Naturwissenschaften 85, 203–218 (1998) c© Springer-Verlag 1998

Atomic Bose-EinsteinCondensates:A Model for MacroscopicQuantum SystemsD. Meschede, V. Gomer

Institut fur Angewandte Physik, UniversitatBonn, Wegelerstrasse 8, D-53115 Bonn

H. Monien

Physikalisches Institut, Universitat Bonn, Nuss-allee 12, D-53115 Bonn

A novel type of macroscopic quantum system has re-cently become available through the experimental real-ization of Bose condensates from neutral atoms. We re-view experimental results and the elementary quantummechanical approach and outline advanced theoreticalconcepts regarding finite size, potentials, dimensional-ity, and interactions.

Introduction: MacroscopicQuantum Systems

Quantum theory is traditionally considered andtaught as a description of microscopic systems inthe first place. Its precision and prediction capacityfor simple physical systems such as the hydrogenatom is without precedent and has remained unchal-lenged for more than 70 years. On the other hand,so-called “macroscopic quantum systems” continueto draw much excitement. When we attribute “quan-

tum character” to an object, it is not only the dis-crete energy level structure of a system with finitedimensions. Perhaps more importantly it is the waveaspect, which allows for interference phenomena ofparticles with mass. Strictly speaking, any condensedsample, even a simple electron gas in a metal, be-longs to this class. The classical appearance, i.e.,visualization in terms of a stochastic ensemble ofpointlike particles, is usually caused by a large num-ber of degrees of freedom available for the motionof such systems, resulting in a rapid dephasing ofquantum correlations at elevated temperatures. Insuch systems quantum interferences can safely beneglected.The attraction of quantum systems that are called“macroscopic” is, in contrast, derived from the factthat quantum correlations of the system are impor-tant or even dominant over conventional forces suchas the Coulomb interaction. Longer known exam-ples include superconductivity and superfluidity ofliquid helium. More recently a novel macrosocpicquantum state of matter has been produced froma dilute gas of bosonic alkali atoms in a magneticcontainer, which is considered to be the first real-ization of a weakly interacting Bose condensate. Itis common to all such systems that their lowest en-ergy level can be described by a single macroscopicwavefunction, i.e., the ground state of this systemis occupied by numerous particles. We restrict theterm “macroscopic quantum state” here to this classof many-particle systems.The first part of this overview is devoted to anaive analysis of thermodynamical properties of abosonic quantum gas. Quantum mechanics is em-ployed through elementary quantum statistics andthrough wavefunction analysis. In the second partwe make a bold attempt to introduce more advanced

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concepts of many-body quantum physics and fieldtheory to a general audience.

Atomic Bose-Einstein-Condensates

Elementary Predictions

In 1925 Albert Einstein [1] showed that for an idealBose gas ofN particles, with massm confined ina rigid container of volumeV , a macroscopic oc-cupation of the ground state can occur at the finitetemperature

Tc =h2

2πkBm

[ 12.612

N

V

]2/3

(1)

wherekB is Boltzmann’s constant andh is Planck’sconstant. He was inspired in this work by Satyen-dra N. Bose [2], who had considered the problemfor a gas of identical photons. This purely quantum-statistical phase transition is a consequence of the in-distinguishability of bosonlike particles, and one ofthe rare examples of a macroscopic quantum state.This mysterious phenomenon has been known asBose-Einstein condensation (BEC) for more than 70years, but it has proven difficult to achieve by ex-perimenters in its pure form.A naive way to “explain” the BEC makes use of thethermal de Broglie wavelength:

Λ =h√

2πmkT(2)

We can rewrite Eq. 1 in terms of the particle numberdensityn = N/V :

nΛ3 = 2.612 (3)

Thus the condition for Bose condensation to occurin a 3D box requires the average distance betweenparticlesn−1/3 to be approximately equal to the deBroglie wavelength.Λ is a quantum measure fordelocalization of a particle, i.e., the appearance ofwavelike properties. AtTc the wave packets describ-ing the particle begin to overlap, indicating the onsetof quantum degeneracy. Add the known preferenceof bosons to bale and one can “understand” the ef-fect.The productnΛ3 is called “phase space density.”Phase space describes the configuration volume(space×momentum) available to a given physicalsystem. In three dimensions it is divided into quan-tum unit cells of volumeh3. A unit cell can be pop-ulated by only one fermionic particle (with half in-teger spin) at a time but by an arbitrary number of

bosonic particles (with integer spin quantum num-ber) constituting a “degenerate quantum system.”The prediction of Einstein is a consequence of theBose-Einstein distribution describing the occupationnumberNε of particles in an energy levelε andgiven by:

Nε =1

exp(ε−µkT

)− 1

(4)

The chemical potentialµ is determined by the con-straint that the total umber of particles in the systemis N :

N = N0 +∑ε/=ε0

Nε → µ(N, T ) (5)

It was recognized by Einstein that the occupationnumberN0 of the lowest energy levelε0 must betreated in a special way in order to avoid divergenceof the sum in Eq. 5. It is therefore separated fromthe rest of the sum. We see from Eq. 4 thatµ ≤ ε0because the occupation number must be positive.At room temperature all real gases behave classi-cally, and one can neglect 1 in the BE distribution(Eq. 4). For a rigid 3D box of radiusr0 one canshow [3] that:

Nε ≤ N0 =(Λ

r0

)3

N � 1 (6)

which means that all states are about equally andmicroscopicallypopulated.It is certainly not surprising that atT = 0 all parti-cles are in the state of lowest energy. The strikingfeature of the BE distribution (Eq. 4) is the possibil-ity of a macroscopicoccupation (of the order of thetotal particle numberN ) of the ground state also atfinite temperature. Consider what happens to a sys-tem ofN bosons held at constant volume when wedecrease the temperature. To maintain the normal-ization condition (Eq. 5) the chemical potentialµmust adjust by increasing towards the ground stateenergyµ → ε0. Below some critical temperatureTcthe number of particles in all states except the low-est one can reach the value ofN only if µ = ε0and then the chemical potentialµ no longer dependson the temperature. [At this point we can draw aclear analogy: For constantµ the BE distributionEq. 4 coincides with Planck’s law describing black-body radiation (µ = 0). The Stefan-Boltzmann lawfor the total radiation energy density scales asT 4,and we can easily estimate the number of photonsfrom N −N0 ∝ E/~ω ∝ Eλ, where typical wave-lengths are scaling asλ ∝ 1/T according to Wien’sdisplacement law. The final result,N − N0 ∝ T 3,

