The cliff a slow atom bounces off

A ball rolling along a table and reaching the edge falls off. It never bounces back from the edge; there is nothing there to push it. Every classical particle meeting a downward step in potential energy behaves the same way: it speeds up as it goes over, and carries on. Tunnelling, the best-known quantum surprise, runs the other way — a particle getting through a barrier it classically could not climb. The reverse surprise is less famous and just as real. A quantum particle approaching a drop can be reflected by it, and if it is slow enough, it almost certainly will be.

This is not an exotic effect confined to textbook potentials. Atoms approaching a surface feel a weak attraction, the van der Waals force, and if they reach the surface they usually stick. Slow enough atoms never reach it. They are turned back by the attraction alone, at distances of many atomic diameters, in a region where the only force on them points towards the surface. It is called quantum reflection, and it has been measured with atoms falling at millimetres per second onto solid surfaces and onto liquid helium.

Reflection without a wall #

What happens where the medium changes worked through a wave meeting a boundary between two media with different properties: some of it reflects, by an amount set by how different the media are. A matter wave sees its potential energy as a medium. In a region of lower potential energy the particle moves faster, its wavenumber is larger and its wavelength shorter. A sudden drop is a sudden change of wavelength, and the wave reflects from it exactly as a wave on a string reflects where a light string joins a heavy one. For a sharp step, with wavenumbers $k_1$ before and $k_2$ after, the reflected fraction is

$$R = \left(\frac{k_1 - k_2}{k_1 + k_2}\right)^2.$$

It does not care whether the step is up or down. A downward step reflects just as well as an upward one of the same size, provided the particle has enough energy to pass the upward one.

The limit at low energy is the surprise. As the particle’s energy falls, $k_1$ goes to zero while $k_2$, set by the depth of the drop, stays finite, and $R$ goes to one. A very slow particle meeting a deep drop is almost entirely reflected. A ball would roll over the edge; a slow enough wave cannot.

The reason is the mismatch of wavelengths. A slow particle has a long wavelength, and the region beyond the drop supports only short ones. The wave cannot change its wavelength abruptly without reflecting, just as a light wave cannot pass from air into a medium of very high refractive index without most of it bouncing back. An electron reaching the surface of a metal from outside meets exactly such a drop, into the metal’s deep potential well, and the strong reflection of slow electrons is part of why low-energy electron beams reflect so effectively from surfaces.

Softening does not help a slow particle #

The obvious objection is that real potentials are smooth. A sharp step is an idealisation, and smooth changes, as the taper that matches every note found for horns, transmit waves without reflection. That is true, but only for waves whose wavelength is short compared with the distance over which the change happens.

The smooth step here, a potential that falls following the shape of a Fermi function over a width $a$, has an exact solution derived in Landau and Lifshitz’s textbook, and it shows the competition clearly. For a fast particle, softening the step over even a fraction of the wavelength the drop corresponds to removes almost all reflection. For a slow particle, the softening has to be compared with the particle’s own, much longer wavelength, and on that scale the drop still looks sudden. As the energy goes to zero, the reflection goes to one for every width. Gentleness is relative to the wave, and a slow enough wave sees every change as a cliff.

One formula for light, sound and atoms #

The step formula is not peculiar to quantum mechanics. Light arriving at a boundary between media of refractive indices $n_1$ and $n_2$, head on, reflects a fraction $((n_1 - n_2)/(n_1 + n_2))^2$ of its power — four per cent at a window, because glass’s index is 1.5 against air’s 1. Sound arriving at a boundary between two materials reflects a fraction set by the same expression with acoustic impedances in place of indices, which is why sound in air hardly enters water at all. A matter wave’s wavenumber plays the role of the refractive index, and the formula is identical. Quantum reflection is the optics of an atom in a medium whose index is set by the potential energy.

That makes the low-energy limit easy to see from the optical side. A slow particle meeting a deep drop is like light in air meeting a material of enormous refractive index: the ratio of the two indices goes to zero or infinity, and the reflection goes to one. Metals reflect light well for a related reason, their effective index for visible light being large and mostly imaginary. And the cure is also borrowed from optics. The layer that makes a reflection vanish found that a coating a quarter of a wavelength thick, with an index between the two media, cancels the reflection at one wavelength. A matter-wave equivalent, an intermediate step in the potential a quarter of a wavelength wide, does the same for atoms of one speed, and graded steps transmit them over a range of speeds, as graded-index coatings do for light.

