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Growth and Collapse of a Bose Condensate with Attractive Interactions C.A. Sackett1, H.T.C. Stoof1,2, and R.G. Hulet1 1Physics Department and Rice Quantum Institute, Rice University, Houston, Texas 77251-1892 2Institute for Theoretical Physics, University of Utrecht, Princetonplein 5, 3584 CC Utrecht, The Netherlands 8 Weconsiderthedynamicsofaquantumdegeneratetrappedgasof7Liatoms. Becausetheatoms 9 have a negative s-wave scattering length, a Bose condensate of 7Li becomes mechanically unstable 9 when the number of condensate atoms approaches a maximum value. We calculate the dynamics 1 of the collapse that occurs when the unstable point is reached. In addition, we use the quantum n Boltzmannequationtoinvestigatethenonequilibriumkineticsoftheatomicdistributionduringand a after evaporative cooling. The condensate is found to undergo many cycles of growth and collapse J before a stationary state is reached. 5 PACS number(s): 03.75.Fi, 67.40.-w, 32.80.Pj, 42.50.Vk ] h c TherecentobservationsofBose-Einsteincondensation of the condensate size near the instability point. e m in weakly interacting gases [1] has enabled a series of Experimentally,itisalsoimportanttounderstandhow beautiful experiments that probe the dynamics of the such a condensate can be formed from a noncondensed - t condensate. The frequency and damping of the collec- thermal cloud by means of evaporative cooling. This a tive modes of a condensate [2], propagation of sound in question was recently addressed by Gardiner et al. in t s a condensate[3],andrecently,the growthofthe conden- the context of the experiments with gases having a > 0 . t sate[4],havebeenreported. Althoughtheseexperiments [14]. Theseauthorsneglectthe coherentdynamicsofthe a m haveused atoms with positive s-wavescattering lengths, condensate and focus instead on the kinetics of conden- we show in this paper that the dynamical behavior of a sation [15]. By treating the noncondensed atoms as a - d negative scattering length gas, such as 7Li, is especially static‘heatbath’withachemicalpotentialthatislarger n interesting,andofferstheopportunitytodirectlyobserve than the ground state energy, they are able to derive a o and study macroscopic quantum tunneling. simple equationfor the growthof the number of conden- c [ A negative scattering length a implies effectively at- sate particles that appears to fit well with experimental 1 tractive interactions. In a spatially homogeneous gas, rinesuthltesc[4a]s.eTohfeatsoammieca7pLpirboaecchau,sheowitevdeore,sdnooets naloltowwofrokr v theseinteractionsleadtoordinaryclassicalcondensation 4 into a liquid or solid, preventing Bose condensation in the collapse of the condensate. Moreover, the process 1 of evaporative cooling leads to dynamical changes in the themetastableregionofthephasediagram[5]. However, 0 noncondensed ‘heat bath’. The study of both these ef- confinement in an atom trap produces stabilizing forces 1 fectsonthedynamicsofthecondensateisthemaintopic 0 that enable the formation of a metastable Bose conden- ofthisLetter. Somepreliminaryresultshavealreadyap- 8 sate, if the number ofcondensedatoms is less than some peared in a recent review article [16]. 9 maximum number N . For a harmonic trapping poten- m t/ tial, Ruprecht et al. [6]showedthat in mean-fieldtheory When collisions can be neglected, the dynamics of a N 0.57l/a,wherel=(h¯/mω)1/2 istheextentofthe the condensate wavefunction ψ for a gas with a < 0 is m m ≃ | | well-described by the nonlinear Schrödinger (or Gross- one-particle ground state in the harmonic trap [7]. For d- the 7LiexperimentsofRef.