The condensate a trap makes

On 5 June 1995, in a laboratory in Boulder, Colorado, a cloud of about two thousand rubidium atoms held in a magnetic trap was cooled to 170 billionths of a kelvin and then released. Photographed as it flew apart, its shadow had a shape nobody had seen in a gas: a broad, rounded cloud with a sharp narrow peak standing up from its centre. The peak was a few thousand atoms all in the same quantum state, the lowest one the trap allowed — a Bose–Einstein condensate, predicted by Einstein in 1925 for a gas and never before made from one. A group at MIT made a larger one from sodium four months later. Eric Cornell, Carl Wieman and Wolfgang Ketterle shared a Nobel Prize for it in 2001.

The liquid that will not slow down found the same statistics behind superfluid helium, but in helium the atoms are packed as tightly as in any liquid and interact strongly, and the condensate is a small, hidden part of a complicated whole. In a trapped gas the atoms are a hundred thousand times further apart and barely interact. The condensation there is close to Einstein’s ideal, and the reason for it can be followed with nothing but counting.

The prediction itself came from a letter. In 1924 Satyendra Nath Bose, in Dacca, sent Einstein a derivation of Planck’s radiation law that counted photons as indistinguishable rather than as labelled individuals. Einstein translated it, had it published, and within months applied the same counting to a gas of atoms. He found that below a certain temperature a fraction of the atoms must collect in the lowest state with no attraction between them to cause it — “a separation without attractive forces”, he wrote — and he was not sure it was physical. For seventy years it was a calculation about a gas that could not be made cold enough without first becoming a liquid or a solid.

How many atoms the excited states can hold #

Atoms of rubidium-87 are bosons: any number of them can occupy the same state, and when they are identical, they are counted as Bose and Einstein counted photons — the gas that nobody counted did it for light — not by labelling atoms but by asking how many occupy each state. The average number in a state of energy $\varepsilon$ is

$$\bar n(\varepsilon) = \frac{1}{e^{(\varepsilon - \mu)/kT} - 1},$$

where $\mu$, the chemical potential, is whatever number makes the total come out to the number of atoms. For a gas of bosons $\mu$ must lie below the lowest state’s energy, or that state’s occupation would be negative. That one constraint is the whole of the physics.

Take the lowest state’s energy as zero. Then $\mu$ is negative, and as it rises towards zero every excited state’s occupation rises, but only up to a limit: with $\mu = 0$ exactly, a state at $\varepsilon$ holds $1/(e^{\varepsilon/kT} - 1)$ atoms, and no more. Summed over every excited state, that limit is the most atoms the excited states can hold at temperature $T$. If there are more atoms than that, $\mu$ cannot rise further — it is pinned just below zero — and the excess has nowhere to go but the lowest state, whose occupation, $1/(e^{-\mu/kT} - 1)$, grows without bound as $\mu$ approaches zero.

Why a trap gives a cube #

How many atoms the excited states can hold depends on how many excited states there are at each energy, and that depends on the container. In a box, the number of states with energy up to $\varepsilon$ grows as $\varepsilon^{3/2}$ — the count of standing waves that fit, as for sound or light in a room. In a harmonic trap, the states are the levels of a three-dimensional oscillator, $\varepsilon = n\hbar\omega$, and the number of ways of sharing $n$ quanta among three directions is $(n+1)(n+2)/2$: the number of states up to $\varepsilon$ grows as $\varepsilon^3$. Summing the excited occupations against that count, the excited states of a trap can hold at most

$$N_{\text{ex,max}} = \zeta(3)\left(\frac{kT}{\hbar\omega}\right)^3$$

atoms, with $\zeta(3) = 1.202$. Setting that equal to the number of atoms gives the transition temperature, $kT_c = \hbar\omega(N/\zeta(3))^{1/3} = 0.94\,\hbar\omega N^{1/3}$, and below it the excited states are full and the ground state holds the rest:

$$\frac{N_0}{N} = 1 - \left(\frac{T}{T_c}\right)^3.$$

The cube is the trap’s signature. A gas in a box condenses with the three-halves law instead, more gradually below the transition, because a box’s states are fewer at low energy and its excited states fill more slowly as the temperature falls. The figure computes the fraction exactly, level by level, rather than from the formula: the chemical potential is found for each temperature by requiring the sum to equal the number of atoms, and the ground state’s share read off. For ten million atoms the exact sum is indistinguishable from the cube. For a thousand it is not. A small gas has its levels spaced coarsely compared with $kT_c$, so the sum is not well approximated by the smooth integral that gives $\zeta(3)$, and the transition is both rounded and moved down. Finite size is not a nuisance here but a measurable shift, and the early experiments with a few thousand atoms saw it.

