The limit that belonged to a simpler atom

In the summer of 1988 a group at the National Bureau of Standards in Maryland measured the temperature of a cloud of sodium atoms held in optical molasses, and did it carefully. They switched the beams off, let the atoms fall and spread, and timed their arrival at a probe beam below: a hot cloud spreads fast and arrives smeared, a cold one arrives in a sharp pulse. The answer was 43 microkelvin, give or take 20.

The theory of the friction made of light put a floor under that number at ħΓ/2kB, which for sodium is 240 microkelvin. A measurement six times below a theoretical minimum is not a small discrepancy to be absorbed into the error bars. Either the thermometer was wrong, or the minimum was not a minimum.

The thermometer was checked, repeatedly, by several groups with different methods, and it was right. What made the result impossible to dismiss was a second observation made at the same time: the atoms got colder as the lasers were tuned further from resonance and as their intensity was turned down. The Doppler theory predicts the opposite of both.

The trend that gave it away #

A floor that is lower than predicted could be a correction. A trend that runs backwards cannot be. The Doppler temperature has a minimum near half a linewidth, and on the far side of it the friction falls faster than the heating: tune further away and every atom is heard more faintly by both beams, and a faint velocity sensor cools badly. At a fixed intensity the predicted temperature rises almost in proportion to the detuning.

The measured temperatures fell. That is what a second mechanism looks like when it is hiding behind a first: a quantity that depends on the knobs differently, so that the knob settings which weaken the known mechanism strengthen the unknown one. Within a year two groups had worked out what it was, Jean Dalibard and Claude Cohen-Tannoudji in Paris and Steven Chu’s group at Stanford, and the answer was contained in an assumption so ordinary that nobody had thought of it as an assumption.

What the simple atom left out #

The Doppler theory treats the atom as having one ground state and one excited state, with light of any polarisation driving the transition equally. No real alkali atom is like that. Sodium’s ground state has angular momentum, which means it has sublevels — orientations of an angular momentum that is not a rotation — and circularly polarised light of one handedness couples those sublevels to the excited state with different strengths.

The second ingredient is already on the page in the light that pulls rather than pushes. Light tuned below a resonance does not only scatter; it shifts the energy of the ground state downward by an amount proportional to the intensity, and a spatially varying shift is a potential the atom moves in. For a single ground state that potential is the gradient force of an optical trap. For a ground state with sublevels, each sublevel is shifted by a different amount, depending on how strongly it couples to the local light.

Now arrange the two beams of a molasses with their linear polarisations at right angles. Where they overlap they add into a standing pattern whose polarisation changes along the axis: circular of one handedness at one point, linear an eighth of a wavelength further on, circular of the other handedness a quarter of a wavelength from the start. The intensity is uniform; the handedness is not.

For the simplest atom that shows the effect, with a spin-½ ground state, the two sublevels see the two potentials drawn: sinusoids with a period of half a wavelength, one the other shifted by a quarter. Where the light is circular of one handedness the sublevel aligned with it is shifted most and sits at the bottom of its well; the other sublevel, which couples weakly there, sits at the top of its hill. A quarter of a wavelength later the roles are exchanged.

Climbing a hill that keeps moving #

The last ingredient is optical pumping. An atom that scatters a photon of one circular polarisation tends to fall back into the sublevel aligned with that polarisation, so scattering redistributes atoms between the sublevels — and the rate at which it moves an atom out of a sublevel is highest exactly where that sublevel couples weakly to the local light. That is the top of its hill. The figure checks the coincidence rather than taking it on trust: the pumping rate out of the upper potential peaks at the same position as the potential itself, to within the resolution of the search.

So an atom moving along the axis climbs a hill, slowing as it goes, and near the top it is most likely to be pumped into the other sublevel — whose potential at that point is at the bottom of a well. The atom finds itself at the bottom of a new hill with the kinetic energy it had at the top of the old one, and starts climbing again. Each such transfer removes one well depth of energy.

Where the energy goes is the same bookkeeping the friction made of light found for the Doppler case, with a different shift. The pumping photon is emitted from a level that sits one well depth lower than the level it was absorbed from, so it leaves bluer than the laser light by exactly that energy. The atom is being cooled by photons that climb out of the light-shift potential it was forced to climb into.

Dalibard and Cohen-Tannoudji named it after the king condemned to roll a boulder up a hill for ever and see it roll back. The naming is apt with one inversion that matters: Sisyphus lost his boulder at the top, and the atom loses its hill.