[8],Nm 1400atoms,which Pitaevskii [17]) equation ≃ n agrees with the measured value. ∂ψ(x,t) ¯h2 2 o i¯h = ∇ +V(x) ψ(x,t) Although a condensate can exist in a trapped gas, it ∂t (cid:18)− 2m (cid:19) c v: is predicted to be metastable and to decay by quantum +T2B(0,0;0)ψ(x,t)2ψ(x,t) . (1) or thermal fluctuations [9–11]. The condensate has only | | i X one unstable collective mode, which in the case of an Heremdenotesthemassof7Li,T2B(0,0;0)=4πa¯h2/m r isotropictrapcorrespondstothebreathingmode[12,13]. is the two-body T-matrix, and a is the scattering length a The condensate therefore collapses as a whole, either by of 27a0 [18]. For the external trapping potential V(x) − thermal excitation over a macroscopic energy barrier,or we take a harmonic potential with an effective isotropic by quantum mechanicaltunneling throughit. The prob- level splitting h¯ω =h¯(ω ω ω )1/3 of 7 nK [8]. Note that x y z ability of forming small, dense clusters is greatly sup- we ignore the mean-field contribution from the noncon- pressed because of the large energy barrier for this pro- densed atoms in this equation [19], because it is nearly cess, compared to that for the breathing mode. This constant over the size of the condensate and therefore suppression can also be understood from the fact that only slightly affects the condensate dynamics. the typical length scale for fluctuations of the conden- This description is semiclassical, and also neglects sate is the healing length, which is precisely of the order quantum and thermal fluctuations. These fluctuations 1 are most easily incorporated by means of the partition Besides the calculation of the decay rates, the above function of the condensate, which is a functional inte- anologyalsoallowsstudyofthe dynamicsofthecollapse ∗ ∗ gral d[ψ ]d[ψ] exp( S[ψ ,ψ]/¯h) over all periodic con- that occurs after the condensate has been driven out of − figurRations of the condensate, with a weight determined its metastable minimum. A typical example is shown by the (Euclidean) action of the nonlinear Schrödinger in Fig. 2. From this figure we see that the condensate equation. It is most convenient to calculate this parti- first collapses with increasing speed along the potential tion function in terms of the density and phase of the hill outside the barrier. However, during the collapse, condensate, defined by ψ = √ρeiχ. The Gaussian inte- the condensate density grows rapidly, thereby increas- graloverthephasefieldχcanthenbeperformedexactly ing the decay rate for inelastic two and three-body colli- leaving only the determination of the functional integral sions. Atoms that inelastically collide acquire subtantial d[ρ] exp( S[ρ]/¯h). energy and are ejected from the trap. Because of these − R Unfortunately, this integral cannot be calculated ex- loss mechanisms, the collapse is arrested when the num- actly. However, since we are primarily interested in the berofcondensateatomsisoftheorderofone. Atomsare dynamics of the unstable breathing mode of the conden- lost so quickly that the density of the condensate always sate,wecanuseavariationalmethod[11,20]andconsider obeys na3 1, which is a necessary requirement for | | ≪ only Gaussian density profiles the validity ofthe nonlinear Schrödingerequation(1). If 1 3/2 x2 there were no inelastic collisions, the condensate would ρ(x;q(t))=N0(cid:18)πq2(t)(cid:19) exp(cid:18)−q2(t)(cid:19) . (2) fnuelelydecdo.llapse [23], and a more complex theory would be Substituting this profile in the action S[ρ], we find that the dynamics of the condensate