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0 1 2 3

0

1

2CV/Nk

T/Tc

N0/N

Fig. 1. Important thermodynamic properties of an ideal Bose gas in a large3D box: fractional number of atoms in the normal and the condensed state(N0/N ) and specific heat (CV /Nk) as a function of temperature

describes a special but very important case: the oc-cupation of the normal phase (particles with nonzeromomentum) of bosonic atoms in a 3D harmonic po-tential which has a density of states identical to thephoton gas (see Eq. 7). Of course, for a photon gasN0 = 0.]Further cooling further decreases the occupation ofall excited states, and all excess particles must gointo N0. Thus for T < Tc the population of theground state with zero momentum becomes macro-scopic. This is called “condensed phase” while theremaining fraction is called “normal phase.” In a ho-mogeneous system, i.e. in a box with infinite walls,the condensed fraction occupies the same volume asthe normal fraction – hence one speaks of “conden-sation in momentum ork-space.”This resembles the behavior of a saturated vapor in-teracting with its solid phase. [It might seem moreobvious to compare the quantum phase transition toa vapor-liquid interface. For this system, however,no long-range order is established at the transitionpoint, as is, in contrast, the case for a crystal or aBose condensate. This situation is also manifestedby the existence of a critical point which shows thata liquid and a vapor have the same type of sym-metry.] Discontinuous behavior of thermodynamicproperties shown in Fig. 1 is also typical of phasetransitions. However, in contrast to usual phase tran-sitions caused by intermolecular interactions, Bosecondensation is a consequence of the wave functionsymmetry alone.

Experimental Efforts

It is easy to estimate from Eq. 1 that for airmolecules (M = 28) and normal density (N/V =1019 cm−3) the BEC condition is met atTc ' 40

Fig. 2. Magnetic trap for paramagnetic atoms (Ioffe-Pritchard trap).Arrows,direction of the current generating the magnetic field. The magnetic fieldis designed to create a field minimum in thecentral region. A gas ofparamagnetic atoms with magnetic moments antiparallel to the magneticfield lines is confined near the axis of the trap. Because magnetic dipoleforces are weak, only slow atoms can be trapped (typically< 10 mK)

mK. At such cryogenic temperatures, however, airand all other conventional substances have long un-dergone one or another type of structural phase tran-sition due to intermolecular forces, resulting in asolid phase. Only helium remains liquid down to thethermodynamic zero point and shows unusual “su-perfluid” behavior below the “lambda point” at 2.17K. Due to strong interactions, however, its proper-ties deviate widely from predictions for a pure Bosecondensate.BEC was indeed considered an unphysical althoughhighly interesting theoretical concept [3] until exper-iments were designed to produce quantum degener-acy unobstructed by strong interparticle interactions.One such procedure is to use a dilute spin polar-ized gas of very cold neutral atoms which are sub-ject to weak interatomic van der Waals interactionsonly. Paramagnetic atoms can be trapped in suitablemagnetic fields, for instance, in the so-called Ioffe-Pritchard trap, shown schematically in Fig. 2. There-fore hydrogen or alkali atoms confined in a magnetictrap had long been predicted [4] to form potentiallya Bose condensate. During the 1980s, laser cooling(see e.g., [5]) was turned into an extremely uselfultool of experimental atomic physics, and it was ap-plied to atomic gases confined in a suitable trap. Un-expectedly low temperatures in theµK range wereobserved and opened the route to dramatically in-creased phase space densitiesnΛ3 (Fig. 3) withinsight of the quantum degeneracy borderline. Notethat the high phase space density does not coincidewith very high particle density of such an ultracoldgas in an atom trap. The latter is of order1012cm−3,which does not exceed the particle density of a good

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Dx/l

1

0.1

0.01

Dv/vRec

10

100

0.01 0.1 1 10 100

Evaporative Cooling

BEC

Recoil Limit

1 Atom

l3

LaserCooling

Fig. 3. Phase space accessible by conventional laser cooling techniques andphase space volume of BEC (qualitatively). Spatial coordinates (∆x) arenormalized to the cooling laser wavelength (λ), and momentum coordinates(v = p/m) to the recoil velocityvrec = h/λm, i.e., the velocity changecaused by a single recoil at photon emission or absorption. With lasercooling density remains always less than 1 atom/λ3, and the recoil velocitysets a practical lower limit to the temperature. This diagram is universallyapplicable to all atoms prepared by laser cooling before evaporative coolingtakes the sample to the quantum condensed phase. Note that the entireshaded “BEC” area contains a single unit cell (=h) of phase space only

vacuum chamber, corresponding to' 10−5 mbarpressure at room temperature.With slow laser-cooled atoms it was learned to storevery cold samples of alkalis for minutes or evenhours in magnetic traps, but in spite of large ex-perimental efforts the critical density according toEq. 3 was not achieved. The presence of laser lightset a lower limit to the temperature through heat-ing processes caused by spontaneous emission. Atthis “recoil limit” the de Broglie wavelength equalsthe cooling laser wavelength,Λ ≈ λ. The limitationof particle densities is caused by multiple scatter-ing of photons (Fig. 3) at the optical density limitn ≈ λ−3.The most decisive breakthrough was achieved bythree groups of experimenters in the United Stateswhen they applied the evaporative cooling technique(developed for cooling hydrogen gas [6], which istechnically inaccesible to laser cooling) to samplesof laser-cooled alkali atoms. Although evaporativecooling is a well-known phenomenon, details of itsimplementation are indeed crucial, since it is associ-ated with a loss of particles: a Bose condensate canbe formed only if the increase in phase space densityoccurs at a rate faster than the loss rate in particledensity [7]. After 3 years of successful operation thismethod is now well established.

The First Atomic Condensates

The first and much publicized step across the bor-derline to a truely degenerate quantum gas of neutralatoms was taken in 1995 by Cornell and coworkersat JILA, Boulder [8], with a gas of spin-polarizedrubidium atoms. In the first experiments samples ofup to104 atoms were analyzed by 2D-image absorp-tion images (or its shadow) when they were releasedfrom the trap (Fig. 4). The condensation was identi-fied through a sudden appearance of a central com-ponent of very high optical – and hence particle –density corresponding to a very narrow momentumdistribution. It rapidly became possible to carry outmeasurements of thermodynamical properties of thecondensed sample. Excellent agreement with theo-retical predictions for the ground state occupationN0 which in a 3D harmonic potential evolves ac-cording to (see “Elementary Predictions”):

N0 = N

(1 −

(T

Tc

)3)(7)

was, for instance, obtained by a measurement of theoccupation number in the JILA group [9] (Fig. 5).When the attractive borderline to the quantum de-generate regime was later also crossed by Ketterleand his group at MIT [10] with a sodium sample,dramatic technical advances were soon obtained inthe preparation of the condensate (now up to107

atoms in only 30 s) and using a nondestructive dark-field imaging method [36]. It was this breakthroughwhich opened the route to the observation of con-densate interference which is described below.A third experiment was carried out at the same timeby the group of Hulet at Rice University with a sam-ple of lithium atoms [12, 13]. Lithium is interestingsince it provides both a bosonic and fermionic iso-tope for comparative studies. Furthermore, bosonic7Li atoms attract each other, and hence the forma-tion of a condensate is completely suppressed in ahomogeneous system, i.e., in a large box. A trappingpotential can exert a stabilizing influence, however,by confinement of the atoms. Increased zero-pointkinetic energies can compensate atomic attraction upto an atomic density depending on the parameters ofthe potential. A condensate of some 1000 atoms wasindeed observed, in fair agreement with theoreticalpredictions.

