The phase a magnet leaves on a path it never touched

The potentials that are not unique makes the vector potential sound like a bookkeeping device: a field from which the real fields are obtained by differentiation, adjustable at will by a gauge change, and therefore not the sort of thing an instrument could ever point at. Classical electromagnetism supports that reading completely. A charged particle in a classical field obeys the Lorentz force, the Lorentz force contains $\mathbf{E}$ and $\mathbf{B}$ and nothing else, and a region where both vanish is a region where nothing happens to a charge.

Send a wave through instead of a particle and the conclusion fails.

Two quantities that behave completely differently #

The figure separates two things that are usually not distinguished because inside a laboratory they usually go together.

One is the field at a point. Outside a long solenoid it is small, and can be made as small as patience and money allow: lengthen the winding, close it into a torus so there are no ends for the lines to leave by, wrap the whole thing in a superconductor thicker than the depth a field gets into a metal. All of that was done, and the field on the electron’s path in the definitive experiments is smaller than anything the shift could be attributed to.

The other is the flux enclosed by a loop. That does not fall off, because it is not a local quantity at all — it is a statement about everything inside the loop, and the loop can be a metre across while the flux it counts sits in a whisker two hundred nanometres wide.

Classical mechanics has no way for the second quantity to matter, since the equation of motion is local and reads only the field where the particle is. Quantum mechanics has a way immediately, because the object being propagated is a wave with a phase, and the phase accumulated along a path is a line integral rather than a local reading.

The number that every encircling loop agrees on #

The phase an electron picks up along a path, in a magnetic field, is $(q/\hbar)$ times the line integral of the vector potential along it. Two paths that leave the same source and meet at the same detector differ in phase by the same integral taken round the closed loop they form.

That closed line integral is where the gauge freedom goes. A gauge change adds the gradient of some function to the potential everywhere, and the line integral of a gradient round a closed loop is zero. So the phase difference between two paths is unchanged by any gauge change whatever, even though the phase along each path separately is not — and only the difference is ever observed.

By Stokes’ theorem the closed integral is the flux through the loop. The figure verifies that directly rather than quoting it: three contours of quite different shapes and sizes give the same number to five decimals, and a contour that does not encircle the winding gives zero to within one part in a million. Both halves matter. The first says the answer depends only on what is enclosed. The second says the field really is absent along the paths, so the electron is not quietly being pushed.

This is the invariant statement the effect makes, and it is worth being careful about, because it is often overstated. The experiment does not show that the vector potential is real in the way a field is real; the potential at a point is still not measurable and still not unique. It shows that a particular gauge-invariant functional of it — its circulation round a closed curve — is measurable, and that it is measurable in a region where the field it differentiates to is zero.

What the fringes do #

The observable is an interference pattern, and its displacement is the measurement.

The arrangement is the electron version of a two-slit experiment — in practice an electrostatic biprism rather than slits, because slits narrow enough would pass almost nothing — with the magnet placed in the shadow between the two beams. Threading flux moves the pattern sideways and does nothing else.

The “nothing else” is the part that identifies the effect. A force would change the trajectories, which changes where the beams land and therefore the envelope of the pattern, and it would change the fringe spacing by changing the angle at which the two beams converge. Neither happens. The envelope stays where it is, the spacing stays what it was, and the fringes slide underneath.

The electron’s energy is unchanged too, which is the same statement in another currency: a magnetic field does no work — that is the force that does no work — and here there is not even a field to do none.

The whole quantum that puts it back #

The periodicity is the sharpest evidence that what is being seen is a phase.

A leak — a residual field on the paths — would give a displacement that grew steadily with the current in the winding, and that depended on how far out the electrons ran. It could be made to look like the effect at one setting. It cannot be made periodic, and it cannot be made to return the pattern exactly to its original position at a particular flux and again at twice that flux.

The flux at which it returns is $h/e$, four femtowebers or so, which is a very small amount of flux and a very ordinary amount of magnet: a whisker a fifth of a micron across carries one quantum at about thirty millitesla. That combination — a tiny absolute flux and an easy field — is what made the experiment possible at all, and it is why the effect was demonstrated with iron whiskers years before the toroidal versions closed the last objection.

The phase, meanwhile, is $2\pi$ per quantum, and it is worth noticing that this brings in Planck’s constant twice. Once because the phase of a matter wave is an action divided by $\hbar$, which is everything has a wavelength again; once because the natural unit of flux for a charge $q$ is $h/q$. The experiment is a comparison of two quantum quantities that were introduced for unrelated reasons.

The period weighs the carrier #

Since the phase is $q\Phi/\hbar$, the period in flux is $h/q$, and the period is therefore a direct reading of the carrier’s charge — with no model of the material, no mobility, no assumption about carrier density, and no Hall geometry.

That turns an interference experiment into a charge meter, and it has been used as one. A metal ring threaded by flux shows a resistance that oscillates with period $h/e$, because the electron waves going round the two ways interfere. A superconducting ring shows $h/2e$, and the factor of two is the pairing — the same two that the two in the flux quantum accounts for from the other direction. More recently the same period-counting has been read on fractionally charged excitations, where nothing else measures the charge at all.

The generality is the point. The argument nowhere used the fact that the wave was an electron’s. Anything with a charge and a phase acquires the shift, and the shift counts the charge in units of the flux quantum it defines.

How the objection was actually closed #

The history is unusually clean, because the effect was predicted, then seen, then disbelieved for twenty years on one specific ground, and then the ground was removed.

The first demonstrations used iron whiskers — single crystals a fraction of a micron across, which grow with the magnetisation along their length and carry a few flux quanta. A whisker laid in the shadow of an electron biprism shifted the fringes, and the shift scaled with the whisker’s cross-section as it should.

