Bose-Einstein Condensation of Atomic Gases

A gas of identical bosons cooled to the degenerate regime does something no classical gas can: a finite fraction of all the atoms collapses into the single lowest-energy single-particle state, and that fraction grows toward one as the temperature falls. The phenomenon follows from Bose-Einstein statistics alone — it needs no interaction between the atoms — and it sets in when the atoms' quantum wavepackets, of size the thermal de Broglie wavelength, grow large enough to overlap. This lesson derives the transition temperature, the temperature dependence of the condensate fraction, and the momentum-space signature by which the 1995 experiments identified it.

The phase-space criterion

At temperature $T$ an atom of mass $M$ has a thermal spread of momenta $\Delta p\sim \sqrt{M{k}_{\mathrm{B}}T}$, and by the uncertainty principle a spatial coherence length — the thermal de Broglie wavelength

Classically, atoms are point-like and ${\lambda }_{\mathrm{dB}}$ is irrelevant. As the gas cools, ${\lambda }_{\mathrm{dB}}$ grows as $T^{-1/2}$, and when it becomes comparable to the mean interparticle spacing $n^{-1/3}$ the wavepackets overlap and the atoms can no longer be treated as distinguishable. The dimensionless measure is the phase-space density

the mean occupation of a phase-space cell of volume $h^3$. Condensation sets in when $\varrho$ reaches order unity; the exact threshold for a uniform ideal Bose gas is ${\varrho }_c=\zeta (3/2)=2.612$, where $\zeta$ is the Riemann zeta function.

The critical temperature

Take an ideal gas of $N$ bosons in a volume $V$. The mean occupation of a single-particle state of energy $\epsilon$ is the Bose-Einstein distribution

with the chemical potential $\mu$ fixed by requiring the occupations sum to $N$. For bosons $\mu <{\epsilon }_0$ always, and as $T$ falls $\mu$ rises toward the ground-state energy ${\epsilon }_0$. Separate the ground state from the rest and count the excited-state atoms with the free-particle density of states $g(\epsilon )=\frac{V}{4{\pi }^2}{\left(\frac{2M}{{\hslash }^2}\right)}^{3/2}{\epsilon }^{1/2}$:

The maximum number the excited states can hold occurs at $\mu =0$ (its ceiling). Set $\mu =0$ and substitute $x=\epsilon /k_{\mathrm{B}}T$:

The integral is ${\int }_0^{\infty }\frac{x^{1/2}}{e^x-1}\mathbf{d}x=\Gamma (3/2)\zeta (3/2)=\frac{\sqrt{\pi }}{2}\zeta (3/2)$. Collecting constants and writing $n=N/V$,

When the actual number $N$ exceeds this ceiling, the excess atoms have nowhere to go but the ground state, and they accumulate there. The transition is where the ceiling equals $N$, i.e. where $n{\lambda }_{\mathrm{dB}}^3=\zeta (3/2)$. Solving for temperature gives the critical temperature

For a dilute trapped gas the numbers are extreme. A rubidium-87 cloud at density $n=2.6\times 10^{13}\mathrm{c}{\mathrm{m}}^{-3}$ condenses at $T_c\approx 160\mathrm{nK}$ — a temperature at which the atoms move at millimetres per second and ${\lambda }_{\mathrm{dB}}$ has grown to hundreds of nanometres, comparable to the spacing between atoms. The gas is a hundred thousand times more dilute than air yet cold enough to be quantum-degenerate.

The harmonic-trap transition

Experiments condense atoms not in a box but in a harmonic trap $V(\vec{r})=\frac{1}{2}M({\omega }_x^2{x}^2+{\omega }_y^2{y}^2+{\omega }_z^2{z}^2)$, and the density of states of a three-dimensional oscillator differs from the free-particle one: $g(\epsilon )={\epsilon }^2/2(\hslash \bar{\omega }{)}^3$ with the geometric-mean frequency $\bar{\omega }=({\omega }_x{\omega }_y{\omega }_z{)}^{1/3}$. Repeating the excited-state count with $\mu =0$ replaces the integral $\int x^{1/2}/(e^x-1)\mathbf{d}x$ by $\int x^2/(e^x-1)\mathbf{d}x=2\zeta (3)$, so the transition condition becomes $N=\zeta (3)(k_{\mathrm{B}}{T}_c/\hslash \bar{\omega }{)}^3$ and

now depending on atom number rather than density. The condensate fraction picks up the larger exponent set by the same density of states,

a steeper onset than the box's $3/2$. For $N=10^6$ atoms in a trap of mean frequency $\bar{\omega }/2\pi =100\mathrm{Hz}$ this gives $T_c\approx 450\mathrm{nK}$, in the range the experiments reach by evaporation. The trap also concentrates the gas at the centre, so the condensate appears there first, as a dense core inside the more diffuse thermal cloud.

