We demonstrate the possibility of energy-selective removal of cold atoms from
a tight optical trap by means of parametric excitation of the trap vibrational
modes. Taking advantage of the anharmonicity of the trap potential, we
selectively remove the most energetic trapped atoms or excite those at the
bottom of the trap by tuning the parametric modulation frequency. This process,
which had been previously identified as a possible source of heating, also
appears to be a robust way for forcing evaporative cooling in anharmonic traps.Comment: 5 pages, 5 figure

C.S. Adams

C.W. Gardiner

G. Modugno

G. Roati

H. Engler

J. Ye

K.L. Corwin

K.M. O’Hara

M.D. Barrett

M.E. Gehm

N. Poli

R. J. Brecha

R. Jáuregui

S. Friebel

T.A. Savard

V. Vuletic

W. Ketterle

English

arXiv

Crossref

Cooling atoms in an optical trap by selective parametric excitation

Poli, Nicola

Modugno, Giovanni

Florence Research

RAPID COMMUNICATIONSPHYSICAL REVIEW A, VOLUME 65, 021401~R!Cooling atoms in an optical trap by selective parametric excitationN. Poli, R. J. Brecha,* G. Roati,† and G. ModugnoINFM, LENS, and Dipartimento di Fisica, Universita` di Firenze, Largo E. Fermi 2, 50125 Firenze, Italy~Received 3 July 2001; published 3 January 2002!We demonstrate the possibility of energy-selective removal of cold atoms from a tight optical trap by meansof parametric excitation of the trap vibrational modes. Taking advantage of the anharmonicity of the trappotential, we either selectively remove the most energetic trapped atoms or excite those at the bottom of thetrap by tuning the parametric modulation frequency. This process, which had been previously identified as apossible source of heating, also appears to be a robust way for forcing evaporative cooling in anharmonic traps.DOI: 10.1103/PhysRevA.65.021401 PACS number~s!: 32.80.Pj, 32.80.LgParametric excitation of atomic motion due to laser inten-sity fluctuations has been recently indicated as one of themajor sources of heating in far-off resonance optical traps~FORTs! @1,2#, and as a limitation in achieving very longtrapping times @3,4#. In this paper, we show that parametricexcitation in a FORT can also lead to a cooling of the trappedsample, since the anharmonicity of the potential allows aselective removal of the most energetic atoms alone.The excitation process has typically been modeled assum-ing a harmonic FORT potential @1,2#. For a modulation of thepotential depth at twice the trap vibrational frequencies, anexponential heating of the trapped sample accompanied bytrap losses has been predicted. The main features of the para-metric heating process predicted by the harmonic theoryhave been experimentally observed @5–7#. Going beyond thebasic trap model, it has been possible to further refine theo-retical calculations through modeling of the anharmonic trap-ping potential used in actual experiments @8#. One of theconsequences of the anharmonicity is a variation of the trapfrequencies with vibrational energy, which implies that amodulation at a single frequency will be resonant with a fewtrap levels only.We present here a detailed study of the temperature of asample of cold atoms in a one-dimensional ~1D! latticeFORT, in which the anharmonicity of the potential is ex-ploited to perform energy-selective excitation and removal ofthe atoms, resulting in either heating or cooling of the re-maining trapped sample. In particular, the cooling processappears to be a robust way to lower the atomic temperaturevia forced evaporation. One of the interesting aspects of thisprocess is that the trap depth need not be reduced to force theevaporation, in contrast to the technique investigated for thefirst time in Ref. @9#, and recently used to achieve Bose-Einstein condensation @10#; the result is a tighter confine-ment of the cooled sample.The trap is created with a beam of single-mode radiationgenerated by a Ti:sapphire laser at lT5787 nm, red detunedby approximately 18 nm from the D lines of potassium. Thelaser beam, after passing through an acousto-optic modulator*Permanent address: Physics Department, University of Dayton,Dayton, OH.†Present address: Dipartimento di Fisica, Universita` di Trento,I-38050 Povo ~Tn!, Italy.1050-2947/2002/65~2!/021401~4!