The lattice no phase can bend

Two laser beams of one colour, crossed, make fringes: parallel sheets of bright and dark, spaced by the wavelength divided by twice the sine of half the angle between them. Three beams crossed in a plane make a pattern of bright spots on a lattice. Four beams in space make a crystal of light — a three-dimensional array of bright points in the region where they overlap — and atoms cooled to microkelvin temperatures feel the light’s intensity as a potential and settle into it, one to a site, as though into the sites of a solid. Those arrays hold the atoms of the most accurate clocks ever built and serve as laboratory models of electrons in metals.

The pattern is made by superposition, and superposition cares about phase. Every beam is reflected from mirrors that tremble at the scale of a wavelength with every footstep and every passing lorry, so the phases of the beams relative to one another wander constantly, and holding them still takes active stabilisation of every mirror. It would seem that the lattice must wander with them — not merely move, but distort, its sites brightening and dimming, its geometry changing from moment to moment. For some numbers of beams that is exactly what happens. For others it cannot happen at all, and the dividing line is a piece of arithmetic that decided how the first optical lattices were built.

Two beams can only slide #

The simplest case is a standing wave: one beam and its reflection, travelling in opposite directions along a line. Their sum is bright where they arrive in step and dark where they arrive opposed, and the alternation repeats every half wavelength — the same arrangement that fits only some notes on a string. If the mirror moves, the reflected beam’s phase changes and the pattern moves with it. The first figure shows three phases: the curves are identical in shape and differ only in position. No choice of phase can make one peak brighter than the next or bring the zeros up from zero, because the only thing the phase does is decide where along the line the two beams are in step.

That is why the fringes of an interferometer can be walked along the screen by moving one mirror, and why their shape stays the same however long they are watched. It is so familiar that it does not look like something that needs explaining, but it is a special property of two beams in one dimension, and it fails the moment a third beam is added along the same line.

With three beams along a line the phases change the shape. The three curves in the second figure are not one curve moved; the heights of their peaks differ, and the pattern within each two-wavelength repeat is rearranged. Nothing in the setup is exotic — three waves at chosen wavenumbers along a line — and yet a phase shift that two beams would have absorbed as a slide now changes what the light looks like.

Why the count decides #

The reason is clearest in what the pattern is made of. The intensity is the square of the sum of the beams’ fields, and squaring a sum of N waves gives, besides the N individual intensities, one cross term for every pair. The cross term for beams $i$ and $j$ is a fringe pattern: a sinusoid in space whose direction and spacing are set by the difference of the two wavevectors, $\mathbf k_i - \mathbf k_j$, and whose position is set by the difference of the two phases, $\phi_i - \phi_j$. The whole pattern is the sum of these fringes. Its shape is decided by how the fringes sit relative to one another, which is decided by the phases.

Now slide the whole pattern by a displacement $\mathbf a$. Each beam’s field at the new position is its field at the old one with an extra phase $\mathbf k_i \cdot \mathbf a$. So a slide is the same thing as adding $\mathbf k_i \cdot \mathbf a$ to every beam’s phase. Adding the same amount to every phase does nothing at all, because only differences enter the cross terms. So of the N phases there are, one is irrelevant from the start and $d$ more — one for each direction the displacement can point in — can be undone by sliding the pattern. What remains is

$$N - 1 - d$$

combinations of phases that no slide can undo. Those, and only those, change the pattern’s shape.

For two beams in one dimension this is $2 - 1 - 1 = 0$: nothing but slides. For three beams in one dimension it is 1, and the second figure’s changes in shape are that one combination. The count needs the beams’ wavevectors to be spread out enough that sliding in each direction actually changes the phases differently — for three beams in a plane, not all along one line — which is the generic case.

For three beams in a plane the slide that undoes a change of phases can be written down. Suppose the second beam’s phase rises by $\alpha$ and the third’s by $\beta$, relative to the first. The pattern shifted by $\mathbf a$ has the second beam’s phase changed by $(\mathbf k_2 - \mathbf k_1)\cdot\mathbf a$ relative to the first and the third’s by $(\mathbf k_3 - \mathbf k_1)\cdot\mathbf a$. Those are two linear equations in the two components of $\mathbf a$, and because the two difference vectors point in different directions they always have exactly one solution. So for any $\alpha$ and $\beta$ there is one displacement that makes the phase-shifted pattern identical to the original, and that is the whole proof that three beams in a plane cannot change shape. With a fourth beam there would be three equations in the same two unknowns, and in general no solution.

