BOSE-EINSTEIN CONDENSATION AND SUPERFLUIDITY IN TRAPPED ATOMIC GASES Sandro Stringari Dipartimento di Fisica, Universit`a di Trento, and Istituto Nazionale per la Fisica della Materia, I-38050 Povo, Italy (February 6, 2008) Bose-Einstein condensates confined in traps exhibit unique features which have been the object of extensive experimental and theoretical studies in the last few years. In this paper I will discuss someissuesconcerningthebehaviouroftheorderparameterandthedynamicandsuperfluideffects 1 exhibited by such systems. 0 0 2 A. Introduction n a TheaimoftheselecturesistoreviewsomekeyfeaturesexhibitedbyBose-Einsteincondensationintrappedatomic J gases, with special emphasis to the phenomena of superfluidity which have been the object of recent experimental 9 and theoretical investigation and which are characterized in a non trivial way by the combined role of the phase of 1 the order parameter and of two-body interactions. 1 Let us start recalling that these systems are characterized, at sufficiently low temperatures, by a complex order v parameter which can be written as 9 9 Ψ (r,t)= n (r,t)eiS(r,t) (1) 2 0 0 1 where n = Ψ 2 is the so called condensate densitypand S is the phase. In general the condensate density does 0 0 | 0 | not coincide with the density n of the gas, because of the occurrence of quantum and thermal effects. However, in a 1 0 very dilute gas, where the quantum depletion is negligible and for low temperatures, such that the thermal depletion / is small, the condensate density coincides in practice with the total density. For this reason one often identifies t a n with n. For the same reason in these systems the condensate density can be measured with good precision 0 m through imaging techniques. Even at higher temperature the condensate density can be extracted experimentally - with reasonable accuracy. In fact, due to the harmonic shape of the external confinement, the thermal component d is spacially separated from the condensate so that the density profile is characterized by a typical bimodal structure n allowingforsystematicexperimentalanalysis. Theorderparameteroftrappedatomicgasesthenemergesasacrucial o quantitynotonlyfromtheconceptualpointofviewlikeinallmany-bodysystemscharacterizedbyabrokensymmetry, c : butalsofromtheexperimentalone. Thisrepresentsakeydifferencewithrespecttosuperfluidheliumwheretheorder v parameter is a very hidden variable which can be measured only under very special conditions. The order parameter i X of a trapped gas can vary in space, time and temperature and characterizes in a direct way most of the physical r quantities that one measures in these systems. a In the first part I will mainly discuss the properties of the modulus of the order parameter. From the theoretical point of view this quantity is interesting because even its shape at equilibrium is very sensitive to the presence of two-body interactions. In the second part I will give more emphasis on the role of the phase, a quantity which plays a crucial role in characterizing the coherence and superfluid features of our systems. Of course the modulus and the phase of the order parameter are deeply related quantities and in many cases their behaviour cannot be discussed in a separate way. Many of the results discussed in these notes are derived in more detail in ref. [1] to which we refer also for a more complete list of references. The discussion on Bose-Einstein condensation presented in these notes is stimulated by several features of funda- mental interest that are worth mentioning. In particular it is important to recall that -BECgasesintrapsprovideauniqueopportunitytostudythetransitionfromthemicroscopictothemacroscpic world. - In these systems BEC shows up both in momentum and coordinate space. -BECgasesprovideanalmostidealrealizationofclassicalmatterwaves,givingrisetonewcoherencephenomena. - Trapped gases are well suited to study rotational superfluid phenomena. 1 B. BEC is a phase transition In the presence of harmonic trapping 1 V = m ω2x2+ω2y2+ω2z2 (2) ext 2 x y z (cid:0) (cid:1) the critical temperature for Bose-Einstein condensation of an ideal gas can be easily evaluated using the standard techniques of quantum statisticalmechanics. To this purpose one has to define the thermodynamic limit. In the case of harmonic trapping the natural choice is to take N and ω 0, with the combination Nω3 kept fixed. → ∞ ho → ho Here ω = (ω ω ω )1/3 is the geometrical average of the three harmonic frequencies. Notice that in this limit the ho x y z system, differently from traditional thermodynamic bodies, is characterized by a non uniform density varying on a macroscopic scale, much larger than the typical interatomic distance. The critical temperature takes the value k T =0.94¯hω N1/3 (3) B c ho while the condensate fraction obeys the law 3 N T 0 =1 (4) N − T (cid:18) c(cid:19) for T <T . Similarly to the case of the uniform gas one predicts a phase transition even in the absence of two-body c interactions. This unique prediction of quantum statistical