There is a further connection to waves with no quantum mechanics in them at all. The pulse that climbs a rope at half of g followed a pulse descending a hanging rope through tension changing a thousandfold without reflection, because the tension changed slowly on the scale of the pulse, and found the reflection appearing only near the free end where the wavelength became comparable to the distance over which tension changed. That is the badlands criterion, in a rope. The free end is a turning point, the badlands of quantum reflection are not, but the condition for reflection — a medium changing by a large fraction within one local wavelength — is the same in both.

Why the low-energy limit is universal #

The fact that every drop reflects everything in the limit of zero energy, whatever its shape, is an instance of a general feature of slow waves. A wave much longer than the region over which the potential changes cannot resolve that region’s shape; it sees only a single summary number. In scattering theory that number is the scattering length, and for a potential that can absorb, like the surface, it is complex. All the low-energy behaviour follows from it: the reflection probability, the phase of the reflected wave, and the linear approach $1 - 4k\beta_4$ to total reflection, in which the coefficient is set by the imaginary part of the scattering length. Eugene Wigner’s threshold laws of 1948 are the general statement: near zero energy, how a wave scatters depends only on the potential’s long-range tail and on a few numbers summarising everything closer in.

That universality is what makes quantum reflection a measurement. The slope of $R$ against speed at low speeds measures the strength $C_4$ of the surface attraction, and experiments with atoms of different kinds on different surfaces have used it to measure the retarded van der Waals force at distances of tens to hundreds of nanometres, where it is otherwise hard to reach. A force that is too weak to measure directly at those distances is read from how often slow atoms bounce.

The surface that pulls #

An atom approaching a solid surface feels an attraction from the fluctuating dipoles of the surface’s electrons, the van der Waals force. Close to the surface it falls as the inverse cube of distance; further out, beyond a distance comparable to the wavelength of the atom’s own transitions, the finite speed of light weakens it to the inverse fourth power, as the attraction that weakens when light is too slow traced in the Casimir–Polder theory. The potential is $-C_4/x^4$ in that outer region, with no barrier, no wall, nothing pushing back. Any atom that reaches the surface is assumed here to stick, or be lost, so the surface absorbs.

The potential has its own natural length, $\beta_4 = \sqrt{2mC_4}/\hbar$, at which the kinetic energy of localisation equals the attraction. Measured in that length, every atom on every surface sees the same problem, and solving the Schrödinger equation for it gives the probability of reflection as a function of the atom’s wavenumber alone.

The result is the same shape as the step’s, for the same reason. Fast atoms, with wavelengths short compared with $\beta_4$, follow the attraction smoothly into the surface and stick. Slow atoms, with wavelengths long compared with $\beta_4$, cannot follow it; the potential falls away too steeply on the scale of their wavelength, and their wave is turned back. At low energies the reflection approaches one linearly, $R \approx 1 - 4k\beta_4$, a law that holds for every potential falling faster than $1/x^2$ and that has been confirmed in experiments. For atoms near solids, $\beta_4$ is a fraction of a micrometre, and appreciable reflection requires atoms moving towards the surface at millimetres per second, which is why the effect needed laser-cooled atoms to be seen.

The first clear observations came with atomic hydrogen bouncing off liquid helium films in the 1980s and with metastable neon atoms reflected from silicon by Fujio Shimizu in 2001, where atoms dropped onto the surface at grazing incidence, with a tiny velocity perpendicular to it, bounced off with a probability that rose as that velocity fell, as the curve says it should.

The badlands #

Where does the reflection happen? There is no turning point — the place where a classical particle would stop and turn around — because the atom speeds up all the way to the surface. The reflection happens where the semiclassical description breaks down.

The semiclassical, or WKB, approximation works whenever the local wavelength changes little over one wavelength, the condition that $|d\lambda/dx|$ is small. Near the surface the attraction is so strong that the atom is fast and its wavelength tiny, changing smoothly. Far out the attraction is negligible and the wavelength is that of the free atom. In between there is a region where the wavelength changes rapidly compared with itself, and there the approximation fails and reflection occurs. Physicists working on the problem named it the badlands. For slow atoms the badlands lie further out and the failure is more severe, which is why they reflect more.

