PHYSICAL REVIEW A, VOLUME 64, 033403

Anharmonic parametric excitation in optical lattices

R. JaureguiInstituto de Fı´sica, Universidad Nacional Auto´noma de Me´xico, Apartado Postal 20-364, Me´xico 01000, Distrito Federal, Mexico

N. Poli, G. Roati, and G. ModugnoINFM–European Laboratory for Nonlinear Spectroscopy (LENS), Universita` di Firenze, Largo Enrico Fermi 2, 50125 Firenze, Italy

~Received 16 March 2001; published 1 August 2001!

We study both experimentally and theoretically the losses induced by parametric excitation in far-off-resonance optical lattices. The atoms confined in a one-dimensional sinusoidal lattice present an excitationspectrum and dynamics substantially different from those expected for a harmonic potential. We develop amodel based on the actual atomic Hamiltonian in the lattice and we introduce semiempirically a broadening ofthe width of lattice energy bands which can physically arise from inhomogeneities and fluctuations of thelattice, and also from atomic collisions. The position and strength of the parametric resonances and theevolution of the number of trapped atoms are satisfactorily described by our model.

DOI: 10.1103/PhysRevA.64.033403 PACS number~s!: 32.80.Pj, 32.80.Lg

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I. INTRODUCTION

The phenomenon of parametric excitation of the motof cold trapped atoms has recently been the subject of sevtheoretical and experimental investigations@1–3#. The exci-tation caused by resonant amplitude noise has been propas one of the major sources of heating in far-off-resonaoptical traps~FORT’s!, where the heating due to spontaneoscattering forces is strongly reduced@4#. In particular, theeffect of resonant excitation is expected to be particulaimportant in optical lattices, which usually provide a vestrong confinement to the atoms, resulting in a large vibtional frequency and in a correspondingly large transferenergy from the noise field to the atoms@1#.

Nevertheless, parametric excitation is not only a sourceheating, but it also represents a very useful tool to characize the spring constant of a FORT or in general of a trapcold particles, and to study the dynamics of the trappedIndeed, the trap frequencies can be measured by intentionexciting the trap vibrational modes with a small modulatiof the amplitude of the trapping potential, which resultsheating@5# or losses@2,6# for the trapped atoms when thmodulation frequency is tuned to twice the oscillation frquency. This procedure usually yields frequencies that safactorily agree with calculated values, and are indeedpected to be accurate for the atoms at the bottom oftrapping potential. From the measured trap frequenciesthen possible to estimate quantities such as the trap dand the number and phase space densities of trapped aWe note that this kind of measurement is particularly imptant in optical lattices, since the spatial resolution of standimaging techniques is usually not enough to estimateatomic density from a measurement of the volume of a sinlattice site.

Recently, one-dimensional~1D! lattices have proved to bthe proper environment to study collisional processes in laand dense samples of cold atoms, using a trapping poteindependent of the magnetic state of the atoms. In thistem, the parametric excitation of the energetic vibratiomode along the lattice provides an efficient way to inve

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Most theoretical studies of parametric excitation rely onclassical@8# or quantum@1# harmonic approximation of theconfining potential. Under certain circumstances thesepressions show quite good agreement with experimentasults @3#. However, general features of the optical latticould be lost in these approaches. For example, a sinusopotential exhibits an energy-band structure and a spreatransition energies, while harmonic oscillators have jusdiscrete equidistant spectrum. Thus, we might expect thatexcitation process may happen at several frequencies,with a non-negligible bandwidth. Such anharmonic effecan be important whenever the atoms are occupying a rtively large fraction of the lattice energy levels. The purpoof this paper is to give a simple description of parametexcitation in a sinusoidal 1D lattice. In Sec. II, we briefldiscuss general features of the stationary states on sulattice. Then, we summarize the harmonic description givin Ref. @1# and extend it to the anharmonic case. By a nmerical evaluation of transition rates, we make a tempodescription of parametric excitation which is compared wexperimental results. We discuss the relevance of broadeof the spectral lines in order to understand the excitatprocess in this kind of system. Some conclusions are giveSec. V.

II. STATIONARY STATES OF A SINUSOIDAL OPTICALLATTICE

The Hamiltonian for an atom in a red detuned FORT i

H5P2

2M1Veff~xW !, ~2.1!

with

Veff~xW !52 14 auE~xW !u2, ~2.2!

