Bose–Einstein Condensation 71

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Bose–Einstein Condensation

A.J. Leggett

Bose–Einstein condensation (BEC) is a phenomenon that occurs in a macroscopicsystem of bosons (particles obeying � Bose–Einstein statistics) at low temperatures:a nonzero fraction of all the particles in the system (thus a macroscopic number ofparticles) occupy a single one-particle state. This would, of course, happen for asystem of distinguishable, noninteracting particles at zero temperature, but in thiscase the phenomenon disappears as soon as the temperature becomes comparableto the energy splitting between the single-particle groundstate and the first excitedstate – a quantity which tends to zero with the size of the system. By contrast,in BEC the macroscopic occupation occurs at all temperatures below a transitiontemperature, usually denoted Tc, which while a function of intensive parameterssuch as density and interaction strength is constant in the thermodynamic limit.

The fundamental reason for the occurrence of BEC lies in the requirement, whichfollows from considerations of quantum field theory, that the � wave function of asystem of identical bosons should be symmetric under the exchange of any two par-ticles. This has the consequence that states that differ only by such an exchangemust be counted as identical, i.e. counted only once. Thus, for example, while for asystem of N distinguishable objects, which must be partitioned between two boxes,the number of ways of putting M of them into one box is given by the familiar bino-mial formula N !/(M!N −M!), for bosons there is exactly one way for each M . Theeffect is to remove the “entropic” factor, which for distinguishable objects militatesagainst putting a large fraction of them in a single one-particle state.

For noninteracting bosons in thermal equilibrium at temperature T a calculationof the average number of particles 〈ni〉 occupying the various single-particle statesi is straightforward and was carried out by Albert Einstein (1879–1955) [1] in 1925on the basis of the statistics derived by Satyendra Nath Bose (1894–1974) [2] a yearearlier:

〈ni〉 = {[exp(εi − μ)/kBT ] − 1}−1 (1)

where μ is the chemical potential,which must be fixed by the condition

∑i

〈ni〉 = N (2)

where N is the total number of particles present. In order to make sense of (1), it isclear that the chemical potential must be negative (we set the lowest single-particleenergy to zero by convention); since the LHS of (2) is an increasing function of μ,it follows that if in it we take the value of 〈ni〉 for μ = 0, the equality must bereplaced by an inequality. Thus, if we were to replace the sum by an integral andintroduce the single-particle density of states ρ(ε) in the standard way, we wouldfind the condition

72 Bose–Einstein Condensation

∫ ∞

0

ρ(ε)dε

exp(ε/kBT )− 1� N (3)

However, if ρ(ε) tends to zero with ε, as happens for a gas in three-dimensionalfree space, this condition cannot be fulfilled below a certain “critical temperature”Tc, which for 3D free space is given by

Tc = 3 · 31n2/3�

2/m (4)

where n = N/V is the density.What then happens for temperatures T < Tc? According to Einstein, while for

the states with εi > 0 the sum can still be legitimately replaced by an integral, thezero-energy state (the single-particle groundstate) must be taken out and handledseparately. In fact, the difference – call it N0 – between the right and left sides of (3),which is proportional to N and for T < Tc is positive, is the number of particleswhich occupy the groundstate. Thus a single state, in this case the single-particlegroundstate, is occupied by a macroscopic number of particles – the phenomenonof BEC. Note that for free particles in d dimensions, BEC does not occur for d � 2,since in this case the LHS of (3) is divergent and the equation is trivially satisfied atany nonzero value of T . For a free gas in 3D the condensate fraction is given by theformula

N0(T )/N = 1− (T /Tc)3/2 (5)

and so tends to 1 as T tends to 0.Since in real life many-particle systems are rarely noninteracting and in addition

may not be in thermal equilibrium, it is desirable to have a more general definition ofBEC. Such a definition was formulated by Oliver Penrose (*1929) and Lars Onsager(1903–1976): If we choose any complete � orthonormal basis (in general time-dependent) of single-particle wave functions χi(r : t), then we can define in thisbasis the single-particle density matrix ρij (t) ≡ 〈a†

iaj 〉(t). Since the matrix ρ̂(t) isHermitian, general theorems guarantee that for any given time t we will be able tofind a basis which diagonalizes it, i. e. such that

ρij (t) = δij 〈ni〉(t) (6)

If one and only one1 of the eigenvalues 〈ni〉 (call the relevant value of i 0 by conven-tion) is of order N while all the rest are all of order 1, then we say that the systempossesses the property of Bose–Einstein condensation (BEC); the quantity 〈n0〉 (of-ten written N0) is called the “condensate number” (so that N0/N is the “condensatefraction”), and the associated eigenfunction of ρ̂(t), χ0(r), is called the “condensatewave function.” Note that in the general case both N0 and χ0(r) may be functionsof time.

1 It is possible, though for various reasons uncommon, for more than one eigenvalue to be oforder N . In this case the system is said to possess “fragmented BEC.”

