Bose–Einstein condensate

A Bose–Einstein condensate (BEC) is a state of matter that can form in a gas of bosons at very low temperatures, in which a macroscopic number of particles occupy the same single-particle quantum state. This collective occupation produces a macroscopic quantum wavefunction and allows quantum effects such as coherence, interference and superfluidity to appear on scales much larger than individual atoms. The phenomenon was predicted by Satyendra Nath Bose and Albert Einstein in 1924–1925 and first observed in dilute atomic vapours in 1995. For this achievement, Eric A. Cornell, Carl E. Wieman and Wolfgang Ketterle were awarded the Nobel Prize in Physics in 2001.

Historical background

In 1924 Bose derived Planck’s blackbody radiation law by treating photons as indistinguishable particles with a new statistical counting procedure. Einstein generalized this method to massive particles obeying what are now called Bose–Einstein statistics, and in 1925 he showed that below a critical temperature a finite fraction of non-interacting bosons must accumulate in the single lowest-energy quantum state. The predicted macroscopic occupation was initially regarded as a mathematical curiosity. In 1938 Fritz London suggested that the superfluidity of liquid helium-4 might be connected to this condensation; later work, especially the Penrose–Onsager criterion of 1956, established a generalized definition of BEC in interacting systems based on the one-body density matrix. However, liquid helium is strongly interacting, making the condensate fraction relatively small and difficult to observe directly. The search for an almost pure, weakly interacting BEC motivated the development of laser cooling and magnetic trapping of neutral atoms, culminating in the 1995 experiments.

Bose–Einstein statistics and critical conditions

Bosons are particles with integer spin that do not obey the Pauli exclusion principle. At thermal equilibrium the mean occupation number of a single-particle state with energy ε is given by the Bose–Einstein distribution:

n_i = 1 / [exp((ε_i − μ)/k_B T) − 1],

where μ is the chemical potential, k_B is Boltzmann’s constant, and T is temperature. For a gas of non-interacting bosons in a three-dimensional box, the total number of particles cannot exceed the number that can be accommodated in the excited states when T is low. The maximum number in excited states is

N_ex,max = V (2π m k_B T)^(3/2) / h^3 × ζ(3/2).

Setting this equal to N gives the critical temperature

T_c = (h^2 / 2π m k_B) [n / ζ(3/2)]^(2/3),

where n = N/V is the number density and ζ is the Riemann zeta function. Equivalently, condensation occurs when the phase-space density satisfies

n λ_T^3 = ζ(3/2) ≈ 2.612,

where λ_T = h / sqrt(2π m k_B T) is the thermal de Broglie wavelength. When the de Broglie wavelengths of particles overlap, their wave nature becomes collective.

For a harmonically trapped ideal gas, the transition temperature is

k_B T_c = ħ \bar{ω} [N / ζ(3)]^(1/3) ≈ 0.94 ħ \bar{ω} N^(1/3),

where \bar{ω} is the geometric mean of the trap frequencies. Above T_c the occupation of the ground state is negligible; below T_c a macroscopic fraction N_0/N = 1 − (T/T_c)^3 appears in the harmonic trap.

In an interacting gas the transition is shifted by interactions, but the essential condition remains that the thermal de Broglie wavelength becomes comparable to the interparticle spacing.

Macroscopic wavefunction and order parameter

A BEC is described by a complex order parameter

ψ(r,t) = sqrt(n_c(r,t)) e^{iφ(r,t)},

where n_c is the condensate density and φ is a phase. All condensed particles share this macroscopic wavefunction. In weakly interacting dilute atomic gases, ψ obeys the Gross–Pitaevskii equation:

iħ ∂ψ/∂t = [ −ħ²∇²/(2m) + V_ext(r) + g |ψ|² ] ψ,

where g = 4πħ² a/m and a is the s-wave scattering length. The nonlinear term represents mean-field interactions. The healing length ξ = 1/sqrt(8π n a) gives the distance over which the condensate density recovers to its bulk value after a local perturbation.

The existence of a macroscopic occupied single-particle state can be defined more generally through the one-body density matrix ρ_1(r, r′). A BEC is present if ρ_1 has an eigenvalue of order N, the total particle number; this is the Penrose–Onsager criterion. The corresponding eigenstate is the condensate orbital. A BEC possesses off-diagonal long-range order: ρ_1(r, r′) does not vanish as |r−r′| → ∞.