mechanics can be now tested experimentally in trapped Bose gases [2,3]. In fact, close to the critical temperature, these gases are so dilute that one can neglect in first approximation the role of the interaction. Estimate (3) has actually guided experimentalists to the right domain of temperatureswhichturnsouttobeoftheorderofµK (typicallythetrappingfrequenciesareoftheorderofafewnK while the number N of atoms varies from 104 to 107). Also the temperature dependence of the condensate fraction turns out to be in reasonable agreement with the theoretical prediction (4) as shown by fig 1. An interesting question is how two-body forces modify the ideal gas estimate (3). Interaction effects can have two different origins. A first one is a genuine many-body effect exhibited by uniform systems where one works at fixed density. This effect is highly non trivialfrom the theoreticalpoint of view [4] but the resulting correctionis expected to be small and difficult to observe in trapped gases. A second effect is the result of mean field forces which, in the case of positive scattering lengths, tend to push the gas towards the external region, thereby reducing the density of the gas. This causes a natural decrease of the critical temperature. The opposite happens if the interaction is attractive. The mean field effect can be easily calculated and the result takes the form [5] δT a c = 1.3 N1/6 (5) T − a c ho where a is the scattering length and a = ¯h/mω is the oscillator length. Typical values for δT /T are a few ho ho c c percent in the available traps and the present accuracy of experimental data does not allow for a quantitative check p of this prediction. By tuning the value of the scattering length to larger values it should be possible to obtain more sizable corrections to the value of the critical temperature. Theabovediscussionhasintroducedafirstimportantcombinationoftherelevantparametersoftheproblem,given by N1/6a/a . This combination fixes the importance of interaction effects on the thermodynamic properties of the ho system. Its origin is easily understood if one calculates the ratio between the interaction energy E gn where int g =4π¯h2a/misthecouplingconstantfixedbythescatteringlengthandnisthedensityofthethermalclou∼devaluated in the center of the trap, and the thermal energy E k T. A simple estimate for the thermal density is given T B by the classical value n N(k T/mω )3/2. Using res∼ult (3) for the critical temperature, one finds that the ratio B ho ∼ E /E , evaluatedatT T , is actually fixedby the combinationN1/6a/a . This shouldnotbe confusedwith the int T c ho ∼ so called Thomas-Fermi combination Na/a which instead characterizes the effects of two body-interactions on the ho zero temperature properties of a trapped condensate. The Thomas-Fermicombination is recoveredby evaluating the ratio between the interaction energy E gn, where n is now the central value of the condensate density, and the int ∼ quantum oscillator energy h¯ω . The dependence on N given by the two combinations discussed above is extremely 0 different and explains why the effects of the interaction at T T are small, while at T 0 they are so large that c ∼ ∼ cannot be treated in a perturbative way. The role of the Thomas-Fermi parameter will be discussed systematically in the next sections. Before concluding this section it is useful to remind that the measurements of the temperature dependence of the condensate density available in trapped Bose gases are more reliable than the ones obtained in superfluid helium [6] 2 (seefig.2). Inheliumthecondensatefractionisnotanaturalobservableandcanbeextractedonlythroughelaborated analysis of neutron scattering data at high momentum transfer. Actually the main evidence for the phase transition in superfluid helium comes from the analysis of other thermodynamic quantities, like the viscosity and the specific heat. C. Order parameter and long range order A key quantity characterizing the long range order of a many-body system is the one-body density matrix [7]: n(r,r′)= Ψˆ†(r)Ψˆ(r′) (6) h i where Ψˆ(r) = aˆ φ (r)+ aˆ φ (r) is the field operator expressed in terms of the annihilation operators relative 0 0 i6=0 i i to a generic basis of single particle wave functions. We have here separated the contribution arising from the lowest P single particle state φ in order to emphasize the effect of BEC. 0 Let us first consider an ideal gas at zero temperature. In this case all the particles occupy the same state φ 0 determined by the solution of the Schr¨odinger equation for the single particle Hamiltonian ¯h2 2/2m+V . The ext density matrix (6) then takes the separable form n(1)(r,r′) = Nφ∗(r)φ (r′) and remains−diff∇erent from zero for 0 0 macroscopicdistances r r′ of the order of the size of the sample (long range order). This is also called first order | − | coherence and is a key consequence of Bose statistics. Atfinitetemperaturesoneshouldincludethethermaloccupationoftheothersingleparticlestatesandeq.