The light that makes a cold cloud repel itself

The friction made of light built a cooling force out of two opposed laser beams tuned a little below an atom’s resonance: an atom moving towards one beam sees it Doppler-shifted closer to resonance and scatters more of its photons, so it is pushed back against its motion. Add a weak magnetic field that grows in every direction from a central point — produced by the pair of coils carrying opposite currents that the two coils one radius apart turned into a uniform field by running them the same way — and the same mechanism pushes an atom back towards the centre wherever it strays. The result, the magneto-optical trap, first demonstrated in 1987, holds a cloud of millions of atoms at a tenth of a millikelvin, and it is the starting point of nearly every experiment with cold atoms.

The atoms the light stops touching ended with a worry that had stood in the way of cooling dense gases by light: a photon scattered by one atom can be absorbed by another. In a cloud of a hundred atoms that is a curiosity. In a cloud of a hundred million it decides the cloud’s shape, and it does so in a way that has an exact and unexpected counterpart elsewhere in physics.

A cloud that stops getting denser #

The first figure shows what is observed. A few thousand atoms sit in the trap as a small Gaussian ball whose size is set by their temperature and the stiffness of the trap — a thermal cloud, 78 micrometres in radius at a hundred microkelvin in a trap whose spring makes an atom oscillate two hundred times a second. Add atoms and the ball does not keep its size while growing denser, as a thermal cloud would. Its central density rises to a ceiling, here 2.5 × 10¹⁰ atoms per cubic centimetre, and stays there. The extra atoms go into a growing shell outside, so that a cloud of a million atoms is a sphere of uniform density 0.21 millimetres in radius and a cloud of a hundred million is a sphere of the same density nearly a millimetre in radius. Each ends in a soft edge of the same width.

Thomas Walker, David Sesko and Carl Wieman measured exactly this in 1990, in clouds of caesium, and reported that the density was independent of the number of atoms over a wide range while the volume was proportional to it. A trap that holds atoms more strongly does give a higher ceiling, and a colder gas does not.

The pull of a shadow and the push of a neighbour #

Two forces between the atoms come from the light, and both were identified before the ceiling was measured.

The first is a shadow. Each atom scatters light out of the beams, so an atom sitting behind another in a beam receives slightly less light from that direction than it would alone. The beam from the opposite side is undimmed, so the net push on the shaded atom is towards the one shading it. Jean Dalibard pointed out in 1988 that this makes the beams squeeze a cloud: atoms near the edge, shaded from inside, are pushed inwards. The second force is the one the atoms the light stops touching worried about. Each atom scatters photons in all directions, and a photon from one atom absorbed by another delivers its momentum along the line between them, away from the first.

Both forces fall as the inverse square of the distance between the atoms, because both are carried by light spreading from a point. Their strengths differ by one factor. The shadow’s pull is set by how strongly an atom absorbs the laser, its cross-section $\sigma_L$, appearing twice: once for the light the first atom removes, once for the second atom’s absorption of what is left. The push is set by $\sigma_L$ for the light the first atom scatters and by $\sigma_R$ for the second atom’s absorption of that scattered light. If the two cross-sections were equal, push and pull would cancel. They are not, because the light an atom rescatters is not a copy of the laser. The laser is tuned below resonance on purpose, while part of the scattered light is shifted to the atom’s own resonance, where a neighbour absorbs it more readily, so that $\sigma_R$ is larger than $\sigma_L$. In the figures the ratio is taken to be 1.3, an assumption of the size the measurements suggest. The net force is then an inverse-square repulsion three-tenths as strong as either part.

It is a very small force by everyday standards and a very large one by the standards of neutral atoms. Two rubidium atoms ten micrometres apart in this trap push each other apart with a force 1.6 × 10²⁴ times their gravitational attraction. Matching it to Coulomb’s law gives the charge each atom would need to carry to repel its neighbour that hard: about a ten-thousandth of an electron’s charge.

A plasma of neutral atoms #

An inverse-square repulsion among particles held by a harmonic well is a familiar problem. It is the problem of electric charges in a uniform background, and it has a known answer. Gauss’s law turns an inverse-square force into a statement about divergence, so the repulsion from a region of density n spreads outward at a rate proportional to n. The trap’s restoring force, growing linearly with distance, has a divergence that is the same everywhere. Where the two balance the density is fixed:

$$n_{\max} = \frac{3\kappa c}{(\sigma_R - \sigma_L)\,\sigma_L\,I},$$

with $\kappa$ the trap’s spring constant and $I$ the total intensity of the beams. The density does not depend on the number of atoms, so a cloud with more atoms must be bigger, and since its density is uniform its radius grows as the cube root of the number. The same calculation gives the uniform density of ions in a Penning trap and of electrons in the plasmas held in such traps, and it describes them with the same equations.

