The atoms the light stops touching

The limit that belonged to a simpler atom followed laser cooling below its first floor, the Doppler limit, by letting atoms climb hills of light that kept moving under them. It ended at a second floor that no scheme of that kind could pass: the kick of a single photon. An atom that absorbs and re-emits light is pushed by the momentum $\hbar k$ of each photon it absorbs and, in a random direction, of each photon it emits. Cooling by scattering means using those kicks to slow the atom down, and a process made of kicks of size $\hbar k$ cannot leave the atom’s momentum known to better than about $\hbar k$.

That floor is real for the mechanism that sets it — the pressure of light that light has a pressure measured on a vane, applied one photon at a time to one atom — and there is a way past it that does not fight it at all. The coldest atoms are not pushed any harder. They are left alone.

The size of one photon’s kick #

For an atom of mass $m$ the kick of one photon of wavelength $\lambda$ changes its velocity by the recoil velocity $\hbar k/m$, with $k = 2\pi/\lambda$. A photon with a momentum found that a photon carries momentum $h/\lambda$ whether or not anything is moving, and for an atom the kick is small but not negligible: three and a half millimetres a second for caesium at 852 nanometres, three centimetres a second for sodium, nine centimetres a second for metastable helium at 1083 nanometres. The corresponding temperature, $T_r = \hbar^2k^2/mk_B$, is the temperature at which a single kick is as large as the whole spread of velocities in the gas: 198 nanokelvin for caesium, 2.4 microkelvin for sodium, about 4 microkelvin for helium.

The reason a scattering gas cannot be colder than this is a statement about random walks. The jiggle that proved atoms followed a particle kicked at random in position; here the kicks are in momentum, and each one moves the atom’s momentum by a step of size $\hbar k$ in a random direction. A friction that pulls the momentum back towards zero can keep the walk from spreading, but it cannot make the spread smaller than one step, because the friction itself is made of the same scattering. The friction made of light built its friction from the Doppler shift, which tells a moving atom from a still one by how far the light’s frequency appears to move; to tell an atom moving at a fraction of the recoil velocity from one at rest, it has only photons to do it with, each of which changes the velocity it is trying to measure by the full recoil. So the floor is not a matter of design. It is set by asking light to cool and to measure at once with the same photons.

A rate that vanishes at rest #

The way out was found by asking the light to measure the velocity and do nothing else. Suppose the scattering rate, rather than the force, depends on velocity — and vanishes exactly at zero. An atom moving at any appreciable speed scatters, and each scattering throws it to a new random momentum. An atom that lands near zero stops scattering, and stays.

The first scheme of this kind uses an atom with two ground sublevels and one excited level, illuminated by two counter-propagating beams of opposite circular polarisation, each of which can carry the atom from one ground sublevel to the excited state. There is then a particular superposition of the two ground sublevels — one with momentum $-\hbar k$ and the other with $+\hbar k$ — for which the two routes to the excited state cancel exactly, so that the atom in it cannot absorb either beam. The cancellation requires the two components to stay in step, and they do only if the superposition as a whole has zero momentum; at a small momentum $p$ the two components drift out of phase at a rate proportional to $p$, and the leak out of the dark state goes as the square of that, as $p^2$. Alain Aspect, Claude Cohen-Tannoudji and collaborators demonstrated it with metastable helium in 1988 and called it velocity-selective coherent population trapping, which says precisely what it does.

The second scheme does the selection with pulses. Two beams of slightly different frequency drive a Raman transition between two ground states, and the Doppler shift makes the transition resonant for one velocity class only. A sequence of pulses tuned to address every velocity class except those near zero pushes atoms towards zero, and an optical pumping step brings them back to the first ground state with a random kick. How sharply a pulse can pick out a velocity is set by its duration, by the same trade between time and frequency that makes a width a lifetime: a pulse lasting a millisecond addresses a band of Doppler shifts about a kilohertz wide, and a longer pulse a narrower one. Pulses with smooth envelopes — the Blackman shape is the standard choice — have frequency spectra with almost nothing in their wings, and the residual excitation of atoms near zero can be made to grow as the fourth power of the velocity rather than the second. Mark Kasevich and Steven Chu demonstrated Raman cooling below the recoil with sodium in 1992.

Waiting with no typical wait #

An atom that lands at momentum $p$ waits a time $1/R(p)$ before scattering again. For the dark state that is proportional to $1/p^2$, and it grows without limit as $p$ approaches zero. The landing is random, so the wait is random too, and its distribution is what decides everything else.