(6)yields n(r,r′)=N φ∗(r)φ (r′)+ n φ∗(r)φ (r′) (7) 0 0 0 i i i i6=0 X wheren =(expβ(ǫ µ) 1)−1arethethermaloccupationnumbersandǫ arethesingle-particleexcitationenergiesof i i i − − theexcitedstatesφ . InthethermodynamiclimittheratioN /N tendstoaconstantvalue(condensatefraction)giving i 0 risetolongrangeorder,whilethesecondterminther.h.s. ofeq.(7)canbereplacedwithanintegralandconsequently vanishes at large distances. The importance of long range order is hence reduced at finite temperature since N is 0 smaller than at T = 0. For a uniform ideal gas the decay of the thermal component is given by n(s) kTm/(4πs) where s= r r′ . Above the critical temperature one has no more long range order (N /N 0) and→the one-body 0 | − | → density matrix decays faster than 1/s at large distances. In the limit T T one finds the gaussian behaviour c n(s)=n(0)exp[ s2mk T/2h¯2]. ≫ B − The effects due to Bose-Einsteincondensationdiscussedabovecanbe alsoderivedby setting aˆ =aˆ† √N . This 0 0 ≡ 0 is the wellknownBogoliubovprescriptionwhich correspondsto ignoringthe non commutativitybetween the particle operators a and a† and treating them like c-numbers. In virtue of the Bogoliubov assumption the field operator 0 0 takes the form Ψˆ(r)=Ψ (r)+ aˆ φ (r) (8) 0 i i i6=0 X where Ψ (r)= N φ (r) (9) 0 0 0 p is a classical field, called the order parameter. The order parameter can be also regarded as the expectation value of the field operator Ψ (r)= Ψˆ(r) (10) 0 h i where the average is taken on a configuration with broken gauge symmetry. In terms of the order parameter the diagonal term of the density matrix can be written as n(r)=n (r)+n (r) (11) 0 T where n (r)= Ψ (r) 2 is the condensate density and n (r)= n φ (r) 2 is the contribution arising from the 0 | 0 | T i6=0 i | i | particlesoutofthecondensate. Inadilute gasn canbesafelyidentifiedwiththe densityofthe thermalcomponent. T P Ingeneraloneshouldnothoweverignorethe factthatevenatzerotemperaturethetotaldensityisdifferentfromthe condensate density since the occupation numbers n with i=0 of eq.(7) differ from zero, giving rise to the quantum i 6 3 depletion of the condensate. An example where quantum depletion plays a crucial role is presented in fig.3 which reports the density profile of a cluster of helium atoms calculated at T = 0 using a correlated basis approach [8]. These systems, differently from atomic gases, are strongly correlated. The figure clearly shows that the contribution of the condensate is only a small fraction (about 10%) and that in this case the measurement of the density profile would not yield any useful information on the condensate density n even at zero temperature. In the calculation 0 of ref. [8] the condensate density has been evaluated by determining the natural orbits which diagonalize the 1-body density matrix (see eq.(7)) D. Equation for the order parameter The equation for the order parameter can be derived starting from the Heisenberg equation for the field operator ∂ ¯h2 2 i¯h Ψˆ(r,t)=[Ψˆ(r,t),Hˆ]= ∇ +V (r,t) ext ∂t − 2m h + Ψˆ†(r′,t)V(r′ r)Ψˆ(r′,t)dr′ Ψˆ(r,t). (12) − Z i ThisisanexactequationifoneusesforV(s)theexacttwo-bodyinteractionbetweenparticlesandholdsnotonlyfor dilute gases, but also for strongly interacting systems like superfluid 4He. A closed equation for the order parameter is obtained if we replace the field operator Ψˆ with the classical field Ψ or, equivalently, with its expectation value 0 (10). This procedure is correct provided a series of assumptions are satisfied: i)Thethermaldepletionshouldbe smallsothatthe thermalfluctuationsofthefieldoperatorcanbe ignored. This implies temperatures much smaller than the critical value. ii)Thesystemshouldbe sodilute thatthe quantumfluctuations ofthe fieldoperatorcanbe ignored. This requires that the gas parameter na3 be much smaller than unity or, equivalently, that the average distance between particles be much larger than the scattering length. iii)ThemicroscopicinteractionV(s)shouldbereplacedbythepseudopotentialg4π¯h2(a/m)δ(s)fixedbythes-wave scatteringlengtha. This impliesthatonlylowenergyfeaturesofthe problemcanbe investigatedwiththis approach. Equivalently, only variations of the order parameter over distances larger than the range of the interaction can be explored. iv) The number of atoms N should be large enough in order to justify the Bogoliubov prescription. Under the hypothesis i)-iv) one obtains the most famous Gross-Pitaevskiiequation [9] ∂ ¯h2 2 i¯h Ψ (r,t)= ∇ +V (r,t)+g Ψ (r,t) 2 Ψ (r,t), (13) 0 ext 0 0 ∂t − 2m | | (cid:18) (cid:19) for the classical field Ψ . This equation provides the basic tool to explore the static and dynamic features of non 0 uniform Bose gases at low temperature. The GP equation shares interesting analogies with the Maxwell equations of classical electromagnetism. In both approaches one provides a description of a many-body problem (atoms anf photonsrespectively)intermsofaclassicalfield(orderparameterandelectromagneticfield). Fromthispointofview a gasofBose-Einsteincondensedatomscanbe regardedasa classical matter wave. With respectto the equations