The figures do not assume this answer. They compute the cloud by requiring that the atoms be distributed as the trap and their temperature demand, each weighted by the Boltzmann factor of the trap’s potential plus the mean repulsion of all the others, and they find the self-consistent profile numerically for each number of atoms. A Gaussian comes out at small numbers and a flat-topped sphere at large ones. The plateau sits at the predicted density to within two per cent and the edge at the predicted radius.

The edge has a width, and that width is the plasma’s Debye length. The long-range force that does not reach found the same length in a gas of real charges: the distance over which mobile charges rearrange to cancel a disturbance, set by the balance between temperature and the strength of the interaction. Here it is about 45 micrometres, and it is why the profiles in the first figure all end in the same soft shoulder rather than a cliff. A cold-atom cloud of this kind is a weakly coupled plasma: the repulsion between neighbours at their typical spacing of two micrometres is less than a thousandth of their thermal energy, so the atoms behave as a fluid rather than a crystal, just as the electrons in a laboratory plasma do.

The ceiling that does not move #

The third figure follows the central density as atoms are added. For small clouds it rises in proportion to the number, exactly as a thermal cloud’s would. Around ten thousand atoms the two curves part. The thermal estimate goes on rising and would put a hundred million atoms at 541 times the ceiling. The computed peak bends over and stays flat. For experiments this was the important finding: the ceiling, not the temperature, set the density of the gas that later cooling had to start from, and it was low. A density of 10¹⁰ per cubic centimetre at a hundred microkelvin is a phase-space density several million times too small for Bose–Einstein condensation. The condensate a trap makes needed that factor, and the first condensates got it by switching the light off and cooling by evaporation in a magnetic trap, where no rescattered light pushed back.

The arithmetic also shows why the clever traps of the 1990s worked. The “dark spot” trap of Wolfgang Ketterle and colleagues, in 1993, blocked the repumping light at the centre of the cloud so that atoms there fell into a state that does not scatter the trapping light. Atoms that do not scatter neither push their neighbours nor are pushed. The density in the dark region rose by more than an order of magnitude above the ceiling, because the formula’s $\sigma_L$ had in effect been set to nearly zero where it mattered.

Why a colder cloud is not a denser one #

The formula for the ceiling has no temperature in it, and that absence is the most counter-intuitive thing about a large trap. For a thermal cloud, cooling is compression: the cloud’s radius is $\sqrt{kT/\kappa}$, so halving the temperature shrinks it by a factor of $\sqrt2$ and nearly triples its density. Sub-Doppler cooling, discovered in 1988, brought magneto-optical traps from a few hundred microkelvin to tens, and for small clouds it raised the density as expected. For large clouds it changed almost nothing, and the plasma picture says why. The density in the plateau is fixed by a balance between two forces that are both independent of how fast the atoms move. Temperature enters only at the edge, where thermal motion lets atoms spill a little way past the radius at which the repulsion and the spring balance — and the Debye length, which measures that spill, grows as the square root of the temperature.

Temperature also decides where the change of regime happens. A cloud is thermal while its thermal radius is larger than the radius a plateau of that number of atoms would need, and a plasma once the plateau’s radius is the larger. The two are equal when the number of atoms is $\tfrac{4\pi}{3}\,n_{\max}\,(kT/\kappa)^{3/2}$: about fifty thousand atoms for the trap in the figures, at a hundred microkelvin. That is why the curves of the third figure part company between ten thousand and a hundred thousand atoms. A colder cloud makes the switch at fewer atoms, because its thermal radius is smaller, and then sits at the same ceiling as a warmer one. Experiments that tracked a trap’s density while varying its number found these two regimes, a temperature-limited one and a multiple-scattering one, and the number at which one hands over to the other.

The practical lesson was a reordering of priorities. To raise the density of a large cloud there is no point in cooling it further. What helps is a stiffer trap, a larger κ, or less light where the atoms are, a smaller $\sigma_L$. Every successful technique for loading dense samples from a magneto-optical trap — compressing it briefly with a stronger field gradient, detuning the beams further for a few milliseconds, darkening its centre — is one of those two moves.

What sets the ceiling #

The fourth figure isolates the factor that makes the effect possible: the difference between how strongly rescattered light and laser light are absorbed. At a ratio of 1.1 the ceiling is three times higher than at 1.3; at 2 it is three times lower. As the ratio falls towards 1 the ceiling rises without bound, and the reason is plain from the forces. With equal cross-sections, every photon of rescattered push is matched by a photon of shadow pull, and the atoms do not interact through the light at all.

