

So one number decides which physics is being seen: $h/e$ means single carriers, $h/2e$ means pairs. That is what makes the 1961 result a measurement of the carrier’s charge and not merely a demonstration of quantisation.

The trap is that $h/2e$ has a second and quite different cause. In a disordered normal metal a wave can return to its starting point by a path and by that path reversed, and the two are automatically in phase because they are time-reverses of one another. Their interference is periodic in the enclosed flux with period $h/2e$, because such a pair of paths encloses the area twice. That effect — Al’tshuler, Aronov and Spivak’s — has nothing to do with pairing whatever, and it means a bare period of $h/2e$ is evidence only when the mechanism is otherwise settled. The 1961 experiments were safe from it because they measured a trapped moment and a transition temperature rather than a resistance.

The crossing point, used deliberately #

The figure of parabolas has a feature that was for forty years a curiosity and is now a device: at every half-integer of applied flux, two winding numbers have exactly the same energy.

At that point the ring has two available states with equal energy and opposite circulating currents — several microamperes clockwise or several microamperes anticlockwise, each carried by an enormous number of electrons. Left alone the ring simply picks one, since nothing connects them: changing the winding number requires destroying the condensate across the ring.

Interrupt the ring with a weak link and that changes. A barrier thin enough for pairs to tunnel through supplies a way for the phase to slip without the condensate being destroyed, so the two states are coupled, and coupling two degenerate states splits them into a symmetric and an antisymmetric combination. The ground state of the ring is then a superposition of a current going one way round and a current going the other — not a mixture of rings, and not an average, but one state of one ring.

That is a flux qubit, first demonstrated around 2000, and it is one of the two or three ways of building a superconducting quantum computer. Two details of its operation are read directly off the parabolas. It is biased at the half-integer because that is where the two states are degenerate and the interesting superposition exists. And it is biased exactly there because the splitting between the two levels has zero slope with respect to flux at the crossing — the parabolas’ slopes are equal and opposite — so to first order the qubit’s frequency does not care about the flux noise that is the dominant nuisance everywhere else on the curve.

The physics being exploited is entirely on this page: an integer that cannot change gradually, two values of it that cost the same energy at one particular field, and a barrier that lets them talk.

Where the model runs out #

The ring has to be thick, or the path has to be chosen carefully. The clean statement — flux exactly quantised — needs a path along which the current vanishes, which exists only inside a superconductor much thicker than the penetration depth. For a thin-walled cylinder the quantity that is exactly quantised is the fluxoid, the flux plus a term in the current, and the flux alone is not.

The whole argument needs a path through the material along which the current vanishes, and in a thin slab there is no such path. Field profiles through slabs of one, three, ten and forty penetration depths show it directly: only in the thick case does the field fall to nothing in the interior. Below that thickness the flux through the ring is not quantised — it is quantised in the fluxoid, a combination of flux and circulating current, and the distinction stops being invisible.

The quantum is a ratio of constants and the measurement is of something else. What is measured is a period in a field multiplied by an area, and the area of a real cylinder is uncertain at the per-cent level. That is why the experiment settles a factor of two convincingly and does not compete as a determination of $h/2e$; the precision determinations of that combination come from the Josephson effect instead, where a frequency stands in for the area.

Type II superconductors admit flux rather than expelling it, and the quanta are then objects rather than conditions. Above a lower critical field the flux enters as an array of vortices, each carrying exactly one quantum, each with a normal core. That is the same quantum doing a different job, and it is what every practical superconducting magnet is built on.

And nothing here says why the carriers pair. The measurement establishes that the charge is $2e$ and is silent about the mechanism, which is a lattice-mediated attraction in conventional superconductors and is still argued about in the high-temperature ones — where, notably, the flux quantum is also $h/2e$, so whatever the mechanism, the carriers there are pairs too.

The ladder from here #

Later rungs on this anchor: the Josephson effect, where two superconductors separated by a thin barrier carry a current that depends on the difference of their phases and nothing else; the SQUID, which uses a pair of such junctions to measure flux to a small fraction of a quantum and is the most sensitive magnetometer there is; type II superconductors and the vortex lattice; and the critical current, set by vortex motion rather than by pair breaking, which is the practical limit on every superconducting device.

The neighbouring ladders are the field that is pushed out, which is the state this quantisation is a property of, the field that cannot get out, which is the classical conservation this strengthens, and the whirlpool that comes in one size, where the same condition with no charge in it quantises a circulation instead.