Collective excitation of a Bose-Einstein condensate by modulation of the atomic scattering length
S. E. Pollack, D. Dries, R. G. Hulet, K. M. F. Magalhaes, E. A. L. Henn, E. R. F. Ramos, M. A. Caracanhas, V. S. Bagnato
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A p r Collective excitation of a Bose-Einstein condensateby modulation of the atomic scattering length
S. E. Pollack, D. Dries, and R. G. Hulet
Department of Physics and Astronomy and Rice Quantum Institute,Rice University, Houston, Texas 77005, USA
K. M. F. Magalh˜aes, E. A. L. Henn, E. R. F. Ramos, M. A. Caracanhas, and V. S. Bagnato
Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo,Caixa Postal 369, 13560-970, S˜ao Carlos-SP Brazil (Dated: May 14, 2018)We excite the lowest-lying quadrupole mode of a Bose-Einstein condensate by modulating theatomic scattering length via a Feshbach resonance. Excitation occurs at various modulation fre-quencies, and resonances located at the natural quadrupole frequency of the condensate and atthe first harmonic are observed. We also investigate the amplitude of the excited mode as a func-tion of modulation depth. Numerical simulations based on a variational calculation agree with ourexperimental results and provide insight into the observed behavior.
PACS numbers: 03.75.Kk, 03.75.Nt, 67.85.De
Collective excitation of a Bose-Einstein condensate(BEC) is an essential diagnostic tool for investigatingproperties of the ultracold quantum state. Fundamen-tal information about condensate dynamics can be deter-mined from observations of collective modes [1–3], includ-ing the effects of temperature [4–6], and the dimensional-ity of the system [7]. In addition, the interplay betweenthese modes and external agents, such as random poten-tials [8, 9], lattices [10, 11], as well as other atoms [12–16]can be investigated. Examining the excitation spectrumof the BEC allows for a detailed comparison with theoret-ical models [17–20], and related quantum systems such assuperfluid helium [21, 22] and superconductors [23, 24].Generically, a collective excitation is generated by themodification of the trapping potential of the conden-sate [25]. One convenient method is to apply a suddenmagnetic field gradient, thereby shifting the center of thetrap and exciting a dipole oscillation about the trap cen-ter. One may also suddenly change the curvature of thetrap to excite quadrupole modes. The lowest-lying m = 0quadrupole mode is characterized by out-of-phase axialand radial oscillations.If the condensate is not the only occupant of the trap,i.e., there exists a thermal component or another speciesof atoms, then the other atoms may also be excitedthrough these processes. The evolution of a collectiveexcitation can therefore be complicated because the mul-tiple components may affect damping or induce frequencyshifts of the oscillation [12–16]. Therefore modulating thetrap, although an extremely useful tool for an isolatedcondensate, can be cumbersome when the system to bestudied is multi-specied. An alternative approach is toexcite the condensate alone, leaving the other occupantsof the trap untouched.In this work, we demonstrate the excitation of thelowest-lying quadrupole mode in a BEC of Li by mod-ulating the atomic scattering length via a magnetic Fes-hbach resonance. In contrast to abruptly changing the scattering length [26], sinusoidal modulation enables thecontrolled excitation of a single mode at a specific fre-quency. In addition, by using this method a coexist-ing thermal component will be minimally excited by themean-field coupling to the normal gas [27, 28]. In thecase of a multi-species experiment, resonant modulationof the scattering length of one species will not necessarilyexcite the others, depending on the details of other Fes-hbach resonances present in the system [29]. Therefore,this technique may be useful for investigating non-zerotemperature effects, and as a powerful diagnostic tool formulti-species ultracold atomic experiments.A trapped BEC at zero temperature may be describedby the three-dimensional cylindrically symmetric Gross- µ m FIG. 1: (color online) Quadrupole oscillation excited bythe modulation of a . A series of in situ polarization phase-contrast images of condensates taken during excitation atΩ = (2 π ) 10 Hz, separated in time by 15 ms. The change inpeak density is nearly an order