ofelectromagnetismtheGPequationhoweverexhibitsimportantdifferences: firstitcontainsanimportantnonlinear term due to interactions (this difference is not crucial since non lineariteies are exhibited also by electromagnetic phenomenaindispersivemedia);asecondimportantdifferenceistheroleplayedbythePlanckconstanth¯. Differently fromtheMaxwellequationstheGPequation(13)infactdepends explicitlyonthisfundamentalconstantofquantum mechanics. This feature is directly connected with the different momentum-energy relationship exhibited in the two cases. For massive particles the relation is quadratic (E =p2/2m), while for photons it is linear (E =cp). When we employ the ondulatory description through the de Broglie prescriptionE =h¯ω and p=h¯k, the relationship bewteen the wavevector and the frequency is still independent on the Planck constant in the case case of photons (ω = ck), while it exhibits an explict dependence on h¯ in the case of atoms (ω = h¯k2/2m). This explains, for example, why interference phenomena with classical matter waves depend explicitly on the value of the Planck constant. E. Ground state configuration If we look for stationary solutions of the form 4 Ψ (r,t)=Ψ (r)exp( iµt/¯h) (14) 0 0 − the Gross-Pitaevskiiequation (13) becomes ¯h2 2 ∇ +V (r) µ+g Ψ (r) 2 Ψ (r)=0. (15) ext 0 0 − 2m − | | (cid:18) (cid:19) This equation can be also derived using the standard variational procedure δ(E µN)=0 (16) − where ¯h2 g E = Ψ 2 +V (r) Ψ 2 + Ψ 4 dr=E +E +E (17) 0 ext 0 0 kin ext int 2m |∇ | | | 2 | | Z (cid:18) (cid:19) is the energy functional of the system. In this equation we have naturally identify the kinetic energy (E ), the kin externalpotential(E )andthe meanfieldinteraction(E )energies. Startingfromthe GPequationonecaneasily ext int derive the useful relationship Nµ=E +E +2E (18) kin ext int for the chemical potential, and the virial identity 2E 2E +3E =0 (19) kin ext int − characterizing the equilibrium configuration. It is also useful to recall the so called release energy E =E +E (20) release kin int whichcoincideswiththe energyofthe systemafterswitchingoffthe trapandishencegivenbythe sumofthe kinetic and interaction energies. It is important to remind here that the quantity E entering the above equations is the kinetic energy of the kin condensate (mean field kinetic energy) and should not be confused with the full kinetic energy of the system. The latter takes in fact contribution also from the particles out of the condensate and, using the notation (7) for the 1-body density matrix, can be written in the form ¯h2 Etotal = dr Ψ (r) 2 + n φ (r) 2 . (21) kin 2m|∇ 0 | i |∇ i |  Z i6=0 X   Whilefordilutegasesthecontribution n ofthenoncondensatecomponenttothetotalnumberofatomsissmall i6=0 i atT =0andone cansafely identifyN with N ,the correspondingcontributionto the kinetic energy(secondtermof 0 P eq.(21))maybemuchlargerthanthecondensatekineticenergy. Forsimilarreasonsthemeanfieldinteractionenergy E shouldnotbe confusedwiththe full interactionenergywhoseevaluationrequiresthe explicitknowledgeofshort int range correlations. The distinction between the mean field and total energies becomes particularly clear and stark in thecaseofauniformgasinteractingwithhardspherepotentials. Inthiscasethetotalkineticenergycoincideswiththe energy of the system, the interaction energy being exactly zero due to the special choice of the potential. Conversely the kinetic energy of the condensate is zero because of the homogeneity of the sample. Notice that the mean field schemefailstoevaluatethefullkineticenergy(21). Thiscanbeseeninthecaseofauniformgaswheretheoccupation number of the single particle states is given by the Bogoliubov expression n = (p2/2m+mc2)/2ǫ(p) 1/2. The p quantity n isevaluableandgivesthequantumdepletionofthecondensate. Viceversathesum −(p2/2m)n p6=0 p i6=0 p exhibits an ultravioletdivergency,showing that the evaluationof the kinetic energy requires the proper knowledge of P P the momentum distribution at momenta of the order of the inverse of the scattering length and cannot be achieved within the mean field scheme. It is finally worth pointing out the key role played by the chemical potential in the formalism of the GP equation. As one can see from (14) the time dependence of the order parameter is fixed by µ and not by the energy. This is a consequence of the fact that the order parameter is not a wave function and that the GP equation is not a Schr¨odingerequationintheusualsenseofquantummechanics. Fromthepointofviewofmany-bodytheorytheorder parametercorrespondstothematrixelementofthefieldoperatorbetweentwomany-bodywavefunctionscontaining, 5 respectively, N and N +1 particles. This implies that its time dependence is fixed by the factor e−i(E(N+1)−E(N))t and hence by the chemical potential µ=∂E/∂N rather than by the energy E. If the Thomas Fermi parameter Na/a is very large the GP equation (13) can be solved analytically. In fact in ho this case the density profile, as a consequence of the repulsive effect of the interactions, becomes so smooth that the kinetic energy term in (13) can be ignored and the density takes the shape of an inverted parabola (we consider here spherical trapping) 1 µ r2 n(r)= (µ V )= 1 (22) g − ext g − R2 (cid:18) (cid:19) where the radius R of the condensate is given by the formula 1/5 a R=a 15N (23) ho a (cid:18) ho(cid:19) andµ=mω2 R2/2. Foranaxiallydeformedtrapthedensityprofileissimplygeneralizedton(r)=(µ/g)(1 r2/R2 z2/Z2) wherheor2 = x2+y2 and the radial and axial radii are related by the equation mω2R2 = mω2Z2−=⊥2µ.