/$20.00 65 0214~AOM!, is focused to w590 mm within a two-lens tele-scope, and then retroreflected to form a vertical standing-wave. The focusing is chosen to yield a large Rayleigh lengthzR53 cm, which results in an almost uniform lattice depthalong the vertical direction. Therefore, the optical potentialclose to the beam waist has the formU~r ,z !52U0 cos2~kz !expS 2 2r2w2D , ~1!where k52p/lT . About 107 atoms are prepared in amagneto-optical trap ~MOT! at a temperature of 60 mK, andtypically 5% of them are transferred to the FORT thanks to acompression procedure described in @11#. After loading theatoms in the FORT, the MOT beams are extinguished bymeans of AOMs and mechanical shutters, and the atoms areleft to evolve in the presence of the trap light only. Thedensity and the radial temperature of the trapped atoms aredetermined using absorption imaging of the atomic cloudafter 1 ms of expansion following release from the FORT.The typical vertical size of the cloud of 700 mm indicatesthat approximately 1800 lattice sites are occupied by an av-erage of 300 atoms each. The axial temperature of the atomsis measured with a time-of-flight ~TOF! technique, by ab-sorption of a sheet of light passing about 1.5 cm below theFORT. The temperature of the trapped atoms, after a 100-msequilibration phase following loading, stays constant duringstorage in the FORT, because the heating due to photon scat-tering is compensated by a slow-evaporative cooling @11#,and the radial and axial temperatures are found to be equalwithin the uncertainty. The lifetime of the FORT is limited toabout 1.2 s mainly by collisions with background gas.The harmonic approximations for the radial and axial fre-quencies of the single lattice site are measured with a para-metric excitation technique, as described below. Typical val-ues for these quantities are vr52p31.3 kHz, va52p3700 kHz, which for the typical temperature T580 mKlead to a density of 1012 cm23. From the measured value ofva , it is possible to estimate the trap depth as U05MlT2 va2/8p2, which corresponds to U05700 mK.We apply the parametric excitation via the control AOMas a small sinusoidal modulation on the power of the trap-ping laser, typically 200 ms after loading the FORT. To mea-sure the spectrum of excitation, we have repeated at different©2002 The American Physical Society01-1RAPID COMMUNICATIONSN. POLI, R. J. BRECHA, G. ROATI, AND G. MODUGNO PHYSICAL REVIEW A 65 021401~R!modulation frequencies a procedure composed of an excita-tion for a period Tex at fixed modulation amplitude e andfrequency V , followed by detection of the number of atomsand radial temperature. For both radial and axial excitation,as shown in Figs. 1 and 2, respectively, we see large reso-nances in the loss of atoms, accompanied by dispersive reso-nances in the temperature of the atoms left in the trap. Theresonances in the trap losses have been observed previouslyin optical lattices @5,8,11#, and their presence is expectedeven if we approximate the FORT as a harmonic potential:when V is close to 2 va /n or 2 vr /n , with n51,2, the at-oms are excited upwards along the ladder of energy levels@1#, and eventually lost from the trap. Note that for the axialexcitation an additional resonance appears close to 4 va , asexpected for a sinusoidal ~i.e., nonharmonic! potential withtight confinement @8#.On the other hand, the observed shape of the resonancesin the temperature of the remaining trapped atoms are notexpected for a harmonic trap @1,2#. Their origin must be re-FIG. 1. Spectrum of the resonances in the trap losses ~opencircles! and in the radial temperature ~solid circles! following aTex550 ms, e530% excitation of the radial vibration. The shadedarea indicates the initial temperature range, and represents the un-certainty in temperature.FIG. 2. Spectrum of the resonances in the trap losses ~opencircles! and in the radial temperature ~solid circles! following aTex5200 ms, e516% excitation of the axial vibration.02140lated to the anharmonicity of the potential, which impliesthat the vibrational frequency changes with atomic energy. InFig. 3, we show the excitation frequency 2 va calculated fora perfectly 1D sinusoidal lattice: while the atoms at the bot-tom of the trap are expected to oscillate at the harmonicvibrational frequency, the most energetic ones have a muchlower frequency, because of the decreased curvature of thepotential. A similar behavior is also expected for the radialoscillation frequency because of the Gaussian shape of theradial potential. Therefore, when the modulation frequency istuned to the red of 2 vr , 2 va or their harmonics and sub-harmonics, it is mainly the most energetic atoms that areparametrically excited and ejected from the trap, resulting ina reduction of the mean energy of the remaining atoms. Incontrast, modulation frequencies resonant with the nominalharmonic trap frequencies excite primarily atoms at the bot-tom of the trap, resulting in a net increase of the mean en-ergy. We, therefore, identify the quantities 2 vr , 2 va withthe position of the main resonances in the temperature spec-tra of Figs. 1 and 2 leading to a heating of the trappedsample.These observations are evidence of theoretical predictionsthat the peak in the resonance of atomic losses in the trapoccurs at a slightly lower frequency than the actual harmonicfrequencies @8#. This kind of information is very useful, sincethe frequencies measured from the trap-loss resonances areused to estimate quantities such as the number density or thephase-space density of atoms in an optical lattice @5,11#.Note that all the temperature measurements reported inthis paper have been performed on the radial degree of free-dom only, since the TOF detection of the axial temperaturewas strongly perturbed by the atoms excited out of the trapas long as 100 ms before release. We have extracted theradial temperature, assuming thermal equilibrium from thewidth of the atomic distribution, which was always well fit-ted by a Gaussian.In the following, we restrict our attention to the excitationof the axial vibrational mode, which is the most energeticand, therefore, requires a smaller perturbation of the trappingpotential @12#, making modeling easier. We also suspect thatFIG. 