Bosenova and three-body loss in a Rb-85 Bose-Einstein condensate
P. A. Altin, G. R. Dennis, G. D. McDonald, D. Döring, J. E. Debs, J. D. Close, C. M. Savage, N. P. Robins
BBosenova and three-body loss in a Rb Bose-Einstein condensate
P. A. Altin, G. R. Dennis, G. D. McDonald, D. Döring, J. E. Debs,J. D. Close, C. M. Savage and N. P. Robins
Department of Quantum Science, Research School of Physics and Engineering,Australian National University, ACT 0200, Australia ∗ (Dated: November 8, 2018)Collapsing Bose-Einstein condensates are rich and complex quantum systems for which quanti-tative explanation by simple models has proved elusive. We present new experimental data on thecollapse of high density Rb condensates with attractive interactions and find quantitative agree-ment with the predictions of the Gross-Pitaevskii equation. The collapse data and measurementsof the decay of atoms from our condensates allow us to put new limits on the value of the Rbthree-body loss coefficient K at small positive and negative scattering lengths. PACS numbers: 03.75.Kk,67.85.Hj
While most experiments with dilute gas Bose-Einsteincondensates have employed atomic species with repul-sive interactions, it has long been known that interestingand exotic physics manifests in attracting condensates.These include macroscopic quantum tunnelling [1], theformation of soliton trains and vortex rings [2, 3], andthe violent collapse and explosion known as the ‘bosen-ova’ [4–6]. The first evidence for the collapse of attract-ing Bose-Einstein condensates was found by Sackett andcoworkers, who analysed the thermal equilibration of asample of Li atoms with negative scattering length thatwas cooled below the critical temperature [7]. Soon af-ter this work, condensate collapse was directly observedin pioneering experiments at JILA [4], which revealed ahost of interesting dynamics and prompted a surge oftheoretical interest [8–15]. More recently, the collapse ofdipolar chromium BECs has been observed, displayingthe striking d -wave symmetry of long-range dipole-dipoleinteractions in excellent agreement with theory [3].However, while initial mean-field analysis of the JILAexperiment using the Gross-Pitaevskii (GP) equation wasable to qualitatively account for most of the experimentalobservations, including the formation of atomic ‘bursts’and ‘jets’ [8, 12–15], further investigation exposed a quan-titative discrepancy between theory and experiment inthe time taken for the condensates to collapse [11]. Thiswas especially puzzling as the short time, low densityphase of the experiment is exactly where the GP equa-tion should be an excellent approximation. This disagree-ment, of about 100%, could not be eliminated by morecomplex quantum field calculations [15, 16], and has ledto the development of competing models for the collapsemechanism [17]. Yet amid the extensive theoretical workon this phenomenon that has continued in recent years,there has been a notable absence of further experimentaldata, and the discrepancy between theory and the Rbexperiment remains unresolved. ∗ Electronic address: [email protected]; URL: http://atomlaser.anu.edu.au/
Here we present the first results on this phenomenonfrom a new Rb BEC machine [18], finding good agree-ment between the measured collapse times and those pre-dicted by a GP model. Although we use the same atom,our experiment has several important differences fromthe original JILA work. Most notably, our condensatesare confined in a purely optical potential, with a homo-geneous magnetic bias field applied to manipulate theinteratomic interactions. In addition, we have measuredcondensate collapses with × atoms in a tighter trap,which together result in an initial density over an order ofmagnitude larger than in Ref. [4]. This leads to shortercollapse times and lower values of the critical scatter-ing length, but should not affect the ability of mean-fieldtheory to describe the evolution of the system. It alsoallows us to investigate three-body recombination ratesin a high density regime where they are the dominantsource of atom loss.Our apparatus for producing Bose-Einstein conden-sates of Rb with tunable interactions has been de-scribed in detail elsewhere [18]. In brief, we employ sym-pathetic cooling using Rb as a refrigerant, initially ina quadrupole-Ioffe configuration magnetic trap and sub-sequently in a weak, large-volume crossed optical dipoletrap. During the final evaporation, a magnetic bias fieldof 167 G is applied to reduce losses due to two-body in-elastic collisions [19, 20]. We can create condensates ofup to Rb | F = 2 , m F = − (cid:105) atoms with a ther-mal fraction below 10% in a trap with harmonic oscilla-tion frequencies ω x,y,z = 2 π × { , , } Hz. Conden-sates form at scattering lengths between a = +50 a and a = +200 a , where a is the Bohr radius. We determinethe scattering length from the applied magnetic bias fieldusing the known parameters of the 155 G Rb Feshbachresonance [21]. The field is