The difference vectors also say what the lattice is. Every fringe in the pattern has a wavevector that is a difference of two beams’ wavevectors, and the pattern is periodic exactly when all those differences are whole-number combinations of $d$ basic ones — the reciprocal lattice of the pattern. With $d + 1$ beams there are exactly $d$ independent differences, so the pattern is always periodic, and its cell is fixed by the beam directions alone.

The same counting appears wherever a pattern is a sum of waves whose origin is arbitrary. Crystallographers meet it as the question of which combinations of the phases of diffracted beams are independent of where the origin of the unit cell is placed, and answer it with the same subtraction. The difference here is that the phases are not unknowns to be recovered from a measurement but disturbances to be survived.

Three beams in a plane #

The case that matters for atoms is three beams in a plane, arranged at 120° to one another.

With $N = 3$ and $d = 2$ the count is zero. Whatever the three phases are, the pattern is one triangular lattice of bright spots, spaced two-thirds of a wavelength apart, with nine times one beam’s intensity at every site. The three panels were drawn with arbitrary phases and then searched for the shift that best maps each onto the first: the best shift reproduces the pattern to within 2 per cent of the peak, and the 2 per cent is the coarseness of the search, not a real difference.

This is the design principle behind the first optical lattices in more than one dimension, published in 1993 by Gilbert Grynberg and colleagues in Paris: use $d + 1$ beams in $d$ dimensions — three in a plane, four in space — and the lattice’s geometry becomes independent of the beams’ phases. A trembling mirror then only moves the lattice. A slow drift moves the atoms with it, which does them no harm; a fast shake heats them, which is a much smaller problem than a lattice whose sites deepen and vanish. The phases never have to be controlled. The atoms that sub-Doppler cooling brought to microkelvin temperatures could be trapped in a crystal whose geometry was guaranteed by arithmetic rather than by vibration isolation.

The same principle builds solid structures. Four non-coplanar beams exposing a light-sensitive resin record their three-dimensional pattern in it; washing away the unexposed resin leaves a porous crystal whose period is set by the wavelength and the beam angles. Because four beams in three dimensions leave no shape-changing combination, the structure’s geometry is immune to the phase jitter during a long exposure, and a drift in the mirrors only shifts it. Such interference lithography was used from 2000 to make the periodic dielectric structures that reflect light from every direction at a chosen colour.

Four beams in a plane #

One beam more and the count is one.

Four beams in a plane at right angles form two counter-propagating pairs. The combination that no slide can absorb is $\phi_1 - \phi_2 + \phi_3 - \phi_4$: the sum of each pair’s phases, one pair’s against the other’s. It measures how the two standing waves the pairs make are timed relative to one another. With the combination at zero, their bright planes coincide at the crossings and add to spots sixteen times one beam’s intensity. At half a turn one standing wave’s field is largest where the other’s is zero, the two no longer add at any point, and the pattern becomes a finer one with half the period along each axis and the peak reduced to half.

Nothing in the geometry distinguishes the two cases. Only the phase combination does, and a mirror moved by a quarter of a wavelength changes it by half a turn. Experiments that built square lattices with four beams in the early 1990s therefore had to measure that combination continuously and hold it with a piezoelectric mirror in a feedback loop — and in return had a knob that changed the lattice’s geometry, which the three-beam lattices could not offer.

How far a mirror has to move #

The size of the disturbance is worth putting numbers to, because it is what makes the count matter in practice. A beam’s phase at the atoms changes by a full turn whenever the path it travels changes by one wavelength. Lattices for atoms are commonly made with infrared light of about a micrometre, so a mirror that moves by a quarter of a micrometre along a beam changes that beam’s phase by a quarter of a turn, and a mirror in a retro-reflected beam, whose path it lengthens twice, by half a turn.

Mechanical vibration on a good optical table moves mounts by tens of nanometres, which is a few hundredths of a turn. Temperature is worse. An aluminium mount ten centimetres long grows by 2.3 micrometres for every kelvin it warms, so an unregulated laboratory whose air drifts by a degree in an hour turns each beam’s phase through a couple of full turns in that hour, independently for each beam.

For a lattice of three beams in a plane, that drift is a slide. The whole triangular pattern wanders by a couple of lattice spacings per kelvin, slowly enough that the atoms in it follow their sites like marbles in a moving egg box, and the experiment notices only if it is imaging the atoms against a fixed reference. For four beams in a plane, the same drift carries the leftover combination through a couple of turns per kelvin, and the pattern cycles through every shape it has — spots, bars, the finer pattern, and back — a few times an hour. Nothing can be done with atoms in a lattice that does that. Hence the feedback loop, and hence the preference, whenever the geometry allows it, for the number of beams that needs none.