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R. JAUREGUI, N. POLI, G. ROATI, AND G. MODUGNO PHYSICAL REVIEW A64 033403

wherea is the effective atomic polarizations andE(x) is theradiation field amplitude. For the axial motion in a sinusoid1D lattice we can take

Hax5Pz

2

2M1V0 cos2~kz! ~2.3!

5Pz

2

2M1

V0

2@11cos~2kz!#.

~2.4!

The corresponding stationary Schro¨dinger equation

2\2

2M

d2F

dz21

V0

2~11cos~2kz!!F5EF ~2.5!

can be written in canonical Mathieu’s form

d2F

du21~a22q cos 2u!F50 ~2.6!

with

a5S E2V0

2 D S 2M

\2k2D 2q5V0

2 S 2M

\2k2D . ~2.7!

It is well known that there exists countably infinite setscharacteristic values$ar% and$br% which, respectively, yieldeven and odd periodic solutions of the Mathieu equatiThese values also separate regions of stability. In particufor q>0 the band structure of the sinusoidal lattice corsponds to energy eigenvalues betweenar andbr 11 @9#. Theunstable regions are betweenbr and ar . For q!1, there isan analytical expression for the bandwidth@9#:

br 112ar;24r 15A2/pq(1/2)r 1(3/4)e24Aq/r !. ~2.8!

The quantities defined above can be expressed in termsfrequencyv0 defined in the harmonic approximation of thpotential

1

2Mv0

25V0

2

~2k!2

2!, ~2.9!

thus obtaining

a5S E2V0

2 D S 4V0

\2v02D ; q5S V0

\v0D 2

. ~2.10!

Thus, the width of ther band can be estimated using E~2.8! whenever the condition (V0 /\v0)2@1 is satisfied. Inthe experiment we shall be working with a 1D optical lattihavingV0;10.5\v0. While the lowest bandr 50 has a neg-ligible width ;10218\v0, the bandwidths for highest lyinglevels r 510, 11, 12, and 13 would, respectively, be 0.0060.1036, 1.52, and 20.56 in units of\v0.

In order to determine the energy spectrum, a variatiocalculation can be performed. We considered a harmoniccillator basis set centered in a given site of the lattice, a

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with frequencyv0. The diagonalization of the Hamiltoniamatrix associated to Eq.~2.4! using 40 basis functions givethe eigenvaluesEn,V0 shown in Table I forV0510.5\v0.According to the results of last paragraph, the eigenvalueand 13 belong to the same band while the band widthlower levels is smaller than 0.11\v0.

III. PARAMETRIC EXCITATION

As already mentioned, parametric excitation of ttrapped atoms consists of applying a small modulation tointensity of the trapping light

H5P2

2M1Veff@11e~ t !#. ~3.1!

Within first order perturbation theory, this additional fieinduces transitions between the stationary statesn and mwith an averaged rate

Rm←n51

T U2 i

\ E0

T

dtT~m,n!e~ t !eivmntU2

5p

2\2uT~m,n!u2S~vmn!; vmn5

Em2En

\,

~3.2!

where

T~m,n!5^muVeffun&5Endnm21

2M^muP2un& ~3.3!

is the matrix element of the space part of the perturbatand

TABLE I. Energy spectrum in units of\v0 obtained from thediagonalization of the Hamiltonian Eq.~2.4! for V0510.5\v0 in aharmonic basis set with the lowest 40 functions. The third colushows the bandwidthss r , Eqs. ~3.14! and ~4.2!, used in the nu-merical simulations reported in Sec. IV.

r Er Er 112Er s r

0 0.494 0.976 0.0141 1.470 0.95 0.0152 2.420 0.923 0.0193 3.343 0.897 0.0254 4.240 0.867 0.0325 5.107 0.837 0.0426 5.944 0.802 0.0517 6.746 0.767 0.0628 7.513 0.727 0.0729 8.240 0.680 0.082

10 8.920 0.624 0.09211 9.544 0.55112 10.095 0.40213 10.497

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ANHARMONIC PARAMETRIC EXCITATION IN OPTICAL . . . PHYSICAL REVIEW A 64 033403

S~v!52

pE0

T

dt cosvt^e~ t !e~ t1t!& ~3.4!

is the one-sided power spectrum of the two-time correlatfunction associated to the excitation field amplitude.

If the confining potential is approximated by a harmonwell, the transition rates different from zero are

Rn←n5pv0

2

16S~0!~2n11!, ~3.5!

Rn62←n5pv0

2

16S~2v0!~n1161!~n61!. ~3.6!

