Quantum

The bounce that comes in fixed heights

A ball dropped on a floor can bounce to any height. A neutron cannot. Slow enough, and resting on a flat mirror in the Earth's gravity, it has a lowest state about fifteen micrometres high, with an energy of 1.41 pico-electronvolts, and a staircase of higher states above it — the first quantum states ever seen in a gravitational field, measured in 2002 with a slit narrower than a hair. Shaking the mirror at 255 hertz drives a neutron from the first state to the second. The staircase is now a laboratory for gravity at distances where nothing else can test it.

15 min read 5 figures Only some values fit Made into an instrument

Assumes: The fall that leaves the mass in the phase · The motion that cannot be stopped

Everything has a wavelength gave every particle a de Broglie wavelength, and the essays that followed sent matter waves through slits, magnetic fields and gratings made of light. The fall that leaves the mass in the phase put gravity into the picture: a neutron wave split into two paths at different heights picks up a phase difference proportional to its mass, which is the one place where the equivalence principle’s “the mass cancels” fails. The grating made of light turned a standing wave of light into a diffraction grating for atoms. In every case the matter wave travelled. It was diffracted or shifted, and then it left.

This essay holds one still. A matter wave confined in a well has only certain allowed energies, as the motion that cannot be stopped found for a particle in a box, and the question is whether the Earth’s gravity, the weakest force there is, can make such a well for anything a laboratory can hold. It can. A neutron resting on a flat mirror is in a well whose floor is the mirror and whose ceiling is the slope of its own weight, and the well has a staircase of states, each with a definite height. They are the first quantum states of any particle in a gravitational field ever observed.

A well made of a floor and a slope

A ball bouncing on a hard floor stays in a region bounded below by the floor and above by the height at which it runs out of energy. Its potential energy is $mgz$ above the floor and infinite below it. Every height is allowed, because the ball’s energy can take any value.

For a quantum particle the same potential allows only certain energies, and the scale on which that happens can be found without solving anything. A particle confined to a height $z$ has a momentum uncertainty of about $\hbar/z$ , which costs a kinetic energy of about $\hbar^2/mz^2$ ; holding it at height $z$ costs a potential energy of about $mgz$ . The first falls as the height increases and the second rises, and the total is least where they are comparable. That fixes a length,

z_0 = \left(\frac{\hbar^2}{2m^2g}\right)^{1/3},

and an energy $E_0 = mgz_0$ . For a neutron in the Earth’s gravity $z_0$ is 5.87 micrometres and $E_0$ is 0.602 pico-electronvolts — a millionth of a millionth of an electronvolt.

The heights a neutron is allowed to bounce to. A neutron above a horizontal mirror in the Earth's gravity: the potential energy mgz (straight line), the first four allowed energies and, drawn on each, the probability density of the neutron's height. The natural length is z₀ = (ħ²/2m²g)^(1/3) = 5.87 μm and the natural energy mgz₀ = 0.602 peV. The allowed energies are 1.41, 2.46, 3.32, 4.08 peV — pico-electronvolts — and a classical neutron with each energy would rise to 13.7, 24.0, 32.4, 39.8 μm. Each density is an Airy function, with n − 1 nodes, leaking a little above the classical turning height and vanishing at the mirror. The lowest state never touches the mirror and never rises above about 25 μm. — Fig. 1 The potential mgz for a neutron above a mirror, the first four allowed energies, 1.41 to 4.08 peV, and the probability density of the neutron’s height in each. The lowest state rises to about 14 μm classically and leaks a little above.

The exact solution of the Schrödinger equation in a uniform slope is the Airy function, the same function that describes the light just inside a rainbow’s bright edge, and the floor fixes which of its forms fit: the wavefunction must vanish at the mirror. So the allowed energies are $E_0$ times the places where the Airy function crosses zero — 2.338, 4.088, 5.521, 6.787 — which for a neutron are 1.41, 2.46, 3.32 and 4.08 peV. The figure draws each with its probability density. The lowest state is a single hump, peaked about eight micrometres above the mirror, falling to zero at the mirror itself and fading away above the classical turning height of 13.7 micrometres. The second has two humps, the third three. None of them touches the floor, and none can sit still on it: a neutron at rest on a mirror is never at rest and never on the mirror.

How the counting of these states relates to the classical action of the bounce, and why the correction to the old quantum rule is three quarters of a unit, is the subject of the quarter cycle a turning point costs, which took the bouncer as one of its examples. Here the interest is in what the staircase looks like and whether it can be seen.