⊥I−n ⊥ ⊥ ⊥ z many available configurations the size of the atomic cloud can become much larger than the oscillator length and reach almost macroscopic values. F. Visibility of the condensate In the introduction we have emphasized the important fact that, due to the harmonic trapping, the condensate density can be separated from the thermal component, a crucial condition for the experimental visibility of the condensate. Due to the importance of this fact it is worth discussing in a deeper way the consequences of two-body interactions. Infacttheargumentusuallyinvoquedisbasedontheidealgasmodelwherethewidthofthecondensate is fixedby the oscillatorlength a = ¯h/mω . On the other handthe width of the thermalcloud is of the orderof ho ho p 1/2 k T B R (24) T ∼ mω2 (cid:18) ho(cid:19) so that, for ”macroscopic”temperatures k T ¯hω , one has R a with the consequentnet separability of the B ho T ho ≫ ≫ two components. In the previous sectionwe have howevershown that interactions are responsible for a huge increase of the size of the condensate whose value, in the Thomas-Fermi limit, is given by eq.(23). One then conclude that interacations will partially reduce the visibility of the condensate by making its size comparable to the one of the thermal cloud. Interactions, by producing a smoother density profile, make the system closer to a uniform gas and consequently reduce the visibility of BEC in coordinate space. In fig 4 we show a typical in situ measurement of the density profile taken at different temperatures [10]. A similar argument holds if one explores the behaviour of the release energy E . For an ideal gas the release energy is fixed by the oscillator energy h¯ω , which is much release 0 smaller than the release energy of the thermal cloud, fixed by k T. This provides a net separation between the two B componentswhichexpandatdifferentvelocities. Thedifferenceissizablyreducedifonetakesintoaccountinteractions which increase significantly the value of E making it comparable to the thermal energy kT. In conclusion the release separability of the condensate from the thermal component is significantly reduced by two-body interactions and requires a careful bimodal fit to the measured profiles (both in in situ and after expansion). While the visibility of the condensate in coordinate space is reduced by two-body (repulsive) interactions, it is converselyenhancedinmomentumspace. Infact,accordingtotheHeisenbergrelationship,thewidthofthecondensate in momentumspace behaveslike ¯h/R andhence becomes smaller andsmalleras R increases,approachingthe typical δ distribution characterizing BEC in uniform systems. While in the absence of interactions the visibility of BEC in coordinate and in momentum space are perfectly equivalent, due to the symmetric role played by the spatial and momentumcoordinatesoftheharmonicHamiltonian,intheThomas-Fermilimitthissymmetryislost. Thepossibility of measuring the momentum distribution of trapped atomic clouds, via inelastic photon scattering experiments [11] opens new possibilities to separate at a deeper level the condensate from the thermal component. In principle such measurements might provide an identification of the quantum depleted part too. In practice the effect of quantum depletion is too small to be observed in the presently available condensates. 6 G. Role of the phase: interference with BEC The phase of the order parameter plays a crucial role in characterizing the interference phenomena exhibited by a Bose-Einsteincondensedgas. Thesimplestexampleistheinterferenceproducedbytwoinitiallyseparatedcondensates which expand and overlap. The corresponding experiment was first carried out at Mit [12]. Let us first consider a single expanding condensate. At large times the phase of the order parameter takes a quadratic dependence, yielding the asymptotic form Ψ (r,t)= Ψ (r,t) eimr2/2h¯t (25) 0 0 | | for the order parameter. It is interesting to notice that the velocity field ¯h v= S (26) m∇ associated with the order parameter (25) coincides with the classical law v = r/t. For large times the velocity then exhibits a fully classical behaviour. However the phase S keeps its quantum nature fixed by the Planck constant. If we now consider two expanding condensates separated by a distance d and we assume that for large times the order parameter is given by a linear combination of the form Ψ = Ψ eim(r+d/2)2/(2h¯t)+eiα Ψ eim(r−d/2)2/(2h¯t) (27) 0 1 2 | | | | we find that the density Ψ 2 acquires a modulation with typical fringes orthogonal to the vector