calibrated by addressing ra-diofrequency transitions between the m F sublevels of the F = 2 manifold; the transition frequency is related tothe magnetic field strength by the Breit-Rabi equation[22]. The magnetic field can be determined in this way towithin 5 mG, which near the zero crossing of the scatter-ing length corresponds to an uncertainty in a of ± . a .To observe condensate collapse, we follow the proce- a r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug dure of Donley et al. , tuning the atomic interactions usingthe Feshbach bias magnetic field as shown in Figure 1(a).First, the scattering length is ramped smoothly from a = +89 . a to a init = +0 . a over 100 ms to produce anear-ideal, noninteracting gas. The magnetic field is thenincreased suddenly ( < µ s) to a value at which the in-teractions are attractive a collapse < , and held there fora time τ before the trap is switched off and the scatter-ing length simultaneously increased to a = +50 a . Thecondensate is allowed to expand ballistically at this valuefor 15 ms, after which the magnetic field is switched off,changing the scattering length to a bg = − a . Follow-ing a futher 5 ms of free evolution, the number of atomspresent is determined by absorption imaging.The number of atoms remaining as a function of evo-lution time at a collapse = − a is shown in Figure 1(b).In agreement with the original work of Donley et al. , weobserve a sudden and delayed onset of atom loss. Thisis explained by density-dependent three-body recombi-nation; when the interactions are made attractive, thecondensate begins to contract slowly and its peak den-sity n increases, although not enough to cause significantthree-body loss. As the condensate shrinks, however, thecontraction accelerates, resulting eventually in a suddenimplosion which increases the density by several orders ofmagnitude. This induces significant recombination losses(the loss rate scales with n ) which ultimately halt thegrowth in density. The subsequent dynamics include fur-ther sporadic local implosions, which effect decay of theatom number in an approximately exponential form. Wehave observed remnant clouds surviving long after thecollapse which contain several times the critical numberof atoms N cr (cid:39) . a ho / | a | , where a ho = (cid:112) (cid:126) /mω ho is theharmonic oscillator length [23] (the critical number for acondensate with a = − a in our trap is N cr (cid:39) ;c.f. Figure 1). Such configurations have been shown toachieve stability through the formation of mutually re-pelling bright solitons [2].The discrepancy between the JILA experiment andtheoretical models concerns one of the most elementalcharacteristics of the bosenova: the time for which theatom number remains constant before the first densityimplosion – the so-called ‘collapse time’ t collapse . Al-though the dynamics after the collapse are predicted tobe complex and may exhibit behaviour beyond mean-field effects, the evolution prior to the first implosionshould be captured in the mean-field approximation, andis determined almost exclusively by the initial densitywhich is experimentally constrained. In particular, it hasbeen noted that t collapse does not depend strongly on thethe three-body recombination rate K [8, 11], which isnot well-determined in the vicinity of the Feshbach res-onance. Despite this, Gross-Pitaevskii (GP) simulationswere found to systematically overestimate the collapsetime measured in the JILA system by almost 100%, adiscrepancy at the σ level given the experimental un-certainties [11].As in previous work, we determine the collapse time by × -4400 100 150 20050050100 (b)(a) trap off image FIG. 1: (a) Manipulation of the scattering length to induceand observe condensate collapse. After a variable evolutiontime τ , the atoms are released from the optical trap simul-taneously with an increase in a from a collapse to +50 a . Thecloud is allowed to expand at this value for 15 ms before themagnetic bias field is switched off. (b) Measured atom num-ber as a function of τ for a collapse = − a . The solid line is afit of the experimental data to equation (1). The atom num-ber remains approximately constant for a time t collapse , beforea sudden onset of loss due to three-body recombination. fitting plots of the measured atom number versus time tothe function N ( t ) = ( N − N f ) exp (cid:20) − ( t − t collapse ) τ decay (cid:21) + N f , (1)for t > t collapse , where N and N f denote the atom num-ber at t < t collapse and t (cid:29) t collapse respectively. Figure 2shows the collapse time determined in this way as a func-tion of a collapse for samples of N = 4 × atoms. Asexpected, the collapse time is shorter for larger | a collapse | ,as stronger attraction between the condensate atoms re-sults in more rapid contraction. The data are in qualita-tive agreement with the original experiment of Ref. [4]and later theoretical work.To ascertain the ability of mean-field theory to quan-titatively reproduce