If landings are spread evenly over momentum, the chance of landing within $p$ of zero is proportional to $p$, and the wait after such a landing is longer than $1/p^2$. So the chance that a given wait exceeds a time $\tau$ falls only as $\tau^{-1/2}$. That is a heavy tail, and its consequence is that the average wait is infinite: every long record contains a wait comparable to the record itself. The diffusion that slows the longer it is watched met exactly this distribution in a molecule that keeps getting stuck, and found that sums of such waits obey not the ordinary central limit theorem but Lévy’s generalisation of it, in which the sum is dominated by its largest term. The record in the figure is typical, not unlucky: a few dozen quick jumps, and then one landing close to zero that holds the atom for half of all the time there is.

The same statistics decide the fate of the whole gas. At any moment, the atoms are divided between those still jumping and those caught in a long wait near zero, and because the waits have no average, the second group never stops growing. There is no rate of escape that balances a rate of arrival.

A distribution that never settles #

The usual end of a cooling process is a steady state — a temperature, at which heating by the random kicks balances cooling by the friction, and a distribution of velocities that has stopped changing. A dark-state gas has no such end.

The peak at zero momentum grows while the light stays on, and narrows as it grows. The reason is in the waits: after the light has been on for a time $\theta$, the atoms still sitting in the peak are those that landed close enough to zero that their wait outlasts $\theta$ — those with $R(p)\,\theta$ less than about one. For a $p^2$ rate that means $|p| \lesssim \hbar k/\sqrt{R_0\theta}$, a width that shrinks without limit. Atoms further out have had time to leak away, and they keep landing at random until they too find a place close enough to zero.

Calling the result cold needs care. The momentum distribution is not a Maxwell–Boltzmann distribution; it is a narrow peak on a broad base, with wings that fall as $1/p^2$ rather than as a Gaussian. It is not stationary, and it does not forget how long the light has been on — a gas that has not lost the memory of its own preparation, which is what ergodicity would have taken from it. The energy that refuses to be shared met a system that kept the memory of where it started because its motion never explored the states it was allowed; here the memory is kept because the exploring never finishes, the longest wait always being comparable to the whole time. What it has is a width, and the width can be translated into an effective temperature by asking what temperature would give a Gaussian of the same width. By that reckoning the peak falls below the recoil temperature almost at once and keeps going.

A width with no floor #

The narrowing follows a power law set by how the rate vanishes. With $R(p) \propto p^\alpha$ near zero, the atoms left in the peak after a time $\theta$ are those with $p^\alpha\theta$ below about one, so the width shrinks as $\theta^{-1/\alpha}$: as the inverse square root of the time for a dark state, as the inverse fourth root for a $p^4$ filter.

Neither line has a floor. After ten thousand scattering times the dark-state peak is a hundredth of a photon’s momentum wide, and squaring that gives an effective temperature ten thousand times below the recoil temperature. Experiments have not gone that far, for reasons the model leaves out — the dark state is never perfectly dark, off-resonant light leaks into it, and the time the light can be left on is limited by the atoms falling out of the beams — but one-dimensional dark-state cooling of helium reached effective temperatures hundreds of times below the recoil. The $p^4$ filter narrows more slowly. That looks like a defect, and in one dimension it is. In three dimensions it is the reverse.

A target that three dimensions make small #

In one dimension, an atom landing at random over a range of momentum $2\hbar k$ wide has a chance of a tenth of a percent of landing within $0.002\,\hbar k$ of zero. In three dimensions the region within $0.002\,\hbar k$ of zero is a small ball inside a large one, and the chance of landing in it is a millionth of the one-dimensional chance. The long waits are just as long, but they are found far less often, and whether they are found often enough decides whether the scheme works at all.

The arithmetic is short. The chance that an atom lands within $p$ of zero is proportional to $p^D$ in $D$ dimensions, and the wait it then has is $1/p^\alpha$, so the chance that a wait exceeds $\tau$ falls as $\tau^{-D/\alpha}$. The exponent $\mu = D/\alpha$ decides the gas’s fate. If $\mu$ is less than one, the waits have no average, the trapped atoms accumulate without limit, and eventually almost all of them are in the peak. If $\mu$ is greater than one, the waits have a finite average after all, a steady state exists, and the fraction of atoms in the peak settles at a value that is small when the peak is narrow. A dark state in one dimension has $\mu = 1/2$ and works beautifully. The same dark state in three dimensions has $\mu = 3/2$: it still produces a sub-recoil peak, narrowing as before, but the peak holds only a couple of per cent of the atoms within a tenth of $\hbar k$ and never more. A $p^4$ filter in three dimensions has $\mu = 3/4$, below one, and gathers atoms without limit again, only slowly.