Where the atoms are #

Below the transition, the atoms are in two very different places at once.

The excited levels hold their maximum, spread thinly over a great many states: each level near $kT$ contains hundreds of atoms but they are shared among hundreds of degenerate states, roughly one atom per state or fewer. The ground level holds sixty-three thousand atoms in a single state. That disproportion — one state with a macroscopic share of the gas, every other state with an atom or less — is what a condensate is, and nothing about the atoms’ forces on each other entered it.

The chemical potential that achieves it sits a few millionths of $kT$ below the ground state. It is a vanishingly small number, and it is the most delicate quantity in the problem: move it by a millionth of $kT$ and the ground state’s occupation changes by tens of thousands of atoms. The rest of the gas does not notice.

Seeing it #

A condensate in its trap is a few micrometres across, too small to image directly with the precision the early experiments needed. They released it instead. Switching off the trap lets every atom fly away with the velocity it had, and after a few milliseconds the cloud has expanded to many times its original size; a pulse of resonant laser light then casts its shadow onto a camera. The shadow’s size records the spread of the atoms’ velocities.

The thermal atoms have velocities spread as the speeds in a still room found for any gas, with a width set by the temperature, and they make a broad, rounded shadow. The condensed atoms are in the trap’s ground state, whose spread of momentum is the smallest the uncertainty principle allows for a wavefunction of its size; they expand far more slowly and make a narrow peak. In a rubidium trap at 100 hertz, the ground state is about a micrometre across and the thermal cloud at the transition about ten micrometres: the peak is ten times narrower than its base.

The bimodal shadow — a sharp peak on a broad base, appearing suddenly as the temperature passes $T_c$ — was the signature the 1995 papers reported, and every condensate since has been diagnosed the same way. The two components can be fitted separately, the base giving the temperature and the peak the number of condensed atoms, which is how the fraction against temperature has been measured and compared with the cube law. Real condensates are not quite ideal: the atoms’ weak repulsion spreads the condensate in the trap into a broader, parabolic shape — drawn in the figure — and shifts the transition by a few per cent, but the bimodal form survives.

One state means one wave #

The atoms in the peak share a single wavefunction, and that is a statement with consequences beyond counting. A single quantum state of a hundred thousand atoms has a phase, the same everywhere in it, as a laser beam does; two such states should interfere when they overlap, as two laser beams do. In 1997 the MIT group made two condensates side by side in a trap split by a sheet of laser light, released them, and photographed them as they expanded into each other. Where they overlapped the shadow was striped with fringes, fifteen micrometres apart — two clouds of atoms that had never touched, producing an interference pattern as clean as the one one arrival at a time built up from single particles, but made in one shot by a hundred thousand atoms at once.

The fringes ruled out a picture in which the “condensate” was merely a dense lump of atoms each going its own way. Atoms each going their own way have no common phase and give no fringes. The interference also made possible the atom laser: a condensate leaking atoms through an outcoupler into a coherent beam of matter, with the same relation to an ordinary atomic beam as a laser has to a lamp. Everything has a wavelength, and here a macroscopic number of atoms have the same one.

Turning it, and finding it superfluid #

A condensate with a single phase responds to rotation the way superfluid helium does. Its flow velocity is the gradient of its phase, so the circulation round any loop is fixed in whole quanta, and a rotating condensate cannot simply turn as a body: it nucleates quantised vortices, each a thread of zero density round which the phase winds once. The whirlpool that comes in one size found exactly this in helium; in a gas condensate, where the vortex cores are large enough to photograph after expansion, a stirred cloud shows a triangular lattice of dozens of holes, first seen in 2001. The lattice is the proof that the condensate is a superfluid and not only a population of one state: an ordinary gas stirred in the same way would simply rotate. And a condensate stirred slowly enough does not respond at all, the rule that the speed below which nothing can be made states for a superfluid — measured in gases by dragging a laser beam through a condensate and finding no heating below a critical speed.

How cold #

The transition temperature contains only the trap’s frequency and the number of atoms. For a million atoms in a trap of 100 hertz it is 451 billionths of a kelvin; for a thousand, 45.

The atom’s mass is absent from the formula, which is surprising if condensation is thought of as the atoms’ quantum wavelengths overlapping — heavier atoms have shorter wavelengths at a given temperature. The resolution is that a trap of given frequency holds heavier atoms in a smaller cloud. Their wavelengths are shorter but their spacing is smaller in proportion, and in a harmonic trap the two effects cancel exactly: sodium and rubidium in traps of the same frequency condense at the same temperature, the rubidium at a higher density.

