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PHYSICAL REVIEW A FEBRUARY 1999VOLUME 59, NUMBER 2
Spatial observation of Bose-Einstein condensation of87Rb in a confining potential
B. P. Anderson and M. A. KasevichPhysics Department, Yale University, New Haven, Connecticut 06520
~Received 1 July 1998!
Bose-Einstein condensation of87Rb has been observed in a vapor cell time-averaged orbiting potential trap,and the trapped condensates have been studied usingin situ absorption and dark-ground imaging methods.Condensates of 33104 atoms were observed after 26 s of evaporative cooling. The evaporative coolingsequence consisted of a combination of cooling with ramping magnetic-field strengths and radio-frequency-induced cooling. The measured sizes, numbers of atoms, and transition temperatures of the condensates areconsistent with theoretical predictions that include the effects of atomic interactions.@S1050-2947~99!50502-7#
PACS number~s!: 03.75.Fi, 05.30.Jp, 32.80.Pj
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The ground-breaking studies of Bose-Einstein condendilute atomic vapors have enabled new tests of theoryweakly interacting degenerate Bose gases@1–3#. Importantexamples of such tests include measurements of condefraction and the mean-field interaction energy@4,5#. In pre-vious experiments, these studies have been based on motum distribution measurements that are extracted fromages of initially condensed samples following an intervalfree-space ballistic expansion. We present complemenmeasurements of these quantities based on analysis of dimages of87Rb condensates, made in a regime wherenumber of condensed atoms was small enough that thenetic energy of the trapped atoms could not be ignoredcomparison with the mean-field interaction energy.
Our experimental approach for condensate producparalleled that of the first JILA experiment@1#. Atoms wereinitially captured from a dilute Rb vapor into a magnetoptic trap~MOT! @6# operating in a modified dark-spot configuration @7,8#. Atoms were subsequently compressed@9#and cooled, optically pumped into theF52, mf52 groundstate, then loaded into a purely magnetic time-averagedbiting potential~TOP! trap @10#. Atoms were further com-pressed by changing the TOP field parameters, andevaporatively cooled@11# to condensation. The condensaand thermal cloud were imaged using absorptive and dground@12# techniques.
Our approach deviated from prior work in the followinrespects. First, the bulk of the evaporation was done byducing the strength of the rotating component of the Tmagnetic field, while maintaining a fixed quadrupole fiegradient, rather than by radio-frequency~rf!-induced evapo-ration. Second, we carefully tailored the evaporationquence through a piecewise optimization of evaporationrameters. This allowed us to obtain efficient evaporation wrelatively small numbers of atoms. Third, we directly imagthe trap with anf /2.5 optical system through high-qualitoptical viewports. Finally, we achieved a performancehancement during the MOT loading period by capturingoms in a hybrid trap consisting of a MOT plus a weak roting magnetic bias field.
An illustration of our apparatus is shown in Fig. 1. A Rpartial pressure was maintained in the main vacuum chamwith a Rb-coated cold finger. Antireflection-coated opticviewports, used to admit laser light, were attached to
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stainless-steel chamber with indium metal seals. T10 cm310 cm32.5 cm main chamber was mounted betwea pair of anti-Helmholtz coils oriented with their symmetaxis parallel to gravity. The coils were capable of producia spherical quadrupole magnetic field with linear gradients300 G/cm~in the radial, or horizontal, direction! at full cur-rent. Four additional coils were used to create the rotatbias field of the TOP trap. The bias field rotated at a fquency of 10.5 kHz in the horizontal plane and couldvaried in strength up to 36 G.
Atoms were initially collected from the background vapinto a MOT. Each of the MOT laser beams had a 1/e2 diam-eter of ;1.2 cm, an intensity of;10I sat, and was tuned 9MHz to the red of the87Rb 5S1/2, F52 to 5P3/2, F853transition (I sat is the saturation intensity for theF52, mf
52 to F853, mf853 transition!. Another laser beam of in-tensity I sat , tuned;5 MHz below theF51 to F852 tran-sition, was used to repump atoms. A 4-mm-diam dark spothe center of this beam, filled with a second beam tunedthe F52 to F852 transition, was used to mitigat
FIG. 1. Schematic view of the experimental apparatus, showthe vacuum chamber~octagon! and imaging optics~not to scale!.The straight lines overlapping the octagon represent the TOP cthat surround the chamber, and the smaller circles represent tevaporation coils. The four dark arrows represent the four Mbeams that lie in the horizontal plane. The beam paths of the pbeam~dark lines! and the light diffracted and refracted by the atom~shaded area! are also shown. The lens at position~1! collects thediffracted and refracted light, which is imaged onto the CCD bylens at position~3!. Position~2! indicates the focus of the unscatered probe light, which is the position of the dark-ground imagbeam block.