3. Calculated spectrum of the axial excitation frequencyfor a 1D optical lattice with U05700 mK.1-2RAPID COMMUNICATIONSCOOLING ATOMS IN AN OPTICAL TRAP BY . . . PHYSICAL REVIEW A 65 021401~R!the large broadening of the radial resonances in Fig. 1 mightcome from a slight ellipticity in the cross section of the laserbeam that makes vr vary in the radial plane, masking inter-esting characteristics of the resonances’ shape.To analyze in more detail the axial resonances, we need torefine the simple 1D model for the dispersion of the axialvibrational frequencies presented in Fig. 3. Indeed, whenconsidering the actual potential of Eq. 1 one would expectthat the axial oscillation frequency va depends also on thetotal energy, and not only on the axial energy. Since theharmonic axial and radial oscillation periods differ by a fac-tor of about 500, it is possible to define a local axial fre-quency that varies during the radial motion of the atoms @13#va~r !52pA2U0 exp~22r2/w2!/MlT2 . ~2!As a result, the discrete excitation frequencies shown in Fig.3 for the 1D case become bands in 3D. The width of theselevels is expected to be negligible for atoms at the bottom ofthe trap, which experience an harmonic potential, and to in-crease for increasing total energy, because of the longer timespent by the atoms in the anharmonic part of the potential.Thus, we expect that there is a broad range of frequencies onthe red of the resonances where the cooling is effective, com-pared with a narrow range of frequencies in resonance whereheating is present. These features seem to be qualitativelyconfirmed by the temperature spectrum in Fig. 2.In addition, the anharmonicity of the potential might beresponsible for a transfer of energy from the radial to theaxial vibrational mode, after a parametric excitation phasehas perturbed the equilibrium distribution. Indeed, after wehave removed some of the axially energetic atoms with 2 msof modulation at V51.8 va , the radial degrees of freedomappear to equilibrate on the time scale of 5 ms, as reported inFig. 4, which is almost an order of magnitude shorter thanwhat expected for a thermalization mediated by collisions@11#. Moreover, we have observed that the equilibration rateFIG. 4. Equilibration of the radial degree of freedom after anexcitation at 1.8 va ~triangles!, with Tex52 ms and e512%, andat 2 va ~circles!, with Tex510 ms and e512%. The solid curvesrepresent the best fit with an exponential equilibration, plus a con-stant heating rate for the lower set of data.02140does not substantially vary when reducing the atomic densityby a factor of 2, indicating that cross-dimensional anhar-monic mixing could actually favor a fast exchange of energybetween the axial and radial modes of individual atoms inthe trap @14#. On the contrary, as also reported in Fig. 4,when the atoms at the trap bottom are excited by tuning V to2 va , the two degrees of freedom equilibrate on a longertime scale of 20–50 ms, which is more likely to be deter-mined by the rate of interatomic collisions alone, and thatindeed shows an almost linear dependence on the atomicdensity, as already observed in Refs. @11,15#. This result isexpected, since the anharmonicity is not relevant for the at-oms at the trap bottom. In conclusion, our heuristic modelseems to predict a relatively fast removal of energy from allthe three vibrational modes of the atoms, once a red-detunedaxial modulation is applied continuosly.We have investigated the time evolution of losses andtemperature for modulation frequencies red detuned from2 va , to get an indication of the efficiency of the selectiveparametric excitation in cooling the atoms in the FORT. Theevolution reported in Fig. 5 was obtained with e512%, V51.6 va , and Tex variable between 5 and 300 ms; the atomnumber and temperature were detected after a 20-ms ther-malization period. The general features of the process dis-cussed above are thus confirmed: at the beginning of theexcitation phase the energetic atoms are very efficiently re-moved from the trap, and very likely the selection is per-formed on the total ~3D! energy because of anharmonic mix-ing. We believe in any case that elastic collisions areresponsible for a continuous replenishing of the high-lyingstates of the trap on a time scale much longer than 5 ms,giving rise to the observed exponential decrease of both atomnumber and temperature. Thus, we can consider this processto be one of evaporative cooling, forced by the parametricexcitation. As shown in Fig. 5, after 150 ms the atom numberand temperature reach a steady-state value, which is con-served even after a further 150-ms period of excitation. Thefeatures just described are very interesting from the perspec-tive of using this technique to cool the atoms in the trap,since an excitation at the proper frequency provides a fastFIG. 