Bose-Einstein condensation in the alkali gases: Some fundamental concepts

Anthony J. Leggett
Rev. Mod. Phys. 73, 307 – Published 24 April 2001; Erratum Rev. Mod. Phys. 75, 1083 (2003)
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Abstract

The author presents a tutorial review of some ideas that are basic to our current understanding of the phenomenon of Bose-Einstein condensation (BEC) in the dilute atomic alkali gases, with special emphasis on the case of two or more coexisting hyperfine species. Topics covered include the definition of and conditions for BEC in an interacting system, the replacement of the true interatomic potential by a zero-range pseudopotential, the time-independent and time-dependent Gross-Pitaevskii equations, superfluidity and rotational properties, the Josephson effect and related phenomena, and the Bogoliubov approximation.

    DOI:https://doi.org/10.1103/RevModPhys.73.307

    ©2001 American Physical Society

    Erratum

    Authors & Affiliations

    Anthony J. Leggett

    • Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080

    • aIt is convenient for the present purpose to include H in the alkalis. However, some of the values quoted as “typical” below do not apply to it.
    • bHowever, the detuning is usually still small enough relative to the resonance frequency that the counter-rotating terms (seeCohen-Tannoudji,op. cit.) can be neglected in the analysis.
    • cIn the following I use “hyperfine” as a shorthand for “hyperfine-Zeeman”; thus two different hyperfine species may differ in the values ofF and/ormF.
    • dThe numbers quoted are for4He at saturated vapor pressure; the numbers at pressures up to freezing, and for the light isotope3He, are similar in order of magnitude.
    • eThe Fermi system3He is believed to become superfluid by forming Cooper pairs, which then effectively undergo BEC; the pairs are somewhat analogous to the alkali atoms in that they possess a hyperfine degree of freedom.
    • fThis tendency may be seen already in the trivial caseN=p=2: for distinguishable particles, states involving double occupation are 50% of the whole, while for bosons they are 66%.
    • gBecause of the critical role in the theory of BEC played by conservation of total particle numberN, it is not entirely obviousa priori that the use of the grand canonical ensemble is justified. However, calculations using the microcanonical ensemble give similar results for largeN (see, for example,Gajda and Rzazewski, 1997).
    • hThe caseD=1 is rather delicate; seeDalfovo et al., 1999.
    • iThe generalization to systems with internal degrees of freedom is made in Sec. V.D.
    • jAn interesting attempt to justify Eqs. (3.14) and (3.13) via the concept of a “phase standard” has been made byDunningham and Burnett (1998) and is discussed inLeggett (2000a).
    • kStrictly speaking, in the limitr the leading term inVat(r) is the electromagnetic interaction between the electron spins, which falls off only asr3. However, for all the alkalis, includingH, the mean field due to this term may be verified to be small compared with the “standard” mean field calculated below.
    • lThis is one respect in whichH is very different:C66.5,ro5 Å,Ec3 K.
    • mIt may be worthwhile to note that the zero-energy Schrödinger equation can be solved explicitly for any potential of the formα/rn (and in particular for the van der Waals potential,n=6) in terms of Bessel and Neumann functions; seeGribakin and Flambaum, 1993, Eq. (13).
    • nIf we attempt an estimate of the kinetic energy on the basis of Eq. (4.2), we find it is linearly divergent, so that at this point we must go back to the true short-range(rr0) behavior ofψ(r). We return to this point in Sec. VIII.C.
    • oHowever, to avoid having to consider the effects of indistinguishability at this stage we continue to assume they can be “tagged.”
    • pBy an obvious extension of the argument, the probability of finding three particles at the same point is six times larger when they are all in different states than when they are all in the same one-particle state. As pointed out byKagan et al. (1985), this leads to a significant reduction of three-body recombination processes in the BEC state, a prediction verified experimentally byBurt et al. (1997).
    • qSee, for example,Mandel and Wolf (1995), p. 752.
    • rStrictly speaking this is an oversimplification: even at this level, the many-body wave function must build in the short-range(rijas) correlations discussed in Sec. IV.B. In the following I implicitly assume that this has been done.
    • sStrictly speaking, theT=0 time-independent Gross-Pitaevskii equation.
    • tSimilar remarks apply to Eq. (5.9) below.
    • uThe quantityni is then constant in time and determined by itst=0 value.
    • vIn words, the condensate atoms are in the lower hyperfine manifold and their spins are everywhere oriented along the direction of the local magnetic field.
    • wNeedless to say, if we accept the concept of spontaneously broken U(1) symmetry (see Sec. III.D), the above derivation may be short-circuited by the replacements, in Eq. (5.26),ψα(r)ψβ(r)ψγ(r)ψδ(r)ψα(r)ψβ(r)ψγ(r)ψδ(r),ψα(r)Ψα(r), etc. I prefer to avoid this “automatic” derivation both for the reasons given in Sec. III.D and because it makes the generalization to general (nonsimple) BEC less transparent.
    • xThe origin of this difficulty is essentially the point noted at the end of Sec. V.B above, namely, that within the Hartree-Fock (Gross-Pitaevskii) approximation the condensate and the normal quasiparticles experience different effective Hamiltonians.
    • yNote that the maximum value ofnas3 isη6/(15)3/5.
    • zAll the ensuing considerations have close analogs in the theory of superconductivity, which indeed is nothing but superfluidity occurring in a charged system; see, for example,Vinen (1969) orLeggett (1995b).