Experimental realization in atomic gases

The first atomic BECs were produced in 1995 in two laboratories. Cornell and Wieman at JILA cooled a gas of rubidium-87 atoms to about 170 nanokelvin, and Ketterle at MIT cooled sodium-23 atoms. The techniques combine laser cooling, which reduces the temperature of an atomic vapour to microkelvin levels, with magnetic trapping and evaporative cooling, in which the most energetic atoms are selectively removed so that the remaining gas thermalizes at lower temperature. Because alkali atoms are bosons and, at low density, remain gaseous rather than solidifying, they are well suited to this approach.

After condensation, the velocity distribution narrows sharply; in time-of-flight absorption imaging this appears as a dense peak of slow atoms above a broader thermal background. Later experiments produced condensates of other bosonic isotopes, including lithium-7, hydrogen, helium-4 in an excited metastable state, cesium, ytterbium, strontium, dysprosium and erbium, as well as molecules. Feshbach resonances allow the strength of interactions to be tuned, and optical traps and lattices offer flexible potentials.

Properties of Bose–Einstein condensates

Coherence and interference

A BEC has a well-defined phase over macroscopic distances. When two condensates are allowed to overlap, they produce high-contrast interference fringes, as observed by Ketterle’s group in 1997. This is a direct demonstration of first-order coherence. An atom laser can be produced by outcoupling a fraction of a trapped condensate; the resulting beam of atoms is coherent, analogous to an optical laser.

Superfluidity and vortices

A weakly interacting BEC is a superfluid: it can flow without dissipation below a critical velocity. The superfluid velocity is proportional to the gradient of the phase, v_s = (ħ/m)∇φ. Because the phase must be single-valued, the circulation around a closed path is quantized in units of h/m. Rotating a BEC can create a lattice of quantized vortices, each carrying one quantum of circulation. These vortices were observed in sodium and rubidium condensates and are a hallmark of superfluidity.

Collective excitations and sound

Small perturbations of a BEC are described by Bogoliubov theory. At long wavelengths the excitation spectrum is linear in wavevector k, corresponding to phonons, with speed c = sqrt(n g/m). At short wavelengths the spectrum becomes free-particle-like. The interplay between interactions and kinetic energy determines the speed of sound and the response to external perturbations. In the Thomas–Fermi regime, the kinetic energy term in the Gross–Pitaevskii equation can be neglected in the bulk, and the density profile mirrors the external trap.

Condensate fraction and fluctuations

In an ideal gas at zero temperature all particles are in the condensate. In an interacting dilute atomic BEC the zero-temperature condensate fraction is reduced by quantum depletion, typically by less than a few percent for alkali gases but much greater in strongly interacting systems such as superfluid helium-4, where the condensate fraction is about 10% at saturated vapour pressure. A BEC also exhibits reduced density fluctuations compared with a classical gas, although the phase and particle number obey an uncertainty relation.

BEC in other physical systems

Bose–Einstein condensation is not limited to dilute atomic gases. Superfluid helium-4 is often described as closely related to BEC, with a condensate fraction of roughly 10%; its strong interactions complicate the simple picture. Superconductors can be regarded as condensates of Cooper pairs, though the pairs are strongly overlapping in conventional BCS superconductors. In the BCS–BEC crossover of ultracold fermionic gases, pairs of fermionic atoms can be tuned from a BCS superfluid to a BEC of tightly bound molecules.

Quasiparticle condensates have also been observed. Exciton-polaritons in semiconductor microcavities have bosonic character at low density and can condense at temperatures much higher than atomic BECs, in some systems near room temperature. Photons in a dye-filled optical microcavity can thermalize and form a two-dimensional BEC. Magnons in magnetic insulators and other quasiparticles can also show condensation. These systems often differ in dimensionality, equilibrium character and interactions, but share the feature of a macroscopic occupation of a single quantum state.

Applications and research directions

Atomic BECs provide a controllable platform for quantum simulation. In optical lattices, atoms in a BEC can realize Hubbard models, allowing study of superfluid–Mott-insulator transitions and quantum magnetism. BEC-based atom interferometers are used in precision measurements of gravity, rotation and fundamental constants, and have been operated in microgravity. BECs are also used to study quantum turbulence, Anderson localization, solitons, topological excitations and analogue gravity. Research continues on dipolar condensates, mixtures, quantum droplets and quantum information applications.