The latter equation was used in@1# to obtain a simple expression for the heating rate

^E&5p

2v0

2S~2v0!^E& , ~3.7!

showing its exponential character. The dependence on 2v0 ischaracteristic of the parametric nature of the excitation pcess. The fact that\ is not present is consistent with thapplicability of Eq.~3.7! in the classical regime.

Classically, parametric harmonic oscillators exhibit resnances not just at 2v0 but also at 2v0 /n with n any naturalnumber@8#. In fact, the resonances corresponding ton52,i.e., at an excitation frequencyv0, have been observed ioptical lattices@2,6#. A quantum description of parametriharmonic excitation also predicts resonances at the samequencies vianth-order perturbation theory@10#. In particular,the presence of the resonance atv0 can be justified with thefollowing argument. According to the standard proceduthe second-order correction to the transition amplitudetween statesun& and um& is given by

Rm←n(2) 5^nuU (2)~ t0 ,t !um&5(

kS 2 i

\ D 2

T~n,k!T~k,m!,

Et0

t

dt8eivnkt8e~ t8!Et0

t

dt9eivkmt9e~ t9!, ~3.8!

with U (2)(t0 ,t) the second-order correction to the evolutioperatorU. Therefore, the transition may be described atwo-step procedureum&←uk&←un&. For harmonic parametricexcitation the matrix element of the space part of the perbation differs from zero just for transitionsun&←un& and un62&←un&. Consider a transition in Eq.~3.8! involving a‘‘first’’ step in which the state does not changeun&←un& anda ‘‘second’’ step for whichun62&←un&. Then resonancephenomena occur when the total energy of the two exctions, 2\V, coincides with that of the second step transitioi.e., for an excitation frequencyV5v0.

These ideas can be directly extended to anharmonictentials: the corresponding transition probability raR(n,m) would be determined by the transition matrT(n,m), by the transition frequenciesvnm , and by the timedependence of the excitatione(t). In general, anharmonic

03340

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transition matrix elementsT(n,m) will be different fromzero for a wider set of pairs (n,m). Besides, the transitionenergies will not be unique so that the excitation procesnot determined by the excitation power spectrum at a singiven frequency 2v0 and its subharmonics 2vo /n. As anexample, the transition energies for the specific potenconsidered in this work are reported in Table I. Therefowithin the model Hamiltonian of Eq.~3.1!, resonance effectscan occur for several frequencies that may alter the shapthe population distribution within the trap. However, in geeral these resonant excitations will not be associated withescape of trapped atoms.

Here we are interested in a 1D lattice; the direct extensof the formalism mentioned above requires the evaluationthe matrix elementsT(n,m) among the different Mathieustates that conform a band. This involves integrals whichour knowledge, lack an analytical expression and requiremerical evaluation. As an alternative, we consider functiowhich variationally approximate the Mathieu functions. Thare the eigenstates of the Hamiltonian Eq.~2.4! in a har-monic basis set of frequencyv0:

un&5(i 51

i max

cniu i &v0. ~3.9!

These states are ordered according to their energy:En<En11 as exemplified in Table I. Within this scheme onobtains a very simple expression forT(n,m)

T~n,m!5Endnm2 (i , j 51

i max

cnicn j

1

2M^ i uP2u j &. ~3.10!

It is recognized that any discrete basis set approximationsystem with a band spectrum will lack features of the orinal problem which have to be carefully analyzed. Anywaalternatives to a discrete basis approach may be cumbersand not necessarily yield a better approach to understangeneral properties of experimental data. While the discrbasis approach is exact for transitions between the lowlevels, which have a negligible width, eigenstates belongto a band of measurabe width should be treated with specare. Thus, we shall assume that matrix elementsT(n,m)involving states with energiesEn and Em , so thatEn2(En2En21)/2<En<En1(En112En)/2 with an analogous expression forEm , are well approximated byT(n,m).

Within this scheme the equations which describe the prability P(n) of finding an atom in leveln, given the transi-tion ratesRm←n , are

P~n!5(m

Rm←n(1) ~P~m!2P~n!! ~3.11!

in the first-order perturbation theory scheme, and the findifference equations

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R. JAUREGUI, N. POLI, G. ROATI, AND G. MODUGNO PHYSICAL REVIEW A64 033403

Pn~ t !5Pn~ t0!1(m

Rm←n(1) ~Pm~ t0!2Pn~ t0!!~ t2t0!