The drawn staircase of the landscape figure is the idealised mechanism. The stochastic run is what it looks like when pumping is a matter of probability rather than of position, and it has three features the idealisation hides. Some pumping events happen away from the hilltops and take energy out of the well depth rather than removing it, which the jagged steps upward show. Every scattering, pumping or not, kicks the atom by a photon’s momentum, so the energy never settles on a single value. And the atom that ends up in a well is not stuck there: a large enough kick lifts it over a hilltop, and the climbing begins again.

A friction set by a delay #

The mechanism has a structure worth pulling out, because it is not special to atoms. The atom’s internal state follows the polarisation of the light it is in, but it follows with a lag — a pumping time. An atom standing still has its sublevel populations in equilibrium with the local light and feels no net force. An atom moving through the pattern carries populations appropriate to where it was, and so it is always in the wrong sublevel for where it is: more often on the uphill side than the downhill one. The force that results opposes the motion, and it is largest when the atom crosses a quarter wavelength in about one pumping time.

That is a relaxation loss. A response that lags a periodic drive absorbs energy from it at a rate that peaks when the drive period matches the relaxation time, which is the liquid that remembers absorbing energy most strongly when it is sheared at the inverse of its relaxation time, and it is the reason a dielectric heats most at the frequency its dipoles can just follow. The atom is the lagging medium, the pattern of polarisation it moves through is the drive, and the friction is the absorption.

One consequence of that picture is decisive for what follows. The friction coefficient for slow atoms comes out independent of the intensity: brighter light makes the hills higher, which strengthens the force, and pumps faster, which shortens the lag, and the two effects cancel. The heating, on the other hand, follows the scattering, which grows with intensity. So the equilibrium temperature — heating over friction — grows in proportion to the light.

A floor that falls with the light #

The runs bear the argument out and add two corrections to it. The temperature is a straight line in the well depth from a hundred recoil energies to eight hundred, with a slope of a third — roughly a third of a well depth of thermal energy, which is the statement that cooling stops mattering once the atom can no longer reach a hilltop. It is a straight line and not a proportion: it has an intercept of about forty recoil energies, which is the heating from the photons’ own recoils, a heating that does not vanish when the light is turned down.

And below a hundred recoil energies the line fails outright. At fifty the atoms are warmer than at a hundred, and at twenty-five warmer again. A well that shallow does not hold an atom with the energy the cooling leaves it, the atom stops being localised on a hill, and the lag argument that produced the friction no longer applies. Measured sub-Doppler temperatures show the same shape — linear in intensity over detuning, with an offset, and a sharp rise when the light is made too weak — and in three dimensions caesium molasses reach a few microkelvin.

That explains the 1988 measurement completely. The temperature depends on the well depth, which is proportional to intensity over detuning, so dimmer beams and larger detunings make colder atoms. The Doppler mechanism is still there, and at a few microkelvin it is irrelevant: its own floor is a hundred times higher.

Two floors, set by different things #

A mechanism that lowers the temperature in proportion to the light cannot do so for ever, even in principle, and the reason is not in any of the potentials. Every cooling scheme that works by scattering photons delivers the last kick with a photon, and that kick has a size.

The two floors contain different things, and the figure is the clearest way to see it. The Doppler limit contains the linewidth, which is a lifetime, and nothing about the atom’s mass; the recoil temperature contains the mass and the wavelength and nothing about the linewidth. For every broad line the first sits hundreds of times above the second, which is the room polarisation-gradient cooling works in — and the minimum of the stochastic runs, 70 recoil energies for caesium, sits a factor of thirty-five above the recoil floor, which is about where real molasses stop.

Strontium’s intercombination line is the case that exposes the logic. Its linewidth is so narrow that the Doppler limit falls below the recoil, and cooling on that line reaches the recoil floor by the ordinary Doppler mechanism with no sublevel tricks at all. The limit a scheme runs into is whichever of its floors is higher, and a scheme is only as good as its worst assumption.

Past the last photon #

The recoil floor is a statement about photons that are absorbed and re-emitted at random, and like the Doppler floor it was treated as fundamental until it was not.

