The fall that leaves the mass in the phase

A feather and a hammer fall together on the Moon, and the fall that does not depend on what is falling has been tested to a part in $10^{15}$ with objects of every composition. The mass that measures a body’s inertia is the mass that gravity pulls on, and so it cancels from every trajectory in a uniform field.

A neutron obeys that rule, and a neutron is also a wave with a wavelength of its own. The trajectory of a wave is where its phases add up, and the phase of a matter wave is the action along its path divided by Planck’s reduced constant. The action contains the mass. So the mass cancels from where the neutron goes and survives in how the wave’s phase accumulates on the way — and a neutron interferometer, which compares the phase along two paths, can see gravity acting through the mass on a quantum phase.

In 1975 Roberto Colella, Albert Overhauser and Samuel Werner did exactly that, with neutrons from a research reactor passing through an interferometer cut from a single crystal of silicon. They tilted the interferometer so that one path ran higher than the other and watched the neutrons move from one detector to the other. It was the first experiment whose result depended both on Newtonian gravity and on the quantum mechanics of the particle it acted on.

Two paths at two heights #

The interferometer is a block of perfect silicon with three thin slabs standing up from a common base. Neutrons diffract at the first slab into two beams, which the second slab turns back towards each other and the third recombines, so the two paths enclose a diamond a few centimetres across. Two detectors count the neutrons leaving the last slab in two directions, and the ratio between them depends on the phase difference between the paths.

With the block level both paths are at the same height. Tilted by an angle $\alpha$ about the incoming beam, one path is lifted above the other, and a neutron on the higher path has climbed against gravity and is slower. At 30° of tilt, for the 10.1 square centimetres the 1975 interferometer enclosed, the effective height difference is 15.8 millimetres, and a neutron of wavelength 1.445 ångströms travelling at 2,738 metres per second is slower on the higher path by 56.5 micrometres per second. Its wavelength is longer by twenty parts in a thousand million.

That is nothing, by any direct measure, and over 3.2 centimetres of path it adds up to 28.7 radians of phase — more than four and a half whole fringes. The number in the figure was computed by integrating the neutron’s local wavenumber along both paths, not from a formula, and agrees with the closed form

$$\Delta\phi = \frac{2\pi\, m^2 g\, \lambda\, A \sin\alpha}{h^2}$$

to a part in ten thousand. The formula depends only on the enclosed area and the tilt, not on the shape of the diamond, which is why the paths can be replaced by a rectangle of the same area for the calculation.

Why a neutron #

The choice of particle was not a matter of convenience, and the reasons say something about how weak gravity is.

A charged particle is hopeless. The electric force on an electron in a field of one volt per metre — far weaker than the stray fields on any laboratory surface — is about twenty thousand million times its weight, so any phase gravity writes into an electron wave is buried under phases from fields nobody can fully control. The neutron has no charge. It has a magnetic moment, which the experiments had to shield, but it feels neither the electric fields of the apparatus nor the image charges of nearby conductors.

A neutron is also heavy for a quantum particle — eighteen hundred times the electron’s mass — and the phase grows as the mass squared. And thermal neutrons from a reactor have wavelengths of an ångström or two, comparable with the spacing of atoms in a crystal, so they diffract cleanly from the planes of a perfect silicon lattice and pass through centimetres of it with little absorption. That combination made possible the interferometer Helmut Rauch and his colleagues first operated in 1974, adapting a design already used for X-rays: three slabs cut from one crystal, so that their lattice planes are aligned to a fraction of an atomic spacing across the whole device, which is what keeps two paths centimetres apart coherent with each other.

The same instrument was used in the same years to show that a neutron’s wavefunction changes sign when its spin is turned through a single full rotation, by precessing the spin on one path in a magnetic field and watching the fringes. Gravity and spinor rotation were measured with the same slab of silicon within a year of each other.

Tilting sweeps the neutrons between detectors #

Tilting the interferometer from one side to the other sweeps the gravitational phase through 48.5 radians, and the neutrons swing between the two detectors seven and a half times. The two detectors’ shares always add to one, because every neutron leaves by one exit or the other; what gravity changes is which.