5. Decrease of the radial temperature ~triangles! and of thenumber of atoms ~circles!, during a parametric axial excitation withe512% and V51.6 va . The increase in relative phase-space den-sity is shown in the inset1-3RAPID COMMUNICATIONSN. POLI, R. J. BRECHA, G. ROATI, AND G. MODUGNO PHYSICAL REVIEW A 65 021401~R!removal of the energetic atoms without heating the coldestones. After switching off the parametric modulation at t5300 ms, we have measured a temperature increase at arate of 34(12) mKs21, compatible with that expected forphoton scattering.From the two sets of data in Fig. 5, we extract the evolu-tion of the relative phase-space density r reported the inset,assuming thermal equilibrium between radial and axial mo-tions. Our experimental value for the parameter that de-scribes the overall efficiency of an evaporative cooling pro-cess @16# isg tot5N fr fNir i50.7~1 !. ~3!This efficiency, which could be limited also by the compara-tively large heating rate due to photon scattering, can becompared to that of a variety of evaporative techniques inoptical and magnetic traps @10,16#. To our knowledge theprocess we have investigated here represents the only way offorcing evaporative cooling in a standard optical trap @17#without a reduction in the trap depth.Our FORT is actually not optimized for evaporation; in-deed we are obliged by the short lifetime of the trap to forcethe removal of atoms on a time scale not much larger thanthe elastic collisional time. Further investigations in a bettervacuum environment, or at a larger atomic density, can pro-vide better tests of the efficiency of this cooling technique.02140Indeed, with a longer time for evaporation one could try touse a modulation frequency on the red of all the axial reso-nances (V, va), to excite only the very energetic atoms andthus increase the cooling efficiency.In conclusion, we have reported the observation of anenergy-selective removal of atoms from a FORT, as a conse-quence of parametric excitation of anharmonic vibrationalmodes. In addition, we have shown that the energy selectiv-ity can be exploited to cool the trapped sample. Since thiseffect arises due to the anharmonicity of the trapping poten-tial, it should be observable in most actual traps for coldparticles. The removal mechanism should work indepen-dently of the internal structure of the trapped particles ~at-oms, molecules or ions!, as it is based on the excitation oftheir external degrees of freedom. The observations reportedhere should provide the impetus for further investigationsand theoretical modeling necessary for characterizing the ul-timate limits and efficiency of this cooling technique; how-ever, it already appears interesting for a fast cooling of anatomic sample in a tight FORT.We thank M. Inguscio for useful suggestions and for acareful reading of the manuscript. We also acknowledge use-ful discussions with R. Ja´uregui, and we thank Scuola Nor-male Superiore, Pisa for the loan of the trapping laser. Thiswork was supported by the ECC under Contract No.HPRICT1999-00111, and by MURST under the PRIN 2000Program. R.B. was partially supported by ASI.@1# T.A. Savard, K.M. O’Hara, and J.E. Thomas, Phys. Rev. A 56,R1095 ~1997!; M.E. Gehm et al., ibid. 58, 3914 ~1998!.@2# C.W. Gardiner et al., Phys. Rev. A 61, 045801 ~2000!.@3# K.M. O’Hara et al., Phys. Rev. Lett. 82, 4204 ~1999!.@4# H. Engler et al., Phys. Rev. A 62, 031402 ~2000!.@5# S. Friebel et al., Phys. Rev. A 57, R20 ~1998!.@6# V. Vuletic et al., Phys. Rev. Lett. 81, 5768 ~1998!.@7# J. Ye, D.W. Vernooy, and H.J. Kimble, Phys. Rev. Lett. 83,4987 ~1999!.@8# R. Ja´uregui et al., Phys. Rev. A 64, 033403 ~2001!.@9# C.S. Adams et al., Phys. Rev. Lett. 74, 3577 ~1995!.@10# M.D. Barrett, J.A. Sauer, and M.S. Chapman, Phys. Rev. Lett.87, 010404 ~2001!.@11# G. Roati et al., Phys. Rev. A 63, 052709 ~2001!.@12# A general result of the theory presented in Ref. @1# is that theexcitation rate is proportional to e2v2.@13# T. Elsa¨sser, Diploma thesis, University of Heidelberg, 2000.@14# We have labeled the ordinate axis of Fig. 4 as ‘‘temperature,’’since we are not able to distinguish an equilibration mediatedby mixing from a thermalization mediated by collisions.@15# V. Vuletic et al., Phys. Rev. Lett. 82, 1406 ~1999!.@16# W. Ketterle and N.J. van Druten, Adv. At., Mol., Opt. Phys. 37,181 ~1996!.@17# An exception is represented by the optical trap investigated inK.L. Corwin et al., Phys. Rev. Lett. 83, 1311 ~1999!.1-4