A staircase that crowds as it climbs

Three wells, three staircases of energy. The first ten allowed energies, each divided by its ground-state energy, for a particle in a box with hard walls (rising as n²), in a spring's parabolic well (evenly spaced, as 2n − 1) and on a mirror under gravity (as the zeros of the Airy function). The gravitational staircase rises most slowly, as n^(2/3): its tenth level is only 5.49 times its first, so its levels crowd together going up: the gap between the first two is 0.75 of the ground energy and between the ninth and tenth 0.38. The shape of a well decides the spacing of its levels: walls that steepen faster than a parabola's spread the levels as they rise, and a uniform slope, which steepens less, crowds them together. — Fig. 2 The first ten energies of three wells, each divided by its ground energy: a box (as $n^2$ ), a spring (as $2n-1$ ) and a mirror under gravity (the Airy zeros, rising as n^(2/3)). The tenth gravitational level is only 5.49 times the first.

The shape of a well decides how its levels are spaced. A box with hard walls spaces them ever wider, as $n^2$ ; a spring’s parabola spaces them evenly; a uniform slope, whose walls open out more slowly than a parabola’s, crowds them together, as $n^{2/3}$ . The tenth level of the gravitational well is only five and a half times the first. That crowding is what makes the higher states hard to separate and the lowest few the ones worth measuring: the gap between the first and second is three quarters of the ground energy, and between the ninth and tenth less than two-fifths.

The heights grow the same way. Each state’s classical turning height is $z_0$ times its Airy zero, so the first four states reach 13.7, 24.0, 32.4 and 39.8 micrometres. A neutron in the lowest state occupies a slab of space about as thick as a human hair, above a mirror it never touches.

Neutrons slow enough to stand

To see any of this, the neutron’s vertical energy must be comparable with a pico-electronvolt, which means a vertical speed of a few centimetres a second. That rules out almost every neutron there is. Neutrons leave a reactor at kilometres a second; those cooled by heavy water and then by liquid deuterium come out at hundreds of metres a second; and a tiny fraction, slowed further by climbing against gravity and by scattering in special converters, emerge as ultracold neutrons, moving at a few metres a second. Such neutrons have so little energy that most surfaces reflect them at any angle of incidence, as though the surface were a wall: the nuclei of a polished glass plate present an average repulsive potential of about a hundred nano-electronvolts, far more than the neutron’s vertical energy. A flat mirror is therefore a floor that a neutron cannot pass through, and the gravitational well is exactly the one drawn.

The experiment that first saw the states was done at the Institut Laue–Langevin in Grenoble in 2002 by Nesvizhevsky and colleagues. Ultracold neutrons moving horizontally at about ten metres a second were sent along a polished glass mirror a few centimetres long, with a rough absorbing plate held parallel above it at an adjustable height. Any neutron whose wavefunction reached the absorber was likely to be lost. The count of neutrons reaching the far end was measured as the gap was opened.

$Neutrons through a slit that only quantum states fit. A model of the experiment that first saw gravitational quantum states of the neutron: neutrons fly horizontally between a mirror and an absorber set at height h above it, and the fraction transmitted is plotted against h. Each of the lowest twelve states is equally populated and is removed at a rate proportional to the share of its probability lying above h. Below about 16 μm almost nothing gets through, because even the lowest state reaches the absorber; the transmission then rises in steps as each state comes to fit — 0.2 per cent at 15 μm, 8.4 at 25, 24.1 at 40. The dashed curve is the classical count of states whose turning height lies below h, which rises smoothly as h^(3/2): the steps are the quantum staircase showing through. The absorption model is a simplification; the measured steps are rounded more than these.$

Fig. 3 A model of the slit experiment: transmission through the gap between a mirror and an absorber at height h, with the lowest twelve states equally populated and each absorbed in proportion to its probability above h. Nothing passes below about 16 μm; the classical count rises smoothly as h^(3/2).

Classically the transmission should rise smoothly from zero: a gap of height $h$ lets through every neutron whose bounce stays below $h$ , and the number of such bounces grows as $h^{3/2}$ . Quantum mechanically nothing gets through until the gap is large enough to hold the lowest state, whose wavefunction extends to about fifteen micrometres; then each further state adds a step as the gap grows past its height. The model in the figure shows the shape, and the measurement showed it too: zero transmission below about fifteen micrometres, where the classical expectation was well above zero, and a rise in rounded steps after. The steps were less sharp in the data than in an idealised calculation, because the absorber’s effect on each state is gradual and because the neutrons arrive with a spread of states; what could not be explained classically was the absence of any transmission through gaps a classical neutron would have crossed easily.

