The cloud that stops shrinking

No two in the same state argues that the exclusion principle is why matter has volume. Electrons cannot share a state, so an atom’s electrons fill shell after shell outwards, and an atom’s size is the size of its outermost filled shell rather than of its innermost. The pressure that is not a temperature applies the same principle to the conduction electrons of a metal: filling states one to a level up to the Fermi energy gives them a kinetic energy corresponding to tens of thousands of kelvin in a wire at room temperature, and a pressure to match. And the mass no cold matter can hold up follows that pressure into a white dwarf, where it holds up a star.

All three arguments are inferences. Nobody has looked at a white dwarf’s electrons and seen them refuse to be compressed; the evidence is that the star has the radius the calculation says it should. In 2001 Randall Hulet’s group at Rice University made the inference visible. They cooled two isotopes of lithium together in one magnetic trap — lithium-7, whose atoms are bosons, and lithium-6, whose atoms are fermions — and photographed both clouds as the temperature fell. The bosons’ cloud kept shrinking. The fermions’ cloud, at about a quarter of its Fermi temperature, stopped. A team at the École Normale Supérieure in Paris saw the same thing the same year.

The cloud that stops shrinking #

A gas held in a harmonic trap, a bowl-shaped potential that pulls each atom back towards the centre with a force proportional to its distance, has a size set by the competition between the trap’s pull and the atoms’ thermal motion. For a classical gas the mean square distance from the centre is proportional to the temperature: halve the temperature, and the cloud’s width falls by a factor of $\sqrt 2$. Cooled indefinitely, a classical cloud would shrink to a point.

The figure compares three ideal gases of a hundred thousand atoms each in the same trap, with temperature measured in units of the Fermi temperature $T_F$ that the fermions would have, and size in units of the Fermi radius $R_F$, the distance from the centre at which a fermion with the Fermi energy would turn round. Above about $T_F$ all three are indistinguishable from the classical line. Below it they part. The bosons follow the classical line down to their condensation temperature, about half of $T_F$ for equal numbers, and then a growing fraction of them collapses into the trap’s single lowest state, whose size is set by the uncertainty principle alone and is a few per cent of $R_F$. The fermions part from the classical line in the opposite direction. Their size bends away above the classical curve, and below a quarter of $T_F$ it has almost stopped changing, approaching $R_F/\sqrt 8$ at absolute zero.

In the Rice experiment the two isotopes cooled side by side at the same temperature, which was the point of mixing them. At high temperature the two clouds were the same size. At the lowest temperatures, the fermion cloud was distinctly larger than a classical cloud at its temperature would be, and larger than the boson cloud beside it, and further cooling made no difference to it.

The energy no cooling can take away #

The size of a trapped cloud and its energy are the same measurement in two forms. For a harmonic trap, the potential energy of each atom is on average exactly half its total energy, so the mean square size is proportional to the energy per atom.

The classical gas carries $3kT$ an atom in a three-dimensional trap, three halves of $kT$ for each direction of motion and as much again in potential energy. The bosons’ energy falls faster than the classical line once they begin to condense, since the atoms in the ground state carry only the trap’s zero-point energy. The fermions’ energy falls more slowly than the classical line and levels off at three-quarters of the Fermi energy. A cloud at absolute zero still carries that much energy per atom, and no amount of further cooling can remove it, because there is no lower state for any atom to drop into: every lower state is already taken.

This is the whole content of Fermi pressure. The atoms in the comparison are ideal: they do not collide, attract or repel at all. The energy at zero temperature is not stored in any force between them. It is the cost of putting a hundred thousand atoms into a hundred thousand different states, which forces most of them into states far above the ground state. Squeeze such a cloud, by tightening the trap, and the states move up in energy, every atom’s with them; that rise is the work done against the pressure.

Filling a trap one atom to a state #

The states of a three-dimensional harmonic trap come in shells. The energies are $(n + 3/2)\hbar\omega$, and the $n$-th shell has $(n+1)(n+2)/2$ states, since that many ways exist of dividing $n$ quanta among three directions.

Two hundred and twenty fermions fill the first ten shells exactly, one atom to every state, up to the shell at $10.5\hbar\omega$. At three-tenths of the Fermi temperature the top shells are partly emptied and the next ones partly occupied, but the bottom shells stay completely full — the Fermi distribution’s rounded step, applied to a trap. Bosons at zero temperature would all occupy the single state of shell 0.