We demonstrate the possibility of energy-selective removal of cold atoms from a tight optical trap by means of parametric excitation of the trap vibrational modes. Taking advantage of the anharmonicity of the trap potential, we either selectively remove the most energetic trapped atoms or excite those at the bottom of the trap by tuning the parametric modulation frequency. This process, which had been previously identified as a possible source of heating, also appears to be a robust way for forcing evaporative cooling in anharmonic traps

Brecha, Robert J.

Roati, Giacomo

University of Dayton

University of DaytoneCommonsPhysics Faculty Publications Department of Physics1-2002Cooling Atoms in an Optical Trap by SelectiveParametric ExcitationNicola PoliUniversità di FirenzeRobert J. BrechaUniversity of Dayton, rbrecha1@udayton.eduGiacomo RoatiUniversità di FirenzeGiovanni ModugnoUniversità di FirenzeFollow this and additional works at: https://ecommons.udayton.edu/phy_fac_pubPart of the Engineering Physics Commons, Optics Commons, Other Physics Commons, and theQuantum Physics CommonsThis Article is brought to you for free and open access by the Department of Physics at eCommons. It has been accepted for inclusion in Physics FacultyPublications by an authorized administrator of eCommons. For more information, please contact frice1@udayton.edu, mschlangen1@udayton.edu.eCommons CitationPoli, Nicola; Brecha, Robert J.; Roati, Giacomo; and Modugno, Giovanni, "Cooling Atoms in an Optical Trap by Selective ParametricExcitation" (2002). Physics Faculty Publications. 6.https://ecommons.udayton.edu/phy_fac_pub/6Cooling atoms in an optical trap by selective parametric excitationN. Poli, R. J. Brecha,* G. Roati,† and G. ModugnoINFM, LENS, and Dipartimento di Fisica, Universita` di Firenze, Largo E. Fermi 2, 50125 Firenze, Italy~Received 3 July 2001; published 3 January 2002!We demonstrate the possibility of energy-selective removal of cold atoms from a tight optical trap by meansof parametric excitation of the trap vibrational modes. Taking advantage of the anharmonicity of the trappotential, we either selectively remove the most energetic trapped atoms or excite those at the bottom of thetrap by tuning the parametric modulation frequency. This process, which had been previously identified as apossible source of heating, also appears to be a robust way for forcing evaporative cooling in anharmonic traps.DOI: 10.1103/PhysRevA.65.021401 PACS number~s!: 32.80.Pj, 32.80.LgParametric excitation of atomic motion due to laser inten-sity fluctuations has been recently indicated as one of themajor sources of heating in far-off resonance optical traps~FORTs! @1,2#, and as a limitation in achieving very longtrapping times @3,4#. In this paper, we show that parametricexcitation in a FORT can also lead to a cooling of the trappedsample, since the anharmonicity of the potential allows aselective removal of the most energetic atoms alone.The excitation process has typically been modeled assum-ing a harmonic FORT potential @1,2#. For a modulation of thepotential depth at twice the trap vibrational frequencies, anexponential heating of the trapped sample accompanied bytrap losses has been predicted. The main features of the para-metric heating process predicted by the harmonic theoryhave been experimentally observed @5–7#. Going beyond thebasic trap model, it has been possible to further refine theo-retical calculations through modeling of the anharmonic trap-ping potential used in actual experiments @8#. One of theconsequences of the anharmonicity is a variation of the trapfrequencies with vibrational energy, which implies that amodulation at a single frequency will be resonant with a fewtrap levels only.We present here a detailed study of the temperature of asample of cold atoms in a one-dimensional ~1D! latticeFORT, in which the anharmonicity of the potential is ex-ploited to perform energy-selective excitation and removal ofthe atoms, resulting in either heating or cooling of the re-maining trapped sample. In particular, the cooling processappears to be a robust way to lower the atomic temperaturevia forced evaporation. One of the interesting aspects of thisprocess is that the trap depth need not be reduced to force theevaporation, in contrast to the technique investigated for thefirst time in Ref. @9#, and recently used to achieve Bose-Einstein condensation @10#; the result is a tighter confine-ment of the cooled sample.The trap is created with a beam of single-mode radiationgenerated by a Ti:sapphire laser at lT5787 nm, red detunedby approximately 18 nm from the D lines of potassium. Thelaser beam, after passing through an acousto-optic modulator~AOM!, is focused to w590 mm within a two-lens tele-scope, and then retroreflected to form a vertical standing-wave. The focusing is chosen to yield a large Rayleigh lengthzR53 cm, which results in an almost uniform lattice depthalong the vertical direction. Therefore, the optical potentialclose to the beam waist has the formU~r ,z !52U0 cos2~kz !expS 2 2r2w2D , ~1!where k52p/lT . About 107 atoms are prepared in amagneto-optical trap ~MOT! at a temperature of 60 mK, andtypically 5% of them