---
title: Bose-Einstein Condensation of Atomic Gases
module: Modern Atomic Physics
moduleNumber: 9
lessonNumber: 3
order: 903
summary: >
  Below a critical temperature a gas of identical bosons places a macroscopic
  fraction of its atoms in the single lowest-energy state. The transition occurs when
  the thermal de Broglie wavelength grows to the interparticle spacing, so the phase-
  space density reaches order unity. The critical temperature follows from the
  Bose-Einstein distribution and the density of states, the condensate fraction grows
  as one minus (T/Tc) to the three-halves, and the condensate reveals itself in
  time-of-flight as a sharp bimodal peak in the momentum distribution. The 1995
  rubidium and sodium experiments realized it in dilute trapped gases.
topics: [Modern Atomic Physics]
sources:
  - book: Foot
    ref: "Ch. 10 — Magnetic Trapping, Evaporative Cooling and Bose-Einstein Condensation; §10.2 The Bose-Einstein Condensation, §10.4 The Condensate Wavefunction"
  - book: Demtröder
    ref: "Ch. 11 — Bose-Einstein Condensation (concepts)"
  - book: Tipler & Llewellyn
    ref: "Ch. 8 — Statistical Physics; the Bose-Einstein condensation section"
draft: false
---

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
$\Delta p \sim \sqrt{Mk_{\mathrm B}T}$, and by the uncertainty principle a spatial
coherence length — the **thermal de Broglie wavelength**

$$
\lambda_{\mathrm{dB}} = \frac{h}{\sqrt{2\pi M k_{\mathrm B}T}}
= \sqrt{\frac{2\pi\hbar^2}{M k_{\mathrm B}T}}.
$$

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**

$$
\varrho = n\,\lambda_{\mathrm{dB}}^3,
$$

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.

> **Definition (Phase-space density).** The number of atoms within a cube of side the
> thermal de Broglie wavelength, $\varrho = n\lambda_{\mathrm{dB}}^3$. A classical gas
> has $\varrho \ll 1$; Bose-Einstein condensation in a uniform gas begins at
> $\varrho = \zeta(3/2) \approx 2.612$. Every cooling stage in a cold-atom experiment
> is measured by how much it raises $\varrho$.

$$
% caption: As the gas cools the thermal de Broglie wavelength grows until the atomic
% wavepackets overlap; condensation begins when the phase-space density n times
% lambda-cubed reaches about 2.6.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % hot: small wavepackets, far apart (left box)
  \draw[black] (-3.4,-1.4) rectangle (-0.4,1.4);
  \node[black, anchor=south] at (-1.9,1.45) {hot: isolated points};
  \foreach \p in {(-3.0,0.9),(-2.2,-0.6),(-1.4,0.5),(-0.8,-1.0),(-2.6,-1.0),(-1.0,0.9)}
    \fill[acc] \p circle (1.6pt);
  % arrow
  \draw[->, black, very thick] (-0.1,0) -- (0.9,0) node[midway, above] {cool};
  % cold: overlapping wavepackets (right box)
  \draw[black] (1.2,-1.4) rectangle (4.2,1.4);
  \node[black, anchor=south] at (2.7,1.45) {cold: overlapping waves};
  \foreach \p in {(1.6,0.9),(2.4,-0.6),(3.2,0.5),(3.6,-1.0),(2.0,-1.0),(2.8,0.9)}
    \draw[acc, thick] \p circle (0.42);
\end{tikzpicture}
$$

## The critical temperature

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

$$
f(\varepsilon) = \frac{1}{e^{(\varepsilon - \mu)/k_{\mathrm B}T} - 1},
$$

with the chemical potential $\mu$ fixed by requiring the occupations sum to $N$. For
bosons $\mu < \varepsilon_0$ always, and as $T$ falls $\mu$ rises toward the
ground-state energy $\varepsilon_0$. Separate the ground state from the rest and count
the excited-state atoms with the free-particle density of states
$g(\varepsilon) = \frac{V}{4\pi^2}\left(\frac{2M}{\hbar^2}\right)^{3/2}\varepsilon^{1/2}$:

$$
N_{\mathrm{exc}} = \int_0^\infty
\frac{g(\varepsilon)\,\d\varepsilon}{e^{(\varepsilon - \mu)/k_{\mathrm B}T} - 1}.
$$