    • aaStrictly speaking, that of the fixed stars; I shall ignore the difference.
    • bbIn fact, the limit of exact cylindrical symmetry needs to be approached with some care; seeLeggett (2000a).
    • ccThere are slight complications, which I shall ignore, concerning the small shift in μ and the fact that (ford/R finite) a meniscus tends to form.
    • ddHowever, in the following I shall implicitly assume that the axis of rotation lies within the container cross section, at least at some values ofz (otherwise the problem becomes rather trivial and uninteresting).
    • eeHowever, the case|n|>1 is not of much practical interest, since such multiply quantized vortices are rather generically unstable against dissociation into several singly quantized ones.
    • ffThis follows, for a Bose system in arbitrary geometry, from the fact that the many-body ground-state wave function can always be chosen real.
    • ggIn general, negative values ofg are apt to lead to collapse in position space (seeDalfovo et al., 1999, Sec. III.C), but for any given geometry it is possible to chooseg small enough that this does not occur.
    • hhThere are further complications when more than one hyperfine species is involved. See Sec. VI.F.
    • iiIt is a matter of purely historical interest that in the original realization of the Josephson effect in superconductivity (a) the “bosons” (Cooper pairs) are composite rather than elementary objects, and (b) the system is coupled to external current leads and therefore cannot be described by the Hamiltonian (7.5) as it stands. The closed problem described by Eq. (7.5) is actually conceptually simpler than the original version (and nowadays can be realized in a superconducting context, e.g., in mesoscopic systems).
    • jjThis choice of definition forM, while technically convenient, means that some of the formulas below, which are written explicitly for evenN, need correction by factors(N1) for oddN. This affects none of the conclusions and will not be noted explicitly below.
    • kkThis situation is to be contrasted with that obtained in traditional condensed-matter systems, where physical considerations generally restrict the motion to the regime when Eq. (7.5) is an excellent approximation.
    • llSubsequent experiments may in fact have attained this regime (seeRohrl et al., 1997).
    • mmIt may be shown that in the WKB limit (B1) theM dependence ofEJ resulting from this effect is much greater than the “kinematic” dependence explicitly appearing in Eq. (7.21) below.
    • nnFor example, the experiments ofHall, Matthews, Wieman, and Cornell (1998) use two pulses of approximately equal amplitude but shifted by a phase φ in the rotating frame. In this caseEJ(t) is of course zero outside the pulses, and we can takeζ(t)0 during the first pulse andφ during the second.
    • ooThis is at least partly due to the anomalously small value of the quantity (a11+a222a12) for87Rb and is not necessarily a generic characteristic of such experiments.
    • ppA possible alternative approach (Javanainen and Ivanov, 1999) is to take over the definition of phase used byPegg and Barnett (1988) to the two-state problem. While this automatically guarantees the Hermiticity ofφ^, the relation to the coherent state (7.14) is not so direct.
    • qqWhile these authors also discuss the status of an angle variable, it is the angle of a coordinate vector rather than of the angular momentum itself, and we therefore cannot simply take over their discussion verbatim.
    • rrThese include, for example,Milburn et al., 1997;Smerzi et al., 1997;Javanainen and Ivanov, 1999;Leggett, 1999a;Marino et al., 1999;Raghavan et al., 1999;Sols, 1999.
    • ssIn the classical approximation.
    • ttAs discussed in Sec. VII.C, during the pulse itself the system is in the Rabi limit.
    • uuThis result is of course only approximate, and in particular ignores important questions concerning the effect of interatomic interactions in the overlap region (which fortunately turns out to be fairly negligible), but it will be adequate for our present purpose.
    • vvNote that had we started with a wave packet (such as the ground state of a harmonic well) that had a finite spread in φ and hence a value of(ΔM)o much smaller thanN1/2, the decoherence time would have been much longer. Cf.Leggett and Sols, 1998;Javanainen and Wilkins, 1997.
    • wwInclusion inĤ of cubic and higher terms in will tend to reduce the amplitude of the recurrences even for an isolated system, but a dimensional estimate shows that the effect is at most of order unity and may be smaller.
    • xxFor related number-conserving formulations see, for example,Girardeau et al. (1959) andGardiner (1997), and for the standard (number-nonconserving) approach seeHuang (1987), Chap. 13.
    • yyTo obtain Eq. (8.13) we replaced matrix elements such asN0(N0+1) byN0. Note that the sum overq now goes overall nonzeroq.
    • zzTo my knowledge, the first place in the literature where this is (implicitly) pointed out is a paper byFetter (1972), which is probably less well known in the BEC community than it deserves to be.
    • aaaProvided it is real, which will be true if the Hamiltonian is invariant under time reversal.
    • bbbI hope to discuss this question further elsewhere.
    • cccFor explicitly number-conserving formulations of the ensuing argument and that of Sec. VIII.E, seeGardiner (1996) orCastin and Dum (1997).
    • dddIn the He literature it is conventional to express the ensuing argument in terms of the superfluid velocityvs(r) defined by Eq. (3.15); the equation corresponding to Eq. (6.12) is the Onsager-Feynman quantization condition (3.17).

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    Vol. 73, Iss. 2 — April - June 2001

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