d and separated 0 | | by the distance ht λ= . (28) md Thisdistancedepends explictlyonthe Planckconstantandwithtypicalchoicesofthe parametersdandtisavisibile length of the order of 10 20µK. The above estimate for the interference effects ignores the role of the interaction − between the two clouds which are responsible, during the expansion, for deviations from the law (27). The basic physics is howeveraccountedby this simple model whichprovidesa useful qualitative estimate ofinterference effects. Itpointsout,inparticular,aconceptuallyimportantquestionconnectedwiththeroleofthe relativephaseofthe two condensates. Suppose that the two condensates are built in an independent way before measurement. Why should the order parameter have the form (27) with a well defined value of the relative phase α giving rise to interference? The question addresses the important problem of the quantum measurement and of the consequent reduction of the wavepacket. The standardtheoreticalpointview,basedonthe traditionalrulesofquantummechanics,predictsthat after measurementthe wavefunction of the whole systemwill be of the form (27) with a relativephase which cannot predicted in advance, unless the condensates were already in a coherent configuration before the expansion. H. Role of the phase: irrotationality of the superfluid flow Acrucialconsequenceofthe existence ofanorderparameterofthe form(1), is the irrotationalityofthe flow. This feature is expressedby eq.(26) from which one deduces that the phase plays the role of a velocity potential. Actually starting fromthe Gross-Pitaevskiiequationfor the orderparameterit is possible to derivethe equationsof motionin a formwhichresemblesthe equationsofirrotationalhydrodynamics. These arethe basic equations forthe describing the motion of a superfluid. The derivation is straightforward in a dilute gas. One starts from expression (1) for the order parameter and derives, using the GP equation (13), a closed set of equations for the density and for the phase or, equivalently, for the velocity field. The equations take the form [13] ∂n +div(vn)=0, (29) ∂t ∂ 1 ¯h2 ¯h S+ mv2+V +gn 2√n =0. (30) ext ∂t 2 − 2m√n∇ (cid:18) (cid:19) The first equation is the usual equation of continuity, while the second one can be regarded as an equation for the velocity potential. Notice that in these equations, which are exactly equivalent to the original Gross-Pitaevskii 7 equation, the Planck constant enters through the so called quantum kinetic pressure term proportional to 2√n. ∇ This term should be compared with the interaction gn given by two-body forces. If the role of interactions is very important the quantum pressure term can be neglected. This happens in the study of the equilibrium profile in the Thomas-Fermilimit. Thequantumpressuretermisnegligiblealsointhetime dependentproblem,providedwestudy phenomena where the density varies in space on lengths scales L larger than the so called healing length ¯h ξ = . (31) s2mgn If L ξ it is immediate to see that the quantum pressure in (30)is negligible and that the same equationreduces to ≫ the simpler Euler-like form ∂ 1 m v+ mv2+V +µ =0 (32) ext ∂t ∇ 2 (cid:18) (cid:19) where µ = gn is the chemical potential evaluated for a uniform body at the corresponding density. The equation of continuity and the Euler equation (32) are called the hydrodynamic equations of superfluids. They are characterized by two important features: the irrotationality of the motion and the crucial role of interactions which suppress the effects ofthe quantum pressure. It is worthnoticing that the ”hydrodyinamic”form of these equations is not the result of collisional processes as happens in classical gases, but is the consequence of superfluidity. It is also useful to recall that the validity of the hydrodynamic equations is limited to the low T regime (at finite temperature one should use the formalism of two-fluid hydrodynamics). In the limit of small temperatures these equations can be used to study also dense superfluids, like 4He as well as Fermi superfluids. Of course in this case one should use the corresponding expression for the chemical potential. It is finally worth mentioning that in the hydrodynamic limit the Planck constant, which entered equation (30) for the velocity field through the quantum pressure term, has disappeared from the equations of motion. Starting from the hydrodynamic equations of superfluids one can explore different interesting problems in trapped BEC gases. These include the propagationof sound and the excitation of collective modes. I. Sound and collective excitations In the limit of small perturbations (linear limit) the hydrodynamic equations (29) and (32) can be reduced to the typical form ∂2 δn= [c2(r) δn] (33) ∂t2 ∇· ∇ characterizing the propagationof sound waves in nonuniform media. Here c(r) is a r-dependent sound velocity fixed by the relationshipmc2(r)=µ V (r). Soundcanpropagatein trappedBosegasesifthe wavelengthλ ofthe wave ext − is smaller than the size ofthe system. Inthe Thomas-Fermilimit this conditioncanbe wellsatisfiedespecially in the elongated direction of the condensate. In the experiments carried out at MIT [14] the axial size Z is a few hundred micrometers while ξ is a fraction of micrometer, so there is wide space for the propagation of sound waves in such samples. Itis alsointeresting to consider the case where the wavelengthλ is smallerthan the axialsize (λ<Z), but larger than the radial size (λ > R ). In this case the nature of the wave is characterized by typical 1-dimensional ⊥ features [15]. If the wavelengthbecomes comparable to the size of the condensate the solutions of eq.