are transferred to the FORT thanks to acompression procedure described in @11#. After loading theatoms in the FORT, the MOT beams are extinguished bymeans of AOMs and mechanical shutters, and the atoms areleft to evolve in the presence of the trap light only. Thedensity and the radial temperature of the trapped atoms aredetermined using absorption imaging of the atomic cloudafter 1 ms of expansion following release from the FORT.The typical vertical size of the cloud of 700 mm indicatesthat approximately 1800 lattice sites are occupied by an av-erage of 300 atoms each. The axial temperature of the atomsis measured with a time-of-flight ~TOF! technique, by ab-sorption of a sheet of light passing about 1.5 cm below theFORT. The temperature of the trapped atoms, after a 100-msequilibration phase following loading, stays constant duringstorage in the FORT, because the heating due to photon scat-tering is compensated by a slow-evaporative cooling @11#,and the radial and axial temperatures are found to be equalwithin the uncertainty. The lifetime of the FORT is limited toabout 1.2 s mainly by collisions with background gas.The harmonic approximations for the radial and axial fre-quencies of the single lattice site are measured with a para-metric excitation technique, as described below. Typical val-ues for these quantities are vr52p31.3 kHz, va52p3700 kHz, which for the typical temperature T580 mKlead to a density of 1012 cm23. From the measured value ofva , it is possible to estimate the trap depth as U05MlT2 va2/8p2, which corresponds to U05700 mK.We apply the parametric excitation via the control AOMas a small sinusoidal modulation on the power of the trap-ping laser, typically 200 ms after loading the FORT. To mea-sure the spectrum of excitation, we have repeated at different*Permanent address: Physics Department, University of Dayton,Dayton, OH.†Present address: Dipartimento di Fisica, Universita` di Trento,I-38050 Povo ~Tn!, Italy.RAPID COMMUNICATIONSPHYSICAL REVIEW A, VOLUME 65, 021401~R!1050-2947/2002/65~2!/021401~4!/$20.00 ©2002 The American Physical Society65 021401-1modulation frequencies a procedure composed of an excita-tion for a period Tex at fixed modulation amplitude e andfrequency V , followed by detection of the number of atomsand radial temperature. For both radial and axial excitation,as shown in Figs. 1 and 2, respectively, we see large reso-nances in the loss of atoms, accompanied by dispersive reso-nances in the temperature of the atoms left in the trap. Theresonances in the trap losses have been observed previouslyin optical lattices @5,8,11#, and their presence is expectedeven if we approximate the FORT as a harmonic potential:when V is close to 2 va /n or 2 vr /n , with n51,2, the at-oms are excited upwards along the ladder of energy levels@1#, and eventually lost from the trap. Note that for the axialexcitation an additional resonance appears close to 4 va , asexpected for a sinusoidal ~i.e., nonharmonic! potential withtight confinement @8#.On the other hand, the observed shape of the resonancesin the temperature of the remaining trapped atoms are notexpected for a harmonic trap @1,2#. Their origin must be re-lated to the anharmonicity of the potential, which impliesthat the vibrational frequency changes with atomic energy. InFig. 3, we show the excitation frequency 2 va calculated fora perfectly 1D sinusoidal lattice: while the atoms at the bot-tom of the trap are expected to oscillate at the harmonicvibrational frequency, the most energetic ones have a muchlower frequency, because of the decreased curvature of thepotential. A similar behavior is also expected for the radialoscillation frequency because of the Gaussian shape of theradial potential. Therefore, when the modulation frequency istuned to the red of 2 vr , 2 va or their harmonics and sub-harmonics, it is mainly the most energetic atoms that areparametrically excited and ejected from the trap, resulting ina reduction of the mean energy of the remaining atoms. Incontrast, modulation frequencies resonant with the nominalharmonic trap frequencies excite primarily atoms at the bot-tom of the trap, resulting in a net increase of the mean en-ergy. We, therefore, identify the quantities 2 vr , 2 va withthe position of the main resonances in the temperature spec-tra of Figs. 1 and 2 leading to a heating of the trappedsample.These observations are evidence of theoretical predictionsthat the peak in the resonance of atomic losses in the trapoccurs at a slightly lower frequency than the actual harmonicfrequencies @8#. This kind of information is very useful, sincethe frequencies measured from the trap-loss resonances areused to estimate quantities such as the number density or thephase-space density of atoms in an optical lattice @5,11#.Note that all the temperature measurements reported inthis paper have been performed on the radial degree of free-dom only, since the TOF detection of the axial temperaturewas strongly perturbed by the atoms excited out of the trapas long as 100 ms before release. We have extracted theradial temperature, assuming thermal equilibrium from thewidth of the atomic distribution, which was always well fit-ted by a Gaussian.In the following, we restrict our attention to the excitationof the axial vibrational mode, which is the most energeticand, therefore, requires a smaller perturbation of the trappingpotential @12#, making modeling easier. We also suspect thatFIG. 