Questions in BEC

Einstein’s initial discovery assumed an ideal quan-tum gas of noninteracting particles in a homoge-neous system of infinite extension. In theoretical in-

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Fig. 4. BEC in a gas of rubidium atoms was first identified by an extremely high optical density in the center of a cooled cloud of atoms. The 2D shadowimages were taken 60 ms after the magnetic trap had been switched off, and the cloud had been allowed to expand ballistically. Therefore the imagerepresents the 2D velocity distribution of atom.From left to right, the three images were taken just before (thermal Gaussian-like distribution) and justafter the appearance of the condensate (two-component cloud with a sharp increase in the peak density), and after further evaporation had nearly leftapure condensate. The condensate fraction is elliptical, indicative that it is highly nonthermal. The temperature can be determined from the noncondensedfraction of atoms (Courtesy E. Cornell [8])

vestigations it was realized early on that a Bose con-densate of weakly interacting particles is decidedlymore interesting than the ideal gas case. The rea-son is that noninteracting particles “do not noticeeach other” – i.e., they cannot have any collectiveexcitations. Theoretical work confirmed that “real”BEC can show surprisingly rich and subtle behav-ior depending on the details of the considered sys-tems. Atomic condensates may provide a new test-ing ground for more profound questions concerningthe many body quantum nature of a Bose conden-sate. Below we illustrate several interesting aspectsof BEC.

Dimensions.In 1967 Hohenberg demonstrated thatBEC can occur only in 3D [14]; this is now known asthe Hohenberg–Mermin–Wagner theorem [15, 16].Why does the intuitively clear picture of overlap-ping de Broglie wavelengths which is successful in3D fail in two or one dimension? Furthermore, theimpossibility of BEC in 2D and 1D turned out tobe valid only for homogeneous systems, that is forsystems confined by rigid boundaries. This was firstpointed out by Widom [17] who showed that BEC

occurs in a 1D gas in the presence of a gravitationalfield.

BEC in a Potential.In all experiments describedabove atoms are confined by a spatially varying po-tential. In this case the coupling between energyand space fundamentally alters the nature of BEC.In general, a trapping potential supports BEC [18]:as the mean energy decreases, the effective volumeavailable to the system also decreases enhancing thegrowth of density. For example, in a power-law po-tential BEC is always possible in two dimensions,while 1D systems were predicted to display BECin traps that are more confining than parabolic [19].Formally it is sufficient to evaluate the density ofstates to resolve the quest for quantum degeneracyin a potential.

Finite Numbers of Particles.The usual theory ofBEC assumes the thermodynamic limit of an infi-nite system. An investigation of an ideal gas witha finite number of particles [20] has shown that thetransition temperature in a 3D harmonic potentialis lower than in the thermodynamic limit, as onewould expect. Surprisingly, however, BEC in a 1Dharmonic potential can also occur ifN is finite.

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1.0

0.8

0.6

0.4

0.2

0.0

N/N0

T/TC

0.0 0.3 0.6 0.9 1.2 1.5

Fig. 5. Fractional occupation numberN0/N of the ground state in a Bosecondensate as a function of temperature. The scale temperatureTc(N ) is thepredicted critical temperature for an ideal Bose gas in a harmonic potential.Solid line, expected occupation number for an infinite number of particles;dashed curve, takes a finite number of atoms into account (see Eq. 7).(Courtesy E. Cornell [9])

Particle interactions.In a real gas interparticle in-teractions are extremely important and can totallychange the behavior of the system, especially in thelow-dimensional case. They are responsible for thefinite compressibility of the condensate and there-fore for the existence of sound modes at long wave-lengths instead of free particle excitations. Obvi-ously the interactions are also very important for theonset of condensation and for the build-up of co-herence in matter waves, which is a field of activeresearch.

Generalized BEC Condition for Potentialsand Lower Dimensions

The conditions for a macroscopic population of theground state can be summarized by a transparentformula which is derived from an analysis of thephase space volume available to the system (see“Appendix”). In d dimensions we find for a systemwith N particles and a de Broglie wavelengthΛ ac-cording to Eq. 2 the simple relationship which is incomforting agreement with our physical intuition:

ΛdN

V ∗ ≈ 1 (8)

and in direct analogy with Eq. 3. The primary focusof this formula is the introduction of an effectivevolumeV ∗ ≈ rd which for an isotropic power-lawpotentialU (r) ∝ rn is determined from the condi-tion kT ≈ U (r). Let us demonstrate how well thisformula works with two examples.

BEC in Trapping Potential.The ground-state pop-ulation fraction forT < Tc is given by (see “Ap-pendix”):

N0

N= 1 − (T/Tc)

η (9)

with η = d/2+d/n, reproducing the original predic-tion by Einsteinη = 3/2 for a homogeneous system(d = 3, n = ∞) and (Eq. 7)η = 3 for a 3D-harmonictrap (n = 2) as in current experiments

Two-Dimensional Systems.When in a homoge-neous 2D system (d = 2, n = ∞) the volumeV ∗is restricted to the surface areaS we immediatelyobtain from Eq. 8 a critical temperature:

Tc =π~

2

2kmN

S(10)

This result agrees perfectly with the transition tem-perature deduced for a so-called quasi-condensatewith short-range correlations (or Kosterlitz-Thoulessphase [21]). Note, however, that our argument allowsan estimate only for the onset of quantum degener-acy. The stability of such a “condensate” must bedetermined from a more elaborate analysis.

Interactions and Sound in the Condensate

Although the occurrence of a phase transition onpure quantum mechanical grounds has captivated theminds of physicists ever since its discovery, Bose–Einstein condensates would be rather boring objectsif it weren’t for the interactions within this appealingstate of matter. In a gas of noninteracting Bose par-ticles the energy structure of the noncondensed sam-ple is preserved, and the only effect of excitations isto remove individual particles from the condensedstate. A quantum condensate of interacting particles,however, can react collectively to an external dis-turbance exhibiting low lying and novel excitationswithin the condensed state. For instance, helium inits quantum fluid state exhibits unusual phenomenarelated to such disturbances which are usually called“sound.”It is indeed due only to the interaction of the con-densed particles, or the finite compressibility, thatdisturbances of, for instance, the density propagatein a nontrivial manner.