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There are strong arguments that the occurrence of BEC should lead to the phe-nomenon of superfluidity (� Superfluidity), so that when the latter phenomenonwas detected, in 1938, in He-II (the phase of liquid 4He below the so-called lambda-temperature, about 2.17 K), it was almost immediately suggested by Fritz Londonthat BEC is occurring in this phase. This conjecture is now almost universallybelieved to be correct, and although the strong and mostly repulsive interatomicinteractions in liquid helium prevent the direct observation of the onset of BECwhich is possible in the alkali gases (see below), it has proved possible (with cer-tain caveats, see e.g. ref. [3]) to observe a nonzero condensate fraction N0(T )/N

by high-energy neutron scattering and other experiments; it increases from zero atthe lambda-temperature to about 8% at T = 0. (By contrast, the superfluid fractionis 100% at T = 0). The strong “depletion” of the condensate fraction relative to itsvalue for the free gas is believed to be due to the strong interactions occurring inthis high-density system.

A second system in which BEC has been achieved is the bosonic atomic alkaligases2. Since (neutral) alkali atoms by definition have an odd number of electrons,odd-A alkali isotopes such as 87Rb, 23Na or 7Li are composed of an even num-ber of fermions and thus behave, as wholes, as bosons; at the densities currentlyrealized the transition temperature Tc to the BEC phase is predicted to be of theorder of a microkelvin, a temperature now relatively easily reached by laser coolingand rf evaporation techniques. These gases are normally held in trapping potentials(generated by magnetic fields or lasers) that are harmonic in form, and in such ageometry the effect of the onset of BEC is spectacular: Above Tc the density dis-tribution in the trap is approximately Gaussian, with a large value of the halfwidth.If the atoms were noninteracting, then below Tc a nonzero fraction would occupythe single-particle groundstate of the harmonic potential, which has a very muchnarrower width. In real life this effect is reduced owing to the repulsive interatomicinteractions, but one still sees a sharp “spike” in the density appear below Tc, see e.g.ref. [4]; this is probably the most convincing evidence that BEC is indeed occurringin these systems as theory confidently predicts.

In contrast to liquid helium, the atomic alkali gases are very dilute, and thus theeffects of the interatomic interactions are generally rather weak and can be handledby perturbation theory. Thus it has been possible to achieve a very good quantitativeunderstanding of the effects of BEC in these systems.3

Primary Literature

1. A. Einstein: On the quantum theory of the perfect gas. Abh. d. Preuss. Akad. Wiss. 1, 3 and 3,18 (1925).

2. S. N. Bose: Heat-equilibrium in radiation field in presence of matter. Z. Phys. 27, 384 (1924).

2 BEC has also been realized recently in a few non-alkali bosonic gases.3 This material is based upon work supported by the National Science Foundation under AwardNo. NSF-DMR-03-50842.

74 Bose–Einstein Statistics

3. W. M. Snow, P. E. Sokol: Density and temperature dependence of the momentum distribution inliquid He 4. J. Low Temp. Phys. 101, 881 (1995).

4. M. R. Andrews, M-O. Mewes, N. J. Van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle: Direct,nondestructive observation of a Bose condensate. Science 273, 84 (1996).

Secondary Literature

5. L. P. Pitaevskii, S. Stringari: Bose–Einstein Condensation. Oxford University Press, Oxford2003.

6. A. J. Leggett: Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed MatterSystems. Oxford University Press, Oxford 2006.

7. C. J. Pethick, H. Smith: Bose Einstein Condensation in Dilute Gases. Cambridge UniversityPress, Cambridge 2003.

Bose–Einstein Statistics

Arianna Borrelli

Bose–Einstein statistics is a procedure for counting the possible states of quantumsystems composed of identical particles with integer � spin. It takes its name fromSatyendra Nath Bose (1894–1974), the Indian physicist who first proposed it for� light quanta (1924), and Albert Einstein (1879–1955), who extended it to gasmolecules (1924, 1925).

Both in classical and in quantum mechanics, the behaviour of systems composedof a large number of particles can be investigated with the help of statistical con-siderations. If all particles obey the same dynamics, and if their interactions can beneglected in a first approximation, one can determine all possible energy states ofa single particle, and then make statistical assumptions on the distribution of theparticles among single-particle states, thus computing the average behaviour of thewhole system. The usual statistical assumption is that all possible states of the many-particle system (i.e. all configurations) are equally probable. As became clear aroundthe middle of the 1920’s, the description of quantum systems of many particles hasto be different from that of classical ones, a fact usually described by referring tothe � indistinguishability of quantum particles as opposed to the distinguishabilityof classical ones. Two kinds of � quantum statistics have been found to play a rolein quantum mechanics: the statistics of Bose–Einstein and that of � Fermi-Dirac.

Let us consider the classical case first, i.e. a system of N identical, noninteractingparticles which are assumed to be distinguishable. The configuration of the systemis determined by indicating which particles are in which states, for example particlea in state 1 and particle b in state 2:

particle a particle bstate 1 state 2 .