The fringes fade as the tilt grows. The neutrons in any real beam have a spread of wavelengths, about one per cent here, and since the phase is proportional to the wavelength, neutrons at the two edges of the spread acquire slightly different phases and their fringes drift apart. At 25° the visibility has fallen to about 0.97 of its value when level. That fading is itself a check on the scaling with wavelength: a phase that did not depend on $\lambda$ would not wash out this way.

Where the mass comes back in #

The formula contains the mass squared, and the reason is instructive. The phase of a matter wave along a path is its action divided by $\hbar$, and for a particle in a uniform field the action is $\int (\tfrac12 m v^2 - mgy)\,\mathrm{d}t$ — mass times something that depends only on the path. The trajectory is where that phase is stationary, and stationarity of $m$ times a function is stationarity of the function: the mass factors out, and every body follows the same path. That is the equivalence principle, recovered from the wave.

The difference in phase between two paths is not a stationarity condition. It is the action difference itself, divided by $\hbar$, and the mass sits in front of it. One factor of $m$ comes from there. The second comes from converting a path length into a time: at a fixed wavelength the neutron’s speed is $h/m\lambda$, so a heavier particle of the same wavelength is slower and spends longer on the higher path.

At the neutron’s wavelength the phase grows from 57.4 radians for a neutron to 904 for a helium atom and 430,000 for a rubidium atom, and falls to a hundred-thousandth of a radian for an electron. At the neutron’s speed the growth is linear. And every one of those particles, over a single 3.2-centimetre leg, falls by the same 0.67 nanometres. The fall is the classical content; the phase is what quantum mechanics adds, and it is a measurement of $m/\hbar$ as much as of $g$.

That is sometimes described as a violation of the equivalence principle, and it is not one. The principle in its operational form says that no local experiment can tell a uniform gravitational field from a uniform acceleration, and that remains true of the phase: an interferometer accelerated upward at $g$ in empty space would show exactly the same fringes, and the equivalence was checked with a neutron interferometer in 1983 by accelerating it rather than tilting it. What the mass dependence shows is that “falls the same way” is a statement about trajectories, and that a wave has properties beyond its trajectory in which the mass is not hidden. It is the gravitational counterpart of the phase a magnet leaves on a path it never touched: a potential that shows up in the phase, where the classical description sees only forces.

What else turns the phase #

A laboratory on the Earth is not an inertial frame, and gravity is not the only thing that changes the phase when the interferometer is reoriented.

While a neutron crosses the interferometer the Earth turns, and the path that runs with the rotation is effectively longer than the one against it. That is the Sagnac effect, familiar from light sent both ways round a turning loop, here for a massive particle. For the interferometer drawn it contributes 1.73 radians, three per cent of the gravitational phase at full tilt. It was measured separately with neutrons in 1979. It can be told apart because it does not depend on the neutrons’ wavelength — a heavier or slower particle spends longer in flight, but the phase from rotation depends on mass and area only — while the gravitational phase is proportional to the wavelength.

The precision experiments that followed found something less tidy. The measured gravitational phase came out about one per cent below the prediction, a discrepancy far larger than the statistical errors. Most of it was traced to the interferometer itself: a silicon block sags under its own weight by amounts that change as it is tilted, and neutron diffraction in a thick perfect crystal carries phase shifts of its own that depend on the geometry. Whether every part of the residual has been accounted for was argued for years, and it is the reason the best measurements of this kind are now made with atoms, where the paths are defined by laser beams rather than by a crystal.

From a test to an instrument #

The neutron experiment established the effect. Its size, 57 radians, is set by a few centimetres of path and twenty microseconds of flight, and there is little room to make either larger with neutrons from a reactor.

Atom interferometers remove both limits. A cloud of laser-cooled atoms is dropped or launched upward, and pulses of laser light act as the beam splitters and mirrors, giving each atom a kick of two photon momenta and later taking it back. The gravitational phase is then the kick’s wavenumber times $g$ times the square of the time between pulses. With rubidium and pulses a tenth of a second apart it is about 1.6 million radians; in a fountain ten metres tall with pulses a second apart, 160 million. Read to a thousandth of a radian, the fountain measures $g$ to a few parts in $10^{12}$.

