The light a full sea will not scatter

No two in the same state introduced the exclusion principle as a rule about where fermions may sit, and the pressure that is not a temperature found it stacking the electrons in a metal into a sea tens of thousands of kelvin deep. The collisions the exclusion principle forbids found its most practical consequence: two electrons deep in that sea cannot collide, because every state either could scatter into is already full, and so a copper wire’s electrons travel hundreds of atoms between collisions. The label three identical quarks forced turned the rule into a demand for a hidden quantum number.

In each of those, the particles that were blocked were the fermions themselves, colliding with one another. This essay blocks something that is not a fermion and not identical to anything in the gas: a photon. An atom in a cold Fermi gas, struck by light, cannot always scatter it, and the reason has nothing to do with the photon. It is that the atom must recoil, and the state it would recoil into is taken.

The recoil a scattered photon leaves #

A photon carries momentum $\hbar k$, as a photon with a momentum established from Compton’s X-rays. When an atom scatters one — absorbs it and re-emits it in a new direction, with the same frequency — the photon’s momentum changes direction, and the difference goes into the atom. If the photon is turned through an angle $\theta$, the atom receives a kick of size

$$\hbar q = 2\hbar k \sin(\theta/2),$$

zero for a photon that carries straight on, $2\hbar k$ for one sent straight back. For a lithium atom and its red resonance light at 671 nanometres, the largest kick sets the atom moving at about twenty centimetres a second. In a warm gas that is nothing: the atoms already move at hundreds of metres a second, and the kick moves one of them from one crowded region of a continuum of states to another. In a gas cooled to a few hundred nanokelvin, where the atoms move at a few centimetres a second, it is not nothing. It is a step in momentum of the same size as the momenta the atoms have.

A sea with the kick drawn on it #

Cool a gas of identical fermions far enough and it becomes a Fermi sea: every state of motion is filled up to the Fermi momentum, and every state beyond it is empty. In momentum space the sea is a ball. An atom inside the ball, kicked by $\hbar q$, would move to a point displaced by $q$.

The figure draws the occupied states as a filled disc and the same disc shifted by the kick. An atom in the overlap of the two would land on an occupied state; the exclusion principle forbids that, and the scattering does not happen. Only atoms in the crescent near the edge of the sea, facing the direction of the kick, can take it. For a kick of 0.8 of the Fermi momentum, a little over half the atoms in the whole sphere can scatter. The rest are not transparent in any ordinary sense — they would absorb and re-emit the light perfectly well in isolation — but the scattering process has no allowed final state.

The share of atoms that can scatter is a function of the kick alone, and for a sphere at zero temperature it is a piece of solid geometry: the fraction of the ball lying outside the ball shifted by $q$,

$$S(q) = \frac{3}{4}\frac{q}{k_F} - \frac{1}{16}\frac{q^3}{k_F^3}, \qquad q < 2k_F,$$

and one for any kick larger than the sea’s diameter. $S(q)$ is the static structure factor of the gas, the same quantity a scattering experiment on a liquid measures to find how its atoms are arranged. For a Fermi gas it says how its momenta are arranged.

The share at every kick #

At zero temperature the curve starts from zero: a vanishingly small kick moves almost every atom onto a state already occupied by its neighbour, and only the atoms right at the surface of the sea can take it. It rises steeply at first, as $3q/4k_F$, and reaches one at twice the Fermi momentum, where the shifted ball no longer overlaps the original at all. Warming the gas blurs the surface of the sea — some states inside it are emptied and some outside it are filled — and the curve rises at small kicks: at a tenth of the Fermi temperature the share at half the Fermi momentum is 0.389 rather than 0.367, at a quarter it is 0.483, and at half it is 0.645. By the Fermi temperature the gas is barely degenerate and the curve is close to one everywhere.

The same function governs the collisions in the earlier argument, where the kick came from another electron and both partners needed an empty state. There the suppression went as the square of the temperature, because both final states had to lie in the thin shell of partly occupied states at the surface. Here only one particle needs a final state, the other being a photon that can go anywhere, so the suppression is linear in the kick and does not vanish at zero temperature. A photon is a probe with no Fermi sea of its own, and that makes the blocking it sees the simplest kind.

Which directions go dark #

A photon scattered through a small angle gives a small kick, and a small kick is blocked most. So the effect does not merely dim the gas; it reshapes the pattern in which it scatters.

