BOSE-EINSTEIN CONDENSATION IN DILUTE ATOMIC GASES A. F. R. de Toledo Piza Instituto de F¶‡sica, Universidade de Sa~o Paulo C.P. 66318, 05315-970 Sa~o Paulo, S.P. Escola Brasileira de Meca^nica Estat¶‡stica Sa~o Carlos, S.P. - 16 a 20 de fevereiro de 2004 Prologue The flve lectures (or Chapters) which follow are intended as a pedagogical, theoretically oriented introduction to the presently very active fleld involving the physics of Bose-Einstein condensation in trapped atomic gases. The lectures have been prepared for the 2004 Brazilian School on Statistical Mechanics,havinginmindanaudienceofgraduateandadvancedundergraduatestudentspossibly,and perhapseventypically,notconcentratinginthesubject. Needlesstosay,thismakesthemquitedistinct fromareviewarticleonthesubject,chie(cid:176)yasconsiderationsofimpedancematchingwiththeintended audience plays the dominant role in the choice and development of topics. This particular School is perhaps a somewhat peculiar setting for these lectures, since the systems which will be considered are orderly enough to be quite well described, for several relevant purposes, even in terms of a single one-body wavefunction. This will be discussed in lecture 2, after the statistical mechanics of the condensation phenomenon for an ideal Bose gas is discussed in lecture 1. A fortunate circumstance in thisconnectionisthatthecondensationofanidealgasistodaynolongerjustasimplifyingidealization, but an experimentally studied (i.e., real) phenomenon (see section 1.2). This will be discussed in some detail in lecture 3, together with the just now fashionable theme of molecular hybridization of dilute Bose-Einstein condensates and Bose-Einstein condensation of diatomic molecules formed in very cold fermionic gases. Lecture 4 will carry us slightly beyond the basic \single one-body wavefunction" description of real condensates, and lecture 5 will introduce experimental results on the optical fracturing of condensates as well as some of the ideas and simplifled models used in connection with such situations. In preparing these lectures, I have tried my best to avoid being trapped in what has been called by the former brazilian minister Pedro Malan in a newspaper article published on page 2 of O Estado de Sa~o Paulo in January 11, 2004 (or was this also a quote?) \some kind of error contract" between someonetryingtotransmitthoughtsandideasandhis\receivers". Thissyndromehasbeensupposedly described by none other than Francis Bacon. It was brought to Malan’s attention (thus flnding its way to his article and flnally to this Prologue) by a book by another economist, Eduardo Giannetti da Fonseca, who identifles Bacon’s The advancement of learning, as the source. As I have not been able to locate the original quote in time, I do my best translating (re-translating?) the brazilian version of the quote to English, certainly not Bacon’s: \Who transmits knowledge chooses to do it so as to enhance belief rather than the possibility of examination, and who receives knowledge seeks rather present satisfaction than the promises of investigation, and thus will rather not doubt than not fail; 1 glory leads the author not to reveal his weaknesses, and laziness leads the disciple not to realize his strength"1. Curious as my sources for these ideas happened to be, the dangers to which they refer are 1Scanning once again Book 1 of Bacon’s The advancement of learning I flnally realized that at least some heavy editing has been involved in these in fact rather loose quotes. The closest, or most relevant, passages I have been able to flnd in Bacon’s original work are transcribed here as they appear in the Renascence Editions \imprint" available on line at the site http://darkwing.uoregon.edu/~rbear/adv1.htm, see especially paragraph 9 of section V: \BOOK 1, IV.12. And as for the overmuch credit that hath been given unto authors in sciences, in making them dictators, that their words should stand, and not counsellors to give advice; the damage is inflnite that sciences have received thereby, as the principal cause that hath kept them low at a stay without growth or advancement. For hence it hath come, that in arts mechanical the flrst deviser comes shortest, and time addeth and perfecteth; but in sciences theflrstauthorgoethfarthest,andtimeleesethandcorrupteth. Sowesee, artillery, sailing, printing,andthelike, were grossly