1(m

Rm←n(2) ~Pm~ t0!2Pn~ t0!!~ t2t0!2, ~3.12!

valid up to second-order time-dependent perturbation thewhenevert;t0. Both sets of equations are subjected tocondition

(n

P~n!51. ~3.13!

Now, according to Eqs.~3.2! and ~3.8!, the evaluation ofRn←m

(r ) also requires the specification of the spectral denS(v). In the problem under consideration, the discrete labm,n are used to calculate interband transitions whichactually spectrally broad. This broadening might ariseonly from the band structure of the energy spectra associwith the Hamiltonian Eq.~2.4!, but also from other sourceswhich we will discuss below. Broad spectral lines canintroduced in our formalism by defining an effective spectdensitySeff(v), which should incorporate essential featurof this broadening without simulating specific featureKeeping this in mind, an effective Gaussian density of staSn(v) is associated to each levelun& of energyEn

Sn~v!51

A2psn

exp2~\v2En!2

2~\sn!2. ~3.14!

The spectral effective densitySeff(vnm) associated to thetransition m←n is obtained by the convolution ofSn(v)with Sm(v) and with the excitation source spectral densS(v). For a monochromatic source the latter is also takena Gaussian centered at the modulation frequency that ointegrated over all frequencies yields the square of the insity of the modulation source. The net result is thatSeff(vnm)has the form

Seff~v!5S0 exp2~v2veff!

2

2seff2

~3.15!

with veff determined by the modulation frequencyV and theenergiesEn andEm . The effective widthseff contains infor-mation about the frequency widths of the excitation souand those of each level.

IV. COMPARISON WITH EXPERIMENTAL RESULTS

We have tested the procedure described Sec. III to moparametric excitation in a specific experiment conductedLENS, Firenze, Italy. In this experiment40K fermionic at-oms are trapped in a 1D lattice, realizing retroreflecting learly polarized light obtained from a single–mode Ti:Saser atl5787 nm, detuned on the red of both theD1 andD2transition of potassium, respectively, at 769.9 and 766.7The laser radiation propagates along the vertical directionprovide a strong confinement against gravity. The laser beis weakly focused within a two-lens telescope to a waist s

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w0.90 mm, with a Rayleigh lengthzR53 cm; the effec-tive running power at the waist position isP5350 mW.

The trap is loaded from a magneto-optical trap~MOT!,thanks to a compression procedure already described in@6#,with about 53105 atoms at a density around 1011 cm23.The typical vertical extension of the trapped atomic cloud,detected with a charge coupled device camera~see Fig. 1!, is500 mm, corresponding to about 1200 occupied lattice siwith an average of 400 atoms in each site. Since the aextension of the atomic cloud is much smaller thanzR , wecan approximate the trap potential to be

V~r ,z!5V0e2(2r 2/w02) cos2~kz!; k52p/l, ~4.1!

thus neglecting a 5% variation ofV0 along the lattice. Theatomic temperature in the lattice direction is measured wittime–of–flight technique and it is about 50mK.

In order to parametrically excite the atoms we modulthe intensity of the confining laser with a fast acousto-opmodulator for a time intervalT.100 ms, with a sine ofamplitude e53% and frequencyV. The variation of thenumber of trapped atoms is measured by illuminatingatoms with the MOT beams and collecting the resulting flurescence on a photomultiplier. In Fig. 2 the fraction of atoleft in the trap after the parametric excitation is reportedvsthe modulation frequencyV/2p. Three resonances in thtrap losses are clearly seen at modulation frequencies of670, and 1280 kHz. By identifying the first two resonancwith the lattice vibrational frequency and its first harmonrespectively, we get as a first estimatev0.2p3340 kHz.As we will show in the following, these resonance are acally on thered region ofv0 and 2v0, respectively, and therefore a better estimate isv0.2p3360 kHz. Therefore theeffective trap depth is, from Eq.~9!, V0.185mK.10.5\v0. Since the atomic temperature is aboutV0/3.5,we expect that most of the energy levels of the lattice hav

FIG. 1. Absorption image of the atoms in the optical lattice, ashape of the optical potential in the two relevant directions.

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ANHARMONIC PARAMETRIC EXCITATION IN OPTICAL . . . PHYSICAL REVIEW A 64 033403

non-negligible population and therefore the anharmonicitythe potential could play an important role in the dynamicsparametric excitation. Note that the third resonance at hfrequency, close to 4v0, is not predicted from the harmonitheory. It is possible to also observe a much weaker renance in the trap losses around 1.5 kHz, which we interto be twice the oscillation frequency in the loosely confinradial direction. However, in the following we will focus ouattention just on the axial resonances.

