The grating made of light

Everything has a wavelength found that every particle carries a de Broglie wavelength, $h/p$, and that it shows only when the particle meets structure on that scale. One arrival at a time found the interference pattern building up particle by particle. The fall that leaves the mass in the phase followed the matter wave through a gravitational field, and noted in passing that modern atom interferometers use pulses of laser light as their beam splitters and mirrors. The phase a magnet leaves on a path it never touched found a phase with no force behind it.

That remark about laser pulses deserves more than a remark, because it hides a reversal. Every diffraction grating in what a thousand slits buy is a piece of matter with a repeating structure, and it diffracts light. A standing wave of light is also a repeating structure — bright bands and dark ones every half wavelength — and a particle crossing it feels the brightness as a potential. So light should diffract matter. Kapitza and Dirac proposed exactly that in 1933, for electrons. This essay follows why the grating works only in whole steps of momentum, why it behaves as two quite different instruments depending on how long it acts, and why a free electron needed a laser a billion times more intense than an atom did.

A grating whose lines are made of brightness #

Two laser beams of the same frequency running in opposite directions make a standing wave. The electric field oscillates everywhere at the light’s frequency, but its amplitude does not: it is largest at the antinodes and zero at the nodes, which alternate every quarter wavelength, so the time-averaged intensity repeats every half wavelength.

A particle moving through that pattern feels the intensity as a potential energy, for a reason that depends on what the particle is. A free electron is shaken by the light’s oscillating field; the shaking carries an energy, averaged over a cycle, proportional to the field’s intensity and inversely to the square of its frequency, and the electron behaves as though it were pushed away from bright regions by a potential of that size. This is the ponderomotive force that held up by a force that averages to nothing met in a rapidly shaken pendulum and a trapped ion. An atom has something better: an internal resonance. Light tuned slightly away from the resonance shifts the atom’s energy by an amount proportional to the intensity, downward for light below the resonance and upward above it — the same shift the light that pulls rather than pushes used to trap a bead in a focus.

Either way the standing wave presents the particle with a potential $V\cos 2kx$, a sinusoidal corrugation of period $\lambda/2$. It is a grating, and the question is what a grating made of a potential does.

Kicks in pairs #

A potential $\cos 2kx$ can change a particle’s momentum only by $\pm 2\hbar k$ at a time. There is a way of seeing this without solving anything. The standing wave is two travelling beams, each carrying photons of momentum $\hbar k$ in opposite directions. The only way the particle can interact with the light without changing its internal state is to absorb a photon from one beam and be stimulated to emit one into the other. Absorbing from the right-going beam and emitting into the left-going one gives the particle $+\hbar k$ and then another $+\hbar k$ of recoil — a kick of $2\hbar k$. The reverse gives $-2\hbar k$. Repeated exchanges give any whole multiple. A photon with a momentum established that each photon carries $\hbar k$; the grating is a machine for passing those momenta through a particle in pairs.

The counting also says what cannot happen. Absorbing a photon from a beam and emitting one back into the same beam changes nothing, and absorbing without emitting would leave an atom excited, which a detuned beam makes improbable. There is a variant that uses exactly that loophole: if the two beams differ in frequency by the splitting between two ground levels of the atom, the exchange can move the atom from one level to the other as it kicks it. The kick is then labelled — every atom that took a pair of photons is in the upper level, every one that did not is in the lower — and the two paths of an interferometer can be read out by counting atoms in each level rather than by resolving their directions. Many atom gravimeters use these labelled kicks; the grating in this essay is the unlabelled version, in which the atom leaves in the state it arrived in.

So the particles leave in a set of discrete directions, one for each number of pairs exchanged. For electrons of 380 electronvolts and infrared light of wavelength 1,064 nanometres, the angle between neighbouring orders is about 120 microradians — the angle a grating with lines 532 nanometres apart would give for waves of the electron’s de Broglie wavelength, 0.063 nanometres.

The thin grating: a flash, and Bessel functions #

If the light acts briefly and strongly, the particle does not move appreciably while it acts. The only effect is to multiply its wave by a phase proportional to the local potential: $e^{-i\theta\cos 2kx}$, where $\theta$ is the potential’s depth times the time it acts, divided by $\hbar$. A wave multiplied by a periodic phase is a sum of plane waves whose amplitudes are the Fourier coefficients of that phase, and for a cosine phase they are Bessel functions.