This is the reason three-dimensional sub-recoil cooling by dark states took seven years after the one-dimensional demonstration, and why the schemes that work best in three dimensions combine a velocity filter with a friction. A friction that pushes the jumping atoms back towards zero after each scattering — the polarisation-gradient cooling of the previous floor, operating outside the dark region — concentrates the landings near zero instead of spreading them evenly, and raises the chance of finding the long waits. The two mechanisms divide the work: the friction brings atoms to within a recoil of rest, and the filter holds the ones that arrive within a fraction of it.

How cold a gas the light keeps touching can be #

Sub-recoil cooling of this kind keeps the light on and keeps scattering photons from every atom outside the peak. In a dilute gas that is harmless. In a dense one it is not, because a photon scattered by one atom can be absorbed by another, giving it a random kick that the filter did not choose, and it was long believed that this reabsorption would stop light from ever cooling a gas as dense and cold as a condensate. The condensate a trap makes found Bose–Einstein condensation at a phase-space density of order one, and for the first condensates that was reached only by switching the light off and cooling by evaporation, which the limit that belonged to a simpler atom described as the second escape from the recoil floor.

Two groups closed most of that gap in the last decade. Florian Schreck’s group in Innsbruck in 2013 cooled strontium to degeneracy using laser cooling on a transition so narrow that its own recoil is larger than its linewidth, keeping the scattering light away from a small dimple in the trap where the condensate formed. Vladan Vuletić’s group at MIT in 2019 used Raman cooling in an elongated optical trap to reach a condensate of rubidium without any evaporation, keeping the scattering rate low and the trap’s geometry such that reabsorbed photons mostly escaped. Both worked by the principle of this page: light that stops touching the atoms it has finished with. The sequence is the one the staircase that never reaches the floor found in magnetic cooling: each method hands its atoms to the next at the floor of its own mechanism, and the last step is never the first one pushed harder.

What the drawings leave out #

The model behind every figure is the simplest one with the right statistics. Each scattering is taken to throw the atom to a momentum chosen evenly over a range of $2\hbar k$, as if recycling were instant and featureless; real recycling has a shape, and it is that shape the friction in the three-dimensional schemes improves. The rate is taken to vanish exactly as $p^2$ or $p^4$; in a real dark state, off-resonant excitation through other levels leaves a small residual rate at zero, which puts a floor under the width after all, and the time the atoms can be illuminated is limited by gravity and the size of the beams. The atoms are independent, with no reabsorption and no collisions.

There is also a quantum feature the figures hide in a choice of variable. In the dark-state scheme the trapped atom is a superposition of two momentum states, $-\hbar k$ and $+\hbar k$, and the momentum in the figures is the momentum of that pair as a whole. A measurement of the cold atoms’ velocities — by letting them fly apart and imaging the cloud — shows not one narrow peak at zero but two, at plus and minus the recoil velocity, each as narrow as the figures’ peak. The coherence between them was itself measured, by interfering the two peaks, as a demonstration that the atoms in the dark state are spread coherently over a distance set by the inverse of the peak’s width — a wavepacket of each atom, many micrometres long.

The domain of the drawings is the one-dimensional and three-dimensional dilute gas, with an ideal velocity filter and random recycling, over ten thousand scattering times. Within it, the statistics are exact consequences of the waiting-time distribution, and the slopes measured from the simulated atoms match the power laws to the precision of the run.

Still open: how dense a gas light alone can cool #

The condensates made by laser cooling alone are small — tens of thousands of atoms at most — and both depended on traps arranged so that the scattered photons left the dense region before they could be reabsorbed. Whether laser cooling can be extended to large, dense samples, to molecules whose many internal levels make dark states leak, or to atoms whose transitions are too broad for a narrow-line trick, is being worked out experiment by experiment. The underlying question is how sharp a velocity filter can be while photons keep arriving at the atoms outside it, and in a dense cloud the answer depends on how the photons the filter discards are prevented from finding atoms it has kept.

The principle underneath is unusual for a cooling method. An atom kicked by ħk at every scattering cannot be held to better than ħk while it scatters; let the scattering switch off near zero momentum, as $p^2$ or $p^4$, and the coldest atoms are simply left alone for waits with no average, so the gas forms a peak that never settles and narrows as the time to the −1/2 or −1/4 power — in three dimensions only if the filter is sharp enough that D/α stays below one. The record low temperatures of laser cooling were reached not by pushing the slowest atoms harder, but by finding a way to stop pushing them at all.