Which particles could do this

How high a quantum bounce is, for things of different mass. The natural height z₀ = (ħ²/2m²g)^(1/3) of the gravitational quantum states above a mirror, on a logarithmic scale, for six particles. It falls as the mass to the power −2/3. Electron: 0.88 mm, ground energy 0.11 peV; neutron: 5.87 μm, ground energy 1.41 peV; hydrogen atom: 5.87 μm, ground energy 1.41 peV; rubidium atom: 301 nm, ground energy 6.21 peV; caesium atom: 227 nm, ground energy 7.16 peV; C₆₀ molecule: 73 nm, ground energy 12.58 peV. An electron's states are a millimetre high but so faint that a stray field of a nanovolt per metre outweighs gravity; an atom's are a fraction of a micrometre, inside the range of the surface's own attraction. The neutron, uncharged and heavy enough, is the particle for which the staircase is both measurable and clean. — Fig. 4 The natural height $z_0$ of the gravitational quantum states for six particles, falling as the mass to the power −2/3: 0.88 mm for an electron, 5.87 μm for a neutron or a hydrogen atom, 301 nm for rubidium, 227 nm for caesium and 73 nm for a $\mathrm{C}_{60}$ molecule.

The natural height falls as the mass to the power two-thirds, so lighter particles have taller staircases. An electron’s would be almost a millimetre high. But an electron’s weight is $9 \times 10^{-30}$ newtons, which an electric field of a twentieth of a nanovolt per metre would match; the stray fields from the patches of differing potential on any real surface are many orders of magnitude larger, and an experiment to drop electrons in a shielded tube in the 1960s found exactly that the stray fields swamped gravity. The neutron’s lack of charge is the whole of its advantage.

Atoms are neutral too, and a hydrogen atom has almost exactly the neutron’s mass and so the same staircase. Heavier atoms have smaller staircases, a fraction of a micrometre for rubidium and caesium, and at those distances a neutral atom is attracted to the surface by the Casimir–Polder force, which near a mirror is far stronger than the atom’s weight and would distort the states beyond recognition. Antihydrogen, whose staircase would test whether antimatter falls the same way as matter, is one of the proposed uses; cold hydrogen, whose staircase matches the neutron’s, is another. The neutron remains the particle for which the staircase is both large enough to resolve and free of the forces that would swamp it.

A shaken mirror and a quantum jump

Counting neutrons through a slit shows that the states exist. Measuring their energies precisely needs spectroscopy: a way of driving transitions between them at a known frequency. The energies are pico-electronvolts, and a pico-electronvolt corresponds to a frequency of about 242 hertz — the frequencies of sound, not of light.

Quantum jumps a shaking mirror can drive. The first five gravitational energy levels of a neutron on a mirror, in pico-electronvolts, with arrows for transitions between them and the frequency at which each lies: (Eⱼ − Eᵢ)/h. 1 → 2: 255 Hz; 1 → 3: 463 Hz; 1 → 4: 647 Hz; 2 → 3: 208 Hz; 3 → 4: 184 Hz. These are acoustic frequencies. Oscillating the mirror up and down at one of them drives the neutron from one state to the other, which is how the levels have been measured as a spectrum since 2011, to a precision that tests gravity and any other force at micrometre distances. — Fig. 5 The first five gravitational levels of a neutron, with the frequencies of transitions between them: 255 Hz from the first to the second, 463 Hz to the third, 647 Hz to the fourth; 208 Hz from the second to the third, 184 Hz from the third to the fourth.

The transition from the first state to the second lies at 255 hertz, from the first to the third at 463, and from the third to the fourth at 184. No photon is needed to drive them, and a photon at those frequencies would be a radio wave thousands of kilometres long that a neutron, with no charge and a tiny magnetic moment, would barely notice. What drives them is the floor. Oscillate the mirror up and down at one of those frequencies, by a few micrometres, and a neutron in the lower state is shaken into the upper one, exactly as an atom is driven between two levels by light at the resonant frequency.

That was done from 2011 by Abele’s group, using neutrons at the same reactor in Grenoble, in a scheme they called gravity resonance spectroscopy. A first mirror and absorber select neutrons in the lowest states; a vibrating mirror drives transitions; a second absorber removes neutrons that have been lifted into higher states; and the count drops when the vibration frequency matches a transition. The resonances appear where the Airy zeros put them, with widths set by how long a neutron spends over the vibrating mirror, a few tens of milliseconds. Their positions have been measured to a precision of parts in a thousand and are being pushed further.

A bounce that has a period only when it is mixed

A neutron in one of these states does not bounce. Its probability density is fixed in time; the humps in the figure stay where they are, and the neutron is no more likely to be found rising than falling. That is what a stationary state means, and it is the strangest part of the picture to anyone who has watched a ball on a floor.

A bounce appears only when two states are mixed. A superposition of the first and second states has a density that sloshes up and down at the difference of their frequencies, 255 times a second, a period of 3.9 milliseconds. A classical neutron dropped from the first state’s turning height, 13.7 micrometres, would hit the floor and return in $2\sqrt{2h/g}$ , 3.3 milliseconds. The two are close and not equal, which is the usual relation between the classical period of an orbit and the frequency of the quantum transition between neighbouring levels on it — close for low levels and converging as the levels rise. Shake the mirror at 255 hertz and the neutron is being pushed in time with the bounce it would have, which is the classical way of saying why a resonance drives the transition.