The condensate fraction

Below $T_c$ the number in excited states is still capped at the ceiling evaluated at the current temperature (with $\mu \approx 0$), so

using the definition of $T_c$ to eliminate the constants. The remainder condenses:

The condensate fraction rises from zero at $T_c$ toward one as $T\to 0$, with an infinite-slope onset characteristic of the transition. (In a harmonic trap the density of states differs and the exponent becomes $3$ rather than $3/2$; the qualitative behaviour is unchanged.)

The macroscopic wave function

The condensed atoms all occupy the same single-particle state, so a single complex function describes them collectively:

the macroscopic wave function or order parameter. Its modulus gives the condensate density and its phase $\varphi$ is common to $10^5$ atoms, the sense in which a condensate is a coherent matter wave. With weak interactions the wave function obeys the Gross-Pitaevskii equation,

a nonlinear Schrödinger equation in which the mean-field term $g|\psi {|}^2$ encodes the $s$-wave scattering length $a_s$. In a trap large enough that the kinetic term is negligible against interactions (the Thomas-Fermi regime), the density follows the inverted trap potential, $n_0(\vec{r})=[\mu -V_{\mathrm{trap}}(\vec{r})]/g$, a smooth parabolic profile quite unlike a thermal Gaussian, cut off at the Thomas-Fermi radius $R$ where $V_{\mathrm{trap}}(R)=\mu$.

Interactions and the healing length

Two length scales govern the interacting condensate. The diluteness parameter$n{a}_s^3$ measures how far the gas is from a liquid: for rubidium ($a_s\approx 5.3\mathrm{nm}$) at $n=10^{14}\mathrm{c}{\mathrm{m}}^{-3}$ it is $n{a}_s^3\approx 1.5\times 10^{-5}\ll 1$, so the atoms are almost always outside each other's range of interaction and the mean-field $g|\psi {|}^2$ captures the physics. The healing length

is the distance over which the wave function recovers from a localized perturbation — the balance point between kinetic energy ${\hslash }^2/2M{\xi }^2$ and interaction energy $gn$. It sets the size of a vortex core and the width of the boundary layer at the condensate edge; for the rubidium numbers above $\xi \approx 0.3\mu \mathrm{m}$, smaller than the cloud but far larger than the atomic scale. That separation of scales is why a dilute condensate is described by one smooth macroscopic wave function rather than by the positions of $10^5$ individual atoms.

The momentum-space signature

The condensate was identified not in position but in momentum. Releasing the trap lets the cloud expand freely; after a time-of-flight $t$ the spatial distribution maps the initial momentum distribution, because each atom flies a distance $\vec{p}t/M$. A thermal cloud expands isotropically into a broad Gaussian of width set by $T$. The condensate, occupying the ground state, has a momentum spread limited only by its finite spatial size (the uncertainty principle), so it expands into a much narrower — and, from an anisotropic trap, anisotropic — peak.

The bimodal profile — a sharp central spike riding on a broad thermal pedestal — is the fingerprint of the condensate, and its emergence below $T_c$ was the evidence in the first two realizations.

The 1995 realizations

Bose-Einstein condensation in a dilute atomic gas was achieved in 1995 after seventy years as a theoretical prediction.¹

• Rubidium-87 at JILA (Boulder): a magnetically trapped cloud evaporatively cooled to about $170\mathrm{nK}$, producing a condensate of roughly $2000$ atoms, observed by the bimodal time-of-flight peak.
• Sodium-23 at MIT: a much larger condensate, of order $5\times 10^5$ atoms, in a trap with an optical plug to block Majorana losses at the field zero.

Both groups saw the condensate appear as a narrow peak growing out of the thermal background exactly as the temperature crossed $T_c$, and the work was recognized with the 2001 Nobel Prize in Physics. Degenerate gases opened experimental access to matter-wave interference, superfluidity in a dilute gas, quantized vortices, and — with Fermi atoms cooled the same way — the crossover between a molecular condensate and a paired superfluid. The clock physics of the next lesson draws on the same trapping and coherence techniques.

The transition is a consequence of statistics that has no analogue for classical particles: identical bosons prefer to share a state, and below $T_c$ that preference becomes macroscopic. The critical temperature marks where the thermal wavelength reaches the interparticle spacing, the condensate fraction measures how far below that mark the gas has been cooled, and the momentum distribution makes the coherent matter wave directly visible.