(33) cannotbe described in termsofalocalizedpropagationofsound,butshouldbedeterminedglobally. Thesesolutionscorrespondtooscillations of the whole system and the discretization of the eigenvalues cannot be ignored in this case. For spherical trapping one can obtain analytic solutions of (33) in the form δn = R Y where ℓ = 0,1,2.. is the angular momentum nrℓ ℓm (in units of h¯) carried by the excitation, n is the number of the radial nodes and R is the radial function to be r nℓ determined by solving the equations of hydrodynamics. The corresponding eigenfrequencies obey the dispersion law [13] ω =ω 2n2+2n ℓ+3n +ℓ (34) HD ho r r r which should be compared with the prediction ωp(2n + ℓ) of the ideal gas model. The role of interactions is ho r particularly evident for the surface excitations (n = 0). In this case the hydrodynamic theory gives √ℓω to be r ho compared with value ℓω of the ideal gas prediction. 0 8 Result (34) reveals that, in the Thomas-Fermi limit Na/a 1, the dispersion relation of the normal modes of ho ≫ thecondensatehaschangedsignificantlyfromthenoninteractingbehavior,asaconsequenceoftwo-bodyinteractions. However it might appear surprising that in this limit the dispersion does not depend any more on the value of the interaction parameter a. This differs from the uniform case where the dispersion law, in the corresponding phonon regime, is given by ω = cq and depends explicitly on the interaction through the velocity of sound. The behavior exibited in the harmonic trap is well understood if one notes that the discretized values of q are fixed by the boundary and vary as 1/R where R is the size of the system. While in the box this size is fixed, in the case of harmonic confinement it increases with N due to the repulsive effect of two-body interactions (see eq.(23): R (Na/a )2/5(mω )−1/2. On the other hand the value of the sound velocity,calculatedat the center ofthe trap, ho ho is g∼iven by c=(Na/a )2/5(ω /m)1/2 and also increases with N. One finally finds that in the product cq both the ho ho interactionparameter and the number of atoms cancel out, so that the frequency turns out to be proportionalto the bare oscillator frequency ω . A similar argument holds also for the surface excitations. In fact in the presence of an ho external force the dispersion relation of the surface modes obeys the classical law ω2 = Fq/m where F = mω2 R is ho the force evaluated at the surface of the system. Since the product h¯qR gives the angular momentum carried by the surface wave, one immediately recovers the dispersion law √ℓω . ho Analytic solutions of the HD equation (33) are available also in the case of axi-symmetric potentials of the form V =(m/2)(ω2r2 +ω2z2). Inthis casethe thirdcomponentofangularmomentumis stillagoodquantumnumber. ext ⊥ ⊥ z For very elongated traps (ω ω ) the frequency of the lowest solution of even parity takes the value [13] 5/2ω z ⊥ z ≪ in excellent agreement with the experimental results of [16]. p Thehighprecisionoffrequencymeasurementsisnotonlyprovidinguswithapowerfultooltocheckthepredictions ofthehydrodynamictheoryofsuperfluids,butcanbealsousedtoexplorefinereffects,likethetemperaturedependence of the collective frequencies and, possibly, beyond mean field corrections. J. Moment of inertia An important consequence of the hydrodynamic theory of superfluids is that the response <L > z Θ=lim (35) Ω→0 Ω to a rotating field of the form H = ΩL , where L = i¯h r is the third component of the angular rot − z z − k k ×∇k momentum operator, is given by the irrotational value P <x2 y2 > 2 Θ= − Θ (36) <x2+y2 > rig (cid:18) (cid:19) of the moment of inertia where Θ is the classical rigid value. This result can be easily derived by rewriting the rig equations of hydrodynamics in the frame rotating with the angular velocity Ω. Eq.