managed at the flrst, and by time accommodated and reflned: but contrariwise, the philosophies and sciences of Aristotle, Plato, Democritus, Hippocrates, Euclides, Archimedes, of most vigour at the flrst and by time degenerate and imbased; whereof the reason is no other, but that in the former many wits and industries have contributed in one; and in the latter many wits and industries have been spent about the wit of some one, whom many times they have rather depraved than illustrated. For as water will not ascend higher than the level of the flrst springhead from whence itdescendeth,soknowledgederivedfromAristotle,andexemptedfromlibertyofexamination,willnotriseagainhigher than the knowledge of Aristotle. And therefore although the position be good, OPORTET DISCENTEM CREDERE, yetitmustbecoupledwiththis,OPORTOEDOCTUMJUDICARE;fordisciplesdooweuntomastersonlyatemporary beliefandasuspensionoftheirownjudgmentuntiltheybefullyinstructed,andnotanabsoluteresignationorperpetual captivity: andtherefore,toconcludethispoint,Iwillsaynomore,butsoletgreatauthorshavetheirdue,astime,which istheauthorofauthors,benotdeprivedofhisdue,whichis,furtherandfurthertodiscovertruth. ThushaveIgoneover these three diseases of learning; besides the which there are some other rather peccant humours that formed diseases: which nevertheless are not so secret and intrinsic but that they fall under a popular observation and traducement, and therefore are not to be passed over. (...) V.6. Another error hath proceeded from too great a reverence, and a kind of adoration of the mind and understand- ing of man; by means whereof men have withdrawn themselves too much from the contemplation of nature, and the observationsofexperience,andhavetumbledupanddownintheirownreasonandconceits. Upontheseintellectualists, whicharenotwithstandingcommonlytakenforthemostsublimeanddivinephilosophers,Heraclitusgaveajustcensure, saying, MEN SOUGHT TRUTH IN THEIR OWN LITTLE WORLDS, AND NOT IN THE GREAT AND COMMON WORLD;fortheydisdaintospell,andsobydegreestoreadinthevolumeofGod’sworks: andcontrariwisebycontinual meditation and agitation of wit do urge and as it were invocate their own spirits to divine and give oracles unto them, whereby they are deservedly deluded. (...) 9. Another error is in the manner of the tradition and delivery of knowledge, which is for the most part magistral and peremptory, and not ingenuous and faithful; in a sort as may be soonest believed, and not easiliest examined. I: is true,thatincompendioustreatisesforpracticethatformisnottobedisallowed: butinthetruehandlingofknowledge, men ought not to fall either on the one side into the vein of Velleius the Epicurean: NIL TAM METUENS, QUAM NE DUBITAREALIQUADEREVIDERETUR;[13]norontheothersideintoSocrateshisironicaldoubtingofallthings; buttopropoundthingssincerelywithmoreorlessasseveration,astheystandinaman’sownjudgmentprovedmoreor less. (...) 11. But the greatest error of all the rest is the mistaking or misplacing of the last or farthest end of knowledge: for men have entered into a desire of learning and knowledge, sometimes upon a natural curiosity and inquisitive appetite; sometimes to entertain their minds with variety and delight; sometimes for ornament and reputation; and sometimes to enable them to victory of wit and contradiction; and most times for lucre and profession; and seldom sincerely to give a true account of their gift of reason, to the beneflt and use of men: as if there were sought in knowledge a couch whereupontorestasearchingandrestless spirit; oratarrasse, forawanderingandvariable mind towalkupanddown withafairprospect;oratowerofstate,foraproudmindtoraiseitselfupon;orafortorcommandingground,forstrife andcontention;orashop,forprofltorsale;andnotarichstorehouse,forthegloryoftheCreatorandthereliefofman’s 2 serious enough to deserve careful consideration. I thus hope that the content of the lectures will be doubted rather than believed, and tried to encourage this by not concealing remaining obscurities. A word should be said on why these lectures, or Chapters, have been written in what amounts to the lingua franca2 of today. The organizers of the School said they intended to publish written versions of the courses in a special issue of the former Revista Brasileira de F¶‡sica, now known as Brazilian Journal of Physics. My reaction to this was that it tended to create a certain con(cid:176)ict, as