Mean-Field Energy

In a gas of noninteracting particles residual veloci-ties are determined by the zero-point kinetic energies

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in the trapping potential only. In contrast, the highdensity of the condensate causes the average inter-molecular potential energy in the condensate to bemuch greater than kinetic energies. Once the con-fining potential is switched off, the correspondinginternal pressure is released and dominates the ex-pansion.At very low relative velocities (“cold collisions”[22]) rubidium and sodium atoms repel each other.Since the interaction is of short range, it is justifiedto describe the atom-atom interaction energy by asimplified pseudopotential:

U (r ) =4π~

2a

mδ(r ) (11)

where theδ function denotes the quasipointlike in-teraction. The relevant “s wave scattering” lengthais positive for87Rb (a = 60−100A [23]) and 23Na(a ' 27 A [24]) but negative for7Li (a = −15 A[25]). The condensate is called “weakly interacting”if the particle densityn satisfies condition:

na3 � 1 (12)

as is the case for the existing atomic Bose conden-sates. In other words, the scattering lengtha remainsvery small compared to interatomic separations.For an individual atom the contribution of all otheratoms in a condensate is proportional to the localdensityn(r ) and gives rise to the mean field energyby:

UMF(r ) =4π~

2a

mn(r ) (13)

and therefore proportional to the densityN/V . Anatomic Bose condensate is formed in a harmonic po-tential which allows the condensate to expand withN and weakens theN dependence ofUMF. ForT → 0 the meanfield energy can also be identifiedwith the chemical potential,µ → UMF.The mean field energy becomes clearly negative forattractive interaction or negative scattering lengtha,which is the case for7Li and prevents BEC fromoccurring in a large homogeneous system. It can,however, be compensated by the zero-point energy~

2/m`20 if the atoms are confined to volume of radius

`0. This limits the maximum number of atoms in themetastable condensate to [26]

N0(max)' `0

|a| (14)

The BEC criterion used above (macroscopic occu-pation of the lowest single particle energy level Eq.

0 Sw0 m kTC

CollectiveExcitations

Single ParticleExcitations

» UMF/N

Fig. 6. Energy scales of a weakly interacting BEC. For a single particleexcitation, or removal of an atom from the condensed state an energyabove the gap is required. The lowest excitation energies (or oscillationfrequencies) below the energy gap always correspond to intrinsic manybody phenomena, or collective excitations

8) is meaningful for noninteracting particles only.In order to account for interactions we introduce awavefunction which relates each atom in the con-densate to the total particle densityn(r ) through:

n(r , t) = N |ψ(r , t)|2 (15)

The Schrodinger equation can then be constructedfrom the the harmonic oscillator potentialV (r ) =mω2

0r 2/2 and an effective single atom potentialcaused by the meanfield energy per atom,UMF ·n(r ) = 4π~

2a/m · |ψ(r )|2:

i~ ∂∂tψ(r )

=(− ~

2

2m∇2 + V (r ) + 4π~2a

m |ψ(r )|2)ψ(r )

(16)

Within the framework of BEC this nonlinearSchrodinger equation is better known as the Gross–Pitaevski equation [27, 28, 29]. An analysis of theminimum energy state shows that in spite of the vol-ume increase in the condensate, caused by increas-ing intermolecular energy the chemical potentialµin a harmonic trap still grows asN2/5 resulting ina mean field energy much larger than but also pro-portional to the oscillator energy~ω0 [30]. In theso-called Thomas-Fermi approximation kinetic en-ergy terms in the Gross-Pitaevski equation (Eq. 16)are completely negelected.In Fig. 6 we summarize the energy scales relevant fora weakly interacting Bose condensate. The averagemean field energyUMF = 4π~

2a/m ∼ µ of Eq. 13 isalso an estimate of the energy required to transfer asingle atom from the condensed to the normal state[3]. This “energy gap” constitutes a threshold forsingle particle effects.

Vibrations of Atomic Condensates

We concentrate on the case where the mean free pathof a condensate atom is much shorter than the exten-sion of the condensate, establishing a local densityequilibrium. Furthermore, atT/Tc � 1 the normal

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Absorption 0% 100%

Fig. 7. Quadrupolar shape oscillation of a cigar like condensate. The oscil-lation period (20–50 ms) agrees well with theoretical predictions from themean field model. (Courtesy W. Ketterle [11])

phase is virtually absent, it can be neglected, and apure condensate oscillation is expected.In an experiment the trapped condensates may be ex-cited by (periodically) changing the magnetic trap-ping fields or by exerting additional forces throughoptical forces derived from a laser beam focused intothe condensate. Oscillations with frequencies in the10–500-Hz domain have been observed [32, 11].It is possible to excite collective mechanical exci-tations of the condensed state which may be com-pared to the normal mode oscillations observed withstanding sound waves in a liquid droplet confinedto a harmonic potential. The simplest motion is thecenter of mass oscillation at the trapping potentialfrequency. This is called “dipole mode” and is infact the lowest frequency excitation – for small dis-placements it does not change shape and hence themean field energy of the object. Shape oscillations ofquadrupolar character have also been observed [32,11, 33]. An example of the strobed shape variationis shown in Fig. 7.A theoretical analyis based on a linearized approxi-mation of the Gross-Pitaevski (Eq. 16) gives an ana-lytic solution in the hydrodynamic limit [30] and ex-cellent numerical agreement with experimental ob-servations for the lower density case [33, 34]. Mean-while, not only standing but also propagating soundwaves have been observed [35], lending still moresupport to the mean field approach. Note, however,that the modification of oscillation frequencies is aresult of hydrodynamic flow and interactions onlyand does not prove the quantum wave character ofthe condensed sample. For an experimental investi-gation of this phenomenon it is necessary to studyinterferences, or correlations of particle densities.