The fraction scattered into order $n$ is $J_n(\theta)^2$. For a weak flash nearly everything goes straight through and a little appears in the first orders. As $\theta$ grows, the first orders fill, peaking at 33.9 per cent each at $\theta = 1.84$; the straight-through beam empties completely at $\theta = 2.40$, the first zero of $J_0$, with every particle deflected; and a stronger flash spreads the particles across many orders at once, the highest occupied order being roughly $\theta$ itself. This is the Raman–Nath regime, named for the two physicists who worked it out in 1935 for the reverse situation, light diffracted by a sound wave in a liquid.

A thin grating is flexible — the flash’s strength sets how the particles are shared — but it is wasteful for any purpose that wants them in one place. At best two-thirds of the particles end up in the pair of first orders, and a third is spread elsewhere.

The thick grating: a slow wave that notices energy #

If the light acts for longer and more gently, something else takes over: energy conservation. A particle in the standing wave can only exchange momentum in steps of $2\hbar k$, but it can only keep an exchange if its kinetic energy afterwards is the same as before, since the light, in steady state, gives it no energy to spare.

The figure draws the free particle’s energy parabola and the steps the light allows. A particle arriving with momentum $-\hbar k$ along the light — half a step — can be carried to $+\hbar k$, which has exactly the same energy; the exchange costs nothing and can go to completion. A particle arriving at any other momentum would have to change its kinetic energy to take a step, and a grating that acts slowly, over many periods of the relevant energy difference, cannot supply it; the exchange does not happen. The allowed arrival is at an angle whose sine is the ratio of $\hbar k$ to the particle’s momentum — exactly Bragg’s condition for a crystal of spacing $\lambda/2$, the condition that decides which reflections an X-ray beam gives from a crystal, with the roles of matter and light exchanged.

At Bragg’s angle only two states are involved, the incoming one and the one reflected, and they have the same energy. The light couples them, and a two-state system with a steady coupling does one thing: it swings back and forth.

The particles swing wholly from the incoming order into the reflected one and back, and the other orders are barely touched. A pulse timed to end when everything has moved is a mirror for matter: every particle leaves in the reflected direction. A pulse half as long leaves half in each, in a definite superposition — a beam splitter. These are exactly the two elements the fall that leaves the mass in the phase needed for an atom interferometer: a splitter to send each atom along two paths, a mirror to bring the paths back together, and a second splitter to recombine them. Gravimeters, rotation sensors and tests of the equivalence principle built from falling clouds of atoms are built from these pulses.

Thin or thick #

Which instrument a standing wave is depends on one comparison: the time it acts against the time the particle takes to notice an energy mismatch of one recoil.

The figure finds, for each duration, the best that any depth of standing wave can do at putting particles into one order. For brief pulses the best is the Raman–Nath maximum, 33.9 per cent, because a brief pulse cannot tell energy-conserving exchanges from others and fills the orders symmetrically. For pulses lasting longer than a few times $\hbar$ divided by the recoil energy the curve climbs to nearly one. The crossover is the matter-wave version of the distinction between thin and thick holograms, and of the one the physics of acousto-optics draws between a thin sound wave, which scatters light into many orders, and a thick one, which scatters it into one. The time scale for an atom such as rubidium is tens of microseconds, which is why atom-interferometer pulses are tens of microseconds long.

A grating that moves, and the recoil that weighs an atom #

A material grating is fixed to its bench. A grating made of light can be set moving simply by giving the two beams slightly different frequencies: the interference pattern then slides along at a speed equal to half the frequency difference divided by the wavenumber, and a small frequency difference makes it crawl. Nothing about the physics changes in the frame moving with the pattern, so the Bragg condition still holds there — which means that in the laboratory the grating reflects only particles moving at one particular velocity, the one that is $\pm\hbar k/m$ relative to the moving pattern. Sweep the frequency difference and the grating picks out one velocity after another. A slow, weak pulse is an extraordinarily narrow velocity filter, and it is how the velocity spread of the coldest atomic clouds is measured.

The same arrangement measures something more fundamental. When an atom is kicked by a pair of photons, it recoils with a speed $2\hbar k/m$, and its kinetic energy changes by an amount fixed by $h$ divided by its mass. That energy change appears as the frequency difference at which the moving grating reflects it, and can be measured, with a chain of pulses that give the atom hundreds of kicks, to a part in ten billion. The measurement delivers the ratio $h/m$ for the atom. Combined with the atom’s mass relative to the electron’s, which is known very precisely from comparisons of ions in traps, and with the Rydberg constant from hydrogen spectroscopy, it gives the fine-structure constant $\alpha$, the dimensionless strength of electromagnetism.

