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dilute bose - einstein condensates ( becs ) are produced and probed using atomic physics tools  @xcite , and their connection to condensed - matter physics is most clear when studying collective excitations  @xcite , as well as the sound propagation  @xcite . much of our understanding of the nature of sound in quantum fluids comes from studies carried out in liquid helium  @xcite . recently , with gaseous condensates , the researchers have turned to these decades - old theories from condensed matter physics to test their validity thoroughly and very successfully  @xcite . nowadays , these versatile experimental system are even being used as fast complex hamiltonian solvers , which may provide `` on demand '' solutions to particular interesting problems in condensed matter physics . the nature of collective excitations in a homogeneous bose gas depends on the hierarchy of three length scales : the excitation reduced wavelength , @xmath1 ; the healing length , @xmath2 ; and the mean free path for collisions , @xmath3 . in a inhomogeneous bose trapped gas , the excitations are influenced by an additional length : the typical condensate size , the thomas - fermi radii . this gives rise to three different excitation regimes : hydrodynamic , collision and collisionless . the first observations of the low - lying excitations in gaseous becs were made at jila  @xcite and at the mit  @xcite in the zero temperature limit . soon after , stringari carried on in depth theoretical studies of the collective modes  @xcite creating the base framework to understand the observations .    in this work , we investigate the collective excitations of @xmath0rb condensates held in a cigar - shaped quadrupole - ioffe configuration ( quic ) trap  @xcite . the collective excitations of the condensate are produced by an additional , time varying , magnetic field gradient superimposed to a quic potential . besides the expected oscillation of the center of mass , we observed the bec undergoing shape oscillations . the frequency of the observed collective excitations were measured and compared to the theoretical predictions . we see evidence of a non - linear dependence on the amplitude of the perturbing field gradient , not yet predicted by the theory . we believe that the study of collective modes is relevant because they present the fundamental features of quantum degenerate gases . furthermore , we would like to better understand the vortex nucleation and the onset of turbulence in perturbed condensates .    in the next sections , we present a brief discussion on the collective modes observed in becs , followed by a description of the experimental setup used to acquire the data . the results are then presented and discussed based on the current theory for the dispersion relations , where good agreement is found for the measured mode frequencies . finally , the conclusions are presented . bose - einstein condensates were object of intensive studies early since they were first produced in gaseous alkali metal samples . the low - lying collective excitations were theoretically investigated and predicted just one year after the initial experimental bec observations  @xcite . back then , stringari  @xcite was able to derive analytic solutions for both isotropic and anisotropic harmonic potentials in the thomas - fermi regime . the normal modes of a bec are classified by quantum numbers @xmath4 , where @xmath5 is the radial quantum number and @xmath6 stand for the total angular momentum and its axial projection quantum numbers , respectively . for cylindrical symmetry ( as in our trap ) , @xmath7 is still a good quantum number , but @xmath8 is not . thus , the normal modes are a superposition of wave functions with the same @xmath7 . stringari discussed an anisotropic harmonic potential with axial symmetry and showed that the lowest @xmath9 modes are coupled excitations of the @xmath10 modes , quadrupolar surface oscillation , and also the @xmath11 symmetries . for a cigar - shaped condensates the lowest modes were predicted at frequencies  @xcite : @xmath12 @xmath13 and @xmath14 in the above equations @xmath15 , @xmath16 are the axial , radial trap frequencies which will be properly defined in the next section , and @xmath17 is the ratio of those two frequencies known as the asymmetry parameter . in the large anisotropy limit @xmath18 , the solutions of eq . [ eq3 ] above are : @xmath19 and @xmath20 . the first , low - lying , shape oscillations associated with the above dispersion relations are : 1 ) the cylindrically symmetric ( slow ) @xmath9 quadrupole mode where the axial ( @xmath21axis ) length oscillates out of phase with the radial lengths ; 2 ) the in - phase ( fast ) , mostly radial , @xmath22 breathing mode ; and 3 ) the non - symmetric radial oscillation @xmath23 quadrupole mode where the two radial lengths oscillate out of phase . the experimental sequence to produce the bec runs as follows . first , @xmath24 becs containing about @xmath25 atoms are formed in the @xmath26 hyperfine state , with a small thermal fraction in a cigar - shaped quic magnetic trap . in order to achieve the degeneracy , the atoms undergo forced evaporation for @xmath27 , going from @xmath28 down to about @xmath29 , following a non analytic curve experimentally determined by optimizing the phase space density and the runaway condition in each step . we typically produce samples with no more than 30 or 35% of thermal fraction , well into the thomas - fermi regime . the measured trapping frequencies are : @xmath30 in the