Later experiments also looked at the heights directly. Track detectors with micrometre resolution, placed at the end of the mirror, recorded where the transmitted neutrons were, and the distribution of heights showed the shape of the lowest few states — a hump with nothing at the mirror and a tail above, rather than the distribution of a classical bouncing ball, which spends most of its time near the top of its bounce, where it moves slowly.

What the staircase can test

A spectrum measured at that precision is an instrument for anything that would change the energies, and the energies depend only on $g$ , the neutron’s mass and Planck’s constant — the potential is gravity’s, and nothing else should be there. Anything else that acted on a neutron within a few tens of micrometres of a surface would shift the levels.

Several kinds of hypothetical force would. Theories in which gravity spreads into extra dimensions at short range, of the kind discussed in the scale that may not be where it looks, predict a departure from Newton’s law below some distance; theories of dark energy that invoke a new field which hides itself in dense matter predict a force that would appear only near a surface and in a vacuum; and a variety of proposed particles would mediate weak forces with a range of micrometres. Each would shift the levels by a calculable amount, and the absence of any shift so far sets limits on each. For some of these the neutron staircase gives the best limits there are at its range, because a neutron has no charge for an electrostatic force to grip, no polarisability to speak of, and is held off the surface by nothing but the mirror.

There is a subtler test as well. The natural height is really $(\hbar^2/2m_i m_g g)^{1/3}$ , with the inertial mass from the kinetic energy and the gravitational mass from the potential. The equivalence principle says the two are equal, and in classical mechanics they cancel from every trajectory, which is the fall that does not depend on what is falling. In quantum mechanics they do not cancel from the energies, and the staircase therefore tests the equality of the two masses in a regime where the particle is a wave with no trajectory at all. So far it holds.

The surprising thing about a pico-electronvolt

The connection worth noticing is between the size of the effect and the size of the apparatus. Gravity is so weak compared with the electromagnetic forces that hold matter together that its quantum effects are usually placed at the Planck scale, sixteen orders of magnitude beyond any accelerator. The quantum states here are not quantum gravity in that sense: the gravitational field is classical, and only the neutron is quantised. But they show that gravity’s weakness, by itself, is no barrier to seeing quantisation in a gravitational potential. What it takes is a particle slow enough that its weight over a micrometre is comparable with its zero-point energy — a neutron at a few centimetres a second — and a floor flat to a fraction of that.

The pico-electronvolt is also the reason the states can be driven mechanically. A transition at a few hundred hertz can be driven by a loudspeaker’s worth of motion, and a resonance a few hertz wide can be resolved with a clock no better than a laboratory’s. A spectroscopy that began with lines in the visible, a hundred trillion times higher in frequency, has reached the acoustic range, and the particle being probed is held in place by the weight of a single nucleon.

What the pictures cannot show

The figures assume a perfectly flat, perfectly reflecting mirror and a uniform gravitational field. A real mirror has roughness of a few nanometres and waviness over its length; the neutron’s reflection is not perfect at its potential step; and the horizontal motion, ignored here, sets how long a neutron spends over each part of the apparatus. The transmission figure is a simplified model of absorption, with equal populations and a loss rate proportional to the probability above the absorber; the real experiments’ populations and absorbers are calibrated separately, and their steps are rounder. The energies are computed for $g = 9.80665$ m/s²; the local value at Grenoble differs in the fourth figure, which the most precise measurements must include.

Still open: how far the staircase can be pushed

The limiting factor is the number of neutrons. Ultracold neutrons are rare, sources deliver tens to thousands per second into an experiment, and each measurement of a resonance needs many hours. Several new sources of ultracold neutrons, based on superfluid helium or solid deuterium converters, are being built to raise the flux by orders of magnitude, and with them the gravity-resonance spectra could reach precisions of parts in a million or better, tightening every limit on short-range forces. Whether a neutron can be stored in a single gravitational state long enough to be observed for seconds — a quantum state of matter held by nothing but a floor and its own weight — and whether antihydrogen can be put on a similar staircase to compare antimatter’s fall with matter’s, are both being worked towards.

The habit worth carrying away is to find the scale before asking whether an effect is small. A particle on a floor under gravity has quantised heights set by $z_0 = (\hbar^2/2m^2g)^{1/3}$ — 5.87 μm for a neutron, with energies of 1.41, 2.46, 3.32 and 4.08 peV that a slit can count and a mirror shaken at 255 hertz can drive. Gravity’s weakness made the energies tiny, and the neutron’s slowness made them reachable.