The first escape was to arrange for the coldest atoms to stop scattering. An atom in a suitable superposition of sublevels can be in a state that the light cannot excite at all — a dark state — and if the dark state is also a state of zero velocity, atoms that random-walk into it stay there while the others keep being kicked. Aspect, Cohen-Tannoudji and collaborators demonstrated it with helium in 1988, reaching half the recoil temperature, and the scheme’s name, velocity-selective coherent population trapping, is a precise description of what it does. The atoms are not cooled below the recoil so much as collected there.

The second escape removed the light. A cloud of atoms in a magnetic trap, precooled by the methods on these pages, is let lose its most energetic members over the rim of the trap; the rest re-thermalise by collisions at a lower temperature, and the rim is lowered again. Evaporation needs no photons and has no recoil floor, and in 1995 it took a cloud of rubidium to 170 nanokelvin, where the atoms’ de Broglie wavelengths overlapped and a Bose–Einstein condensate formed — the regime in which everything has a wavelength stops being a remark about small numbers.

It is worth noticing the shape of the whole sequence, because it recurs. A limit is derived; it holds for the model it was derived for; a measurement beats it by exploiting something the model left out; a new limit appears for the new mechanism. The staircase that never reaches the floor is the same pattern in magnetic cooling, where each technique hands over to one with a lower floor. What does not change is that each floor is real for the mechanism it belongs to.

Where the model stops #

The atom is the simplest one that shows the effect. A spin-½ ground state and a spin-3/2 excited state give two sublevels and two potentials. Sodium, rubidium and caesium have many more, with a hyperfine structure that makes the potentials more complicated and the numerical factors different; the mechanism — potentials that depend on polarisation, pumping that lags them — is the same, and so is the scaling with intensity over detuning.

The polarisation arrangement is one of two. Beams with opposite circular polarisations also cool below the Doppler limit, by a different mechanism in which the atom’s orientation rather than its sublevel populations lags a polarisation that rotates along the axis. The scaling is similar and the picture of a hill is not.

The atom is treated as a classical particle with a quantum internal state. That is the semiclassical approximation, and it is good while the atom’s wavepacket is much smaller than a well. At a few tens of recoil energies of depth it is not, the atom is spread over several wells, and a full quantum treatment — which gives the atom a band structure in the periodic potential, like an electron in a crystal — replaces the stochastic particle. The failure of the straight line at low depth is partly the physics and partly the model announcing its own edge.

There is no magnetic field. Sub-Doppler cooling depends on degenerate sublevels, and a field of a fraction of a gauss splits them by more than the light shift. The laboratories that first saw 43 microkelvin had to null the Earth’s field to see it at all, and a molasses in an uncancelled field quietly returns to the Doppler limit.

What the pictures cannot show #

The landscape figure draws two potentials and an atom on one of them at a time, and hides that between pumping events the atom is in a definite sublevel only in the stochastic model’s bookkeeping. The quantum state is generally a superposition of the two, and the pumping that the model treats as a jump is a rate at which coherence is destroyed.

Nor does any figure show what the atoms look like when they are cold enough. A few tens of recoil energies of depth, with atoms at a third of it, is a gas whose members spend most of their time sitting in individual wells half a wavelength apart — an array of atoms held in place by nothing but the interference pattern of two laser beams. That array is an optical lattice, and it became a laboratory in its own right: a crystal whose spacing, depth and dimensionality are set by optics, in which atoms play the part electrons play in a solid and meet the gaps a repeat opens in their own motion. The picture of a moving hill is the moment a crystal of light first appeared, and none of these drawings has the resolution to show it.

Still open: how cold light can make a gas that it keeps touching #

The recoil floor was escaped by arranging for the coldest atoms to stop interacting with the light, and by removing the light entirely. Neither is cooling by light in the sense of this page, and the question they leave is whether there is a scheme that goes on scattering photons and still reaches below the kick of a single one.

The candidates are all schemes in which the scattering is made to depend steeply on velocity near zero — so that an atom which has been kicked out of the coldest state is kicked back far more readily than one inside it is kicked out. Raman cooling with carefully shaped pulses and cooling on transitions whose linewidth is narrower than the recoil both do this, and both reach temperatures below the recoil in some directions. Whether a steady, scattering, three-dimensional scheme can do it is an argument about how narrow a velocity filter can be made while photons keep arriving, and it is not settled by any figure here.

The habit worth carrying away is the one the 1988 measurement enforced twice. When an experiment beats a limit, look first at the trend rather than the value. A value below a floor can be an error; a dependence on the knobs that runs backwards cannot be, and it points directly at the assumption that has to go.