Matter Waves

When “immaterial” optical waves are discussed weare not surprised when their nature becomes man-ifest through the occurrence of interference. Whenit comes to matter waves, however, the concept ofinterference still seems to be at odds with our imag-ination of particles.Indeed, the well-known quantum-mechanical anal-ogy between light and matter, first recognized by

de Broglie and manifested in diffraction experimentswith electrons, has very severe limitations. For ex-ample, there can be no classical field theory for elec-trons [36]. To see this, we can consider a scalar field

ψ = Aeiφ (17)

with a phaseφ and an amplitudeA normalized onsuch a scale thatA2 = N is the number of quantaper field mode. Then the uncertainty relation is

δN · δφ ≥ 1 (18)

If we want to treat the field as a classical quantitythe uncertainty in the phase should be small com-pared to unity, and the uncertainty inN should besmall compared toN itself. According to Eq. 18,however, this requires thatN � 1. If we recall thatN is the number of quanta per mode, we see that aclassical field description is forbidden for fermionsby the Pauli principle. This difficulty does not ap-ply to a boson field and one can verify that a laserbeam is usually intense enough (the photon numberis macroscopic!) to allow amplitude and phase to beknown with good precision [37]. The uncertainty ofthe particle number in the coherent state poses noproblems for photons. Application of this concept tomaterial particles such as bosonic atoms left a ratheruncomfortable feeling, however.Recent experiments carried out at MIT demonstrateunambigously that coherent matter waves do existat a macroscopic scale. Let us examine experimentswhich have begun to explore the true quantum natureof an atomic Bose condensate, or the existence ofan amplitude and a macroscopic phase according toEq. 17.

Interfering Bose Condensates

Dramatic experimental progress in both preparingand observing Bose condensates has enabled theMIT group to record an image of two interferingcondensates of Na atoms [38]. For this experimentthe cigar shaped-magnetic potential was cut into twohalves by means of a focused far-off resonant laserbeam which allowed synchronous preparation of twoneighboring but independent condensates. The sam-ples were then released from the trap and interferedon expansion. They overlap horizontally, and inter-ference fringes with a spacing of about 15µm, anda contrast of 20–40% are clearly visible in Fig. 8.Note that the image is derived from a single eventof condensate ejection since the relative phase ofthe condensates varies randomly from shot to shot.This experiment provides the first direct image of

210

Absorption 50%0%

1 mm

Fig. 8. Interference pattern of two Bose condensates. The fringes are due toperiodic density variations of the interfering condensates. Since the patternhas a random global phase in each event, it must be recorded in a singleshot. This experiment demonstrates interference of matter waves beyondsingle particle self interference. (Courtesy W. Ketterle [38])

matter waves interference. Earlier experiments al-ways relied on self-interference of individual parti-cles transmitted through an appropriate matter wavebeam splitter and recombined – a mind-bogglingphenomenon by itself, but clearly on the microscopicscale!The observed interference wavelength can be under-stood in terms of relative atomic motion. When thetwo samples are released from two points separatedby a distanced, they freely fall and expand trans-versely at the same time. After a timet has elapsedcondensate portions moving atv = d/t overlap andhave a relative de Broglie wavelengthΛ = ht/md.The interference pattern observed agrees well withthis rough estimate and also with more detailed the-oretical analysis [41].

A Coherent Matter Wave Generator

A first step towards generation of propagating coher-ent matter waves was also taken by the MIT group.Using radiofrequency excitation it is possible to ex-tract smaller portions of the condensate, as shown inthe strobed image of Fig. 9.The question of whether two coherent light fieldsemanating from two independent laser sources caninterfere at all was debated very intensely shortlyafter the laser was invented – and rapidly settledas well [40]. The observation of interference fromtwo independent atomic Bose condensates has al-ready prompted the term “atom laser” in analogyto a conventional laser, even though in contrast tothe optical laser robust and compact instruments arenot yet available and this designation is still underdebate [41]:

An atom laser is a device which generatesan intense coherent beam of atoms through

0 Density scale (arbitrary units) 1

Fig. 9. Pulses of coherent matter in free fall. The samples have been ex-tracted periodically from the stored condensate (top) by radiofrequencyexcitation. Their coherence is demonstrated by the experiment presented inFig. 8. (Courtesy W. Ketterle).

a stimulated process. It does for atoms whatan optical laser does for light. The atomlaser emits coherent matter waves whereas theoptical laser emits coherent electromagneticwaves. Coherence means, for instance, thatatom laser beams can interfere with eachother... .

Beyond the Pedestrian’s Approach toBEC

In the preceding sections we analyze the phase spacevolume available for a quantum system dependingon dimension or potential shape, yielding estimatesfor the onset of quantum degeneracy in a Bose gasof atoms. Intuitive guidance for wave mechanicaleffects is introduced by an analogy with classicalwaves which are described by an amplitude and by aphase. There is no consideration above of the stabil-ity of such a system, however, which is determinednot only by the total energy of a system but also by

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the entropyS. This competition is usually describedby the so-called Helmholtz free energyF :

F = E − TS (19)

which is minimized in thermal equilibrium. We donot analyze the entropyS of a Bose condensate indetail but rather give a direct way to calculate thefree energy which is an example of athermody-namic potential.It is the purpose of this section to introduce thereader to these important but more subtle aspectsof BEC, mostly on the basis of analogies. Let us trythe unspeakable and describe some of the abstractconcepts fascinating real theorists.

Fundamental Concepts of Many-Body Quan-tum Physics

Formally the transition from classical to quantumstatistical mechanics involves replacement of classi-cal quantities such as the energyE by their opera-tor analogue, in this case the HamiltonianH. Thestate of a microscopic system is then given by theexpected value of the operator for a quantum statecharacterized by a wavefunctionψ. For instance, theenergy of a quantum state is calculated from:

E =∫ψ∗Hψ (20)

A thermodynamic system consists of an enormousnumber of particles. It is no longer the state of ev-ery individual particle that we are interested in. Itis rather the knowledge of quantities such as tem-perature or pressure of the sample which can be an-alyzed by statistical methods. For this purpose weuse the density matrixρ = exp(−βH)/Z, whichdescribes the occupation probability of individualquantum states, and whereβ = 1/kBT is the inversetemperature. The expectation value of the operator:

Z = Trace{exp(−βH)} (21)

is called partition function (in GermanZustands-summe), it is a sum running over all quantum statesand assures the correct normalization of the densitymatrix. The partition function is intimately relatedto the free energy (Eq. 19) of a physical system:

F = −kBT ln(Z) (22)

Once the partition function is known, all thermody-namic parameters can be derived from it.Time evolution of a microscopic quantum state is de-scribed by the Schrodinger equation, and there exists

a close formal analogy between the equation describ-ing the time evolution of the density matrix:

i~∂

∂tψ = Hψ → ~

∂

∂τρ = −Hρ (23)

The only formal difference is that the time in theSchrodinger equation must be replaced by an imag-inary “time,” t → −iτ , which has the unit of aninverse energy and runs from 0 toβ.A very elegant approach to analyzing the evolutionof a quantum system is the method of path integrals.The transition amplitude of the microscopic systemfrom an initial point(xi, ti) to a final point(xf , tf )can be expressed by a path integral over all pathsleading fromxi to xf :

U (xf tf , xiti) = 〈xf |e i~H(tf −ti)|xi〉

=∫ f

iD [x(t)] e

i~S[x(t)] (24)