our experimental data, we have per-formed numerical simulations for the parameters of oursystem using the Gross-Pitaevskii equation for the con-densate wave function Ψ : i (cid:126) ∂ Ψ ∂t = (cid:20) − (cid:126) m ∇ + V trap + 4 π (cid:126) am | Ψ | − i (cid:126) K | Ψ | (cid:21) Ψ , (2)where V trap is the confining potential. Three-body recom-bination is modelled by the phenomenological inclusionof an imaginary loss term proportional to the three-bodyloss rate coefficient K (which differs by a Bose statisticalfactor of from the coefficient for noncondensed atoms).This term leads to loss proportional to the cube of theatomic density: ∂∂t (cid:90) | Ψ | d r = − (cid:90) K | Ψ | d r , (3)and entails the assumption that the products of recombi-nation collisions leave the trap without interacting withthe remaining atoms. As three-body processes dominateat the high densities relevant to this experiment [8], we donot include the effect of two-body inelastic collisions. Tomake the computation tractable, in integrating equation(2) we assume a cylindrically symmetric trap with oscil-lation frequencies ω z,ρ = 2 π × { , } Hz, such that themean trap frequency ¯ ω matches that of our crossed dipole
10 0 5 10 15 20 252345
FIG. 2: Collapse times as a function of scattering length for a init = +0 . a and N = 4 × atoms. The data points rep-resent experimental values from measured decay curves suchas that shown in Figure 1(b), with error bars denoting thestatistical uncertainty in the fit of equation (1). The solidline is the result of GP simulations for our experimental pa-rameters, and shows good quantitative agreement with theexperimental data. The dotted lines represents the variationin the simulated collapse time due to experimental uncertain-ties in a init , a collapse , ¯ ω and N . The dashed vertical line showsthe critical scattering length for collapse at this atom num-ber, below which the condensate’s kinetic energy stabilizes itagainst implosion. trap (the collapse time has been found to be relatively ro-bust to asymmetry in the trapping potential [11]). Thesimulation includes the 100 ms magnetic field ramp from a = +89 . a to a init = +0 . a , but we neglect the expan-sion of the condensate in our simulation, as the densityspikes which trigger the recombination losses cease oncethe interactions are made repulsive.The results of this simulation for N = 4 × and a init = +0 . a are overlaid with the experimental datain Figure 2 (solid line). For these simulations the three-body loss coefficient was scaled with a collapse as K =8 × − a cm /s [13]. The dotted lines show the vari-ation in the simulated collapse time due to the combinedexperimental uncertainties in the initial scattering lengthand the trap frequencies, as well as run-to-run numberfluctuations of 20%. The simulations show good quanti-tative agreement with the experimental data.It should be noted that the 100 ms ramp of the scat-tering length from a = +89 . a to a = +0 . a is nottruly adiabatic. Our GP simulations show that the rampexcites breathing mode oscillations, despite the durationof the ramp exceeding the mean trap oscillation periodby a factor of 3. The oscillation is predominantly alongthe weak trapping axes, and has an amplitude of ap-proximately 10% of the condensate radius. It occursbecause, although a is varied smoothly, the condensatesize does not depend linearly on the scattering length– in the Thomas-Fermi limit, the radius scales as a / .This excitation accelerates the contraction of the con-densate, decreasing t collapse by approximately 15%. Theeffect is included in the simulations shown in Figure 2.It could be reduced by tailoring the magnetic field rampto ensure that the condensate radius decreases smoothly.Our simulations show that ramping a / smoothly over100 ms reduces the breathing mode oscillations to below1%, and causes the collapse times to be indistinguishablefrom those for a condensate that is in the ground stateimmediately prior to the collapse.We now turn our attention to the possible systemat-ics which may affect the agreement between theory andexperiment. The source of the largest experimental un-certainty in our system is the oscillation frequencies ofthe crossed dipole trap, which vary during the evapo-ration to BEC as the intensity of the trapping laser isreduced. For technical reasons, we cannot directly mea-sure the trap frequencies at the end of the evaporation.Instead, we make several measurements at higher inten-sities, which we fit to an analytic model of the dipolepotential including the effect of gravity in the verticaldirection. The model is further constrained by knowl-edge of the intensity I at which gravity overcomes thedipole potential and the trap vanishes. Due to the strongdependence of the vertical trap frequency on the laser in-tensity near I , and the variation in the intensity itself,we estimate an uncertainty in ¯ ω of 10%, predominantlyin the vertical direction. As the peak density of a non-interacting