symmetry axis , and @xmath31 in the radial direction . the radial trapping frequency depends on the offset field , @xmath32 , at the trap bottom and the magnetic field gradient in the radial direction , which can be written as : @xmath33 where @xmath34 , @xmath35 and @xmath36 are the magnetic quantum number , the bohr magneton , and land g - factor for the hyperfine state @xmath37 , and @xmath7 is the atomic mass . the axial trapping frequency is determined by the axial curvature @xmath38 through @xmath39 . the values of @xmath32 , @xmath40 , and @xmath38 are determined based on the geometric parameters of the quic coils and the electric current running through them . for our system , the calculated values are @xmath41 , @xmath42 , @xmath43 , and @xmath44 . correspondingly , the theoretical results of the radial and axial trapping frequencies are @xmath45 and @xmath46 , respectively , for a current of @xmath47 through the coils . finally , we have @xmath48 , @xmath49 , and the anisotropy parameter is @xmath50 . to induce the collective excitations , an additional sinusoidal magnetic field gradient is turned on and superimposed on the atoms just after the bec is produced . the external gradient is a quadrupolar field generated by an anti - helmholtz coil pair set with symmetry axis closely parallel to the weak trap axis . the coil pair is not perfectly aligned to the condensate axis ( @xmath51 ) , so its field gradient , near the trapping region , has components parallel to each trap eigen - axis . a superposition of shape morphing , trap bottom displacement and axial rotation is coupled to the bec . additional , sinusoidal field gradients going from zero up to @xmath52 , along the vertical axis , were applied to the trapped atoms inducing the onset of the collective modes . the extra gradient oscillating at @xmath53 was kept on for six extra periods of trap oscillation and then switched - off . in fig.[fig1 ] , we present a typical set of the perturbed bec samples imaged for different hold times , nearly completing a full oscillation cycle . [ ht ! ] we then kept the bec held in the trap for different time intervals , @xmath54 , ranging from @xmath55 to @xmath56 after which we observe it by performing a resonant time - of - flight ( tof ) absorption imaging , after @xmath57 free fall to avoid pixel saturation and increase the accuracy . center - of - mass position of the cloud as well as its axes sizes and orientations are recorded and plotted as a function of the @xmath54 , from where we extract both frequency and amplitude of the collective modes . using the technique explained in the previous section , we are able to observe dipolar , quadrupolar , breathing and also the scissors modes in our becs . for small excitation amplitudes , we observed the tilting about the gravity direction ( @xmath58 ) . such periodic axial tilting is generated by a sudden rotation of the confining trap , and it is known in the literature as scissors mode . it was theoretically studied by gury - odelin  @xcite and others and will not be discussed here . here , we will focus on the dipolar and the quadrupolar modes only .    after being perturbed , the bec center - of - mass always oscillate in the dipolar mode  @xcite , into the trap . this mode corresponds to the full oscillation at the trap frequency in each direction . we have also observed the onset of the quadrupolar mode  @xcite . since the longitudinal and transverse directions do not oscillate exactly at @xmath59 out - of - phase , we believe that the higher order modes may also be onset . moreover , the condensate s long axis cyclic tilting was observed and priviously reported  @xcite , see fig.[fig1 ] . the observation of the scissors mode  @xcite shows that the perturbing field gradient is able to cause the bec to rotate . [ ht ! ] the study of the low - lying collective excitations was carried on by following the center - of - mass ( cm ) motion and the aspect ratio ( ar ) of the becs as a function of the hold time . for small amplitudes , the cm is usually located near the center of the atomic density distribution , which is also the geometric center of the bec . but , in some situations , at large perturbing amplitudes ( @xmath60 ) , we observed slosh dynamics taking place , clearly displacing the thermal and condensed fractions in opposite , out - of - phase , cm motion  @xcite . it is important to mention that the dipolar mode preserves the ar when compared to an unperturbed bec . therefore , the ar analysis are used only to study the higher order modes ( quadrupolar and breathing ) .    in fig.[fig2](a ) , we present the results acquired by directly tracking the center - of - mass ( cm ) motion as a function of the hold time for four different perturbing amplitudes . one may note that , the perturbing energy not only increases the cm displacement from the rest position , but also shifts the phase of the sinusoidal motion . fig.[fig2](b ) presents the ar evolution with the hold time , measured for the quadrupolar mode . again , note the phase shift introduced in the ar evolution as the perturbing energy amplitude raises from @xmath61 ( 10% ) to @xmath62 ( 70% ) . the data shown in figs.[fig2](a ) and ( b ) are fitted by simple sinusoidal functions from where we determine the oscillation frequency of the collective modes and their amplitudes . the evolution of the cm and the ar as a function of the perturbing amplitude is shown in fig.