Reaching these temperatures took two techniques. Laser cooling, the friction made of light, takes a vapour from room temperature to tens of microkelvin, until the recoil of individual photons — the limit that belonged to a simpler atom — stops it. The last factor of a hundred comes from evaporation: the trap’s walls are lowered so that the most energetic atoms escape, and the remainder, after colliding, rethermalise at a lower temperature. Each atom lost carries away more than the average energy. Evaporation throws away most of the gas — the 1995 condensate began as millions of atoms and ended as two thousand — but it reaches the nanokelvin regime where the cube law takes over.

The heat a transition needs #

A phase transition shows itself in the heat capacity, and the trapped gas has a sharp one.

Below $T_c$ the energy is all in the excited atoms, and as the temperature rises, heat goes both into giving them more energy and into lifting more atoms out of the condensate: the heat capacity rises as $T^3$, reaching more than three times the classical value at the transition. Above $T_c$ there is no condensate to draw atoms from, the heat only spreads an ordinary gas, and the heat capacity falls towards three per atom — $\tfrac32$ for the motion and $\tfrac32$ for the potential energy in the trap, the equipartition share of a three-dimensional oscillator. The fall at $T_c$ is a near-discontinuity for an ideal trapped gas, sharper than the cusp a box gives, and for a hundred thousand atoms it is already well formed. When the energy of a trapped rubidium cloud was measured against its temperature in 1996, the change of slope at the transition was there, and it was the thermodynamic evidence that what had been made was a phase, not merely a dense centre.

Bosons, and the fermions that cannot #

The counting argument depends on the atoms being bosons. A gas of fermions — atoms of potassium-40 or lithium-6, cooled the same way — cannot put two atoms in one state, as no two in the same state found for electrons, and as it is cooled it fills the trap’s levels one atom per state from the bottom up, with a sharp top, a Fermi energy, and a pressure that has nothing to do with temperature. Such degenerate Fermi gases were made in 1999. Pairing the fermions — by tuning their interactions with a magnetic field so that two fermions bind into a boson — produces condensates of pairs, and the crossover between tightly bound molecules that condense like Einstein’s bosons and loosely bound pairs that superconduct like electrons in a metal has been mapped in these gases since 2004.

What the figures leave out #

The figures treat the atoms as ideal: no interactions, a perfectly harmonic trap, thermal equilibrium at every temperature. Real alkali atoms repel each other weakly. The repulsion broadens the condensate in the trap from the ideal ground state’s Gaussian into a parabola several times wider, lowers the transition temperature by a few per cent, and changes how the condensate expands when released; the image figure draws the condensate with that parabolic shape and a radius growing as the fifth root of its atom number, but its thermal cloud is ideal. Traps are not isotropic: most are elongated, with two frequencies, and the formula uses their geometric mean. And the heat capacity is a calculation; the measurements give energy against temperature with uncertainties that blur its peak. The domain is a dilute gas of bosons at temperatures of tens to hundreds of nanokelvin, where interactions are a correction rather than the cause.

Still open: what interactions do to the transition itself #

The ideal gas’s transition temperature is a counting result; the shift interactions give it is not, and it has been surprisingly hard to pin down. In a uniform gas, weak repulsion raises the transition temperature, by an amount proportional to the scattering length that took calculations from the 1950s to the 2000s to settle — the leading coefficient was finally computed by numerical simulation around 2001 — and in a trap the larger effect, a lowering from the repulsion’s spreading of the cloud, competes with it. Measurements in uniform box traps, built since 2013 with walls of light, have begun to test the uniform-gas result directly, and what happens to the transition’s critical behaviour as interactions grow is an active field. The ideal result, that counting alone condenses a gas, stands underneath all of it.

The excited levels of a harmonic trap can hold at most $\zeta(3)(kT/\hbar\omega)^3$ bosons, so below $kT_c = 0.94\,\hbar\omega N^{1/3}$ — 451 nanokelvin for a million atoms at 100 hertz — the excess has nowhere to go but the single lowest state, whose share grows as $1 - (T/T_c)^3$, 63 per cent of $10^5$ atoms at 0.7 $T_c$; it shows as a narrow peak rising from the released cloud’s broad shadow, and as a heat capacity that peaks at three times the classical value. A crowd of noninteracting atoms collects in one state because there is no room for it anywhere else.