the scope, tone and purpose of course notes difier a lot from that of material usually printed even in special issues of periodicals. The intention having been maintained, my option has been to stick to the scope, tone and purpose of course notes while borrowing the use of the lingua franca from the periodical literature. Which might, after all, also have some pedagogical efiect. A. F. R. de Toledo Piza, Feb. 8, 2004. estate. Rut this is that which will indeed dignify and exalt knowledge, if contemplation and action may be more nearly andstraitlyconjoinedandunitedtogetherthantheyhavebeen; aconjunctionlikeuntothatofthetwohighestplanets, Saturn,theplanetofrestandcontemplation,andJupiter,theplanetofcivilsocietyandaction: howbeit,Idonotmean, when I speak of use and action, that end before-mentioned of the applying of knowledge to lucre and profession; for I amnotignoranthowmuchthatdivertethandinterrupteththeprosecutionandadvancementofknowledge,likeuntothe golden ball thrown before Atalanta, which while she goeth aside and stoopeth to take up, the race is hindered; Declinat cursus, aurumque volubile tollit. (...)" (Feb. 28, 2004) 2Lingua Franca; a composite language made up of Italian and the various languages of western Asia, used in the Levant by foreign traders and natives of that region. (Webster’s New Twentieth Century Dictionary of the English Language, Unabridged. Rockville House Publishers, Inc., New York, 1964). 3 Contents 1 Condensation of ideal bosons in a trap 6 1.1 Grand-canonical quantum statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Ideal Bose gas in a trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 \Quasi one-dimensional" harmonic trap . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Uniform vs. trapped ideal Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.3.1 One-dimensional free gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4 Relevant parameters and orders of magnitude . . . . . . . . . . . . . . . . . . . . . . . 21 2 Non-ideal dilute Bose gas 23 2.1 Simple efiective interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 Efiective mean-fleld (Gross-Pitaevski) approximation . . . . . . . . . . . . . . . . . . . 27 2.2.1 The uniform gas and the \healing length" . . . . . . . . . . . . . . . . . . . . . 31 2.2.2 The Thomas-Fermi regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Spin-dependent efiective interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.1 Gross-Pitaevski and Thomas-Fermi treatments . . . . . . . . . . . . . . . . . . 36 2.4 Gross-Pitaevski limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 Roles for atomic structure 44 3.1 Atom-atom resonance scattering and the efiective two-body interaction . . . . . . . . . 44 3.1.1 Efiects of internal structure dynamics . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.2 Single closed channel bound state and resonance scattering . . . . . . . . . . . 49 3.1.3 Resonant phase shift in low energy scattering . . . . . . . . . . . . . . . . . . . 51 3.2 Feshbach resonances and molecular condensates . . . . . . . . . . . . . . . . . . . . . . 54 3.3 Molecules in atomic condensates and hybrid atomic condensates by stimulated transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.3.1 Molecule formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3.2 Hybrid atomic condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 4 One step beyond the Gross-Pitaevski description 66 4.1 Bogoliubov’s quasi-particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.1.1 Easy route to the quasi-particles . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1.2 Ground state depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Self-consistency looks worse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2.1 Di–culties with the contact efiective interaction . . . . . . . . . . . . . . . . . 77 4.3 Elementary excitations on worse looking richer foundations . . . . . . . . . . . . . . . 78 5 Many mode traps, localization and interference 82 5.1 Quick survey of phenomena with some interpretation hints. . . . . . . . . . . . . . . . 82 5.2 Simple models for split condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3 Results for a two-well system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.3.1 Semi-classical domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3.2 Quantum domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.4 Many site one dimensional arrays, periodic boundary conditions . . . . . . . . . . . . . 