As discussed in Sec. III, the overall width of the excittion assumed for our model system could play an importrole in reproducing essential features of experimental dSince the source used in the experiment has a negliglinewidth, it is necessary to model just the broadening ofatomic resonances. The spread of the transition energiesto the axial anharmonicity is reported in Table I, while tbroadening of each energy level, due to the periodic chater of the sine potential, is estimated using Eq.~2.8!. We nownote that the 1D motion assumed in Sec. II is not complevalid in our case, since the atoms move radially alongGaussian potential. Since the period of the radial motionabout 500 times longer than the axial period, the atomsan effective axial frequency which varies with their radposition, resulting in a broadening of the transition frquency. Other sources of broadening are fluctuations oflaser intensity and pointing, and inhomogeneities alonglattice. We also note that elastic collisions within the trappsample, which tend to keep a thermal distribution of the tlevels population, can contribute to an overall broadeningthe loss resonances. Since it is not easy to build a modelinvolves all these sources, we introduce semiempiricallyeffective broadening for ther th level @see Eq.~3.14!#. Rec-ognizing that the width could be energy dependent we csidered the simple expression

s r25l1S Er

V0D p

1l0 ~4.2!

FIG. 2. Experimental spectrum of the losses associated to pmetric excitation of the trap vibrational modes. For the low ahigh frequency regions two different modulation amplitudes of 2and 3%, respectively, were used.

03340

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ta.leeue

c-

lyaiseel-eedpfatn

-

for several values of the constantsl1 , l0, and p. When p50, i.e., for a constant value of the bandwidth we wereable to reproduce the general experimental behavior repoin Fig. 2. The best agreement between the simulation andexperimental observations is obtained forl050.0002, l1

50.0135, in units ofv02, andp53. Similar results are also

obtained for slightly higher~lower! values ofl0,1 togetherwith slightly higher~lower! values of the powerp. In Table Ithe resulting widths are shown for the lower 12 levels. Nothat we have intentionally excluded levels 11, 12, andfrom the calculation, since their intrinsic width is so largthat the atoms can tunnel out of the trap along the latticemuch less than 100 ms@11#. However, the inclusion of theslevels proved not to change the result of the simulation sstantially.

A comparison of experimental and theoretical resultsmade in Fig. 3; the abscissa for the experimental databeen normalized by identifyingv0 with 2p3360 kHz. Asalready anticipated, the principal resonance in trap lossespears atV.1.85v0. This result follows from the fact thathe excitation of the lowest trap levels does not result inloss of atoms, as it would happen for a harmonic potentOn the contrary, the most energetic atoms, which havvibrational frequency smaller than the harmonic one, are eily excited out of the trap. The asymmetry of the resonancwhich has been observed also in@2#, is well reproduced inthe calculations and it is further evidence of the spread ofvibrational frequencies. The first interesting result obtainby our study of parametric excitation is therefore the corrtion necessary to extract the actual harmonic frequency fthe loss spectrum. For the specific conditions of the presexperiment, we find indeed that the principal resonancethe trap losses appears atV.1.85v0. However, the calcu-lation shows that the resonance is nearby this position fothe explored values ofl0,1 andp also for deeper traps, up tV0525\v0, and therefore it appears to be an invariant ch

ra-FIG. 3. Experimental~circles! and theoretical~lines! fraction of

atoms left in the trap after parametric excitationvs the modulationfrequency. The continuous line corresponds to the numerical igration of the first order perturbation theory equations~3.11! andthe dashed line to the numerical integration of the finite differensecond order perturbation theory equations~3.8!.

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R. JAUREGUI, N. POLI, G. ROATI, AND G. MODUGNO PHYSICAL REVIEW A64 033403

acteristic of the sinusoidal potential.The result of the numerical integration of Eqs.~3.12! re-

ported in Fig. 3 reproduces relatively well the subharmoresonance, which in the harmonic case would be expectev0. On the contrary, both experiment and calculation shthat the actual position of the resonance isV.0.9v0. Thefact that the calculated resonance is broader than the exmental one could be due to an overestimation of the broening of the high-lying levels of the lattice when using tsimple model of Eq.~4.2!, which could also explain thesmall disagreement between theory and experiment onred of the other two resonances in Fig. 3. In addition, it mbe mentioned that the accuracy of these results is restriby the finite difference character of Eqs.~3.12! and by thefact that some noise sources which have not been inclucould be resonant at a nearby frequency. In particular, asible modulation of the laser pointing associated to the intsity modulation is expected to be resonant atV5v0 in theharmonic problem@1#, and it could play an analogous roleour sinusoidal lattice.