An atom scatters unpolarised light with a pattern proportional to $1 + \cos^2\theta$, strongest forward and backward and weakest at right angles. Multiply it by the share of atoms able to take each kick and the forward lobe is cut away: at twenty degrees only twenty-nine per cent of the atoms can scatter, and at a few degrees almost none. The backward lobe, whose recoil is the largest the photon can give, survives intact when the Fermi momentum is no larger than the photon’s. A cloud of degenerate fermions lit from one side therefore throws light back towards the source nearly as a classical gas would and is dark in the directions close to the beam.

This is also why the experiments are hard. The forward direction is where a classical cloud scatters most strongly and where its scattering is easiest to collect, and it is exactly where the blocking is strongest. And for the whole effect to be large, the Fermi momentum must be comparable with the photon’s — a sea whose diameter is a few times the photon’s kick — For spin-polarised lithium the Fermi momentum, $(6\pi^2 n)^{1/3}$, equals the photon’s at a density of $1.4 \times 10^{19}$ atoms per cubic metre — a hundred thousand times thinner than air, and dense for a gas this cold — where the Fermi temperature is about three microkelvin, and the gas must be cooled to a small fraction of that.

How dense, how cold #

Adding up the scattering over all directions gives the rate per atom, relative to what a classical gas of the same atoms would scatter. While the sea is much smaller than the photon’s momentum, almost every kick lands outside it and the gas scatters normally. As the density rises the Fermi momentum grows as its cube root, and the scattering per atom falls: to 0.78 of the classical value when the Fermi momentum equals the photon’s, and to 0.47 when it is twice the photon’s, at a twentieth of the Fermi temperature. Warmer gases fall less. At half the Fermi temperature the gas with equal momenta still scatters at 0.84.

A trapped cloud is densest at its centre, so the centre is dimmer per atom than the edges, and an image of the scattered light, divided by an image of where the atoms are, is a map of local degeneracy. That is the measurement three groups reported in 2021, in the same issue of Science: with lithium at MIT and at the University of Otago, and with strontium at JILA. Each compressed a spin-polarised Fermi gas until its Fermi momentum approached the photon’s and saw the scattered light drop below what the number of atoms predicted, by amounts of order tens of per cent, in the way the ideal-gas calculation describes. The effect had been predicted in the early 1990s and then waited thirty years for gases dense and cold enough.

Warming the dimness away #

The suppression fades as the gas is warmed, smoothly and without a threshold. That is worth noticing, because the onset of quantum degeneracy in a Fermi gas has no phase transition, unlike the Bose–Einstein condensation of a gas of bosons. Nothing sudden happens as a Fermi gas is cooled through its Fermi temperature; the occupations of the low states rise steadily towards one and the blocking rises with them. The dimming is a continuous thermometer, and since it depends on the occupations rather than on how the atoms are moving, it reads the degeneracy directly where a time-of-flight image of the gas has to infer it from a fit to the shape of the cloud.

A gas quieter than random #

The blocking at small angles has a second reading, and it is the more surprising one. Light scattered through a small angle by a gas measures how the number of atoms in a region fluctuates: a perfectly uniform medium scatters nothing sideways, and what makes air scatter at all, as Einstein and Smoluchowski showed, is that the number of molecules in any small volume fluctuates from moment to moment. For a classical gas the fluctuations are those of independent placements, with a variance equal to the mean number: the Poisson rule.

The structure factor at small kicks is exactly the ratio of the actual variance to that Poisson value, and for a Fermi gas it is small: at a temperature $T$ well below the Fermi temperature, $S(0) = 3T/2T_F$. A degenerate Fermi gas is quieter than randomness. Its atoms cannot pile up in one region and leave another empty, because piling up means putting more atoms into the same small volume, which needs more states of large momentum, which are expensive; the gas is held uniform by the same statistics that hold up a metal’s electron sea. In real space this is the exchange hole of the force with no force in it: each fermion is surrounded by a region its identical neighbours avoid, and a gas of particles that avoid one another fluctuates less than a gas of particles placed at random.

That suppression was measured before the dimming was, in 2010, by counting atoms in small regions of degenerate lithium clouds in many repeated images and finding the variance below the mean, by up to half, in proportion to how cold the gas was. The dimming of scattered light and the quietness of the atom number are the same structure factor read in two ways: one at large kicks through the light, the other at small kicks through the counts.