The objection was the ends. A whisker is a finite magnet and the lines that go up its inside come back down its outside, which is the field outside the solenoid as a nuisance rather than as a subject. The field on the electron’s path is small, and small is not zero, and a sceptic could always maintain that the shift was an ordinary deflection by a leaked field too weak to measure directly. The figure above makes the size of the complaint visible: the enclosed flux falls by a few per cent between the winding and four radii out, and every one of those lines crosses somebody’s path.

The answer was to build a magnet with no ends. A toroid has no exterior field at all in principle, because every line closes inside it; the experiments of the 1980s used a permalloy torus a few microns across, plated in niobium and cooled below its transition so that the Meissner effect — the field that is pushed out — confined whatever leaked, and covered in copper so that no electron could pass through the metal. The electron beam went through the hole and round the outside, the fringes in the hole and the fringes outside were compared with each other in the same exposure, and the relative shift was half a fringe when the torus held half a quantum and none when it held a whole one.

That last detail is what settled it. Flux in a superconducting ring is quantised, so the torus could only hold whole or half quanta depending on the fluxoid state it froze into, and the experiment did not have to be trusted to set a continuous value correctly. It measured a discrete alternative, and got the discrete answer.

The same phase, in a wire #

The effect stopped being an exhibit and became an instrument when it turned up in ordinary conduction.

A metal ring smaller than the distance over which an electron keeps its phase behaves as a two-path interferometer for every electron that crosses it, and its resistance oscillates as flux is threaded, with period $h/e$. That is the effect above with the vacuum replaced by a disordered metal, and it works because disorder scatters the phase of a path without randomising the difference between two paths that see the same scatterers.

There is a second oscillation at $h/2e$ in the same rings and it has a different origin from the superconducting one. It comes from pairs of paths that are time-reverses of each other — the same loop traversed both ways — which pick up twice the enclosed flux between them and survive averaging over disorder when the $h/e$ term does not. So a measurement on many rings shows the halved period and a measurement on one shows both, and telling the two apart is a diagnostic for how much averaging an experiment is unknowingly doing. It is the same lesson where the interference goes draws about which superpositions survive.

What is actually non-local about it #

There is a temptation to say the electron feels the field at a distance, and it is worth resisting precisely.

Nothing propagates from the solenoid to the electron. The interaction is entirely described by a potential that is defined at every point of the path, so the equation of motion is as local as any equation of motion in physics. What is non-local is the conclusion: the observable phase difference is a property of a whole closed curve and cannot be assembled from readings of the field taken along it, because the field along it is zero.

That is a topological statement rather than a dynamical one. The plane with the solenoid removed is not simply connected, and a loop in it is characterised by how many times it goes round the hole. The phase shift depends on the loop only through that number, which is why three contours of different shapes gave the same answer and the one beside gave nothing.

The classical limit is instructive here rather than reassuring. A phase difference is invisible in classical mechanics, so the effect vanishes as $\hbar \to 0$ — but not by getting smaller. The shift per unit flux grows without bound as $\hbar$ falls, and the fringes it shifts get closer together at the same rate, so what disappears is the possibility of resolving them. The classical world does not lack the phase; it lacks the interference that would display it.

Where the model stops #

The flux is treated as fixed and external. The solenoid’s current is assumed unaffected by the electrons going past, which is fine for a beam and not fine in a mesoscopic ring, where the same physics runs backwards and a persistent current appears in the ring in response to the flux.

Everything here is a phase difference between two paths. The single-path phase is not gauge invariant and is not measurable, so any sentence about “the phase the electron acquired” is shorthand for a comparison. The figures respect this: every number drawn is a closed-contour quantity.

The interference is treated as perfect. A real beam has an energy spread and a finite source size, both of which reduce the fringe visibility, and the shift becomes unmeasurable long before it becomes small. The experimental difficulty in the 1960s was never the phase; it was maintaining coherence across a beam wide enough to go round a magnet.

And the electrostatic partner is not drawn at all. There is a matching effect in which two paths are held at different electric potentials for a while with no field on either, and the phase difference is the time integral of the potential difference. It is harder to do and was done much later, and everything above about locality applies to it identically.

And the shift is computed for a stationary flux. Changing the flux while the electron is in flight adds an electric field by induction, and the clean separation between “a phase with no force” and “a force” stops being available. The effect as stated belongs to a magnet that is already there and stays there, and every careful version of the experiment holds the flux fixed for exactly that reason.

What the pictures cannot show #

The contour figure draws the vector potential’s circulation and cannot draw the potential itself in any meaningful way, because there is no single potential to draw — a gauge change would redraw every arrow and leave every number in the figure alone. The honest picture of what is measurable is the list of numbers, not a field of arrows.

The fringe figures draw an ideal two-path interference with unit visibility. A real biprism pattern sits under a bright background from electrons that missed both paths, and the shift is extracted by fitting rather than by looking. The figure shows what is being fitted, not what is on the screen.

Where the ladder goes next #

The matter-waves ladder began with everything has a wavelength and continued with one arrival at a time, where the interference is shown to be a property of single particles rather than of beams. This rung asks what the wave’s phase responds to and finds that it responds to something the particle never met. The rungs after it: the Berry phase, of which this is the earliest and simplest example, where the geometry doing the remembering is in the space of parameters rather than in the laboratory; the persistent currents that the same phase drives round a normal-metal ring; and the fractional statistics that appear when the flux is attached to the particle itself.

The habit worth carrying away is that a quantity can be unobservable and still be the right variable. The vector potential is not measurable and its circulation is, and a theory written in terms of fields alone has no way to say what the electron responded to — which is the first place in physics where the potentials stopped being a convenience.