96 5 Chapter 1 Condensation of ideal bosons in a trap 1.1 Grand-canonical quantum statistics 1. Prolegomena: state vectors vs. density operators. States of thermal equilibrium of many- particle quantum systems cannot be represented by state vectors on the account of the fact that they do not correspond to deflnite microscopically deflned states, but rather correspond to an incoherent distribution over the various possible such states. In this context, incoherent means that any interfer- ence efiects involving difierent states in the distribution are precluded. One needs therefore a suitably extended way of describing quantum states in order to be able to deal with thermal equilibrium states. The required extension is provided by the notion of density operators. Consider flrst how this notion arises in the particular situation in which the state of the system can be described in terms of a state vector. If the microscopically deflned state of a many particle quantum system is characterized by a normalized state vector ˆ , ˆ ˆ = 1 (or, equivalently, by a wavefunction ˆ(r ;:::;r ) 1 N j i h j i · r ;:::;r ˆ ,where r ;:::;r isaDiraceigenketofthepositionoperatorsforthevariousparticles), 1 N 1 N h j i j i then it is characterized as well by the density operator, deflned in this case as the projection operator ‰ ˆ ˆ · j ih j or, equivalently, by the (N-particle) density matrix ‰(r ;:::;r ;r0;:::;r0 ) r ;:::;r ˆ ˆ r0;:::;r0 1 N 1 N · h 1 Nj ih j 1 Ni · ˆ(r ;:::;r )ˆ⁄(r0;:::;r0 ): · 1 N 1 N Thenormalizationofthestatevectortranslatesintothepropertythatthedensityoperator, ormatrix, has unit trace, i.e. Tr‰ d3r ::: d3r ˆ(r ;:::;r )ˆ⁄(r ;:::;r ) = ` ˆ ˆ ` = 1 1 N 1 N 1 N j j · h j ih j i Z Z j X 6 where the vectors ` constitute an arbitrary orthonormal base in state vector space. In this case j fj ig the operator ‰ is clearly an idempotent (i.e., ‰2 = ‰) self-adjoint operator. Its only possible eigenvalues are therefore 1 and 0. The eigenvector corresponding to the eigenvalue 1 is clearly the state vector ˆ itself (this is therefore how the original state vector ˆ can be retrieved, given the corresponding j i j i idempotent density operator ‰), while the eigenvalue 0 is highly degenerate, since any state-vector orthogonal to ˆ is an eigenvector associated to this eigenvalue. Note moreover that the expectation j i value of some given observable O in the state ˆ can be obtained directly from the density operator j i as TrO‰ ` O‰ ` = ` O ˆ ˆ ` = ˆ ` ` O ˆ = ˆ O ˆ ; j j j j j j · h j j i h j j ih j i h j ih j j i h j j i j j j X X X the last step following from the completeness of the orthonormal base ` . In a similar way one j fj ig can also verify the important cyclic property of the trace TrO‰ = Tr‰O which is in fact more general, in that it does not depend on the particular form of ‰. Therequiredgeneralizationofthewaytocharacterizestatesofquantumsystemsinordertoinclude the needed incoherent distributions of state vectors consists in replacing the idempotency condition ‰2 = ‰ by the weaker condition that ‰ is a non-negative self-adjoint operator with unit trace. The non- negativity condition means that all the eigenvalues are non-negative, i.e. they are positive or zero. These density operators can be conveniently written in terms of their eigenvalues and normalized eigenvectors R as j j i ‰ R = p R ; p 0; ‰ = R p R : (1.1) j j j j j j j j i j i ‚ ¡! j i h j j X The unit trace condition is now expressed as p = 1. The idempotent density operators are clearly j j particular cases of this more general class, which include moreover positive linear combinations of P many (possibly even an inflnite number of) orthogonal projection operators. The unit trace condition (actually the condition that the trace is flnite is of course su–cient) in fact restricts a great deal the spectrum of the more general density operators: by virtue of the Hilbert-Schmidt theorem it is guaranteed to be a purely discrete spectrum; and when the number of non-zero eigenvalues is inflnite, they can only have zero as an accumulation point. Average values of observables in states described by these density operators are also calculated in terms of a trace, which gives now TrO‰ = O R p R = p R O R : (1.2) j j j j j j j i