of magnitude from the mostcompressed to the most extended condensate. The horizontalscale of these images has been stretched by a factor of two forclarity. There is negligible excitation of the dipole mode. Pitaevskii equation [30] i ~ ∂∂t ψ = − ~ m ∇ ψ + V ψ + 4 π ~ am | ψ | ψ, (1)where m is the atomic mass, the trapping potential is V = (1 / m ( ω r r + ω z z ) with ω z ( ω r ) the axial (radial)trapping frequency, a is the s -wave scattering length, andthe density is given by n = | ψ | . It is convenient to in-troduce the anisotropy parameter λ = ω z /ω r . We usea variational approach to solve this equation and deter-mine the frequencies of the lowest-lying modes. Using aGaussian ansatz and minimizing the corresponding en-ergy functional, we derive the following coupled differ-ential equations for the dimensionless axial and radialhalf-widths u z and u r of the condensate [31]¨ u r + u r = 1 u r + Pu r u z ¨ u z + λ u z = 1 u z + Pu r u z , (2)where the interaction parameter P = p /π ( N a/l r ),with l r = p ~ /mω r the radial harmonic oscillator size,and N is the number of condensed atoms. We solveEq. (2) in the case of harmonic motion of the Gaussiansizes about their equilibrium values. The frequencies ofthe lowest-lying quadrupole oscillation is [31] ω Q = ω r √ (cid:20) (cid:0) λ − P , (cid:1) − q (1 − λ + P , ) + 8 P , (cid:21) / , (3)where P i,j = P/ (4 u i r u j z ) with u z and u r the equilib-rium axial and radial sizes, respectively. The frequencyof the m = 0 breathing mode is the sum of the two termsin Eq. 3 rather than the difference, and is a factor ofabout 60 higher in frequency for our experimental param-eters. For the case of a highly elongated trap ( λ ≪ P ≫ ω Q = ω z p / P → ω Q → ω z , as ex-pected.We may also use Eq. (2) to determine the time evo-lution of the size of the BEC [33]. In particular, we areinterested in the dynamics associated with the modula-tion of a using a Feshbach resonance, which has beenproposed previously [34, 35]. In a magnetic field B , thescattering length near a Feshbach resonance may be de-scribed by a ( B ) = a BG (cid:18) − ∆ B − B ∞ (cid:19) , (4)where a BG is the background scattering length, ∆ is theresonance width, and B ∞ is the resonance location. Weconsider a time-dependent magnetic field of the form B ( t ) = B av + δB cos(Ω t ) , where Ω is the modulation frequency. Using this form for B , the result of expanding Eq. (4) to first order in thesmall quantity δB ≪ | B av − B ∞ | is a ( t ) ≃ a av + δa cos(Ω t ) , (5)where a av = a ( B av ) and δa = a BG ∆ δB ( B av − B ∞ ) . This expression for a is substituted into Eq. (2), and afourth-order Runge-Kutta method is used to solve thesystem of equations numerically. The results may be di-rectly compared with those from experiment.Our methods for creating an ultracold gas of Li havebeen described previously [36, 37]. We confine atoms inthe | , i hyperfine state of Li in an optical trap and usea set of Helmholtz coils to provide an axially orientedbias field. We determine the radial trapping frequencyby parametric heating to be ω r = (2 π ) 235(10) Hz, andthe axial trapping frequency by dipole oscillation to be ω z = (2 π ) 4 . a ∼ a , we have N ∼ × atoms with a condensate fraction > B av = 565 G (where a av ∼ a ). For these experimen-tal values, the dimensionality parameter is λP ≈ . λP ≪
1) [38]. At this point we oscillate themagnetic field with a modulation depth of δB = 14 G,corresponding to δa ∼ a . We use in situ polarizationphase-contrast imaging to obtain the density distributionof the condensate [39] to which we fit a Gaussian charac-terized by 1 /e axial and radial half-widths.In Fig. 1, we show pictures taken with Ω = (2 π ) 10 Hz.A quadrupole oscillation of the cloud is readily observ-able. The large oscillation amplitudes considered here ex-tend over approximately 1 mm of the optical trap. A har-monic approximation of the trapping potential about thetrap center is less than 10% in error over this range. Thesize of the cloud in Fig. 1 as a function of time is mod-eled well by the variational calculation, consistent withnegligible anharmonic contributions. Furthermore, weobserve no damping of the quadrupole mode over manyoscillation periods, consistent with a negligible thermalfraction.Results from the variational calculation show that dur-ing the excitation if Ω < ω Q , then the axial and radialsizes of the cloud follow the change in a : growing as a increases, and shrinking as a decreases. This in-phasebehavior of both the axial and radial sizes of the cloud isexpected for adiabatic changes in a , and therefore shouldnot be confused with the high-lying m = 0 breathingmode for low frequencies. However when Ω > ω Q , theradial size follows nearly in-phase, while the axial sizelags behind the radial size by half a period—an out-of-phase oscillation. In both cases when the excitation isstopped, the cloud undergoes free quadrupole oscillations