R938 ©1999 The American Physical Society
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PRA 59 R939SPATIAL OBSERVATION OF BOSE-EINSTEIN . . .
light-assisted, density-dependent collisional losses. Thlosses were further reduced by loading the MOT with a we(;5 G) rotating field component~created using the TOPcoils described above!, in addition to a quadrupole field gradient of 5 G/cm, lowering the atomic density in the centregion of the trap. Operating at a vapor-pressure-limitedeloading time constant of;90 s, we typically loaded;108
atoms in a;150-s interval.Following this loading stage, atoms were compress
then cooled, before being transferred into a purely magntrap. To compress atoms, the rotating bias field was turoff, the hole in the center of the repumping beam was filwith repumping light tuned to theF51 to F852 transition,the trapping frequency was detuned to 42 MHz below renance, and the strength of the spherical quadrupole fieldramped to;8 G/cm. After this 100-ms compression stagthe quadrupole field was turned off and the atoms wcooled in optical molasses for 1 ms. The molasses bewere then extinguished, and the atoms were opticpumped into theF52, mf52 Zeeman sublevel by turninon a;7 G rotating bias field and subjecting the atoms to firesonant, circularly polarized, optical pulses of intensityI50.1I sat , each pulse having a 16-ms duration. The pulseswere synchronized with the rotation rate of the field, aoccurred when the bias field direction was collinear withpropagation direction of the pumping beam.
The optically pumped atoms were then transferred ithe TOP trap. At the end of the optical pumping cycle, tspherical quadrupole field was snapped onto a radial gradof Bq8555 G/cm, while the TOP field was simultaneousestablished at a strength ofBrot510 G ~500-ms switchingtime for each field!. The field strengths were then linearramped over a 900-ms interval toBq85275 G/cm andBrot
536 G. The number of atoms in the magnetic trap at tpoint was;107 and the phase-space density of the clowas ;231026 @13#. The parameters used for the transwere optimized to maximize the phase-space density ofmagnetically trapped atoms.
Atoms were compressed and evaporatively cooled abeing loaded into the TOP trap by reducing the strengththe rotating field. For a TOP trap, the trap curvature}Bq8
2/Brot , so reduction inBrot increases the trap curvaturand hence compresses atoms. The trap depth, on thehand, scales asBrot , and thus decreases with decreasBrot . The combination of these two effects enables efficievaporative cooling by reducingBrot , since the compressiorealized with reduction in trap depth enhances the twparticle collision rate and speeds up the ensemble thermzation rate.
The rotating TOP field strength was ramped down in mtiple stages. The duration and change inBrot for each seg-ment of the TOP ramp were optimized for maximum phaspace density increase~Fig. 2!. We measured phase-spadensity using standard absorptive imaging techniques, asscribed below. Over the course of the 24-s TOP-induevaporation sequence, the loss in number of atoms wfactor of ;100, while the phase-space density increanearly six orders of magnitude to;1021 @14#.
The density and temperature of the trapped atoms wdetermined through absorption imaging. Our imaging geo
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etry is illustrated in Fig. 1. Atoms were irradiated with upfour pulses of low-intensity (;1023I sat) light resonant~ornearly resonant! with the optical transition. Each 16-ms, cir-cularly polarized, pulse was flashed on synchronously wthe TOP field when the TOP field direction was parallel wthe probe propagation axis. The shadow cast by absorpfrom the atomic cloud was imaged onto a cooled charcoupled-device~CCD! camera. The magnification of our imaging system was 7.060.3. The calculated diffractionlimited resolution was;2.6mm (1/e radius!, while themeasured resolution was 3.960.3mm. This measuremenwas madein situ by imaging small ensembles of atomstight trapping potentials, and independently by imaging ttargets.
The intensity in the image plane is proportional to2exp@2a(x,y)#, where a(x,y)5*n(r )s dz is the opticaldepth of the atom cloud,n(r ) is the cloud density at positionr , ands is the optical scattering cross section. We analyzthe images by first extractinga(x,y) from the images, thenfitting a(x,y) to a Gaussian distribution. Cloud temperaturwere extracted from the fitted widths ofa(x,y), and densi-ties inferred from optical depths and widths. The total atonumber was obtained from the density and size measments. For the highest phase-space densities, the trappotential was first adiabatically relaxed to ensure thatpeak optical depth was on the order of unity. The phaspace density in the unrelaxed potential was inferred by sing from those obtained in the relaxed potentials@15#.