That gives the clearest contrast with the quarter cycle a turning point costs, where the semiclassical approximation failed at a turning point, the place where the particle’s speed fell to zero, and the price of passing it was a fixed phase shift. Quantum reflection is a failure of the same approximation in a region where the particle is speeding up. The two are the two ways the wave picture can depart from the particle picture: at a turning point, where a classical particle must reflect, and in the badlands, where a classical particle never would.

A standing wave in front of the surface #

The wave itself shows the reflection directly.

An incoming wave alone would have a constant size; an incoming wave plus an equal reflected wave would have true zeros every half wavelength. The near-zeros here, about a twentieth of the maxima, are what 82 per cent reflection looks like. The pattern begins well away from the surface, many times $\beta_4$ out, which says again that the reflection happens far from the surface itself. The atoms that are reflected never come within reach of the surface they are bouncing off.

A mirror for atoms that touches nothing #

Quantum reflection has been proposed and tested as a way of making mirrors for atoms. Atom optics needs mirrors, as light optics does, and the usual atom mirrors use laser light or magnetic fields to push atoms away from a surface. A quantum-reflection mirror needs only a surface and slow atoms: the attraction itself does the reflecting, and the atoms never touch anything. Experiments have shown that structuring the surface — ridges, posts, a surface that is mostly empty space — reduces the attraction and raises the reflection, since a weaker attraction means a shorter $\beta_4$ and a lower $k\beta_4$ at the same speed.

A related effect matters for antimatter. Antihydrogen atoms falling slowly onto a surface are annihilated if they reach it. Calculations show that quantum reflection should turn back a large fraction of very slow antiatoms before they reach the surface, which would allow antihydrogen to bounce, as ordinary neutrons do in the bounce that comes in fixed heights, and so allow its gravitational quantum states to be measured. That is one of the routes being developed to measure precisely how antimatter falls, the question the antiatom that had to be weighed anyway left open.

What the picture cannot show #

The models are one-dimensional: a particle moving straight at a step or a flat surface. Real atoms usually arrive at grazing incidence, and only the motion perpendicular to the surface matters for reflection, which is how experiments achieve the millimetres-per-second perpendicular speeds without having to slow the atoms to a crawl. The surface potential is taken as a pure $-C_4/x^4$ everywhere, with perfect absorption at the surface; a real atom–surface potential crosses over to $-C_3/x^3$ close in, and has a repulsive wall at the surface itself, which the absorbing boundary replaces on the assumption that atoms reaching the surface are lost. The low-energy law $R \approx 1 - 4k\beta_4$ is insensitive to these details, but the reflection at moderate energies depends on them.

Real surfaces are not flat on the scale of $\beta_4$, are often contaminated, and can have patches of charge that add their own long-range forces. Atoms interact with each other as well as with the surface, which matters for dense clouds such as Bose–Einstein condensates, whose reflection from surfaces has been studied and is modified by the atoms’ interactions. The step calculations use units in which the step’s depth sets the scale; mapping them onto any particular system needs that system’s own parameters.

Still open: how close to perfect an atom mirror of nothing can be #

The low-energy law says reflection approaches one only as the speed approaches zero, so a practical quantum-reflection mirror trades speed for reflectivity. Surfaces structured to weaken the attraction help, and some experiments have reported reflectivities of tens of per cent at accessible speeds, but how far structuring can go — whether a surface can be made so tenuous that atoms at centimetres per second reflect with near certainty — and whether such mirrors can be made flat and stable enough to preserve an atom’s phase for interferometry, remain under investigation. The physics is a single line of algebra in the limit; making a good mirror of it is not.

A matter wave reflects from a drop in potential as a wave reflects from any change of medium, $\left(\frac{k_1 - k_2}{k_1 + k_2}\right)^2$ for a sharp step, and as the particle slows the reflection tends to one for every shape of drop, since smoothness is judged against a wavelength that grows without limit; an atom approaching an absorbing surface that only attracts it, $-C_4/x^4$, is reflected with $R \approx 1 - 4k\beta_4$, 82 per cent at $k\beta_4 = 0.05$, in a region far from any turning point where the semiclassical picture fails. The habit worth carrying is to measure a change against the wavelength of whatever crosses it, because a slow enough wave finds a cliff in every slope.