50 100 (a) S i z e ( µ m )
100 200 (b)
FIG. 2: (color online) Axial (red circles) and radial (bluesquares) 1 /e sizes of a condensate during and after excita-tion with a av ∼ a and δa ∼ a , where (a) Ω = (2 π ) 3 Hzor (b) Ω = (2 π ) 10 Hz. The natural oscillation frequency is ω Q = (2 π ) 8 . ∼ µ m, which limitsaccurate determination of the radial sizes. The frequency ofthe breathing mode is on order 470 Hz and therefore no effectsof this mode are present. with the axial and radial sizes π out-of-phase. Charac-teristic data for these two regimes are shown in Fig. 2and reasonably agree with the variational results. Wefit the time evolution of the size of the cloud after ex-citation and determine the free quadrupolar oscillationfrequency to be ω Q = (2 π ) 8 . λ = 0 .
021 and P ≈
15. Similar agreementbetween measured and predicted quadrupole frequenciesin the dimensional crossover regime have been previouslyobserved [7]. The amplitude of this free oscillation is de-pendent on the duration of excitation as well as the phaseof the driving force at the time when the excitation isstopped. Therefore, care must be taken when comparingdata with theoretical predictions of the amplitude duringthe free oscillation.A less parameter-dependent measure is to observe the F r a c t i ona l A m p li t ude Modulation Frequency (Ω / 2π) (Hz)
FIG. 3: (color online) Fractional amplitude of the drive (redcircles) and quadrupole (blue squares) modes during excita-tion at frequency Ω with a modulation depth δa ∼ a , where a av ∼ a . The solid lines are results from the variationalcalculation with no adjustable parameters. The oscillation isnotably asymmetric for fractional amplitudes of order 1 andlarger, as shown in Fig. 2(b). amplitude of the oscillation during excitation. As can bediscerned from Fig. 2, during excitation the size of thecondensate oscillates at both the drive frequency and thenatural quadrupole frequency. Beating between these fre-quencies modulates the instantaneous deviation from theunperturbed size. An ideal method for determining theenergy in the driven mode and the excited quadrupolemode separately is to use Fourier analysis [40]. Thismethod can be experimentally difficult, however, giventhe required number of coherent oscillations needed toaccurately resolve the sinusoidal peak in the Fourier spec-trum. Instead, we assume that the system can be de-scribed by the linear combination of sinusoids at theknown frequencies Ω and ω Q . After driving an excita-tion for 0.5 s, we fit the observed condensate axial sizeto the following expression during an additional 0.5 s ofexcitation: u ( t ) = u + u Ω sin (Ω t + Φ) + u Q sin ( ω Q t + φ ) , (6)where u is the equilibrium 1 /e size, u Ω and u Q are theamplitudes of the drive and quadrupole modes, respec-tively, and Φ and φ are the respective phases. The frac-tional amplitudes u Ω /u and u Q /u are shown as func-tions of Ω in Fig. 3. As expected, there is a resonantenhancement in both the quadrupole and drive modeswhen Ω = ω Q . In addition, we see a parametric enhance-ment when Ω ≈ ω Q . The dip at exactly 2 ω Q is due todestructive interference between the drive mode and theresonantly excited mode. This interference structure isobserved in both the data and the simulation.Larger drive amplitudes push the oscillations into thenonlinear regime where the amplitude of oscillation is no A x i a l s i z e ( µ m ) Time (s)
FIG. 4: (color online) Quadrupole oscillation driven by alarge amplitude excitation, δa ∼ a where a av ∼ a , nearresonance Ω = (2 π ) 9 Hz. The non-sinusoidal behavior leadsto fragmentation of the condensate during the compressionstages when the axial size becomes smaller than the axialharmonic oscillator size of ∼ µ m. The image at right wastaken during the compression stage at t = 0 .