The final stage of evaporative cooling was accomplishwith rf magnetic fields, enabling precise control of the eergy of this evaporative cut. The rf field was establishusing two coils mounted just outside the chamber agatwo windows. The rf frequency was linearly ramped downa 2-s cooling interval. The final temperature depended onrf frequency at the end of the ramp.
The threshold rf frequency for condensate formationpended onBrot and the number of atoms in the trap, and wmeasured by changing the final frequency of the sweep@16#.In order to obtain reliable images of the condensate,Bq8 wasadiabatically relaxed following this rf sweep. This reducthe optical depth of the cloud and increased its size.typically reduced the trap-spring constants by a factor of3
FIG. 2. Evaporation characteristics vs time for the TOP evarative cooling ramp. The horizontal axes indicate the time omeasurement, referenced to the beginning of the TOP field evration ramp.
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over a 0.4-s interval. The elastic collision rates were henough that the condensate was expected to maintainmodynamic equilibrium with the normal fraction over thduration of this expansion.
Figure 3 shows images above and below the condensathreshold. The images were analyzed by fitting two twdimensional Gaussian distributions toa(x,y). These Gaussians fit a wide distribution, corresponding to the normal~un-condensed! fraction at the wings of the observed atom clouand a narrow distribution, corresponding to condensed atat the center of the atom cloud. We extracted the optdepth and size for both the normal and condensed fractof the cloud from these fits. We then corrected the measuwidths for the finite resolution of our imaging system. Ttemperature (T) was determined from the size of the normcloud. The number of atoms in the condensate (N0) and inthe normal fraction were inferred from the optical depth asize of each component. The total number of atoms (N) wasdetermined by summing the number in each component
As an independent check on our imaging we alsoserved the transition using the dispersive dark-ground iming technique@12#. In this case, the image forms as a resof the lensing of the light as it refracts on its passage throthe atomic cloud. The image is observed by blockingnonrefracted beam at its focus in the imaging system,described in Ref.@17#. For this imaging technique, the optcal intensity at the image plane is proportional tof2(x,y),where f(x,y) is the dispersive phase shift for a ray thpasses through the cloud at transverse position (x,y). Thisexpression is strictly valid forf(x,y)!1 in the far-detunedlimit. For the conditions of our images, the maximum valufor f approach 1 rad. We analyzed these images by fitGaussians tof(x,y), then extracted temperature, numband condensate fraction as described above.
For a noninteracting Bose gas in the thermodynamic li(N→`), the condensate fraction isN0 /N512(T/T0)3,with a transition temperature ofkBT05\v̄@N/z(3)#1/3,
FIG. 3. Absorption images and corresponding optical-dencross sections showing the BEC phase transition. Three diffefinal rf values of the rf evaporative cooling ramp are represen~a! a frequency of 1.80 MHz, which was just above the condention threshold;~b! 1.75 MHz, which was just below the thresholand ~c! 1.68 MHz, which was far enough below the thresholdproduce a nearly pure condensate~normal fraction is not resolv-able!. In ~a!, a single Gaussian distribution fits the data well. F~b!, the fit of a Gaussian distribution to the wings of the normfraction is shown.
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where v̄ is the geometric mean of the trap oscillation frquencies@18#. A more refined theory that includes the effecof weak interactions is given in Ref.@19#. These theories arecompared with our data in Fig. 4, which showsN0 /N vsT/T0 for three data sets. The first and second sets were tausing absorptive imaging, while the third set was taken usthe dark-ground imaging technique. The trap strength~ascharacterized by the radial oscillation frequencyv') wasv'52p38 rad/s for the first set, andv'52p313 rad/s forthe second and third sets@20#. In this analysis,N0 /N, T0 ,andT were extracted directly from the images@21#. In orderto avoid potential systematic biases in the absorption dwe captured these images at several probe detunings~rangingfrom 250 MHz to 210 MHz).