The higher order resonance around 3.5v0 observed in theexperiment is also well reproduced by the calculations baon first-order perturbation theory. Note that a simpler aproximation to the confining potential by a quartic potentVQ(z)5k2z21k4z4 would yield a resonance around 4v0and not 3.5v0. However, it is possible to understand quatatively one of the features of this resonance considerinquartic perturbation of the forme(t)VQ to a harmonic poten-tial. In this case the ratio of the transition rates at the 2v0and 4v0 resonances is set by Eqs.~3.2! to

uT~n62,n!u2/uT~n64,n!u2}V0

2

v02

. ~4.3!

This result can qualitatively explain the absence of the cresponding high-order resonance in theradial excitationspectrum~see Fig. 2!: since the radial trap frequency isfactor of 500 smaller than the axial one, the relative strenof such radial anharmonic resonance is expected to bepressed by a factor of (500)2. In conclusion, high harmonicresonances, which certainly depend on the actual shapthe anharmonic potential, are expected to appear only ifspring constant of the trap is large.

In Fig. 4 theoretical and experimental results for the elution of the total population of trapped atoms at the resonexciting frequencyV52v0 are shown. Although there issatisfactory agreement between the model and the exment, we notice that experimental data exhibit a differrate for the loss of atoms before and after 100 ms. Tchange is probably due to a variation of the collision ratethe number of trapped atoms is modified, which cannoteasily included in the model. A comparison of the expemental evolution of the trap population with and withomodulation shows the effectiveness of the excitation procin emptying the trap on a short time scale.

We have also simulated the energy growth of the trapatoms due to the parametric excitation, which is reportedFig. 5. Our calculations show a nonexponential energy

03340

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crease in contrast with what is expected in the harmoapproximation, Eq.~3.7!. The fast energy growth at shotimes is related to the depopulation of the lowest levewhich are resonant with the 2v0 parametric source. The saturation effect observed for longer times is due to the fact tthe resonance condition is not satisfied for the upper levso that they do not depopulate easily.

V. CONCLUSIONS

We have studied both theoretically and experimentallytime evolution of the population of atoms trapped in a 1sinusoidal optical lattice, following a parametric excitatioof the lattice vibrational mode. In detail, we have presentetheoretical model for the excitation in an anharmonic pottial, which represents an extension of the previous harmomodels, and we have applied it to the actual sinusoidaltential used to trap cold potassium atoms. The simulat

FIG. 4. Theoretical~continuous line! and experimental~tri-angles! results for the evolution of the population of trapped atoat the resonant exciting frequencyV52v0. The circles show theevolution of the population in absence of modulation.

FIG. 5. Calculated evolution of the average energy oftrapped atoms during parametric excitation atV52v0.

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ANHARMONIC PARAMETRIC EXCITATION IN OPTICAL . . . PHYSICAL REVIEW A 64 033403

seems to reproduce relatively well the main features of bthe spectrum of trap losses, including the appearance of rnances beyond 2v0, and the time evolution of the total number of trapped atoms.

By comparing the theoretical predictions and the expmental observations the usefulness of a parametric excitaprocedure to characterize the spring constant of the trapbeen verified. Although the loss resonances are redshand wider than expected in the harmonic case, the latharmonic frequency can be easily extracted from the expmental spectra to estimate useful quantities such as thedepth and spring constant.

We have also emphasized the need for modelingbroadening of bands with a negligible natural width in ordto reproduce the observed loss spectrum. In a harmmodel this broadening is not necessary since the equidisenergy spectrum guarantees that a single transition en

ys

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03340

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characterizes the excitation process. We think that mosthe broadening in our specific experiment is due to the fthat the actual trapping potential is not one dimensional,also to possible fluctuations and inhomogeneities of thetice.

To conclude, we note that the dynamical analysis we hmade can be easily extended to lattices with larger dimsionality, and also to other potentials, such as Gaussiantentials, which are also commonly used for optical trappi

ACKNOWLEDGMENTS

We acknowledge illuminating discussions with R. BrechThis work was supported by the European CommunCouncil ~ECC! under Contract No. HPRI-CT-1999-0011and by MURST under the PRIN 1999 and PRIN 2000 Pgrams.

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