h j h j j i j j X X This average appears thus as a weighted average of quantum expectation values in the state vectors R , with weights p . A standard interpretation of this is that the density operator ‰ describes an j j j i 7 ensemble of systems, in which the (classical) probability of flnding a system in state R is p . A j j j i particularlyrelevantpropertyofthestaterepresentedbythedensityoperator‰,whichgivesameasure of fragmentation of the trace into the array of classical probabilities is its entropy S (sometimes called the von Neumann entropy), deflned as S = k Tr(‰ ln‰) = k p lnp : (1.3) B B j j ¡ ¡ j X For k > 0 this is a non-negative quantity which vanishes in the limit of an idempotent density opera- B tor. Iftheconstantk istakenastheBoltzmannconstant,S hasunitsofthestandardthermodynamic B entropy. Afurther stepisstillneededwhenonewishestousegrand-canonicalmethods,asitisoftenthecase in the context of Bose-Einstein condensation. In this case the number of particles in the many-particle system under consideration is not flxed and must be seen as an observable. The way to accommodate this is to use the language of \second quantization" (see, e.g., ref. [1], Chapter 7). The state vectors nowresideina\Fockspace", oroccupationnumberspace, inwhichanumberoperatorcanbedeflned. The basic objects representing states of thermal equilibrium of quantum many-particle systems are thus to be taken as positive self-adjoint operators of unit trace in Fock space. 2. Grand-canonical equilibrium density operator. We now consider speciflcally a system of many identical bosonic atoms characterized by a hamiltonian H. For the purposes of the formal devel- opments to be undertaken at this point, this may include interactions between atoms, e.g. represented by a suitable two-body potential v(r ;r ), in addition to an external, one-body conflning potential j l representing the trap. The state of the system is described by a density operator ‰ in Fock space. In this space we have also a number operator N, and the hamiltonian is represented by an operator which commutes with N. This means that one could adopt a canonical formulation by restricting the treatment to the N-particle sector of the Fock space, which is closed under the action of the number- conserving hamiltonian H, but it will be convenient to allow for states in which neither the number of atoms nor the energy are sharply deflned. The problem we are set to solve is to determine the density operator which makes the entropy an extremum,withprescribedaverage valuesfortheenergyandnumberofatoms. Thedensityoperatoris written in the form (1.1), so that the average values of the hamiltonian H and of the number operator N are given respectively by (see eq. (1.2)) TrH‰ = p R H R and TrN‰ = p R N R : j j j j j j h j j i h j j i j j X X The entropy is expressed as in eq. (1.3), provided the weights satisfy the condition p = 1 and the j j states R are normalized, R R = 1. These subsidiary conditions can all be taken care of in terms j ji h jj ji P 8 of Lagrange multipliers fl, fi, ‚ and · , so that the variational condition for the thermal equilibrium j state is – k p lnp fl p R H R fi p R N R ‚ p + · R R = 0: B j j j j j j j j j j j j 0¡ ¡ h j j i¡ h j j i¡ h j i1 j j j j j X X X X X @ A Variation of p gives j k lnp +k +fl R H R +fi R N R +‚ = 0 (1.4) B j B j j j j h j j i h j j i while variation of R leads to j h j · j p (flH +fiN) R = · R or (flH +fiN) R = R † R ; (1.5) j j j j j j j j j i j i j i p j i · j i j which shows that the states R are eigenstates of flH +fiN, so that this operator and ‰ are simul- j j i taneously diagonal, with fl R H R +fi R N R = † . Straightforward algebra now gives, from j j j j j h j j i h j j i (1.4), p = e¡1¡‚=kBe¡†j=kB j which, using the unit trace condition to evaluate the flrst exponential, leads to e¡†j=kB p = : j e¡†j=kB j P The denominator of this expression can be written as Tr e¡(flH+fiN)=kB , so that the resulting form for the density operator is ‡ · ‰ = jjRjie¡†j=kBhRjj = e¡(flH+fiN)=kB : PTr e¡(flH+fiN)=kB Tr e¡(flH+fiN)=kB ¡ ¢ ¡ ¢ The meaning of the Lagrange multipliers ‚, fl and fi in macroscopic terms can be found by com- paring the statistical expression for the entropy which results from imposing the variational condition (1.4), namely S = k +‚+flTrH‰+fiTrN‰; B with the corresponding thermodynamic expression 1 S = (› U +„N); ¡T ¡ 9