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

$$
N_{\mathrm{exc}}^{\max}
= \frac{V}{4\pi^2}\left(\frac{2Mk_{\mathrm B}T}{\hbar^2}\right)^{3/2}
\int_0^\infty \frac{x^{1/2}\,\d x}{e^x - 1}.
$$

The integral is $\int_0^\infty \frac{x^{1/2}}{e^x - 1}\d x
= \Gamma(3/2)\,\zeta(3/2) = \tfrac{\sqrt\pi}{2}\,\zeta(3/2)$. Collecting constants and
writing $n = N/V$,

$$
N_{\mathrm{exc}}^{\max} = V\,\zeta(3/2)\left(\frac{Mk_{\mathrm B}T}{2\pi\hbar^2}\right)^{3/2}
= \frac{V\,\zeta(3/2)}{\lambda_{\mathrm{dB}}^3}.
$$

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**

$$
k_{\mathrm B}T_c = \frac{2\pi\hbar^2}{M}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}
\approx 3.31\,\frac{\hbar^2 n^{2/3}}{M}.
$$

> **Definition (Bose-Einstein condensation).** The macroscopic occupation of the
> single lowest-energy single-particle state that occurs in an ideal Bose gas below
> the critical temperature $T_c$, at which the phase-space density reaches
> $\zeta(3/2)$. Below $T_c$ the ground-state population $N_0$ is a finite fraction of
> $N$; it is a phase transition driven by quantum statistics, not by interactions.

For a dilute trapped gas the numbers are extreme. A rubidium-87 cloud at density
$n = 2.6\times10^{13}\ \mathrm{cm^{-3}}$ 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
$V(\vec r) = \tfrac12 M(\omega_x^2 x^2 + \omega_y^2 y^2 + \omega_z^2 z^2)$, and the
density of states of a three-dimensional oscillator differs from the free-particle
one: $g(\varepsilon) = \varepsilon^2/2(\hbar\bar\omega)^3$ 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 $\int x^{1/2}/(e^x-1)\,\d x$
by $\int x^2/(e^x-1)\,\d x = 2\zeta(3)$, so the transition condition becomes
$N = \zeta(3)\,(k_{\mathrm B}T_c/\hbar\bar\omega)^3$ and

$$
k_{\mathrm B}T_c = \hbar\bar\omega\left(\frac{N}{\zeta(3)}\right)^{1/3},
\qquad \zeta(3) = 1.202,
$$

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

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

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

$$
N_{\mathrm{exc}}(T) = V\,\zeta(3/2)\left(\frac{Mk_{\mathrm B}T}{2\pi\hbar^2}\right)^{3/2}
= N\left(\frac{T}{T_c}\right)^{3/2},
$$

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

$$
\frac{N_0}{N} = 1 - \frac{N_{\mathrm{exc}}}{N}
= 1 - \left(\frac{T}{T_c}\right)^{3/2},
\qquad T < T_c.
$$

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.)

$$
% caption: Condensate fraction versus temperature for a uniform ideal Bose gas: zero
% above Tc, rising as one minus (T over Tc) to the three-halves below it, reaching
% unity at absolute zero.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (5.4,0) node[right] {$T/T_c$};
  \draw[->, black] (0,0) -- (0,3.2) node[above] {$N_0/N$};
  % y=1 gridline
  \draw[dashed, black] (0,2.6) -- (5.0,2.6);
  \node[black, anchor=east] at (-0.05,2.6) {1};
  % Tc tick at x=4
  \draw[black] (4.0,-0.08) -- (4.0,0.08);
  \node[black, anchor=north] at (4.0,-0.1) {1};
  % curve: N0/N = 1 - (T/Tc)^1.5 for x in [0,4], scaled: x-> T/Tc = x/4, y scaled by 2.6
  \draw[acc, very thick, domain=0:4, samples=140, smooth]
    plot (\x, {2.6*(1 - (\x/4)^1.5)});
  % zero line above Tc
  \draw[acc, very thick] (4.0,0) -- (5.0,0);
  \node[acc, anchor=south west] at (0.2,2.1) {condensate};
\end{tikzpicture}
$$

