The friction made of light

A sodium atom in a warm vapour moves at a few hundred metres a second. Shine a laser on it and it absorbs photons, each carrying a momentum light has a pressure puts a number on, and each absorption pushes it along the beam. A single beam is therefore a way to push atoms, and pushing is not cooling. A pushed gas is a moving gas, and every atom in it is still jostling about at the same speed relative to its neighbours.

Point a second beam the other way, at the same frequency and the same intensity, and at first sight things are worse. The atom is now pushed both ways at once. At rest the two pushes cancel exactly, which is correct and useless.

The trick is in the word at rest. An atom that is moving does not see the two beams at the same frequency, and if the laser is tuned a little below the atom’s resonance, the beam the atom is running into is Doppler-shifted up towards resonance and the beam behind it is shifted further away. The atom scatters more light from the beam ahead than from the beam behind. The net push points against its motion, whichever way it is moving.

Two pushes that cannot agree #

The two dashed curves are the same function reflected. Each is the scattering rate of a two-level atom driven by one beam, multiplied by one photon’s momentum, and the rate is a resonance curve in the detuning the atom actually experiences — the laser’s detuning minus the Doppler shift $kv$ for the beam coming towards it, plus it for the beam going away. The beam pushing towards positive $x$ is resonant with atoms moving towards negative $x$, into it, and pushes them back; its mirror image does the same from the other side.

Adding them gives the solid curve, and near the origin it is a straight line. Expanding the two resonance curves to first order in $v$, the force is

$$F = -\alpha v, \qquad \alpha = -8\hbar k^2 s\,\frac{\delta/\Gamma}{\left(1 + s + 4\delta^2/\Gamma^2\right)^2},$$

with $\delta$ the detuning (negative, below resonance), $\Gamma$ the natural linewidth and $s$ the intensity of each beam in units of the saturation intensity. The figure measures the slope of the drawn curve at the origin and compares it with that expression; they agree to a part in a million. A force proportional to velocity and opposed to it is exactly what a viscous fluid exerts on a ball moving through it, and the arrangement was named accordingly when it was first built at Bell Laboratories in 1985: optical molasses.

Two numbers come straight off the curve. The first is the damping time, the mass divided by twice $\alpha$, which for sodium at these settings is 17.5 microseconds: a sodium atom’s velocity relaxes in about the time a sound wave takes to cross a millimetre. The second is the velocity at which the force is strongest, 3.1 metres per second. Beyond it the atom has been Doppler-shifted so far that both beams are off resonance, the grip weakens, and a fast enough atom goes straight through. A molasses captures only atoms already moving at a few metres a second — which is a fact about where the atoms have to come from, and a later section returns to it.

The shape of the curve is the same shape the three ways of coming to rest begin from: a restoring response linear near equilibrium, a limit on how much of it there is, and a timescale set by the ratio of an inertia to a damping. What is unusual is the medium. There is nothing here to rub against. The friction is made entirely of photons, and it acts on an atom in a vacuum chamber that contains, apart from the light, almost nothing at all.

A friction has to be paid for #

If that were the whole story the atoms would stop. A linear friction with nothing opposing it takes every velocity to zero exponentially, and the temperature of the gas would fall without limit.

It does not, and the reason is in the phrase a photon’s momentum. The friction is an average over photons, and photons are not delivered as an average. Light arrives in lumps, each absorption is one kick of $\hbar k$ along a beam, and each spontaneous emission that follows is another kick of the same size in a random direction. The average of the absorption kicks is the friction. The spread of the absorption kicks, and the whole of the emission kicks, average to nothing and add up in square: they are a random walk in velocity, and a random walk heats.

That figure has no force in it. Nothing was told that the atoms should slow down; each one simply absorbs from whichever beam the rates say, is kicked, emits, is kicked again, and does this about twelve hundred times in four hundred microseconds. The temperature falls because the beam ahead is chosen slightly more often than the beam behind, and it stops falling at the point where the bias in the choices removes energy exactly as fast as the randomness of the kicks adds it.

The balance can be written down. The heating is the rate at which the mean square momentum grows, which is a recoil squared times the number of kicks per second: two kicks per scattering, from both beams together. The cooling is the rate at which the friction removes it, which is twice $\alpha/m$ times the mean square momentum. Setting the two equal and dividing by the mass gives

$$k_B T = \frac{\hbar\Gamma}{8}\,\frac{1 + s + 4\delta^2/\Gamma^2}{|\delta|/\Gamma}.$$

The simulation was not given that expression and lands within three per cent of it, which is the statistical scatter of twelve hundred atoms.

The surprising part is that this is not a new result at all. A friction and a random force supplied by the same agent, in balance at a temperature, is the jiggle that proved atoms: a pollen grain in water slowed by viscosity and kicked by molecular impacts, where Einstein saw in 1905 that the diffusion constant of the kicks must equal the friction coefficient times $k_BT$ because the water is at a definite temperature. Here the argument runs backwards. The light has no temperature to begin with; it has a friction and a diffusion, and their ratio defines one. The atoms settle into a thermal distribution at a temperature the laser never had, in exactly the way the temperature a molecule does not have ties a fluctuation to a response. Optical molasses is Brownian motion with the fluid replaced by two beams of light.