Two such experiments, one with caesium in 2018 and one with rubidium in 2020, gave $\alpha$ with uncertainties below a part in a billion — as precise as the value from the electron’s magnetic moment, which rests on a completely different chain of theory. The two atom-recoil values disagree with each other by more than five times their combined uncertainty, by a little over a part in a billion, and the disagreement has not been explained. The number that sets the size of every atom and the strength of every chemical bond is, at that level, measured best by timing how fast an atom moves after a grating of light has kicked it.

A billion times the light #

Kapitza and Dirac proposed the effect for electrons, and it was observed with electrons last. The reason is how weakly a free electron responds to light.

The ponderomotive energy of a free electron in light is tiny: an electron in infrared light of ten gigawatts per square metre gains a potential energy of only about $10^{-26}$ joules, and a 380-electronvolt electron crosses a beam a quarter of a millimetre wide in twenty-two picoseconds. The product, divided by $\hbar$, is the phase the grating imprints, and it reaches one radian only at about $3 \times 10^{12}$ watts per square metre. An atom near resonance is shifted enormously more by the same light — for sodium a gigahertz from its yellow line the resonance makes the shift some seventy thousand times larger than a free electron’s in the same intensity — and a thermal atom crossing the beam at a kilometre a second spends ten thousand times longer in it. Sodium atoms need about $3 \times 10^3$ watts per square metre, a third of a watt per square centimetre, which a modest continuous laser supplies.

So the effect was seen first with atoms. David Pritchard’s group at MIT diffracted a beam of sodium atoms with a near-resonant standing wave in 1986, observing the Bessel-function populations of the thin grating, and demonstrated Bragg scattering of atoms from light two years later. Electrons needed a pulsed laser. Experiments in the 1980s saw electrons scattered by intense standing waves in a regime where the pattern was classical rather than a diffraction pattern, and the diffraction itself — discrete orders at the angles the Bragg condition predicts, at the intensity the theory requires — was reported by Daniel Freimund, Kayvan Aflatooni and Herman Batelaan in 2001, sixty-eight years after the proposal.

Where the model stops #

The figures treat the standing wave as a perfect cosine, infinitely wide, switched on and off abruptly, and the particle as having no internal structure beyond a fixed response. A real laser beam has a Gaussian profile, so particles crossing it see a potential that rises and falls smoothly, which softens the distinction between thin and thick and adds a spread of interaction strengths across the beam. An atom close to resonance can absorb a photon from one beam and emit spontaneously in a random direction rather than into the other beam; each such event gives a random kick and destroys the coherence of the diffraction, which is why the light is tuned away from resonance, trading strength for cleanliness. Two-state swinging at Bragg’s angle is exact only when the potential is weak compared with the recoil energy; stronger potentials leak particles into further orders, as the figure’s one per cent shows. And particles with a spread of velocities meet the Bragg condition to different degrees, so a real beam is never perfectly mirrored.

What the pictures cannot show #

The figures show populations — how many particles end up in each direction — and not the phases, which are what an interferometer uses. The light imprints its own phase on every particle it kicks, the phase of the standing wave at the particle’s position, and that is why a pulse can read out where a falling atom was. The figures do not show the particle’s position in the standing wave at all, only its momentum, because in the diffraction problem the two are exchanged by a Fourier transform and the momentum picture is the one that shows discreteness. And they cannot show the historical fact that most of what the Kapitza–Dirac grating is used for — interferometers for inertial navigation, gravity surveys and fundamental constants — relies on atoms rather than the electrons it was proposed for.

Still open: the heaviest thing a grating of light can diffract #

Molecules have been diffracted by standing light waves as well as by material gratings: fullerenes of sixty carbon atoms around 2000, and since then molecules of tens of thousands of atomic mass units in interferometers whose gratings are standing waves of ultraviolet light. A standing wave has an advantage over a material grating for large particles: it has no walls for them to stick to and cannot be clogged. The question these experiments ask is how massive and complex an object can be put into a superposition of two paths a measurable distance apart before something — collisions with gas, emission of thermal radiation, or some unknown mechanism that would modify quantum mechanics for large masses — destroys it. The record keeps rising, and no experiment has yet found a mass at which the pattern disappears for a reason other than a known source of disturbance. Whether one exists is open.

The habit worth carrying away is to ask of any periodic structure what it can exchange. A standing wave of light can change a particle’s momentum only in pairs of photon kicks, and whether it spreads the particles over many orders or sends them all one way depends on whether it acts too briefly to notice energy or long enough to insist on it. Kapitza and Dirac proposed it as a curiosity; slowed down and made gentle, it became the mirror and the beam splitter from which every atom interferometer is built.