(36) can be also written in the form Θ = ǫ2Θ where ǫ = (ω2 ω2)/ω2+ω2) is the deformation of the trap in the plane of rotation and we have irr x− y x y used the Thomas-Fermi results for <x2 > and <y2 >. The quenching of the moment of inertia with respect to the rigid value was confirmed in a series of experiments of the 60’s caried out on superfluid helium, providing an independent measurement of the superfluid density [17]. In this contextitis worthmentioningthatthe superfluiddensityofsuperfluidheliumexhibits averydifferentbehaviour with respect to the condensate density, especially at low temperature. In particular at T = 0 the superfluid density coincides with the total density while the condensate density is only a small fraction ( 10%). ∼ A challenging question is how to measure the moment of inertia of a trapped gas where the direct measurement of theangularmomentumL isnotfeasibleduetothedilutenessofthesample. Aninterestingpossibilityisprovidedby z the fact that, if the deformation of the trap is different from zero, the quadrupole and rotational degrees of freedom are coupled each other. This is well understood by considering the exact commutation relation [H,L ]=im(ω2 ω2)Q (37) z x− y which explicitly points out the link between the angular momentum operator L and the quadrupole operator Q = z x y . Since the quadrupole variable can be easily excited and imaged one expects to obtain information on the i i i rotational properties of the system by investigating the quadrupole modes excited by the operator Q. An important P caseisthesocalledscissorsmode. Thismodecorrespondstotherotationofadeformedcondensatewhoseinclination angle θ oscillates in time. The oscillation can be induced by a sudden rotation of the confining trap with respect to 9 theinitialequilibriumconfiguration. Itisimportanttopointoutthatthefrequencyofthisoscillationdoesnotvanish with ǫ as one would expect for a classical system. In fact in a superfluid both the restoring force parameter and the mass parameter (moment of inertia) behave like ǫ2 so that the frequency keeps a constant value when ǫ 0. The → hydrodynamic theory of superfluids predicts the value [18] ω = ω2+ω2 (38) HD x y q in excellent agreement with the experimental results recently carried out at Oxford [19] (see fig.5). Conversely in a classical gas one predicts the value ω = ω ω . Also these values have been tested experimentally with high x y | ± | accuracy by exciting the rotation of the gas at high temperatures, well above T . c K. Quantized vortices Quantized vortices are one of the most spectacular manifestations of superfluidity (we will not discuss here other importantmanifestationsofsuperfluidity,likethereductionofviscosityrecentlyexploredatMIT[20]). Theexistence of quantized vortices is the combined consequence of the behaviour of the phase of the order parameter, which fixes the irrotationality of the velocity field, and of the non-linearity of the equations of motion which is a crucial the consequence of two-body interactions. In the Gross-Pitaevskii theory a quantized vortex can be regarded as a stationary solution of eq.(13) of the form Ψ (r,t)=e−iµteiφΨ (r) (39) 0 v where φ is the azimuthal angle and Ψ is a real function obeying the equation v ¯h2 2 ¯h2 m ∇ + + (ω2r2 +ω2z2)+gΨ2(r ,z) Ψ (r ,z)=µΨ (r ,z). (40) − 2m 2mr2 2 ⊥ ⊥ z v ⊥ v ⊥ v ⊥ (cid:20) ⊥ (cid:21) The velocity field associated with the order parameter (39) takes the form v=(h¯/m) φ=Ωˆ r/r2 where Ωˆ is the ∇ × unitvectoralongthez-thdirection. Itsatisfiestheirrotationalityconstrainteverywhereexceptalongthevorticalline and gives rise to a total angular momentum given by L =N¯h. (41) z The ”centrifugal” term in 1/r2 of eq.(40) originates from the peculiar behaviour of the velocity field of the vortical ⊥ configuration. This term is responsible for the vanishing of the condensate density Ψ 2 along the z-th axis and v | | produces the so called vortex core, whose size is fixed by the healing length. In fig.6 we plot a typical vortical profile obtained by solving the Gross-Pitaevskiiequation (40). Quantizedvorticesareexcitedstatesofthesystembuttheyshouldnotbeconfusedwiththeelementaryexcitations discussed in sect.I . From the many-body point of view the vortex correspond to a state where all the N particles of thesampleoccupythenewsolutionoftheGPequation. Theexcitationenergyandtheangularmomentumassociated with the vortexare consequently”macroscopic”quantities while the excitationenergyandthe angularmomentumof elementary excitations are of the order, respectively, of the trapping frequencies and of a few units of h¯. The excitation energy E of the vortex, given by the difference between the vortex and the ground state energies, v can be calculated starting from the solution of the GP equation (40 ) and using expression (17) for the energy. For large N samples the calculation can be done analytically. One finds [21] 4π ¯h2 0.67R ⊥ E = n Z ln (42) v 0 3 m ξ where n is the value of the density in the center of the trap. It is worth noticing that the factor in front of the 0 logarithm is proportional to the so called column density dzn(x,y,z) = (4/3)Zn calculated along the symmetry 0 axis (x = y = 0). As we will see later this identification is useful for the calculation of the the energy of a vortex R line displaced from the symmetry axis. Result (42) allows for an estimate of the critical angular velocity needed to generate an energetically stable vortex. In fact, calculating the energy H ΩL of the sample in the frame rotating z − with angular velocity Ω, one easily finds that the vortex is the lowest energy configuration if the angular velocity Ω exceeds the critical value 10