The temperature the balance picks #

The formula has a minimum, and the minimum is the point of it.

Too close to resonance and the friction is feeble: both beams are nearly resonant for an atom at rest, a small velocity changes their rates hardly at all, and the heating from the large scattering rate wins. Too far from resonance and the friction collapses for the opposite reason, because neither beam is heard. In between, the ratio of heating to cooling passes through a minimum, and for weak beams it sits at $|\delta| = \Gamma/2$ and has the value

$$T_D = \frac{\hbar\Gamma}{2k_B}.$$

That is the Doppler cooling limit, and the figure finds it by searching the curve rather than by being told it: the coldest detuning comes out at 0.500 linewidths for weak beams and at $\sqrt{1+s}/2$ in general, with the minimum temperature $T_D\sqrt{1+s}$, both matching the algebra to the precision of the search.

The expression is short enough to read clause by clause. $\hbar\Gamma$ is the energy width of the spectral line — the width that is a lifetime turned into joules — and dividing by $k_B$ turns it into a temperature. The limit is the linewidth written in kelvin. A laser with a sharp frequency cannot produce a velocity sharper than the atom’s own line lets it discriminate, and the atom’s line is as sharp as its excited state is long-lived.

It is worth being clear what the intensity does, because it is the obvious knob and it turns the wrong way. Brighter beams mean a larger friction and a proportionally larger heating, so the ratio does not change while the beams are weak; once they approach saturation the friction stops growing while the scattering, which still doubles with every doubling of the kicks, catches up. The second curve in the figure is the result. More light does not buy a colder gas, and past a point it buys a warmer one.

The mass that is not in it #

The Doppler limit contains a linewidth and two constants. It does not contain the mass of the atom or the wavelength of the light, and both absences are worth a figure.

The mass drops out for a reason that is physical rather than algebraic. A heavier atom is slowed less by each photon, so it takes more of them to damp it — and it is kicked less by each photon, so it takes more of them to heat it. The friction coefficient is a force per velocity and contains no mass; the diffusion is a momentum squared per time and contains no mass either. Their ratio is an energy, and the mass only enters when that energy is converted into a velocity. A rubidium atom at the limit is moving more slowly than a potassium atom at the limit, by the square root of the mass ratio, and the two are at the same temperature.

The linewidth enters because it sets both the resolution and the rate. A broad line scatters quickly and is a coarse velocity detector; a narrow line discriminates finely and scatters slowly. The same trade appears in any detector whose bandwidth is fixed by a decay, and here it produces a temperature floor that runs over four decades on the figure — from strontium’s broad blue line at 732 microkelvin to its forbidden red one at 180 nanokelvin. That last point is so low that it is below the recoil of a single photon, which is a problem of its own and which the formula cannot see: the whole derivation assumed that one kick is small compared with the width of the distribution, and a floor that says otherwise is a floor the model has walked past.

What the kicks leave behind #

A temperature is a statement about a distribution, and the distribution the photons produce can be read directly off the same run.

It is a Gaussian, and the reason it is a Gaussian is the reason any velocity distribution is. Each atom’s final velocity is a sum of many small independent kicks, weighed down by a linear friction that forgets the old ones exponentially, and a sum of that kind has a normal distribution whatever the shape of the individual kicks. That is the same mechanism that makes the speeds in a still room Maxwellian, and the same one the walk that comes home follows through for diffusion. The atoms in a molasses have never collided with one another. They are thermal because their history is a long sum.

The twenty kicks are the number that says how good the description is. Twenty recoils across the distribution means each photon moves an atom by a twentieth of the spread, which is small enough for the smooth friction to be a fair average and large enough that the discreteness is visible in a histogram with forty bins. At the strontium line’s 180 nanokelvin the ratio would have fallen below one, and the picture of a smooth force with a little noise on it would have become a picture of an atom that is stopped and started by single photons.

Getting the atoms there #

The capture velocity of three metres a second means that a molasses can only cool atoms that are already very slow, and a vapour at room temperature has almost none. The atoms have to be slowed first, and the obvious way is one strong beam pointed straight at an atomic beam coming out of an oven.

The arithmetic of the deceleration is startling. At full saturation an atom scatters half a linewidth’s worth of photons per second, which for sodium is thirty million, and each takes three centimetres a second off its velocity. That is a million metres per second squared, a hundred thousand times gravity, and at twice saturation it is still sixty thousand. On paper a 700 m/s atom stops in forty-one centimetres.

The figure shows why it does not. The laser is resonant with atoms at 700 m/s, and an atom slowed to 690 m/s is Doppler-shifted by ten metres a second’s worth of frequency — more than a linewidth — and has stopped hearing the light. The beam does what it can to the atoms near the tuned velocity and then releases them, a little slower, into a pile-up just below the hole. A single-frequency beam is a velocity-selective push, and the selection is what makes it useless for slowing.

