We report sensitive detection of parametric resonances in a high-density
sample of ultracold $^{87}Rb$ atoms confined to a far-off-resonance optical
dipole trap. Fluorescence imaging of the expanded ultracold atom cloud after a
period of parametric excitation shows significant modification of the atomic
spatial distribution and has high sensitivity compared with traditional
measurements of parametrically-driven trap loss. Using this approach, a
significant shift of the parametric resonance frequency is observed, and
attributed to the anharmonic shape of the dipole trap potential

Balik, S.

Havey, M. D.

Win, A. L.

English

arXiv

We report sensitive detection of parametric resonances in a high-density sample of ultracold 87Rb atoms confined to a far-off-resonance optical dipole trap. Fluorescence imaging of the expanded ultracold atom cloud after a period of parametric excitation shows significant modification of the atomic spatial distribution and has high sensitivity compared with traditional measurements of parametrically driven trap loss. Using this approach, a significant shift of the parametric resonance frequency is observed and attributed to the anharmonic shape of the dipole trap potential. 2009 The American Physical Society

Old Dominion University

Old Dominion UniversityODU Digital CommonsPhysics Faculty Publications Physics2009Imaging-Based Parametric Resonance in an OpticalDipole-Atom TrapS. BalikOld Dominion UniversityA. L. WinOld Dominion UniversityM. D. HaveyOld Dominion University, mhavey@odu.eduFollow this and additional works at: https://digitalcommons.odu.edu/physics_fac_pubsPart of the Atomic, Molecular and Optical Physics Commons, and the Optics CommonsThis Article is brought to you for free and open access by the Physics at ODU Digital Commons. It has been accepted for inclusion in Physics FacultyPublications by an authorized administrator of ODU Digital Commons. For more information, please contact digitalcommons@odu.edu.Repository CitationBalik, S.; Win, A. L.; and Havey, M. D., "Imaging-Based Parametric Resonance in an Optical Dipole-Atom Trap" (2009). PhysicsFaculty Publications. 89.https://digitalcommons.odu.edu/physics_fac_pubs/89Original Publication CitationBalik, S., Win, A. L., & Havey, M. D. (2009). Imaging-based parametric resonance in an optical dipole-atom trap. Physical Review A,80(2), 023404. doi:10.1103/PhysRevA.80.023404Imaging-based parametric resonance in an optical dipole-atom trapS. Balik, A. L. Win, and M. D. Havey*Department of Physics, Old Dominion University, Norfolk, Virginia 23529, USAReceived 15 May 2009; published 6 August 2009We report sensitive detection of parametric resonances in a high-density sample of ultracold 87Rb atomsconfined to a far-off-resonance optical dipole trap. Fluorescence imaging of the expanded ultracold atom cloudafter a period of parametric excitation shows significant modification of the atomic spatial distribution and hashigh sensitivity compared with traditional measurements of parametrically driven trap loss. Using this ap-proach, a significant shift of the parametric resonance frequency is observed and attributed to the anharmonicshape of the dipole trap potential.DOI: 10.1103/PhysRevA.80.023404 PACS numbers: 37.10.Gh, 37.10.JkParametrically driven processes are ubiquitous in natureand occur in a range of classical and quantum systems 1–5.In a parametrically driven system, a parameter may be har-monically varied in time. For a one-dimensional harmonicoscillator with a characteristic frequency o, such excitationgenerates resonances at frequencies p=2on , where n is aninteger n=1,2 , . . . A familiar example is the inverted pendu-lum, which may be stabilized against decay of small oscilla-tions by appropriately driving the pivot point. One importantrole of parametric excitation in atomic physics is its influ-ence on ultracold atoms confined in magneto-optical 6,magnetic 7–10, and optical dipole traps 11–15. For situ-ations where long-lived traps are desirable, excitation due tonoise in the trap parameters normally heats the atoms andthus leads to atoms being expelled from the trap. Under somecircumstances parametric driving of a system with an anhar-monic potential can also cool the atom cloud 7–9. Thelifetime of the trap is limited by technical noise in any trapparameters 12–14, two important ones for optical dipoletraps being the pointing stability and the intensity noise char-acteristics of the trapping lasers. In addition to this deleteri-ous effect, parametric resonance is a useful and widely em-ployed tool for characterizing the shape of the trap, includingthe harmonic frequencies of the trap 14. This is importantin experiments where the spatial shape of the trapping poten-tial is needed for interpretation of the results. In experimentsin our laboratory, we are interested in obtaining a high den-sity of atoms in an optical dipole trap, and in knowing thepeak density as well as possible. This requires that the num-ber of