The same blocking in a semiconductor #

The surprising part is not that this happens in a quantum gas. It is that it is the same effect that decides what colours a heavily doped semiconductor absorbs. A photon above the band gap of a semiconductor is absorbed by lifting an electron from the valence band to the conduction band. Dope the semiconductor so heavily with donors that the bottom of the conduction band fills with electrons, and the states that photon would put an electron into are taken. The absorption edge moves up, to the energy needed to reach the first empty state above the Fermi level. It is called the Burstein–Moss shift, it is measured in tenths of an electronvolt, and it contributes to why the conducting oxides on a touchscreen, doped until they conduct like poor metals, still let visible light through.

In both, an optical property that seems to belong to each atom or each electron turns out to belong to the occupation of the states the process needs. The product doping cannot move found that a doped semiconductor’s electron and hole populations obey a mass-action law fixed by the band gap; filling the conduction band past the point where that law holds is what makes it transparent. And the effect has a mirror image. In a gas of bosons, the occupation of a final state does not forbid the transition but enhances it, by one plus the number already there. A Bose–Einstein condensate lit by a laser scatters light into the directions that put atoms into already-occupied recoil states, and the scattering grows on itself into bright, narrow beams, as was seen in 1999. Fermions dim; bosons amplify; the photon carries no statistics of its own into either.

Why it is hard to see in anything but an atom gas #

The calculation behind the figures treats the atoms as an ideal gas that scatters light one atom at a time. A real dense cloud does more than that, which is why the experiments needed care. At the densities required, the atoms are close enough that the light one scatters is scattered again by its neighbours, the dipoles of nearby atoms interact through the light field, and the cloud’s index of refraction bends the beam. These collective effects change the scattered light by amounts comparable with the blocking. The 2021 experiments separated them by tuning the light far from the atoms’ resonance and comparing spin-polarised clouds, which are blocked, with mixtures of two spin states at the same density, in which each atom’s recoil competes only with atoms of its own spin and the blocking is weaker. What remained was the exclusion principle acting on light.

Why the sky is blue and the sunset is not found that air scatters light because its molecules are randomly placed, and the cross-section that forgets the colour found the cross-section for one electron. Neither asked what the scatterer’s recoil needed. For air at room temperature it needs nothing that is not available, which is why the per-molecule cross-section can simply be multiplied by the number of molecules. That multiplication is an assumption about occupations, and it fails in the coldest gases there are.

What the pictures cannot show #

The figures assume an ideal gas of identical fermions in a uniform box, with no interactions, scattering light far from resonance one atom at a time. They leave out the trap, which makes the degeneracy vary across the cloud; the collective and multiple scattering that dense clouds produce; and the interactions between atoms, which in the experiments were made small by using a single spin state but are not zero. They treat the photon’s momentum as fixed by its wavelength and the atom as recoiling freely, which ignores the light shift of the atoms’ states by the probe itself. And they show no noise: the measured suppression is a small difference between two large numbers of photons, and the 2021 results came with uncertainties of several per cent.

Still open: blocking an atomic clock #

A single fermion’s transition is not changed by the sea around it, but the light it emits must still leave the atom in some state of motion, and in a deep enough sea every choice is blocked. Calculations suggest that in a sufficiently degenerate and dense gas the spontaneous emission of an excited atom is itself slowed, since the ground-state recoil states it would decay into are full — a lengthening of an excited state’s lifetime by the exclusion principle. Optical lattice clocks, which hold thousands of fermionic strontium or ytterbium atoms in identical motional states, already rely on the exclusion principle to suppress collisions between atoms that would otherwise shift their frequency. Whether the suppression of emission can be seen directly, and whether it can be used to make an excited state live longer than its natural lifetime, has been proposed and not yet demonstrated.

The habit worth carrying away is to ask of any process what final state it needs, and whether that state is free. An atom scattering a photon must recoil into a new state of motion; in a degenerate Fermi gas the share of atoms whose recoil state is empty is the structure factor $S(q) = 3q/4k_F - q^3/16k_F^3$ at zero temperature, so the gas scatters less light per atom, removes its forward lobe first, and dims more the denser and colder it is. Nothing in the atom has changed. Its neighbours have taken the room it needed.