is equivalent to the dy- ∗ namics of a fictitious particle with mass m = 3N0m/2 in the unstable potential [21] 2 2 3h¯ 3 2 2 N0 ¯h a U(q)=N0(cid:18)4mq2 + 4mω q − √2π m|q3|(cid:19) . (3) 0.5 The rate of decay for both quantum tunneling and 1500 thermal fluctuations can now easily be calculated within q/l this formalism [11] and are shown in Fig. 1. For large N0 750 numbers of condensate atoms, these collective decay mechanisms are much faster than the decay caused by 0 2.725 2.750 inelastic two and three-body collisions, since the energy ωt barrier out of the metastable minimum vanishes as N0 0.0 approaches Nm. For experimentally relevant tempera- 0 1 2 3 tures of 300 to 500 nK, the collective decay processes ωt dominate for N0 greater than about 1100 atoms. FIG. 2. Typical evolution of the condensate during collapse. 101 Theaboveremarkspertaintothedynamicsofthecon- densate in the absence of collisions with noncondensed 1: 0 nK 2: 100 nK atoms, but so far all experiments producing BEC have 3: 200 nK 4: 300 nK relied on evaporative cooling, which requires such colli- -1s) 100 56:: 450000 nnKK sions. Therefore, to investigate the kinetics of condensa- e ( tion in a trapped gas, we use the Boltzmann transport at equation in a way similar to the treatment of evapora- R y tive cooling in Refs. [24,25]. In this approach, we de- a Dec10-1 1 fineasemiclassicaldistributionfunctionf(x,p),withthe 2 5 4 3 number of atoms at position x with momentum p being 6 dxdpf(x,p)/(2π¯h)3. The evolution of the distribution function is then given by [26] 10-2 750 1000 1250 1500 d N f(x,p,t)=I(x,p) Γ(x)f(x,p) , (4) 0 FIG.1. Decay rate of the condensate as a function of the dt − number of condensate particles at 0, 100, 200, 300, 400, and 500nK. Thedashedlineshowsthedecayduetoinelastictwo where the derivative on the left is the total time deriva- and three-body collisions [22]. tive. The effect of elastic collisions is given by 2 σ I(x,p1)= 4π4m¯h3 Z dp2dp3dp4 lmoweaend-fitoeldgrionwte.rFacrtoimonthcaeuvsaersiaqtitoonadleccarelcauselataisonNw0eisfianld- δ3(p1+p2 p3 p4)δ(p21+p22 p23 p24) that q = 0.67l just before the collapse. The error in the × − − − − collisionrate is then a factorof 4,andcollisionterms for [f3f4(1+f1)(1+f2) f1f2(1+f3)(1+f4)], (5) × − otherlow-lyingstateswillbeincorrectbysimilar,though where the factors (1 +f ) account for the Bose statis- smaller, amounts. Although these errors are significant, i tics of the atoms. Here fi stands for f(x,pi) and they only occur for N0 relatively close to Nm. For in- σ = 8πa2. Inelastic collisions, which lead to a loss of stance at N0 0.8Nm, the error is a factor of 2. As will ≃ atoms from the trap, are described by Γ(x) = G1 + be seen below, the condensate number is greater than G2 dpf(x,p)/(2π¯h)3, where the terms with G1 and 0.8Nm for only a small fraction of time, so the approx- G2Rreflectthelossofatomsdue,respectively,tocollisions imations are useful, but for N0 ≃ Nm the quantitative withhotbackgroundgasatomsandtoinelasticcollisions predictions of the model will be somewhat inaccurate. between trapped atoms. The possibility for the condensate to collapse is in- As for the case of a classical gas [24,25], Eq. (4) can cludedinthemodelusingthedecayratesgiveninFig.1. be simplified by taking the motion of the atoms to be A random number is chosen to determine whether a de- ergodic, so that an atom with a given energy will sam- cayoccursduringanintegrationtimestep,andwhenone ple each available cell in x,p phase space with equal does the condensate number is set to zero reflecting the { } probability. The