HereD [x(t)] is a generalized integration variabledemanding an integral over all the possible pathsthat a particle can take. The action for a given pathS[x(t)] is defined by:

S =∫ tf

tidtL[x(t)] (25)

whereL is the Lagrangian of the system, with con-tributions of the kinetic energy and of the relevantpotentials:

L =m

2

(dx

dt

)2

− V (x) (26)

For example, the partition function,Z, for a sin-gle particle can be viewed as the trace of the timeevolution operator in imaginary time if we write thepartition function (Eq. 21) as a trace over the posi-tion eigenstates:

Z = Trace(e−βH) =∫

dx〈x|e−βH |x〉 (27)

To carry out the analogy we express the transitionamplitude again as a path integral over all possibletrajectories of the system:

U (xfτf = β, xiτi = 0) = 〈xf |e 1~H(τf −τi)|xi〉

=∫ f

iD [x(τ )] e− 1

~S[x(τ )] (28)

In analogy to (Eqs. 25 and 26), the microscopic ac-tion S and the Lagrangian are now given by:

S =∫ τf =β

τi=0dτL[x(τ )] (29)

212

and

L =m

2

(dx

dτ

)2

+ V (x) (30)

We can now implement this method for many-bodysystems by replacing the microscopic Lagrangian(Eq. 30) by the Lagrangian of the macroscopic sys-tem. In the path integral we must sum over all pos-sible paths of the entire physical system. This taskmay be simplified by the use of an appropriate rep-resentation.

Coherent States and Expectation Values

The coordinate representation of the wave functionturns out to be an awkward description for the many-body system. Fortunately, there exists a more usefulbasis, the basis of coherent states which is illustratedin more detail in the next section. A coherent state,|φ〉, can be defined as the eigenstate of the annihi-lation operator for particle fields,ψ:

ψ |φ〉 = ψ |φ〉 (31)

The basis of coherent states,|φ〉, is characterizedby their eigenvalues,ψ, which for bosons are justcomplex numbers. For the evaluation of expectationvalues of many-body systems also “adjoint” coher-ent states|φ > with eigenvalueψ are used insteadof the complex conjugate of the wave function usedfor single particle systems. The coherent state repre-sentation of the path integral is obtained by insertingthe basis of the coherent states at every step into thetime evolution.The final form of the partition function is then:

Z =∫

D[ψ, ψ

]e−Smac(ψ,ψ)/~ (32)

where the macroscopic actionSmac is given in termsof the coherent statesψ and ψ by:

Smac(ψ, ψ) =∫ β

0dτ Lmac(ψ, ψ) (33)

The advantage of this approach is that we can infact evaluate any thermal expectation of a physicalobservableA value from a path integral:

〈A〉 =∫D[ψ ψ

]e−Smac(ψ,ψ)/~A(ψ, ψ) (34)

Properties of Coherent States

Before we proceed to describe properties of a Bosecondensate let us illustrate properties of the macro-scopic wave functions (ψ, ψ) introduced in the pre-vious paragraph. Matter is described in terms ofquantum fields which are associated with field oper-ators, ψ(r), ψ†(r′). The particle density operator isn(r) = ψ†(r)ψ(r), similar to the relationship of theintensity and the amplitude of a light field. Extend-ing the analogy to light field interference we mayalso define a (first order) correlation function throughρ(r, r′) = 〈ψ†(r)ψ(r′)〉 which reproduces the densityfor r′ = r. Note that this quantity is measurable fora single sample only if it is split and recombined,for instance, by two beam interference in a Michel-son type interferometer. Alternatively, two indepen-dently prepared samples may be brought into inter-ference, and exactly this was demonstrated in thematter wave interference experiment at MIT [38].A normal fluid is disordered at large distances, andhence there is no correlation except for the imme-diate vicinity of particles (at|r − r′| → 0). In aquantum liquid such as superfluid helium, however,correlations do exist at large separations, promptingPenrose and Onsager [42] to introduce the conceptof “off-diagonal” long range order where the densitycorrelation function factorizes in a special way:

for |r − r′| → ∞ :normal fl. 〈ψ†(r)ψ(r′)〉 → 0quantum fl. 〈ψ†(r)ψ(r′)〉 = 〈ψ†(r)〉〈ψ(r′)〉 /= 0

(35)

In particular it does not vanish even at large sepa-rations, and it is this criterion which is now takenas the most satisfactory definition of a macroscopicquantum state such as a Bose condensate [43]. Co-herent states are then an obvious choice for a formaldescription of such systems.The coherent state representation of the path integral(Eqs. 32, 34) is particularly suited for describing the“off-diagonal long range order.” Coherent state wavefunctions are also widely used to describe propertiesof a laser light field [37].The order is characterized by a complex order pa-rameter,ψ, which has amplitude and phase. Thephase is a macroscopic dynamic variable of the sys-tem and posseses real physical meaning. The numberoperator,N , and the phase operator,φ, are conjugatevariables, with expectation values obeying the un-certainty relation of Eq. 18, as one can easily derivefrom the definition of the coherent states. It is fur-thermore possible to reconstruct a state with a fixednumber of particles,N , from the coherent state of

213

the condensateψcond =√ρe−iφ by averaging over

the phase of the wave function:∫ 2π

0

dφ

2πe−iφN |ψcond〉 =

√ρN

N|N〉 (36)

Mean-Field Approximation

Our focus is directed towards an imperfect Bose gaswith coupling constantg = 4π~a2/m, in a potentialU (r ), governed by a chemical potentialµ, and ki-netic energyεp = p2/2m. In this case the actionSis given by:

S =∫ β

0dτ∫d3r

{~ψ

∂

∂τψ +

~2

2m|∇ψ|2

+(U (r ) − µ)|ψ|2 + g2|ψ|4

} (37)

The important contributions to the partition function,Z, according to Eq. 32 come from configurationswhere the action,S, is small. The mean-field ap-proximation is obtained by expanding the actionSto quadratic order inψ andψ around such a station-ary point. The free energy,F , is then given by:

F = −kBT ln (∫D[ψ ψ

]e−S(ψ,ψ)/~)

≈ kBT S(ψMF , ψMF

)/~ (38)

In Fig. 10 the potential contribution to Eq. 37 cor-responding to the free energy in this case is shownas a function of the order parameterψ:The stationary point in time and space of the action,S, is determined by a nonlinear Schrodinger equa-tion for the field,ψ, and recovers the stationary caseof the Gross-Pitaevski (Eq. 16):

− ~2

2M∇2ψ + (U (r ) − µ)ψ + g|ψ|2ψ = 0 (39)

There are two solutions to this equation. The first isψ = 0. This is the normal state, no coherent statewith finite amplitude|ψ| is occupied. The secondhas a finiteψ. The mean-field solution,ψ, possessesa well-defined global phaseφ which is arbitrary butfixed. The original action (Eq. 37) is invariant undera global change of the phase

(ψ → ψeiϕ

).