## The macroscopic wave function

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

$$
\psi(\vec r) = \sqrt{n_0(\vec r)}\;e^{i\phi(\vec r)},
\qquad n_0 = |\psi|^2,
$$

the **macroscopic wave function** or **order parameter**. Its modulus gives the
condensate density and its phase $\phi$ 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**,

$$
i\hbar\frac{\partial\psi}{\partial t}
= \left[-\frac{\hbar^2}{2M}\nabla^2 + V_{\mathrm{trap}}(\vec r)
+ g\,|\psi|^2\right]\psi,
\qquad g = \frac{4\pi\hbar^2 a_s}{M},
$$

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 $n = 10^{14}\ \mathrm{cm^{-3}}$ it is
$n a_s^3 \approx 1.5\times10^{-5} \ll 1$, 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**

$$
\xi = \frac{1}{\sqrt{8\pi n a_s}}
$$

is the distance over which the wave function recovers from a localized perturbation —
the balance point between kinetic energy $\hbar^2/2M\xi^2$ 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 $\xi \approx 0.3\ \mathrm{\mu m}$,
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.

$$
% caption: Time-of-flight momentum distribution across the transition: above Tc a
% single broad thermal Gaussian; below Tc a sharp narrow condensate peak rises on top
% of the shrinking thermal pedestal, the bimodal signature.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % left: above Tc, broad only
  \draw[->, black] (-3.9,0) -- (-0.4,0) node[right] {$p$};
  \draw[black, thick, domain=-3.6:-0.6, samples=120, smooth]
    plot (\x, {1.4*exp(-2.3*(\x+2.1)*(\x+2.1))});
  \node[black, anchor=south] at (-2.1,1.4) {above $T_c$};
  % right: below Tc, narrow peak on broad pedestal
  \draw[->, black] (0.6,0) -- (4.1,0) node[right] {$p$};
  \draw[black, thick, domain=0.7:3.9, samples=120, smooth]
    plot (\x, {0.7*exp(-2.3*(\x-2.3)*(\x-2.3))});
  \draw[acc, very thick, domain=1.5:3.1, samples=140, smooth]
    plot (\x, {2.4*exp(-22*(\x-2.3)*(\x-2.3)) + 0.7*exp(-2.3*(\x-2.3)*(\x-2.3))});
  \node[acc, anchor=south] at (2.3,2.35) {below $T_c$};
  \node[black, anchor=west] at (3.0,0.5) {thermal};
\end{tikzpicture}
$$

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.[^history]

- **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 $5\times10^5$ 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](/atomic-physics/modern-atomic-physics/optical-clocks-precision)
draws on the same trapping and coherence techniques.

$$
% caption: The cold-atom pipeline as a climb in phase-space density: a MOT and
% molasses raise n-lambda-cubed toward ten-to-the-minus-six, and evaporative cooling
% carries it the last six orders of magnitude across the degeneracy threshold at 2.6.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.6,0) node[right] {phase-space density};
  \draw[->, black] (0,0) -- (0,3.0) node[above] {progress};
  % stages as rising staircase
  \fill[acc] (1.0,0.5) circle (2.4pt); \node[acc, anchor=south] at (1.0,0.6) {MOT};
  \fill[acc] (2.6,1.1) circle (2.4pt); \node[acc, anchor=south] at (2.6,1.2) {molasses};
  \draw[acc, very thick, ->] (3.2,1.3) -- (6.4,2.4);
  \node[acc, anchor=south east] at (4.4,2.25) {evaporation};
  \fill[black] (6.8,2.6) circle (2.6pt);
  \node[black, anchor=south] at (6.8,2.7) {degeneracy};
  \draw[black, dashed] (6.8,0) -- (6.8,2.6);
  \node[black, anchor=north] at (6.8,-0.05) {threshold};
\end{tikzpicture}
$$

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.

[^history]: **Foot**, _Atomic Physics_, §10.2, gives the ideal-gas derivation of $T_c$ and the condensate fraction and reviews the 1995 experiments: M. H. Anderson et al. (rubidium, JILA), _Science_ **269**, 198 (1995), and K. B. Davis et al. (sodium, MIT), _Phys. Rev. Lett._ **75**, 3969 (1995). The 2001 Nobel Prize went to Cornell, Ketterle, and Wieman. Phase-space threshold $\zeta(3/2) = 2.612$ and the density-of-states integral are standard results of Bose-Einstein statistics.