For large numbers the top occupied energy is the Fermi energy, $E_F = (6N)^{1/3}\hbar\omega$, which for a hundred thousand atoms is eighty-four times the trap’s level spacing. For lithium-6 in a typical magnetic trap, with frequencies of a few hundred hertz, that is a Fermi temperature of about a microkelvin. Getting well below it means cooling the atoms to a few hundred nanokelvin, and doing it without being able to use the method that works for every boson.

Cooling something that will not collide #

The standard way to cool a trapped gas below what lasers can reach is evaporation. The most energetic atoms are allowed to escape over the rim of the trap; the rest collide among themselves and rethermalise at a lower temperature. Evaporation depends on collisions. And at low temperatures identical fermions in the same spin state do not collide.

The collisions the exclusion principle forbids gives the reason for electrons in a metal, where the states a collision could scatter them into are full. For cold atoms there is a second, simpler reason. At low energies, collisions happen only in the simplest partial wave, in which the two atoms’ relative wavefunction is symmetric under exchange; identical fermions must have an antisymmetric wavefunction and so cannot collide that way at all. Below a millikelvin or so, a gas of identical fermions stops thermalising.

The answer, in both 2001 experiments, was to cool the fermions sympathetically: evaporate the bosons, which collide with each other freely and also with the fermions, and let the fermions cool by contact with the bosonic bath. Lithium-7 and lithium-6 cooled together. That made the comparison in the figure possible at the same time as it made the experiment possible. The two clouds were measured at a common temperature, set by the bosons, and the bosons themselves, which in lithium-7’s case attract one another and could form only small condensates before collapsing, served as the thermometer for the fermions as well as their refrigerator.

The shape of a full cloud #

A classical cloud in a harmonic trap has a Gaussian density profile, densest at the centre and falling smoothly outward. A degenerate Fermi cloud cannot be densest at the centre beyond a certain limit, since the number of atoms in any small region is limited by the number of states available there.

At absolute zero the density at each point is that of a uniform Fermi gas with a local Fermi energy equal to $E_F$ minus the trap’s potential energy there, and the cloud ends sharply at the Fermi radius, where that local Fermi energy reaches zero. Integrated through two directions, as the figure draws it, the profile is $(1 - x^2/R_F^2)^{5/2}$: flatter on top and steeper at the edge than a Gaussian of the same width. At a quarter of the Fermi temperature the edge has softened but the flat top remains.

This is how temperatures well below $T_F$ are measured. The width alone stops changing, so a fit of the width gives no information; but the shape of the shoulders does change, and a fit of the full Fermi–Dirac profile to the image gives $T/T_F$ down to about a tenth. Deborah Jin’s group at JILA, which made the first degenerate Fermi gas of atoms in 1999 with potassium-40, used the profile and the energy measured from a cloud’s expansion to show the departure from classical behaviour; the side-by-side picture of fermions and bosons came two years later.

The one rule a crowd cannot break #

The three statistics differ, at bottom, in how many atoms may occupy a single state, and that is the clearest place to see them part.

The occupation of the lowest cell of phase space — the state at the centre of the trap with zero momentum — is, for a classical gas, the fugacity $z$, which grows without bound as the gas is cooled at fixed number. That is the sign the classical description has failed: once $z$ approaches one, the gas has as many atoms as states in the most crowded region, and quantum statistics take over. Bosons take more than one, without limit: the occupation $z/(1-z)$ diverges as $z$ approaches one, and at the condensation temperature it does. Fermions take at most one: $z/(1+z)$ approaches one from below and never reaches it. At a quarter of the Fermi temperature the lowest cell is 95 per cent full.

So the fermions’ refusal to shrink and the bosons’ collapse are one fact about counting, read in opposite directions. The condensate a trap makes traces the bosons’ half: the excited states can hold only so many atoms at a given temperature, and the excess has nowhere to go but the ground state. The fermions’ half is the mirror image. Each state can hold only one, so the atoms that would crowd into the low states classically are forced up into high ones, and the cloud is larger than it would otherwise be.

How the size depends on the number #

The Fermi radius can be written in terms of the trap’s own natural length, the size of its quantum ground state, $a = \sqrt{\hbar/m\omega}$. Since $E_F = (6N)^{1/3}\hbar\omega$, the radius at which an atom of that energy turns round is $R_F = \sqrt{2}\,(6N)^{1/6}\,a$. A degenerate Fermi cloud is larger than the ground state by a factor that grows only as the sixth root of the number of atoms: a hundred thousand atoms make a cloud about thirteen times the ground state’s size, and a hundred million make one about forty times. Doubling the number of atoms widens the cloud by twelve per cent. The atoms pack in more and more densely as more are added, each new one going into a state a little higher than the last.