1. Spectrum of the resonances in the trap losses ~opencircles! and in the radial temperature ~solid circles! following aTex550 ms, e530% excitation of the radial vibration. The shadedarea indicates the initial temperature range, and represents the un-certainty in temperature.FIG. 2. Spectrum of the resonances in the trap losses ~opencircles! and in the radial temperature ~solid circles! following aTex5200 ms, e516% excitation of the axial vibration.FIG. 3. Calculated spectrum of the axial excitation frequencyfor a 1D optical lattice with U05700 mK.RAPID COMMUNICATIONSN. POLI, R. J. BRECHA, G. ROATI, AND G. MODUGNO PHYSICAL REVIEW A 65 021401~R!021401-2the large broadening of the radial resonances in Fig. 1 mightcome from a slight ellipticity in the cross section of the laserbeam that makes vr vary in the radial plane, masking inter-esting characteristics of the resonances’ shape.To analyze in more detail the axial resonances, we need torefine the simple 1D model for the dispersion of the axialvibrational frequencies presented in Fig. 3. Indeed, whenconsidering the actual potential of Eq. 1 one would expectthat the axial oscillation frequency va depends also on thetotal energy, and not only on the axial energy. Since theharmonic axial and radial oscillation periods differ by a fac-tor of about 500, it is possible to define a local axial fre-quency that varies during the radial motion of the atoms @13#va~r !52pA2U0 exp~22r2/w2!/MlT2 . ~2!As a result, the discrete excitation frequencies shown in Fig.3 for the 1D case become bands in 3D. The width of theselevels is expected to be negligible for atoms at the bottom ofthe trap, which experience an harmonic potential, and to in-crease for increasing total energy, because of the longer timespent by the atoms in the anharmonic part of the potential.Thus, we expect that there is a broad range of frequencies onthe red of the resonances where the cooling is effective, com-pared with a narrow range of frequencies in resonance whereheating is present. These features seem to be qualitativelyconfirmed by the temperature spectrum in Fig. 2.In addition, the anharmonicity of the potential might beresponsible for a transfer of energy from the radial to theaxial vibrational mode, after a parametric excitation phasehas perturbed the equilibrium distribution. Indeed, after wehave removed some of the axially energetic atoms with 2 msof modulation at V51.8 va , the radial degrees of freedomappear to equilibrate on the time scale of 5 ms, as reported inFig. 4, which is almost an order of magnitude shorter thanwhat expected for a thermalization mediated by collisions@11#. Moreover, we have observed that the equilibration ratedoes not substantially vary when reducing the atomic densityby a factor of 2, indicating that cross-dimensional anhar-monic mixing could actually favor a fast exchange of energybetween the axial and radial modes of individual atoms inthe trap @14#. On the contrary, as also reported in Fig. 4,when the atoms at the trap bottom are excited by tuning V to2 va , the two degrees of freedom equilibrate on a longertime scale of 20–50 ms, which is more likely to be deter-mined by the rate of interatomic collisions alone, and thatindeed shows an almost linear dependence on the atomicdensity, as already observed in Refs. @11,15#. This result isexpected, since the anharmonicity is not relevant for the at-oms at the trap bottom. In conclusion, our heuristic modelseems to predict a relatively fast removal of energy from allthe three vibrational modes of the atoms, once a red-detunedaxial modulation is applied continuosly.We have investigated the time evolution of losses andtemperature for modulation frequencies red detuned from2 va , to get an indication of the efficiency of the selectiveparametric excitation in cooling the atoms in the FORT. Theevolution reported in Fig. 5 was obtained with e512%, V51.6 va , and Tex variable between 5 and 300 ms; the atomnumber and temperature were detected after a 20-ms ther-malization period. The general features of the process dis-cussed above are thus confirmed: at the beginning of theexcitation phase the energetic atoms are very efficiently re-moved from the trap, and very likely the selection is per-formed on the total ~3D! energy because of anharmonic mix-ing. We believe in any case that elastic collisions areresponsible for a continuous replenishing of the high-lyingstates of the trap on a time scale much longer than 5 ms,giving rise to the observed exponential decrease of both atomnumber and temperature. Thus, we can consider this processto be one of evaporative cooling, forced by the parametricexcitation. As shown in Fig. 5, after 150 ms the atom numberand temperature reach a steady-state value, which is con-served even after a further 150-ms period of excitation. Thefeatures just described are very interesting from the perspec-tive of using this technique to cool the atoms in the trap,since an excitation at the proper frequency provides a fastFIG. 