52 s and showsfragmentation of the condensate, the field of view is 25 µ m × µ m. F r a c t i ona l A m p li t ude Fractional Modulation Depth
FIG. 5: Fractional amplitude of oscillation, when Ω ≈ ω Q , asa function of the fractional modulation depth δa/a av after 1 sof excitation. The solid curve is the result from the variationalcalculation. longer linearly dependent on the modulation depth. Thefirst noticeable effect is the non-sinusoidal behavior of theoscillation seen in Fig. 2(b). In Fig. 4 we show the resultof driving the system near resonance with a fractionalmodulation depth δa/a av = 1. As the amplitude of thedriven oscillation grows, it eventually becomes compara-ble with the original condensate size. At this point, thesize of the condensate cannot become smaller, and there-fore, the oscillation becomes increasingly asymmetric. Inthis manner, the rate of energy transfer into the oscilla-tion will decrease. At even larger amplitudes, we observethat the condensate appears to fragment [35]. Similarlooking results have been observed when modulating theradial confinement of a cigar-shaped BEC, where Fara-day waves may be excited [41]. By fitting the excitationspectrum (Fig. 3) to a skew Lorentzian (where the peakexcitation frequency, width, amplitude, and skewness are fit parameters), we determine the maximum fractionalamplitude of oscillation of the quadrupole mode after 1 sof excitation. Our experimental results for the ampli-tude as a function of modulation depth are presented inFig. 5 along with results from the variational calculation,which show good agreement with no adjustable parame-ters. We note that for the smallest of modulation depthsinvestigated, we only observed oscillation of the conden-sate when the drive frequency was near resonance. Inaddition, we found a roughly linear scaling of the instan-taneous amplitude of the quadrupole oscillation with theduration of the excitation in this regime. Furthermore,Fig. 5 conveys to us that the data in Fig. 2, which wasdriven at a fractional modulation depth of ∼ ∼ δω/ω Q ≈ . A z for small A z , where A z ≡ u Q /u is the fractional am-plitude of the axial size [40]. The data shown in Fig. 3has A z ∼ ∼
8% for A z = 2 with a shallow amplitude dependence at larger A z . Whereas a shift of 10% was observed in the oscil-lation of Fig. 4, we were not able to resolve frequencyshifts by fitting Lorentzians to the excitation spectra forour data at low A z shown in Fig. 5. There is an addi-tional frequency shift due to non-zero temperature [4–7]which is estimated to be negligible given our low tem-peratures and interaction strength [43]. Even though wecan neglect temperature effects in the measurements pre-sented here, our method of excitation of the quadrupolemode may be used to study these effects in further detailin regimes of stronger interactions. In addition, goingto large values of a will facilitate investigations of be-yond mean-field effects on the collective modes of a Bosegas [44], complementary to those observed in a Fermi gasat the BCS-BEC crossover [45–47].In this work, we have experimentally demonstrated theexcitation of the collective low-lying quadrupole mode ofa dilute Bose gas by modulating the atomic scatteringlength. Our observations are supported by variationalcalculations of the time dependent Gross-Pitaevskii equa-tion assuming a Gaussian trial wavefunction. Using thisformalism we find good agreement with our experimentalresults. Temporal modulation of the scattering length, asafforded by Feshbach resonances, provides an additionaltool for exciting collective modes of an ultracold atomicgas. This method is quite attractive in circumstanceswhere excitation of the condensate by other means, suchas trap deformation, is unavailable. 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