distribution function f(x,p) can then rapid loss shown in Fig. 2. Also, evaporative cooling is be expressedintermsofthe energydistributionfunction included by setting f(E)=0 for E >Ec(t), where Ec(t) f(E), through the relation is the experimentally settable cutoff energy. f(x,p)= dEδ(H(x,p) E)f(E) (6) Z − 1500 where H(x,p) = p2/2m+V(x) is the classical Hamil- tonian. The differential equation for f(E,t) is derived by substituting (6) into (4), as is described in detail in 1000 Refs. [24,25]. The only difference here is the use of Bose statistics,whichrequirestheinsertionoffactors1+f(Ei) N0 as in Eq. (5). Although the semiclassical approximation results in a continuous function f(E), the quantum na- 500 ture of the trapped gas can be recovered by restricting f(E) to values E =(n+3/2)h¯ω. n The above model neglects the effect of the mean-field 0 interaction energy. Per atom, however, this interaction 0 10 20 30 40 energy is limited to about h¯ω by the stability criterion t (s) arising from a being negative, which limits the effect of FIG.3. Typical evolution of condensate number N in re- 0 the mean-field energy on the kinetics of the gas. The sponse to evaporative cooling. The gas initially consists of accuracy of the approximations can be checked by con- 4×105 atoms at 500 nK. During the first 5 seconds of evo- sidering collisions between condensate atoms, since the lution,roughly40%oftheatomsareremovedbyevaporative mean-field interaction is largest in the condensate and cooling, after which evaporation is halted. the semiclassical approximation is least accurate for the groundstateofthetrap. Wecomparetheratesforinelas- Theresponseofthecondensatenumbertoevaporative tic collisions between condensate atoms, G2 dxρ(x)2, coolingisshowninFig.3. Inthiscase,Ec(t)consistsofa whichdepend onlyonthe density. TheexactcRondensate 5-secondperiodofrapidcooling,afterwhichevaporation density is determined from Eq. (1), using the variational is halted and the gas allowed to equilibrate. The subse- method of Eq. (2). In contrast, the semiclassical density quent behavior shows the condensate alternately filling derived from Eq. (6) is and collapsing, until finally the phase space density of the gas is too low to reach N0 Nm. The slow final 2 2 N0 x2 1/2 decay is due to the trap losses. ≃ ρ(x)= 3 . (7) (cid:18)3π(cid:19) (cid:18) l3 (cid:19)(cid:18) − l2(cid:19) As shown in the figure, the time for which the oscillations in N0 persist is anomalously long compared to From these distributions, the ratio of the exact and the elastic collision time τ = 1/ nσv 0.8 s. In or- c approximate collision rates is 1.2(l/q)3. When interac- der to investigate this phenomenohn furith≃er, the density tions are small, then q l, and the error caused by the profiles generated by the model were fit to equilibrium ≃ semiclassical approximation is only 20%. However, the Bose-Einstein distributions at various times. The val- 3 ues of χ2 resulting from the fits are plotted in Fig. 4. [1] M.J. Anderson et al., Science 269, 198 (1995); C.C. After reaching degeneracy, the curve shows an approxi- Bradley et al., Phys. Rev. Lett. 75, 1687 (1995); K.B. mately exponentialapproachto equilibrium,with atime Davis et al., Phys.Rev.Lett. 75, 3969 (1995). constant of about 10τ . In contrast, the same test per- [2] D.S. Jin et al., Phys. Rev. Lett. 77, 420 (1996); M.-O. c Mewes et al., Phys. Rev. Lett. 77, 988 (1996); D.S. Jin formed on a nondegenerate cloud initially prepared in a et al.,Phys.Rev. Lett.78, 764 (1997). nonequilibrium state yields an equilibration time of 5τ . c [3] M.R. Andrews, D.M. Kurn, H.