atoms, atom temperature, and shape of the trappingpotential be well known. A main goal of these experiments isto study the electromagnetic analog of Anderson localization16–19 in an ultracold gas.In this paper we present experimental results on paramet-ric excitation of a sample of ultracold 87Rb atoms confined inan optical dipole trap. Such studies are frequently done bymeasuring the relative number of atoms that are removedfrom the trap as excitation parameters are varied. In thepresent case we measure instead modification of images ofthe ultracold cloud following parametric excitation and asubsequent period of free expansion. Analysis of the imagesprovides a more sensitive measure of the effects of paramet-ric excitation, including clear evidence of the frequency shiftof the parametric resonance frequency due to trap anharmo-nicity 8,9,20. In the following we provide a brief descrip-tion of the experimental apparatus. This is followed by pre-sentation of the measured images and their analysis.A schematic of the experimental apparatus is shown inFig. 1. In the figure, the central part of the experimentalapparatus is a magneto-optical trap MOT confining ultracold 87Rb atoms. The MOT is a vapor-loaded trap formed ina vacuum chamber with a base pressure 10−9 Torr. The sixMOT beams are derived from an external cavity diode laserECDL with the grating arranged in Littrow configuration.The master diode laser is frequency locked to a saturationabsorption feature produced in a Rb vapor cell. The laserpower is increased by injecting the output into a slave laser,thus providing more than 20 mW of light in trapping laserbeams of cross-sectional area 2 cm2. The slave output isswitched and spectrally shifted as required with an acousto-optical modulator AOM to a frequency about 18 MHz be-low the 87Rb F=2→F=3 trapping transition. The repumperlaser is also an ECDL of the same design as the MOT laserand is locked to the F=1→F=2 hyperfine transition. Therepumper delivers a beam of maximum intensity0.6 mW /cm2 and is delivered along the same optical pathas the trapping laser beams. Repumper switching is con-trolled with an AOM.The cold atom sample is initially produced in the higherenergy F=2 hyperfine level. Direct absorption imaging mea-surements of the peak optical depth on the F=2→F=3transition yielded, for this sample, bo10 in a Gaussian ra-*mhavey@odu.eduFIG. 1. Schematic drawing of the experimental apparatus. In thefigure QUEST stands for quasi electrostatic trap, and chargecoupled device CCD represents charge coupled device. Drawingnot to scale.PHYSICAL REVIEW A 80, 023404 20091050-2947/2009/802/0234044 ©2009 The American Physical Society023404-1dius of ro=0.45 mm 21. This corresponds to a total num-ber of atoms 3.2107 and a peak number density 2.21010 atoms /cm3. However, the main goal is to transfer thetrapped atoms to a carbon-dioxide CO2 laser-based opticaldipole trap. The 100 W CO2 laser operates at 10.6 m. Thelaser is focused to a radial spot size of 55 m, and acorresponding Rayleigh range of zR750 m. The CO2 la-ser focal zone is overlapped with the MOT trapping region,while the application of the CO2 beam itself is controlled bya 40 MHz AOM, which directs the first-order diffractedbeam to the MOT. After the atom sample is formed in theMOT, the CO2 laser is applied and the sample is compressedand loaded into the quasi electro static trap QUEST. This isdone by detuning the MOT master laser 60 MHz to the low-frequency side of the trapping transition, while simulta-neously lowering the repumper intensity by an order of mag-nitude. The resulting temporal dark spot MOT loads theatoms into the lower-energy F=1 hyperfine component, withabout 15% of the MOT atoms transferred to the QUEST. It isimportant to note that this transfer efficiency is measuredafter a QUEST holding period of about 1 s, during which theatomic sample collisionally evolves toward thermal equilib-rium. Measurements of the QUEST characteristics, after thehold period, by absorption imaging, parametric resonance,and the measured number of atoms transferred show asample with peak density about 61013 atoms /cm3 and atemperature of Toc=65 K. The 1 /e lifetime of the confinedatoms is greater than 5 s, limited by background gas colli-sions. The residual magnetic field in the sample area, whenthe MOT quadrupole field is switched off, is estimated to bea few mG.Here we are concerned with parametric excitation of at-oms confined to the QUEST and the sample characteristics.The sample is excited by modulation of the CO2 powerwhich, for a Gaussian focused beam, determines both axialand radial harmonic trap frequencies. Here these are about1.25 kHz for radial excitation and 50 Hz for axial excitation;imposition of acoustic modulation on the CO2 laser AOM inthe range 0–10 kHz is then sufficient to drive the fundamen-tal parametric resonances. The characteristics of the modula-tion are the modulation frequency f , the modulation time T,and the modulation depth h. Detection