The four-beam figure also shows why the answer is not simply “more beams, more interference, more trouble”. The counting says there is exactly one combination to control, not three. A slide absorbs two of the three relative phases no matter how many beams there are, and what an experimenter faces is the remainder: exactly one feedback loop for four beams in a plane, two for five, and no more.

The count, and five beams #

The whole rule fits in a table.

The table’s highlighted rows are the lattices that need no phase control: two beams in one dimension, three in two, four in three. They are the smallest numbers of beams that make a pattern periodic in every direction of the space, which is why they are the arrangement of choice whenever the phases cannot be held. Every beam beyond the minimum adds one combination of phases that must be either controlled or tolerated.

Controlling it is one route, and avoiding it is another. The standard cubic lattice for atoms uses three pairs of counter-propagating beams, six in all, which by the count would leave two combinations free. It does not, because the three pairs are given frequencies differing by tens of megahertz or perpendicular polarisations. Cross terms between beams of different frequencies oscillate at the difference frequency and average to zero over any time an atom can respond to, and cross terms between perpendicular polarisations vanish outright. What remains is three independent standing waves, one along each axis, each of which is the two-beam case of the first figure. Their phases only slide each set of planes along its own axis, and the cubic lattice is safe.

With five beams in a plane the count is two, and the pattern has two ways to change shape.

Five beams at 72° make a pattern that is not a lattice at all. The wavevector differences point in directions related by fivefold rotation, and no periodic pattern in a plane can have fivefold symmetry, so the spots are ordered without ever repeating: a quasiperiodic pattern. Its two free phase combinations rearrange the spots locally without moving the pattern as a whole. Optical patterns of this kind have been used since 1997 to trap cold atoms in order without periodicity. For a quasiperiodic pattern the two shape-changing combinations turn out to be the pattern’s other kind of motion: shifts in directions the plane does not contain. The count is the same subtraction, but what the leftover combinations mean is richer, and it belongs to the mathematics of ordered non-repeating structures rather than to superposition.

A count that assumes too much #

The subtraction treats each beam as a perfect plane wave, and real beams are not.

Beams have finite width and curved fronts. A focused beam’s phase varies across it, and near a focus the fronts curve, so the relative phases of the beams differ from one part of the overlap to another. A three-beam lattice stays triangular everywhere, because the count is local, but its sites are displaced slightly from a perfect lattice across the region, and at the edges of the overlap the pattern fades with the beams.

Amplitudes and polarisations matter as much as phases. The count says which changes of phase alter the shape. A change in one beam’s intensity or polarisation alters the cross terms’ sizes rather than their positions, and that changes the shape whatever the number of beams: a three-beam lattice whose beams have unequal powers still has sites on a triangular lattice, but its sites are shallower and its barriers between them lower. Holding the intensities steady is a separate problem the count does not address.

Light is a vector. The fringes of two beams interfere fully only if their polarisations are parallel. Beams crossing at 120° in a plane can be polarised perpendicular to the plane, and then every pair interferes completely; otherwise the pattern has polarisation that varies across the lattice as well as intensity, and atoms with more than one ground state feel the polarisation too. Some of the most useful lattices exploit exactly that, which is another way of adding structure the scalar count does not see.

Still open: how much of a lattice can be set by light alone #

A lattice made by $d + 1$ beams has its geometry fixed by the wavevectors, so every change of lattice means changing beam directions. With extra beams whose phases are held, the geometry becomes a control parameter: switching lattices mid-experiment, or making lattices with two sites per cell whose relative depths are set by a phase. Experiments have used this to make lattices shaped like graphene and like the kagome net and to switch between them, and to make lattices whose phases are modulated in time so that atoms tunnelling between sites behave as though a magnetic field were applied to them. How far these designs can be pushed — how many independent controls a lattice of light can carry before the phase noise on each ruins what the others do — is limited by exactly the counting drawn here, one feedback loop per leftover combination, and is being explored now.

The habit worth carrying away is to count what a symmetry absorbs before worrying about what is uncontrolled. A disturbance that the system’s own symmetries can undo is not a disturbance to its shape, and the number of those that remain — here, the phases minus one minus the dimensions — is the number of things that genuinely have to be held still.