The repair is to make the resonance follow the atoms down. Sweeping the laser frequency upward in step with their deceleration works for a pulse of atoms. The method that works continuously, introduced by Phillips and Metcalf in 1982, keeps the laser fixed and passes the atoms through a magnetic field whose strength varies along the path, so that the Zeeman shift of the atomic levels cancels the changing Doppler shift at every point. The shape of that field is a square root in distance, because constant deceleration makes the velocity a square root of the distance remaining — the same arithmetic the size the light cannot blow away uses for a grain, run in the opposite direction. The atoms come out of a metre of such a solenoid at a few tens of metres a second, slow enough for a molasses to catch.

Where the energy goes #

The force figure raises a question the other figures answer: an atom that ends up colder has lost kinetic energy, and absorbing light is supposed to add energy rather than remove it.

The resolution is in the frequency of the light that leaves. An atom moving at $v$ towards a beam absorbs a photon at the laser frequency, which in its own frame is higher by $kv$. It re-emits, at low intensity, at the frequency it absorbed in its own frame, and in a random direction, so on average the emitted light carries away the laser frequency plus the Doppler shift. The scattered light is bluer than the light that came in, by exactly the energy the atom’s motion supplied. The laser beam loses nothing, the fluorescence gains a little, and the atom pays the difference out of its kinetic energy. The same bookkeeping a photon with a momentum applies to a single Compton collision is here applied thirty million times a second.

That reading says something useful about what else can be cooled. Any system that absorbs light at one frequency and re-emits it, on average, at a higher one loses energy in the process, and the principle has been used to refrigerate solids: a crystal doped with ions whose fluorescence comes out bluer than the pump has been laser-cooled to cryogenic temperatures with no moving parts and no cold reservoir. The Doppler shift is one way of making the outgoing light bluer. It is not the only one.

Where the two-level atom stops describing a real one #

Real atoms are not two-level systems. Sodium’s ground state is split into two hyperfine levels 1.77 gigahertz apart, and an atom scattering on one of them eventually falls into the other and goes dark. Every working molasses therefore carries a second laser frequency whose only job is to pump the atoms back, and the cooling that follows is the two-level result only if that repumping is fast enough to be ignored.

The model is one-dimensional, with the spontaneous recoil taken along the axis. In three dimensions, with three pairs of beams, emission goes in every direction and the heating per axis changes by a numerical factor; the structure of the result — a friction, a diffusion, and a floor at a temperature of order $\hbar\Gamma/k_B$ — does not.

The beams are added as independent pushes. That is the low-intensity approximation, and at high intensity the two beams interfere in the atom, saturate each other, and produce a standing wave with forces the separate-beam picture does not contain. The figures keep the intensity at a tenth of saturation for the ensemble, where the approximation is good.

And the atom is assumed to have one ground state. That is the assumption the measurement found wrong. The first careful temperature measurements of a sodium molasses came in far below the limit derived here, which is the kind of failure a model announces only by being beaten, and whose explanation lies in exactly the sublevel structure this treatment throws away.

What a force curve hides #

The force figure draws an average and cannot draw what is averaged. An atom in molasses is not decelerating smoothly along the solid line; it is being struck tens of millions of times a second and drifting, in a trajectory that looks like noise with a slight bias, and the second figure’s jagged ensemble temperature is the closest the drawings come to showing it.

Nor does any figure show position. The force depends only on velocity, so an atom cooled to the limit is not held anywhere: it random-walks through the overlap of the beams, with steps so short that it takes a large fraction of a second to wander out, which is why the name molasses stuck and why the atoms can be photographed as a glowing ball. A trap is a different object. It needs a force that depends on where the atom is, and the magneto-optical trap supplies one by adding a weak magnetic gradient that tunes the atoms towards whichever beam pushes them back to the centre. The friction drawn here is half of that device, and the other half is invisible in a picture with velocity on the axis.

Still open: the gas that came out colder than this #

In 1988 a group at the National Bureau of Standards measured the temperature of a sodium molasses carefully, by releasing the atoms and timing how long they took to fall through a probe beam, and found 43 microkelvin. The limit derived on this page is 240. A measurement six times below a theoretical floor is not an experimental error to be corrected: the theory is the thing that has failed, and the question is which of its assumptions was doing the damage.

The candidates are listed in the section above, and the answer turned out to be the most innocent-looking of them — that the atom has a single ground state, so that the light’s polarisation does not matter. A beam that pulls rather than pushes has already shown that light exerts a force on an atom through the energy shift it produces as well as through the momentum it delivers. Put those shifts into a standing wave whose polarisation changes every quarter wavelength, give the atom more than one ground sublevel, and a mechanism appears that makes the atom climb a hill before the light moves it to the bottom of the next one — a cooling that has nothing to do with the Doppler shift and whose floor is set by a different quantity entirely.

The habit worth carrying away from this page is the one the 1988 measurement enforced. A limit derived from a model is a statement about the model. The Doppler limit is exact for a two-level atom, and nature’s atoms were not obliged to be two-level; the right response to a floor is to ask which simplification it depends on, because the floor moves when that simplification is removed — and here it moved by a factor of six, and then by a hundred.