condensate scales as ¯ ω / , this correspondsto an estimated uncertainty in n of 7%, which in turnproduces an uncertainty in the simulated collapse timeof approximately 15%. This is not sufficient to explainthe inconsistency between our results and the JILA ex-periment.We must also consider the possibility of a systematicerror in our determination of atom number. N is cal-culated using the theoretical optical cross-section by in-tegrating the optical depth of an absorption image. Weimage the atoms on resonance with circularly polarizedlight and apply a small bias magnetic field along theimaging direction to provide a quantization axis. Thecalculation therefore makes use of the resonant cross-section and saturation intensity of the cycling transition | F = 3 , m F = ± (cid:105) → | F (cid:48) = 4 , m F (cid:48) = ± (cid:105) . As a result,our measured value of N is a lower bound: any errorsin the polarization or detuning of the imaging light, orin the alignment of the quantization field, will reduce themeasured atom number. We estimate the uncertainty in N due to these effects to be less than 5%. Furthermore,if the atom number were undercounted then correctingfor this would decrease the simulated collapse times, as ahigher initial density speeds up the contraction. This ef-fect therefore also cannot explain the disparity betweenour results and the original experiment, for which GPsimulations over estimated the collapse times.Although t collapse is only weakly dependent on thethree-body loss coefficient K , the shape of the losscurves is affected by varying this parameter. The valuesof K used in simulations of the original JILA experimentranged from K = 2 × − cm /s [8] to K = 2 × − cm /s [11]. Several authors also considered a relationshipbetween the loss coefficient and the scattering length ofthe form K ∼ | a | for a < [14, 24], with Bao et al. deducing K = 2 . × − a cm /s [13]. We find thatthese values cannot reproduce the shape of our measuredloss curves.Figure 3 shows the results of GP simulations using val-ues of K between × − cm /s and × − cm /soverlaid with experimental data for a collapse = − . a .At higher loss rates, the high initial density of our sam-ple causes significant loss during the contraction of thecondensate in the simulation during the moments leadingup to the collapse. In fact, for K > − cm /s thisinitial loss is so great that there is no sudden implosion ofthe condensate and no discernible elbow in the loss curve[29]. In order to obtain the abrupt onset of loss thatwe observe in the experiment, a three-body loss rate of K ≤ × − cm /s at a = − . a is required. Froma similar analysis of the a collapse = − a data shownin Figure 1(b) we find K ≤ × − cm /s at thatscattering length. These limits are more than an orderof magnitude below most of the values used to simulatethe original experiment. Assuming a scaling with | a | ,they imply K (cid:46) × − a cm /s. In this regime, lossafter the initial implosion is caused by intermittent localdensity spikes between which three-body loss is negligi-ble. This was first predicted by Saito and Ueda [5] evenbefore the JILA experiment. These discrete implosions FIG. 3: Comparison of experimental and simulated collapsedata for a collapse = − . a . The data points show the mea-sured atom number N (normalized to N ) as a function ofevolution time τ at a < , and the lines represent GP simula-tion results with different values of the three-body loss coef-ficient K . A value of K ≤ × − cm /s is necessary toreplicate the sudden onset of loss detected in the experiment. result in the numerous plateaus apparent in the simulatedloss curve, although the scatter in our experimental data– caused primarily by run-to-run fluctuations in atomnumber – is too large to observe these directly.We have investigated the inelastic loss rates furtherby measuring the depletion of our condensates over timewith positive scattering lengths, at which the condensatesare stable. The rate at which atoms are lost due to two-and three-body inelastic collisions depends on the densityprofile of the condensate. In the limit that a → , thedensity is given by the modulus squared of the groundstate harmonic oscillator wavefunction, and the loss rateequation ˙ N /N = − (cid:80) i K i (cid:104) n i − (cid:105) becomes: ˙ N = − N/τ − η K N − η K N , (4)where τ represents the one-body loss rate, η =(2 πa ho ) − / and η = ( √ πa ho ) − . In the Thomas-Fermilimit N a/a ho (cid:29) , the condensate density takes on theshape of the confining potential and the loss rate equa-tion evaluates to: ˙ N = − N/τ − γ K N / − γ K N / , (5)with γ = 15 / / [14 πa / a / ho ] and γ =5 / / [56 π / a / a / ho ] . It should be noted thatthese expressions are valid only when the loss rate issmall compared with the trap frequencies K i (cid:104) n i − (cid:105) (cid:28) ¯ ω ,so that the atomic density profile does not changesignificantly.Figure 4 shows the number of atoms remaining as afunction of time in condensates with a = 