17.3 Bose-Einstein condensation

Bose-Einstein Condensation

Bose-Einstein condensation (BEC) is a phase transition in which a macroscopic number of bosons collapse into the single lowest-energy quantum state. It bridges statistical mechanics and quantum mechanics: the same Bose-Einstein distribution function that describes photon or phonon statistics also predicts this dramatic collective phenomenon when a boson gas is cooled below a critical temperature.

Properties of Bose-Einstein condensation

Because every particle occupies the same ground state, a BEC behaves as a single, coherent quantum object on a macroscopic scale. Several striking properties follow from this:

• Macroscopic quantum coherence. All particles share the same quantum phase, so quantum effects normally hidden at atomic scales become directly observable. Superconductivity (in paired-electron systems) and superfluidity are both manifestations of this coherence.
• Superfluidity. The condensate can flow without viscous dissipation. Liquid helium-4 below the lambda point ($T_\lambda \approx 2.17\,\text{K}$) is the classic example, though dilute-gas BECs show the same behavior.
• Matter-wave interference. Two overlapping condensates produce interference fringes, just as two coherent laser beams do. This confirms that the condensate is described by a single macroscopic wavefunction with a well-defined phase.

Critical temperature for condensation

The Bose-Einstein distribution gives the mean occupation number of a single-particle state with energy $E$:

$f(E) = \frac{1}{e^{(E - \mu) / k_B T} - 1}$

Here $\mu$ is the chemical potential, $k_B$ is Boltzmann's constant, and $T$ is the temperature. As $T$ drops, $\mu$ rises toward zero (for a non-interacting gas in a box). When $\mu \to 0$, the ground-state occupation diverges and condensation begins.

Setting $\mu = 0$ and summing over excited states gives the critical temperature:

$T_c = \frac{2\pi\hbar^2}{m k_B}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}$

where $\hbar$ is the reduced Planck constant, $m$ is the particle mass, $n$ is the number density, and $\zeta(3/2) \approx 2.612$ is the Riemann zeta function evaluated at 3/2.

A few things to notice in this formula:

• $T_c$ increases with particle density $n$. Pack more bosons into the same volume and condensation happens at a higher temperature.
• $T_c$ decreases with particle mass $m$. Heavier particles have shorter thermal de Broglie wavelengths at a given temperature, so they need to be colder before their wavefunctions overlap enough to condense.
• For dilute alkali gases at typical lab densities ($n \sim 10^{13}\,\text{cm}^{-3}$), $T_c$ falls in the nanokelvin range, which is why extreme cooling techniques are required.

Experimental realization of condensates

BEC was first achieved in 1995 by Eric Cornell and Carl Wieman at JILA (Boulder, CO) using a gas of rubidium-87 atoms cooled to about 170 nK. They combined laser cooling to slow the atoms and magnetic evaporative cooling to reach temperatures below $T_c$. Wolfgang Ketterle at MIT independently produced a sodium BEC shortly after. All three shared the 2001 Nobel Prize in Physics.

Since then, condensates have been produced in many atomic species:

• Alkali metals: rubidium-87, sodium-23, lithium-7
• Other systems: atomic hydrogen, metastable helium-4, and even molecular condensates

Applications of BEC span several areas:

• Precision measurement. Atom interferometers based on BECs achieve extreme sensitivity, with applications in atomic clocks and tests of fundamental physics (e.g., equivalence principle tests, gravitational wave detection proposals).
• Quantum simulation. Optical lattices loaded with condensates can mimic solid-state systems, letting researchers study quantum phase transitions (like the superfluid-to-Mott-insulator transition) in a highly controllable setting.
• Quantum information. The coherence of a condensate is being explored for quantum computing and quantum cryptography protocols, though practical devices remain an active research frontier.

Condensates vs. classical states