We characterized the strength of the mean-field intertion @22# through direct observation of the size and shapethe condensate. In the Thomas-Fermi limit, where the cdensate kinetic energy is small compared with the mean-fienergy, the condensate aspect ratio is 81/2 and the densityprofile is parabolic. For smaller numbers of atoms, whkinetic energy is no longer negligible, a variational solutiof the nonlinear Schro¨dinger equation can be used to predthe size and shape of the condensed cloud. A simple vational ansatz for the ground state is a Gaussian wave func@22#. With this assumption, one finds that the aspect ravaries continuously between that of a harmonic-oscillaground state, 81/4, and the Thomas-Fermi value of 81/2 as thenumber of atoms increases. For the data shown in Fig.~c!the observed condensate aspect ratio is 2.360.3, signifi-cantly less than the Thomas-Fermi prediction of 2.82. Terror is dominated by uncertainty in the resolution of timaging system. With a measured value ofN0;3613104
atoms andv'52p313 rad/s, the aspect ratio predicted byvariational calculation using a Gaussian trial function is 217% larger than the observed value. The predicted radial
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FIG. 4. Condensate fraction as a function ofT/T0(N). The filledcircles represent data taken with absorption imaging at astrength ofv'52p38 rad/s, the open circles represent data takwith absorption imaging at a trap strength ofv'52p313 rad/s,and the triangles represent data taken with dispersive dark-groimaging at a trap strength ofv'52p313 rad/s. Each data poinrepresents a single image. The shot-to-shot variance of thenumberN was approximately 30%. The solid curve is the expecbehavior of the condensate fraction with the temperature insemi-ideal gas limit~see Ref.@19#!, and is not a fit to the data. Thdashed curve illustrates the expected behavior of a noninteracgas.
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of 7.8 mm is slightly less than our observed radial size9.260.6mm. Note that the observed shift in the aspect rabetween the condensed and normal cloud is an indepensignature of condensation, analogous to the anisotropicmentum distributions observed in the initial Bose-Einstcondensation~BEC! studies. For comparison, the normfraction aspect ratios of the images shown in Figs. 3~a! and3~b! are ;2.8. ~The expected normal fraction aspect ratdetermined by the trap equipotential surfaces, is 81/2.)
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In conclusion, we have demonstrated Bose-Einstein cdensation of87Rb and have shown that direct imaging tecniques can be used to obtain quantitative comparisonstheory for relativity small numbers of atoms. We expect futher study of finite systems to provide crucial insights inthe role of quantum fluctuations in the formation of the codensate.
This work was supported by the NSF and the ONR.
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@1# M. Andersonet al., Science269, 198 ~1995!.@2# C. Bradleyet al., Phys. Rev. Lett.75, 1687~1995!.@3# K. Davis et al., Phys. Rev. Lett.75, 3969~1995!.@4# J. Ensheret al., Phys. Rev. Lett.77, 4984~1996!.@5# M. Meweset al., Phys. Rev. Lett.77, 416 ~1996!.@6# E. Raabet al., Phys. Rev. Lett.59, 2631~1987!.@7# W. Ketterleet al., Phys. Rev. Lett.70, 2253~1993!.@8# M. Andersonet al., Phys. Rev. A50, R3597~1994!.@9# W. Petrichet al., J. Opt. Soc. Am. B11, 1332~1994!.
@10# W. Petrichet al., Phys. Rev. Lett.74, 3352~1995!.@11# N. Masuharaet al., Phys. Rev. Lett.61, 935 ~1988!.@12# M. Andrewset al., Science273, 84 ~1996!.@13# Phase-space density isnl th
3 , wherel th5A2p\2/mkBT is thethermal de Broglie wavelength, andn is the atomic density.
@14# This corresponds to an overall efficiency@defined as ln~phase-space density increase!/ln~number lost!# of 2.8.
@15# Adiabatic changes in the trap potential by a factork change thetrap density byk3/4 and temperature byk1/2, for a harmonicthree-dimensional trapping potential. See W. Ketterle andvan Druten, Phys. Rev. A37, 181 ~1996!.
.
@16# We were also able to achieve condensation solely throughnipulation of the TOP field strength, without the use of ancooling ramp. However, this method was not as efficient ashybrid sequence described in the text.
@17# See, for example, J. Goodman,Introduction to Fourier Optics~Mc-Graw Hill, New York, 1968!.
@18# S. R. de Grootet al., Proc. R. Soc. London, Ser. A203, 266~1950!.
@19# M. Naraschewski and D. Stamper-Kurn, Phys. Rev. A58,2423 ~1998!.
@20# These frequencies were inferred from the measured magnfield strengths.
@21# To determine the dependence ofN ~and thusT0) on T for thedata of Fig. 4, we first fit a line toN vs T data extracteddirectly from our images and then used the fitted functioform to compute the normalized condensate temperatureT/T0 .This suppressed the;10% shot-to-shot variance in the nomalizing parameterT0 .
@22# See, for example, G. Baym and C. Pethick, Phys. Rev. L76, 6 ~1996!.