The way the phase arises in an atom interferometer makes the equivalence principle almost visible. Each laser pulse imprints the local phase of the light onto the part of the atom it kicks, and that phase is set by where the atom is when the pulse arrives. Three pulses, a time $T$ apart, read the atom’s height three times: at the first split, at the mirror pulse in the middle, and at the final recombination. The interferometer’s phase is the wavenumber times the first height, minus twice the middle one, plus the last — a second difference of the trajectory, which for a body in free fall is exactly $-gT^2$. What the atoms record is their own acceleration relative to the laser’s wavefronts, which are fixed to the apparatus. Accelerate the apparatus upward instead of letting gravity pull the atoms down and the second difference is the same, which is why an atom gravimeter cannot, even in principle, tell the two apart. The mass has dropped out here because the laser, not the atom’s own motion, sets the phase; it is still in the recoil velocity that separates the two paths and in every correction beyond the leading term.

Atom gravimeters are now instruments rather than demonstrations, used in geophysical surveys and to measure the gravitational constant — a counterpart to clocks that measure a height, reading the potential’s slope where the clocks read the potential itself. And they have returned to the question the neutron experiment opened. Dropping two isotopes of rubidium together in the same fountain and comparing their phases tests whether atoms of different mass fall the same way, and has done so at the level of a part in $10^{12}$. In 2022 an atom interferometer measured a phase shift from the gravitational potential of a nearby mass in a region where the force from that mass was negligible — the gravitational version of the effect a magnet leaves on a path it never touches.

What the calculation takes as given #

Gravity is uniform across the interferometer. Over a few centimetres the change in $g$ is a part in $10^{8}$, and it matters only for the atomic fountains, where the atoms rise metres and the gradient of gravity — the part of the field free fall cannot remove — contributes a phase of its own that has to be modelled and removed.

The neutron is a point particle with no internal structure that gravity could act on. It has a magnetic moment and a spin, and in the presence of stray magnetic fields both add phases that the experiments had to shield against.

The wavelength spread is Gaussian and one per cent wide. The fading of the fringes depends on the real spectrum of the beam, and a spread with long tails washes the fringes out differently. The shape drawn is illustrative, and the phase at each wavelength is what the measurement actually tests.

Nothing but gravity and rotation acts on the paths. The crystal’s sag and the dynamical phases of diffraction are left out of every figure, although they are the largest systematic effects in the real measurement.

Two paths drawn where the neutron took neither #

The diamond in the first figure is drawn with its height difference exaggerated enormously: a millimetre-scale difference on a few centimetres of path, drawn as a third of the width. More importantly, it draws two paths as though the neutron took one or the other. The interference is the evidence that each neutron does neither; the paths are a bookkeeping device for the two amplitudes whose phase difference the detectors read, and the fringes exist only because no measurement determines which was taken.

The mass chart puts particles on one axis that could never share an interferometer. An electron in a silicon crystal is not a free particle, a rubidium atom cannot diffract from a slab of silicon, and a $\mathrm{C}_{60}$ molecule at the neutron’s wavelength would be moving far too slowly to cross anything in a reasonable time. The chart is a statement about how the phase scales, not a menu of experiments.

Still open: whether gravity itself can carry quantum superposition #

Every experiment described here puts a quantum particle in a classical gravitational field — the Earth’s, or a test mass’s — and every one of them is consistent with gravity being described classically while matter is described quantum mechanically. That the phase depends on $m/\hbar$ says that matter is quantum. It says nothing about whether the gravitational field is.

Proposals made since 2017 would test that directly. Two small masses, each put into a superposition of two positions, would interact only through gravity; if gravity can be in superposition, the two masses would become entangled, and if it cannot, they would not. The masses need to be large enough to produce a measurable gravitational interaction in the time their superpositions survive — about ten picograms each, held in superpositions hundreds of micrometres wide for a few seconds — which is many orders of magnitude beyond anything achieved. Whether gravity can entangle matter is therefore an open experimental question, and the neutron interferometer is where the thread that leads to it began.

The habit worth carrying away is to ask where a quantity cancels. The mass cancels from the equation of motion and not from the action, and so it cancels from trajectories and not from phases. Any wave-like description of matter will show the mass wherever it compares two paths, and the equivalence principle survives as a statement about what trajectories look like, which is all it was ever tested as before 1975.