Below 1 the sign would reverse. The shadow would win, and the cloud would feel an attraction falling as the inverse square of distance: a gas of particles that gravitate. Dalibard’s original analysis considered exactly this case and noted that it would make a cloud unstable to collapse, as a self-gravitating gas is. The disturbance that grows instead of travelling found that instability in clouds of gas between stars, where a region larger than a critical length collapses because its own gravity beats its pressure. Real magneto-optical traps sit on the other side of the line, because the rescattered spectrum is always partly on resonance, and it is the repulsive version, the plasma, that is observed. Theorists have proposed arrangements of laser beams that would give cold atoms an inverse-square attraction instead. A cold cloud that held itself together by an inverse-square force would be a laboratory model of a star.

The analogy with stars runs the other way too. The brightness a mass cannot exceed set the outward push of a star’s own radiation against its gravity, and both of those also fall as the inverse square, so their ratio is the same at every distance and a single luminosity decides which wins. In the trap the same structure appears with the roles exchanged. The binding comes from a spring, the repulsion from scattered light, and the outcome is a ceiling on density rather than on brightness.

Where the argument stops #

The plasma picture rests on each photon being rescattered at most once on its way out, so that the force between atoms is a sum over pairs. That requires the cloud to be optically thin: a beam crossing it must be only slightly dimmed. The last figure measures the cloud against that requirement. Once the density is capped, the optical depth through the centre grows only as the radius, and for this trap it reaches one at about ninety million atoms. Beyond that a typical photon is scattered several times before it escapes, the beams reaching the centre are weaker than those at the edge, and the forces are no longer pairwise. The cloud then does things the plasma model does not describe. In large traps it has been seen to break into rings and orbiting clumps, driven by beams that no longer balance once they have been dimmed by the cloud itself, and to oscillate unstably. The light now diffuses out through the cloud in a slow random walk, the multiple scattering that the light that takes a hundred thousand years to leave traced through the Sun on a vastly larger scale.

The model has smaller simplifications too. It treats $\sigma_R$ as a single number, where the rescattered spectrum has a central peak and side bands and its absorption depends on the local intensity. It treats the trap as a perfect spring, which a real magneto-optical trap is only near its centre. And it takes the temperature to be uniform, where atoms in a dense cloud are heated by the extra kicks of reabsorbed photons, so that the temperature itself rises with density — a second effect working in the same direction as the repulsion.

What the pictures cannot show #

None of the figures shows a photon. The forces are averages over very many scatterings, and the inverse-square law holds for the average push of one atom’s light on another, not for any single exchange. Each exchange is a random kick that heats the cloud as well as pushing it, and the heating appears in the figures only through the temperature put in by hand. The figures also leave the shadow force with the same inverse-square form as the push, which holds for its divergence — the quantity that fixes the density — but not for its direction: the shadow acts along the beams rather than along the line between the atoms. The ceiling depends only on the divergence, which is why the simplification is harmless here, but a calculation of the cloud’s shape in a trap with unequal beams would need the full form.

Still open: how dense a cloud light alone can hold #

The ceiling of 10¹⁰ to 10¹¹ atoms per cubic centimetre has been raised by darkening the centre, by compressing briefly while the light is weak, and by using narrow transitions whose scattered light is far less likely to be reabsorbed. Each attack reduces $\sigma_L$ where the atoms are densest, which the formula says is the way. Light alone has now cooled gases to condensation in shaped traps where reabsorbed photons mostly escape. What remains unsettled is how far a gas can be both dense and continuously illuminated: in a dense cloud the light-induced interactions are not only the pairwise forces drawn here but collective shifts of the atoms’ resonance and superradiant or subradiant scattering, which alter $\sigma_R$ and $\sigma_L$ themselves as the density rises. Whether there is a fundamental ceiling on the density a continuously lit gas can reach, or only a ceiling for each scheme of illumination, is not known.

The habit worth carrying away is to recognise a law by its form. Light rescattered between trapped atoms pushes them apart with an inverse-square force — 1.6 × 10²⁴ times their gravity, as if each carried 1.0 × 10⁻⁴ of an electron’s charge — so a magneto-optical trap fills like a plasma in a harmonic well, at one density, $3\kappa c/[(\sigma_R - \sigma_L)\sigma_L I]$ = 2.5 × 10¹⁰ per cm³ here, out to a radius that grows as the cube root of the number, until at about 9 × 10⁷ atoms the cloud is too opaque for the argument to hold. Any force that falls as the inverse square, whatever carries it, fills a harmonic well the same way.