[fig3](a ) . on the one hand , the cm oscillation amplitude proportionally increases with the excitation amplitude starting from zero , as expected . on the other hand , the evolution of the ar clearly show different evolution slopes , depending on perturbing amplitude range . and it displays an interesting non - linear dependence near @xmath63 . a small region around @xmath64 presents a interesting non - linear behavior , rapidly evolving from the initial to the final ( steeper ) slope , strongly suggesting the emergence of a different regime for perturbing amplitudes lager than @xmath63 . we do not have yet a theoretical explanation for this observation . fig.[fig3](b ) presents the changes on the dipolar and quadrupolar oscillation frequencies as the perturbing amplitude increases . we measured @xmath65 for the dipolar oscillations , compared to @xmath16 predicted . and , @xmath66 for the fast oscillation , quadrupolar mode , which is to be compared to the theoretical prediction : @xmath20 . the observed modes were previously studied and analytic expressions for the dispersion relations were theoretically derived  @xcite . the agreement found between our results and the predicted frequencies , eqs.[eq1 ] to [ eq3 ] , presented in section 2 is very good . the results show almost no change with the frequency for amplitudes up to about @xmath63 and a slight decrease ( @xmath67 ) for larger amplitudes . similar observations were seen before  @xcite . we expected to see a constant evolution along the full range and believe the slight change may attributed to nonlinear interactions coupling the condensate normal modes  @xcite . for nearly pure condensates , the thermal excitations should have no effect at all and , to the best of our knowledge , there is still no theoretical explanation for a possible slow down of the collective modes frequency of a trapped bec . [ ht ! ] we believe though , that the different behavior observed in between the cm oscillation and the shape oscillation ( quadrupolar ) might be due to way we perturb the bec . the addition of a small amplitude , time - varying , tilted magnetic field gradient will mostly displace the cm of the bec . but , once the axial and the radial trapping gradients are a factor of two different , it is enough to induce rotation and shape oscillations . this observation is in agreement with results presented in figs.[fig1 ] - [ fig3 ] . therefore , small perturbation results in cm displacement with little effect on the bec shape . as the perturbation amplitude increases , the condensate is pushed farther away from the trap bottom . in a quic trap , the bec needs to be displaced away from the trap center , up to regions with steeper field curvature ( see fig.2(d ) in the original paper  @xcite ) , where the unbalanced compression is likely to be effective inducing shape oscillations . that is consistent with the near resonant perturbation used to drive our bec . moreover , the quic traps are known to be asymmetric along the weak axis direction due to the presence of a single ioffe coil , making it easier for the condensate to reach the non symmetric boundaries of the trapping potential . this also seems consistent with the observation of a change in the curvature shown in fig.[fig3 ] , where the bec breaks free from the inner harmonic ( symmetric ) trapping region and travels through the outermost , anharmonic , trap region . these outer trap regions present much stepper field gradient and curvature making them suitable to produce shape oscillations in a bec by unbalanced compression . beyond that , we believe that the steep gradient regions might be the inducing mode coupling and and even the dephasing observed in fig.[fig2 ] . this complex coupling dynamics seems particular to our experimental system . it may also be the key to the better understand the vortex nucleation and the turbulence emergence observed in previous experiments  @xcite . in contrast , the presented results are in good agreement with the theoretical predictions except for the observed threshold for the quadrupolar shape oscillations . we have studied the excitations induced in cigar - shaped condensates produced in a quic trap , near zero temperature . collective modes were excited by sinusoidally varying currents running through additional coils , externally set for this purpose . the excitation was not axially symmetric , and so the @xmath68 modes were excited too . dipolar oscillations were readily induced and used to accurately measure the trap frequencies . at least , two additional shape oscillations were set on and studied . the higher frequency , primarily , radial breathing mode was excited and its frequency was measured , in very good agreement with the predicted theoretical value . finally , we have observed an interesting , nonlinear , onset dependence on the minimum excitation amplitude needed to start changing aspect ratio .
we report the experimental observation of the collective excitations induced in a magnetically trapped @xmath0rb bose - einstein condensate . 
 low - lying mode excitations were studied by tracking the condensate s center - of - mass displacement , and its aspect ratio as a function of the hold time in the trap . 
 we were able to partially control the modes onset by modulating the amplitude of the additional field gradient used to excite the bec . 
 the measured excitation frequencies were found to be in good agreement with the literature . 
 we have also found that the modulation amplitude was able to change the phase of the center - of - mass oscillation . finally , an interesting , non - linear dependence was observed on the condensate aspect ratio as a function of the perturbing amplitude which induces the quadrupolar mode .