4. Equilibration of the radial degree of freedom after anexcitation at 1.8 va ~triangles!, with Tex52 ms and e512%, andat 2 va ~circles!, with Tex510 ms and e512%. The solid curvesrepresent the best fit with an exponential equilibration, plus a con-stant heating rate for the lower set of data.FIG. 5. Decrease of the radial temperature ~triangles! and of thenumber of atoms ~circles!, during a parametric axial excitation withe512% and V51.6 va . The increase in relative phase-space den-sity is shown in the insetRAPID COMMUNICATIONSCOOLING ATOMS IN AN OPTICAL TRAP BY . . . PHYSICAL REVIEW A 65 021401~R!021401-3removal of the energetic atoms without heating the coldestones. After switching off the parametric modulation at t5300 ms, we have measured a temperature increase at arate of 34(12) mKs21, compatible with that expected forphoton scattering.From the two sets of data in Fig. 5, we extract the evolu-tion of the relative phase-space density r reported the inset,assuming thermal equilibrium between radial and axial mo-tions. Our experimental value for the parameter that de-scribes the overall efficiency of an evaporative cooling pro-cess @16# isg tot5N fr fNir i50.7~1 !. ~3!This efficiency, which could be limited also by the compara-tively large heating rate due to photon scattering, can becompared to that of a variety of evaporative techniques inoptical and magnetic traps @10,16#. To our knowledge theprocess we have investigated here represents the only way offorcing evaporative cooling in a standard optical trap @17#without a reduction in the trap depth.Our FORT is actually not optimized for evaporation; in-deed we are obliged by the short lifetime of the trap to forcethe removal of atoms on a time scale not much larger thanthe elastic collisional time. Further investigations in a bettervacuum environment, or at a larger atomic density, can pro-vide better tests of the efficiency of this cooling technique.Indeed, with a longer time for evaporation one could try touse a modulation frequency on the red of all the axial reso-nances (V, va), to excite only the very energetic atoms andthus increase the cooling efficiency.In conclusion, we have reported the observation of anenergy-selective removal of atoms from a FORT, as a conse-quence of parametric excitation of anharmonic vibrationalmodes. In addition, we have shown that the energy selectiv-ity can be exploited to cool the trapped sample. Since thiseffect arises due to the anharmonicity of the trapping poten-tial, it should be observable in most actual traps for coldparticles. The removal mechanism should work indepen-dently of the internal structure of the trapped particles ~at-oms, molecules or ions!, as it is based on the excitation oftheir external degrees of freedom. The observations reportedhere should provide the impetus for further investigationsand theoretical modeling necessary for characterizing the ul-timate limits and efficiency of this cooling technique; how-ever, it already appears interesting for a fast cooling of anatomic sample in a tight FORT.We thank M. Inguscio for useful suggestions and for acareful reading of the manuscript. We also acknowledge use-ful discussions with R. Ja´uregui, and we thank Scuola Nor-male Superiore, Pisa for the loan of the trapping laser. Thiswork was supported by the ECC under Contract No.HPRICT1999-00111, and by MURST under the PRIN 2000Program. R.B. was partially supported by ASI.@1# T.A. Savard, K.M. O’Hara, and J.E. Thomas, Phys. Rev. A 56,R1095 ~1997!; M.E. Gehm et al., ibid. 58, 3914 ~1998!.@2# C.W. Gardiner et al., Phys. Rev. A 61, 045801 ~2000!.@3# K.M. O’Hara et al., Phys. Rev. Lett. 82, 4204 ~1999!.@4# H. Engler et al., Phys. Rev. A 62, 031402 ~2000!.@5# S. Friebel et al., Phys. Rev. A 57, R20 ~1998!.@6# V. Vuletic et al., Phys. Rev. Lett. 81, 5768 ~1998!.@7# J. Ye, D.W. Vernooy, and H.J. Kimble, Phys. Rev. Lett. 83,4987 ~1999!.@8# R. Ja´uregui et al., Phys. Rev. A 64, 033403 ~2001!.@9# C.S. Adams et al., Phys. Rev. Lett. 74, 3577 ~1995!.@10# M.D. Barrett, J.A. Sauer, and M.S. Chapman, Phys. Rev. Lett.87, 010404 ~2001!.@11# G. Roati et al., Phys. Rev. A 63, 052709 ~2001!.@12# A general result of the theory presented in Ref. @1# is that theexcitation rate is proportional to e2v2.@13# T. Elsa¨sser, Diploma thesis, University of Heidelberg, 2000.@14# We have labeled the ordinate axis of Fig. 4 as ‘‘temperature,’’since we are not able to distinguish an equilibration mediatedby mixing from a thermalization mediated by collisions.@15# V. Vuletic et al., Phys. Rev. Lett. 82, 1406 ~1999!.@16# W. Ketterle and N.J. van Druten, Adv. At., Mol., Opt. Phys. 37,181 ~1996!.@17# An exception is represented by the optical trap investigated inK.L. Corwin et al., Phys. Rev. Lett. 83, 1311 ~1999!.RAPID COMMUNICATIONSN. POLI, R. J. BRECHA, G. ROATI, AND G. MODUGNO PHYSICAL REVIEW A 65 021401~R!021401-4

Cooling Atoms in an Optical Trap by Selective Parametric Excitation

Cooling atoms in an optical trap using selective parametric excitation

Cooling atoms in an optical trap by selective parametric excitation

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