That slow growth is the opposite of what happens in a star held up by the same pressure. In a trap the confining force is fixed — the trap’s spring constant does not depend on how many atoms it holds — so adding atoms adds pressure against a constant pull, and the cloud grows. In a white dwarf the confining force is the star’s own gravity, which grows with its mass, and the confinement outgrows the pressure: a cold star of degenerate electrons has a radius that shrinks as the cube root of its mass, until at the Chandrasekhar mass it can hold itself up no longer. Self-gravitating matter has other surprises of the same kind, such as the ball of gas that heats up as it cools. The atoms in the trap and the electrons in the star are filling states by the same rule; what differs is whether the bowl they fill depends on how full it is.

It also explains why the exclusion principle is often described, loosely, as a repulsion. Two identical fermions brought towards the same point must occupy different states, and the price in kinetic energy rises as they are squeezed together, exactly as if they repelled. The force with no force in it traces that appearance in the exchange interaction between electrons, where it accounts for chemistry and magnetism. In the trap there are no electrons and no charges, and no potential between the atoms; there is only the energy of the occupied states, and it behaves, for the size of the cloud, exactly as a pressure would.

Fermi pressure in other places #

The same physics, with electrons instead of atoms, holds up white dwarfs; with neutrons, neutron stars, whose radius is set by how stiffly their neutron Fermi sea resists compression, which the sound speed a neutron star needs turns into a constraint on nuclear matter. In atomic nuclei, protons and neutrons fill separate Fermi seas, and the cost of filling one higher than the other is the largest part of why stable light nuclei have equal numbers of each, and why the masses of nuclei of one mass number lie on the parabola that decides which way a nucleus decays. In each of those cases the particles interact strongly and the pressure is mixed with the interactions; the trapped lithium-6 cloud was the cleanest system in which Fermi pressure could be seen alone, because the atoms barely interact at all.

That cleanliness turned into a tool. A magnetic field near a Feshbach resonance can tune the interactions between lithium-6 atoms in two different spin states from attractive to repulsive at will. With strong attraction the fermions pair into molecules, which are bosons and condense; with weak attraction they form loosely bound pairs, like the electrons of a superconductor. The degenerate Fermi gas of 2001 became, within a few years, a system in which the crossover between those two kinds of superfluid could be studied continuously. The fermions that refused to shrink were the starting point for that.

Atoms that never touch, in a perfect bowl #

The atoms are ideal, with no interactions at all; real lithium-7 atoms attract one another weakly, which limited Hulet’s condensates to about a thousand atoms before they collapsed, and real lithium-6 atoms in one spin state interact only through weak higher partial waves. The trap is taken as perfectly harmonic and isotropic, with all frequencies equal; real magnetic traps are elongated, and the sizes then differ along each axis by the ratio of frequencies, without changing the comparison. The bosons and fermions are drawn with equal numbers and in the same trap with the same frequency, which makes the comparison clean but is not what the experiment had: the isotopes differ in mass and number, and their clouds’ Fermi and condensation temperatures differ accordingly. The condensate’s size is that of an ideal gas’s ground state; an interacting condensate is larger. The profile at a quarter of $T_F$ is computed in the local-density approximation, which treats each small region of the trap as a uniform gas, accurate for large numbers of atoms.

Still open: how cold a Fermi gas can be made #

The coldest degenerate Fermi gases now reach about a twentieth of their Fermi temperature in the bulk, and colder in optical lattices where the atoms are rearranged one by one. Below that, the thermometry itself becomes the limit: the profile’s shoulders change less and less as the temperature falls, and the distribution approaches its zero-temperature shape too closely for a fit to tell the difference. Measuring temperatures of a hundredth of $T_F$ — where theories predict exotic magnetic and superfluid phases in lattices of fermions — requires thermometers that read the few atoms still excited above the Fermi surface rather than the many below it, and how best to build them is an active question.

The habit worth carrying away is to ask how many particles a state will hold. A trapped classical cloud shrinks as the square root of its temperature; a Bose cloud condenses below half the Fermi temperature into the ground state; a Fermi cloud stops at $R_F/\sqrt 8$ with three-quarters of the Fermi energy in every atom — not because anything pushes the atoms apart, but because each state takes one. The photograph of 2001 is a photograph of a counting rule.