of the result of exci-tation is made in two ways, each based on measurements onan image of the QUEST following a 3 ms period of freeexpansion. The expansion period allows the density of thesample to be reduced sufficiently that the QUEST becomesoptically thin, so that measurement of light scattered fromthe sample is proportional to the number of atoms in thesample. In the first more traditional method, loss of a certainnumber of atoms is made through measurement of the totalintensity of light scattered from the sample. This trap lossmethod measures the survival probability of atoms in thetrap. In the second method, and the one we focus on here, theaverage peak intensity in the central region of the image ismeasured; this is proportional to the survival probability ofatoms more localized spatially in the harmonic region of thetrap, even after a time T of parametric resonance. The mainmeasured quantities are fluorescence images of the expandedatomic cloud, these being recorded for different h, T, and f .Characteristic results are shown in Fig. 2, where the imagesshow a clear increase in the axial and radial Gaussian radiiand a loss in peak intensity as T is increased at fixed h andf =2.5 kHz. We also draw attention to the change in shape ofthe cloud with increasing T. This is due to heating of theatom sample while it is confined and consequent expansionin the weakly confining axial direction.We begin our analysis by showing in Fig. 3a the excita-tion spectrum in the spectral vicinity of the fundamental ra-dial resonance. Two sets of data are displayed. In the first, weshow the total number of trap atoms surviving parametricexcitation, while in the second we show depletion of signalfrom the central zone of the atom cloud image. The differ-ences between the two are striking. First, even for the quitelarge modulation depth of 15% the total trap loss, as a frac-tion of the entire signal level, is clearly weaker than thedepletion signal. The second main feature is that the trap losssignal is shifted by −0.31 kHz. Both aspects of the totaltrap loss signals represent significant experimental disadvan-tages if one wants to characterize the lower-energy portion ofthe optical dipole trap potential. We attribute the frequencyshift to the anharmonic nature of the trapping potential; asthe average amount of excitation energy of atoms in the trapincreases, the resonance harmonic frequency for those atomsdecreases, shifting the resonance to lower values. Becausethe resonances can be intrinsically quite broad, this shifts theoverall response to lower frequencies. One can envision theprocess as parametric evolution of an atom distribution lo-calized deep in the trap to a warmer one which is morebroadly distributed over the attractive part of the optical di-pole potential. This leads to a “freezing out” of the paramet-ric resonance condition and to a depletion of the number ofatoms in the deepest part of the atom trap. We emphasize thatT= 5 msI0 = 4.5(1) kcountsr0= 280(20) µmT= 20 msI0 = 3.9(1) kcountsr0= 310(20) µmT= 60 msI0 = 2.7(1) kcountsr0= 370(20) µmT= 100 msI0 = 1.9(1) kcountsr0= 390(20) µmT= 150 msI0 = 1.4(1) kcountsr0= 410(20) µmf = 2.5 kHzh = 0.15FIG. 2. Color online Typical images of the expanded cloud ofultracold 87Rb atoms, after a period of parametric excitation. In allcases the cloud is permitted to expand for 3 ms prior to imaging. Iois the peak intensity in the central region of the image. ro is theGaussian radius of the image in the radial vertical direction, whichprovides a spatial scale for the images.BALIK, WIN, AND HAVEY PHYSICAL REVIEW A 80, 023404 2009023404-2similar results are obtained for a range of h and T but that theshift of the resonance to lower frequencies becomes morepronounced as h is increased. Within the range of our data,the resonance positions are insensitive to T. We illustrate thepoint further in Fig. 3b, which shows the main axial reso-nance for the same two measurement approaches used in Fig.3a, but for a larger modulation depth of 20%. In this data,there is no clear resonance for the trap loss signal, but adistinct one appearing at about 105 Hz for the depletion sig-nal. The spectral location of this resonance is within the ex-perimental uncertainty, where it is expected based on thespectral location of the fundamental radial mode at 2.5 kHz,and assuming a trap formed at the focus of a Gaussiantrapping beam.We now turn our attention to the role of the modulationtime T, which represents the amount of time that the trapdepth is modulated before the atoms are released. We pointout that this time scale is short compared to the 5 s hold timeof the atom sample. The variation with T of the intensity atthe peak of the image, for several modulation frequencies, isshown in Fig. 4a. There we see that the peak image inten-sity appears to decrease exponentially for a range of f . Thisdecrease is evident from the smallest T=5 ms, and continuesfor a factor of 60 in T. In Fig. 4b the variation in the