The mean-field solution,ψ, picks out a certain di-rection of the phase angleφ, which is called spon-taneous symmetry breaking, where in the languageof group theory it isU (1) group symmetry of phasetransformations, which is violated. The phase of themean-field,ψ, is coherent over the entire system.

Fig. 10. Thermodynamic potential (free energy) as a function of the orderparameterψ. Note that it corresponds to the potential term in Eq. 37. Thelowest energy state has a single selected phase (arrow) which is valid for theentire condensate. This situation has been named “spontaneous symmetrybreaking.” Variations in the amplitude are called “longitudinal excitation,”fluctuations of the phase “transverse excitation.” This designation can betraced back to quantum descriptions of ferromagnetism, a phase transitionswith analogies to BEC

Fluctuations

In the previous section we discuss the mean-field ex-pectation value for the order parameter. The questionis whether the mean-field solution is stable. Sincethis depends on the spatial dimension of the system,we discuss here a general homogeneous system inddimensions. The order parameter can be destroyedby thermal or quantum fluctuations. To obtain thecontribution of the fluctuations we expand the ac-tion S (Eq. 37) to second order in the fieldsψ andψaround the saddle point solutionψMF = |ψMF |eiφ.Choosing the (arbitrary) phase,φ = 0, the longitudi-nal, ψL, and transverseψT , fluctuations change themean field toψMF → ψMF +ψL + iψT . The fieldψ“moves” in a potential depicted in Fig. 10.Obviously there exist two kinds of excitations. Onecorresponds to longitudinal fluctuations,ψL, of theorder-parameter which have a gap, and another cor-responds to a global rotation of the phase,ψT , andhas no gap in the long-wavelength limit – so-calledGoldstone modes. We can see this formally if weexamine the the action of the fluctuations:

S =∫ β

0dτ∫ddr

12

×{(

∂∂τψL)2

+(∇ψL)2+∆2ψ2L

+(∂∂τψT)2

+(∇ψT )2}

(40)

where∆ is the mean-field value of the fieldψ andψL, and ψT are the coordinates of the longitudi-nal and transverse fluctuations, respectively. From

214

Eq. 40 we can read off the dispersion of the longi-tudinal and transverse mode:

ω2L(q) = ∆2 + q2 (41)

ω2T (q) = q2 (42)

The longitudinal mode corresponds to oscillationsin the amplitude of the order parameter. The orderparameter is robust since a change in magnitude re-quires energy which causes a finite energy gap forthe longitudinal mode. Excitations of the transversemode, however, cost very little energy and thereforebegin without a gap. In this case the wave numberq may be understood as the lengthscale over whichthe phase of the condensate varies in the sample.

Stability of the Condensate

The mean-field “condensate” is stable if small per-turbations due to thermal noise or quantum fluctu-ations do not significantly change the state of thesystem. This stability of the system (the amount of“noise”) can be investigated by determining the cor-relation function of the fluctuations, which are dom-inated at sufficiently low temperatures by transverseexcitations (Eq. 42) without energy gap. Their con-tribution can be estimated from an analysis of theirdensity of states inq space:

〈ψT (r , τ )ψT (0,0)〉 ∼∫

(dq)de−iqr

q2(43)

If this correlation function diverges small externalfields – or noise – have a dramatic effect on the phaseof the system. For spatial dimensionsd > 2 the in-tegral always converges atq = 0. The Goldstone-modes disorder the system but result in a finite re-duction in the order parameter only. In spatial di-mensions ofd = 2 or less the effect of fluctuationsis more dramatic. They can disorder the system com-pletely, which is the basic content of the Hohenberg-Mermin-Wagner theorem [14–16].Another way to understand the existence of a lowercritical dimension is to investigate the competitionbetween entropy and energy in the free energy, ac-cording to Eq. 46. Let us for this purpose draw onanother analogy with ferromagnetism. The minimumenergy state of a simple isotropic ferromagnet ischaracterized by domains which have internally ho-mogeneous magnetization but different orientations.Energy is stored in the walls separating these do-mains. In analogy, a Bose condensate may consistof domains having constant phase which are sepa-rated by domain walls. The energy associated with

twisting the phase locally is parametrized by a phasestiffness,ρ0, which has the units of energy per lengthsquared. The energy of such a domain wall scaleswith the linear dimension of the systemL as:

E ∼ ρ0Ld−2 (44)

[The phase varies continuously over the domainwall. This is different from an “Ising” domain-wallwhereE ∼ ρ0L

d−1 as naively expected.] On theother hand, we might place the domain wall any-where. The number of possible configurations isLd

and therefore the entropy connected with the domainwall is:

S ∼ d lnL (45)

The free energy of the system with the domain wallis approximated by:

F ∼ ρ0Ld−2 − Td lnL (46)

For d > 2 the first term always dominates the freeenergy for low enoughT , and the ordered state isstable. Ford < 2 the second term, entropy, domi-nates and the ordered state is disordered by domainwalls. Hohenberg [14] and Mermin and Wagner [15,16] made these statements more precise and showedthat the lower critical dimension for a homogeneoussuperfluid is 2.

Confined Systems

All experiments to date have been carried out in a3D potential. According to our estimate, the influ-ence of fluctuations due to the low energy “phase”mode leads to disorder (or entropy) but is not suf-ficient to destroy phase coherence. Moreover, con-finement due to the trapping potential causes eventhe lowest phase mode excitation to have a finitefrequency,ω, the trap frequency, due to space quan-tization. Therefore BEC is actually easier to achievein a potential because entropy, which is already dom-inated by energy, is further suppressed. This is alsotrue in moderately anisotropic traps [44].However, we can easily imagine experimental sit-uations where the atomic clouds are put into a re-duced dimensionality. In strongly anisotropic trapsthe condensate stability question becomes more in-teresting again. If for example one spatial extension,say, in thez direction, becomes much larger thanthe other two, the trap frequencies,ωz, depend con-tinuously on the “wave number,”qz. The spectrumhas a discrete structure for the perpendicular quan-tum numbers (ω⊥ � ωz). If the spectrum of thelowest branch is dense enough (the linear extension

215

of the cloud in thez direction being very large), thelong-range order can be destroyed again due to phasefluctuation [45]. This can be also derived from theobservation that the 1D harmonic trap and the 2Dhomogeneous system have an identical density ofstates. The 1D system may give experimental accessto an extreme system where the scattering lengtha isgreater than the transverse extension of the system.When in this situation the particles of the conden-sate can no longer pass each other, the spectaculardifference of bosons and fermions disappears! Thisentanglement of dimensionality, potential shape andquantum statistics and many other unanswered ques-tions are now a subject of increasing activity in the-ory and experiment.