0 and a = -2 -1 -2 -3 -1 × × FIG. 4: Measurements of inelastic losses in Rb condensates.The data points show the atom number as a function of holdtime in the optical trap for condensates with a = 0 and a =+37 . a . The solid lines are fits of the solutions of (4) and(5) to the experimental data, assuming K = 0 (solid) and K = 0 (dashed). Although the contributions of two- andthree-body processes cannot be distinguished in this manner,these fits may be used to place upper bounds on the values of K and K . +37 . a . The lines plot the best-fit solutions to (4) and(5) respectively, assuming that the loss is entirely due totwo-body (solid) or three-body (dashed) inelastic colli-sions. It is difficult to separate the contributions of two-and three-body loss purely from the shape of the decaycurve, as has been noted in previous work [19]. Nonethe-less, attributing all of the measured loss to two- or three-body processes allows us to place an upper limit on thevalue of K and K at these scattering lengths. Figure5(a) shows these upper bounds for scattering lengths be-tween and +100 a . In our system, N a/a ho (cid:39) a/a andwe use the Thomas-Fermi approximation (5) except at a = 0 . The error bars represent the statistical uncertain-ties in the fits; we assign an additional systematic errorof 10% to incorporate the uncertainty in ¯ ω .Theoretical calculations suggest that the recombina-tion rate should vary strongly with the two-body elasticscattering cross-section, with several authors predicting auniversal K ∼ a scaling in the zero-temperature limit[25–27]. Our observations are consistent with a strongsuppression of the recombination rate at the zero cross-ing of the s -wave scattering length, with the measuredupper bound K ≤ (3 . ± . × − cm /s at a = 0 an order of magnitude below that for a > +50 a , andmore than three orders of magnitude below the loss ratefar from the Feshbach resonance, K = 7 × − cm /s[19].We can combine this latest data with previous mea-surements of the two-body loss rate to further constrainthe three-body recombination coefficient. Figure 5(b)shows the locus of possible K , K values for which thesolution to (4) best fits our experimental loss curve at a = 0 . In Ref. [19], Roberts et al. measured K nc (cid:39) . × − cm /s for thermal clouds in the vicinity of a = 0 , corresponding to a value of K (cid:39) . × − cm /s for condensed atoms. This matches our measured upper bound of K ≤ (1 . ± . × − cm /s. Coupledwith this result, our data is consistent with a value of thethree-body loss coefficient K ≤ − cm /s. Ref. [26]predicts K (cid:39) × − cm /s at a = 0 . In comparison,the three-body loss coefficient for Rb is K = 6 × − cm /s [28].In conclusion, we have presented new experimentaldata on the collapse of Rb Bose-Einstein condensateswith attractive interactions in an optical dipole trap. Ourresults qualitatively match those of the original JILAbosenova experiment, but in addition agree quantita-tively with GP simulations. We find that a lower value ofthe three-body loss coefficient K than was used in sim-ulating the original experiment is needed to reproduce -29 -28 -27 -26 -16 -15 -14 -13 -20 40 80 10020 60 -16 -15 -14 -13 -31 -30 -29 -28 (b)(a) FIG. 5: (a) Upper bounds on K (open circles) and K (filledcircles) as a function of scattering length, calculated from fitsto the solutions of (4) and (5). The error bars represent sta-tistical uncertainties. (b) Locus of two- and three-body losscoefficients for which the solution to (4) fits the experimentaldata for a = 0 shown in Figure 4. The x and y interceptscorrespond to the upper bounds shown in (a). Assuming K (cid:39) . × − cm /s [19], the data suggest a three-bodyloss coefficient K ≤ − cm /s. the sudden onset of loss that we observe. We have alsoanalysed the decay of atoms from our condensates andthereby placed further constraints on the three-body losscoefficient K at small positive scattering lengths. We ex-pect that this work will inform future experimental and theoretical investigations of this rich quantum system.We thank J. Hope, M. Johnsson, S. Szigeti and M.Hush for helpful discussions. This work was supportedby the Australian Research Council Centre of Excellencefor Quantum-Atom Optics. [1] M. Ueda and A. J. Leggett, Phys. Rev. Lett. , 1576(1998).[2] S. L. Cornish, S. T. Thompson, and C. E. Wieman, Phys.Rev. Lett. , 170401 (2006).[3] T. Lahaye, J. Metz, B. Fröhlich, T. Koch, M. Meister,A. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi, andM. Ueda, Phys. Rev. Lett. , 080401 (2008).[4] E. A. Donley, N. R. Claussen, S. L. Cornish, J. L.Roberts, E. A. Cornell, and C. E. Wieman, Nature ,295 (2001).[5] H. Saito and M. Ueda, Phys. Rev. Lett. , 1406 (2001).[6] H. Saito and M. Ueda, Phys. Rev. A , 043601 (2001).[7] C. A. Sackett, J. M. Gerton, M. Welling, and R. G. Hulet,Phys. Rev. Lett. , 876 (1999).[8] H. Saito and M. Ueda, Phys. Rev. A , 033624 (2002).[9] S. K. Adhikari, Phys. Lett. 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