totaltrap loss with T for the same modulation frequencies isshown. In contrast to the peak intensity, the total intensitydecreases linearly with T, and further shows a threshold be-low which minimal variation is measured. We qualitativelyinterpret these results as follows: the atoms in the trap areinitially in thermal equilibrium at a temperature of approxi-mately 65 K. While the modulation is applied the averageenergy of the atoms is expected to increase. Further, the 87Rbelastic collision rate is quite large at the experimental condi-tions, so the atomic cloud is expected to stay in approximatethermal equilibrium. This temperature should increase as Tincreases. Even if no atoms were thus removed from the trap,the peak intensity would still decrease, for both the spatialvolume occupied by the atom cloud, and the rate of freeexpansion are increased as temperature is increased. How-ever, as the trap depth is finite, some of the atoms havesufficient energy to leave the trap. This leads to a decrease inthe integrated image intensity and an additional decrease inthe peak image intensity. These arguments also explain whythere is a threshold in the appearance of trap loss; the aver-age amount of energy deposited per atom in the system byparametric excitation must be sufficiently large that a signifi-cant number of them have a total energy larger the trap depthbefore loss is observed. Measurements of the Gaussian widthof the images as in Fig. 2 with increasing T provide a directmeasure of about 800 K /s for the heating rate at constanth and f , confirming this interpretation. We also point out thatthere is an important relationship between the results of Figs.4a and 4b: when the decrease in the total number of at-oms in the trap is accounted for by the data in Fig. 4b, theFIG. 3. Color online a Fundamental radial parametric reso-nances for h=0.15 and T=200 ms. b Fundamental axial paramet-ric resonances for h=0.20 and T=200 ms.FIG. 4. Color online a Peak survival probability as a functionof T for several different parametric resonance excitation frequen-cies. Note the exponential decay with time for longer modulationtimes. b Relative survival probability as a function of T for severaldifferent parametric resonance excitation frequencies. Note the lin-ear decay with time for larger T. h=0.15.IMAGING-BASED PARAMETRIC RESONANCE IN AN… PHYSICAL REVIEW A 80, 023404 2009023404-3remaining decrease seen in Fig. 4a scales quantitatively asthe mean cloud temperature Tc3/2, as is expected for the tem-perature dependent decrease in the peak intensity of a Gauss-ian atom cloud.Finally, we suggest optimum conditions for observingparametric resonance, at least for the range of conditionsexplored here. First, measurement of the Gaussian radius ofthe expanded sample is much more sensitive than measuringtrap loss, as indicated earlier. In fact, for h0.1, we do notobserve convincing trap loss signals. Second, a useful em-pirical scaling is that the cloud temperature which deter-mines the squared Gaussian width, is given by Tc=Toc+CfTh, where Cf is a frequency dependent parameter5 mK /s at the peak of the fundamental resonance. Wepoint out that increasing T or h does not indefinitely increasethe mean temperature and size of the atom cloud. As pre-dicted in 12,13, above a certain critical amount of energydeposition 35% of the trap depth, the temperature of theatom cloud remains constant, additional energy going intoremoving atoms from the trap. As shown elsewhere 22, thisinterpretation is supported by measurements similar to thosereported here.We acknowledge the financial support of the National Sci-ence Foundation Grant No. NSF-PHY-0654226.1 A. B. Pippard, The Physics of Vibration Cambridge UniversityPress, Cambridge, England, 1978.2 H. Steven Strogatz, Nonlinear Dynamics and Chaos AddisonWesley, Reading, MA, 1994.3 A. Kaplan, M. Andersen, N. Friedman, and N. 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At.,Mol., Opt., Phys. 42, 95 2000.15 C. W. Gardiner, J. Ye, H. C. Nagerl, and H. J. Kimble, Phys.Rev. A 61, 045801 2000.16 M. D. Havey, Contemp. Phys. 50, 587 2009.17 M. D. Havey and D. V. Kupriyanov, Phys. Scr. 72, C302005.18 D. V. Kupriyanov, I. M. Sokolov, C. I. Sukenik, and M. D.Havey, Laser Phys. Lett. 3, 223 2006.19 E. Akkermans, A. Gero, and R. Kaiser, Phys. Rev. Lett. 101,103602 2008.20 R. Jáuregui, N. Poli, G. Roati, and G. Modugno, Phys. Rev. A64, 033403 2001.21 For a Gaussian atom distribution of size ro and peak density nonr=noe−r2/2ro2. The total number of atoms is N= 23/2noro3and the peak optical depth is b0=2n00r0. 0 is the weak-field resonance light scattering cross section.22 S. Balik, A. Win, and M. D. Havey unpublished.BALIK, WIN, AND HAVEY PHYSICAL REVIEW A 80, 023404 2009023404-4

Imaging-Based Parametric Resonance in an Optical Dipole-Atom Trap

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Imaging-based Parametric Resonance in an Optical Dipole Atom Trap

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