Summary

The novel Bose quantum systems now available ex-perimentally are exerting a strong impact on bothatomic and condensed matter physics. On the onehand, we must conclude that there have been no realsurprises so far; theoretical predictions have provenwell founded. On the other hand, atomic Bose con-densates have just begun to open a unique experi-mental opportunity with unanticipated flexibility incomparison with liquid helium. Magnetic confine-ment completely isolates the system from any wallsleading to a most refined situation where the kineticsof the sample undergoes pure self-evolution; designof the magnetic trap allows extremely anisotropicsituations to be achieved, yielding lower than three-dimensionality; evaporative cooling control allowsthe number of particles to be varied. Furthermore,atomic Bose condensates have allowed us to peekinto the possibility of having a coherent matter wavegenerator at hand – an intellectually and technicallyhighly attractive tool with consequences which can-not be foreseen today. No doubt BEC will shapemany physics laboratories in the future, where ex-tended work on condensed matter phyics with meth-ods of atomic physics is the next step.

We thank Eric Cornell and Wolfgang Ketterle for providing their figures

for this review.

Appendix: Density of States and QuantumDegeneracy

For an atom trap that produces energy-level spacingswhich are microscopic compared to the mean energy(kT � εi+1−εi), the sum in Eq. 5 can be evaluatedby converting it into an integral:

N = N0 +∫ ∞

0Nερ(ε)dε. (47)

[εi+1−εi is of the order of the energy of the first ex-cited state which for a power-law potential is givenby h2/mr2

0. For example, for a box the validity ofthe semiclassical approximation is equivalent to thecondition r0 � Λ, and for a harmonic potential tokT � ~ω.] Here the ground-state energy is taken tobe zero. The density of statesρ(ε) depends on thetrapping potentialU (r ) and the number of degrees offreedom. To visualize the interplay between the di-mensionality and the potential shape let us considera simple example of an ideal Bose gas in an isotropicpotentialU (r) in d dimensions. Noting that the num-ber of states equals the phase space volume dividedby hd, we calculate the density of states from:

ρ(ε)dε =∫V ∗(ε)

(dp)d(dr)d

hd

∝ h−d∫ r∗(ε)

0pd−1dprd−1dr

=(h2

2m

)−d/2 ∫ r∗(ε)

0

√(ε− U (r))d−2rd−1drdε

(48)

where V ∗(ε) is the volume available for thesystem at the energyε, and r∗ is defined byU (r∗) = ε. Choosing a power-law potentialU (r) =(h2/mr2

0)(r/r0)n, we immediately obtain:

ρ(ε)dε ≈(r0

Λ

)2η ( ε

kT

)η−1 dε

kT(49)

with η = d/2+d/n. From Eqs. 49 and 47 the scalinglaw for the critical temperatureTc can be found bytakingN0 = 0 andµ = 0:

kTc ∝ h2

mr20

[N

I

]1/η

(50)

Equation 50 also gives us the ground-state popula-tion fraction forT < Tc:

N0

N= 1 − (T/Tc)

η (51)

reproducing the 3D special case of Eq. 7. Thusthe deeper the potential well (i.e., the higher thevalue ofh2/mr2

0), the higher the value ofTc. The“confinement power” of the potential, defined as−N−1Tc(dN0/dT )T=Tc = η [18], can be seen todepend on the shape of the potential but not on itsstrength. The integral in Eq. 50:

I =∫ ∞

0

xη−1dx

ex − 1(52)

216

is equal toΓ (η)ζ(η) for η > 1 and diverges oth-erwise. Thus the condition for BEC (Tc > 0) inthermodynamic limit (see below) is given by:

d >2nn + 2

(53)

We see that an ideal Bose gas confined in a box(n → ∞) displays BEC only in 3D [14–16]. In aharmonic potential, (n = 2) the BEC condition canbe fullfilled only in 3D or 2D [19].It can be also shown [18] that the heat capacityC(T ) [The heat capacity is defined byC(T ) =∂E(T )/∂T with the total energy of the system givenby E(T ) =

∫∞0 εNερ(ε)dε.] is discontinuous atTc

for d/n + d/2 > 2. If d/n + d/2 ≤ 2, C(T ) iscontinuous atTc, but ∂C(T )/∂T is discontinuous.In general,C(T ) is larger for a power-law potentialthan for a rigid wall container. This is because in-creasing the energy of the gas requires work againstthe confining potential.

BEC in a Finite System

The reason of the “BEC failure” forη ≤ 1 mustbe found in the usual thermodynamic limit, whichassumesN → ∞ and r0 → ∞ with N/r2η

0 finite.For a real system of finite size the lower limit di-vergence in Eq. 52 is unphysical. Let us considera particular caseη = 1 corresponding to a 1D har-monic potential and also to a 2D box. IntroducingE1/kT as the lower limit of the integral Eq. 52 withE1 of the order of the first excited state, we obtainfor the transition temperature [the exact results [7]are: E1 = ~ω/2 for a 1D-harmonic potential andE1 = h2/(8Mr2

0) for a 2D box]:

N ∝ kTcE1

ln

(kTcE1

)(54)

and for the condensate fraction:

N0

N= 1 − T ln(kT/E1)

Tc ln(kTc/E1)(55)

with the logarithmic terms becoming negligible forlarge particle number in accordance with Eq. 51 forη = 1. Thus the predicted impossibility of BEC inlow-dimensional systems is an artifact of the thermo-dynamic limit, which does not apply to the situationrealized in atoms traps.The N -finite effects are less dramatic for systemsshowing BEC in the thermodynamic limit (η > 1).As an illustration let us consider a 3D harmonic po-tential (η = 3 andr2

0 = ~/mω) whose BEC condition(Eq. 50) takes the form:

N ∼(kTc~ω

)3

(56)

Obviously, this is the first term of the expansion inlarge parameterkT/~ω (see above). The next termshould scale as(kT/~ω)2, giving us the first correc-tion to the critical temperature:

T 0c − TcT 0c

∼ N−1/3 (57)

An exact calculation [7] shows that forN = 103 thetransition temperatureTc is less than 7% the usualresultT 0

c extrapolated fromN = ∞.It is interesting to rewrite Eq. 50 as:

N ∼(kTcU0

)d/nrd0

(MkTch2

)d/2

. (58)

Introducing an effective volumeV ∗ ∼ rd of thesystem which is determined bykT ∼ U0(r/r0)n,that is, V ∗ ∼ rd0(kT/U0)d/n and using for the deBroglie wavelengthΛ2 ∼ h2/MkT , we obtain thegeneral condition for quantum degeneracy:

ΛdN

V ∗ ∼ 1 (59)

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