-J. Miesner, D.S. Durfee, The relatively slow approach to equilibrium for the de- C.G. Townsend, S. Inouye, and W. Ketterle, (unpub- generate case is presumably caused by the small phase- lished). space volume of the condensate and the fact that due [4] W. Ketterle, privatecommunication. to the limit N0 < Nm, the stimulated Bose scattering [5] ForthezerotemperaturecaseseeL.D.LandauandE.M. factors in Eq. (4) cannot become as enormous as they Lifshitz, Statistical Physics (Pergamon, London, 1958). might in the a > 0 case. Furthermore, the oscillations Nonzero temperatures are discussed in H.T.C. Stoof, in N0 will persist until the distribution is very close to Phys. Rev.A 49, 3824 (1994). equilibrium, since the total number of atoms in the trap [6] P.A. Ruprechtet al., Phys.Rev.A 51, 4704 (1995). is much greater than N . [7] Thisisthezerotemperatureresult.Theeffectofnonzero m temperaturesistoreducethisnumbersomewhat.Seefor more details M. Houbiers and H.T.C. Stoof, Phys. Rev. 1.0 A 54, 5055 (1996). [8] C.C. Bradley, C.A. Sackett, and R.G. Hulet, Phys. Rev. Lett. 78, 985 (1997). 0.8 [9] Yu. Kagan, G.V. Shlyapnikov, and J.T.M. Walraven, Phys. Rev.Lett. 76, 2670 (1996). 0.6 [10] E.V. Shuryak,Phys.Rev.A 54, 3151 (1996). 2 [11] H.T.C. Stoof, J. Stat. Phys.87, 1353 (1997). χ [12] R.J. Dodd et al.,Phys. Rev.A 54, 661 (1996). 0.4 [13] K.G.SinghandD.S.Rokhsar,Phys.Rev.Lett.77,1667 (1996). 0.2 [14] C.W. Gardiner et al., Phys.Rev.Lett. 79, 1793 (1997). [15] H.T.C. Stoof, Phys. Rev.Lett. 78, 768 (1997). 0.0 [16] C.A.Sackett,C.C.Bradley,M.Welling,andR.G.Hulet, 0 10 20 30 40 Appl. Phys.B 65, 433 (1997). Time (s) [17] L.P.Pitaevskii,Sov.Phys.JETP13,451(1961)andE.P. Gross, J. Math. Phys. 4, 195 (1963). FIG. 4. Equilibration of degenerate gas. Using the same [18] E.R.I.Abrahamet al.,Phys.Rev.Lett.74,1315(1995). conditions as in Fig. 3, at each time the density distribution [19] V.V.Goldman,I.F.Silvera,andA.J.Leggett,Phys.Rev. isfittoanequilibriumdensitydistribution,andtheresulting B 24, 2870 (1981). valueof χ2 plotted. [20] V.M. Pérez-Garciá et al., Phys. Rev. Lett. 77, 5320 (1996). Note, finally, that the evolution in Fig. 3 represents [21] If the densityfield is expanded as only one possible outcome of evaporative cooling, and that because of the stochastic nature of the collapse a ρ(x;q(t))+ ′Cnℓm(t)e−x2/2q2(t)φnℓm(x;q(t)) , givenevolutionisnotrepeatable. Experimentalmeasure- Xnℓm ments of N0 in the degenerate regime should, however, whereφ (x;q)aretheusualharmonicoscillatorstates nℓm exhibit large fluctuations between 0 and Nm. Observa- with frequency ωq = ¯h/mq2, and the sum does not in- tionofsuchfluctuationsandmeasurementoftheirstatis- cludethestatesφ andφ ,wecansystematicallyim- 000 100 tics would provide confirmation of the collective nature proveuponthisresultbyexpandingtheactionS[ρ]upto of the collapse. If sufficiently low temperatures can be quadratic order in Cnℓm and integrating out the coeffi- reached, these fluctuations would be evidence of macro- cientsdescribingthedynamicsofallthestablecollective scopic quantum tunneling. We therefore believe that a modes. [22] A.J. Moerdijk and B.J. Verhaar, Phys. Rev. A 53, R19 dilute Bosegaswitha<0presentsunique opportunities (1996). for studying the dynamical properties of a condensate. [23] See also, L.P. Pitaevskii, Phys. Lett. A 221, 14 (1996). [24] O.J. Luiten, M.W. Reynolds, and J.T.M. Walraven, Phys. Rev.A 53, 381 (1996). [25] C.A. Sackett, C.C. Bradley, and R.G. Hulet, Phys. Rev. A 55, 3797 (1997). [26] See for instance K. Huang, Statistical Mechanics, (John Wiley and Sons, NewYork,1987). 4