analysis of dynamical tunneling experiments with a bose ...73356/uq73356.pdftrapsr,td + vsr,td + n...

Analysis of dynamical tunneling experiments with a Bose-Einstein condensate

W. K. Hensinger,1,2,* A. Mouchet,3,† P. S. Julienne,2 D. Delande,4 N. R. Heckenberg,1 and H. Rubinsztein-Dunlop11Centre for Biophotonics and Laser Science, Department of Physics, The University of Queensland, Brisbane,

Queensland 4072, Australia2National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

3Laboratoire de Mathématique et de Physique Théorique (CNRS UMR 6083), Avenue Monge, Parc de Grandmont,37200 Tours, France

4Laboratoire Kastler-Brossel (CNRS UMR 8552), Université Pierre et Marie Curie, 4 place Jussieu, F-75005 Paris, France(Received 26 September 2003; revised manuscript received 5 April 2004; published 15 July 2004)

Dynamical tunneling is a quantum phenomenon where a classically forbidden process occurs that is prohib-ited not by energy but by another constant of motion. The phenomenon of dynamical tunneling has beenrecently observed in a sodium Bose-Einstein condensate. We present a detailed analysis of these experimentsusing numerical solutions of the three-dimensional Gross-Pitaevskii equation and the corresponding Floquettheory. We explore the parameter dependency of the tunneling oscillations and we move the quantum systemtowards the classical limit in the experimentally accessible regime.

DOI: 10.1103/PhysRevA.70.013408 PACS number(s): 42.50.Vk, 32.80.Pj, 05.45.Mt, 03.65.Xp

I. INTRODUCTION

Cold atoms provide a system which is particularly suitedto study quantum nonlinear dynamics, quantum chaos, andthe quantum-classical borderland. On relevant time scalesthe effects of decoherence and dissipation are negligible.This allows us to study a Hamiltonian quantum system. Onlyrecently dynamical tunneling was observed in experimentswith ultracold atoms[1,2]. “Conventional” quantum tunnel-ing allows a particle to pass through a classical energy bar-rier. In contrast, in dynamical tunneling a constant of motionother than energy classically forbids one to access a differentmotional state. In our experiments atoms tunneled back andforth between their initial oscillatory motion and the motion180° out of phase. A related experiment was carried out bySteck, Oskay, and Raizen[3,4] in which atoms tunneled fromone unidirectional librational motion into another oppositelydirected motion.

Luter and Reichl[5] analyzed both experiments calculat-ing mean momentum expectations values and Floquet statesfor some of the parameter sets for which experiments werecarried out and found good agreement with the observedtunneling frequencies. Averbukh, Osovski, and Moiseyev[6]pointed out that it is possible to effectively control the tun-neling period by varying the effective Planck’s constant byonly 10%. They showed one can observe both suppressiondue to the degeneracy of two Floquet states and enhancementdue to the interaction with a third state in such a small inter-val.

Here we present a detailed theoretical and numericalanalysis of our experiments. We use numerical solutions ofthe Gross-Pitaevskii equation and Floquet theory to analyze

the experiments and to investigate the relevant tunneling dy-namics. In particular we show how dynamical tunneling canbe understood in a two and three state framework using Flo-quet theory. We show that there is good agreement betweenexperiments and both Gross-Pitaevskii evolution and Floquettheory. We examine the parameter sensitivity of the tunnelingperiod to understand the underlying tunneling mechanisms.We also discuss such concepts as chaos-assisted andresonance-assisted tunneling in relation to our experimentalresults. Finally predictions are made concerning what canhappen when the quantum system is moved towards the clas-sical limit.

In our experiments a sodium Bose-Einstein condensatewas adiabatically loaded into a far detuned optical standingwave. For a sufficient large detuning, spontaneous emissioncan be neglected on the time scales of the experimentss160 msd. This also allows us to consider the external de-grees of freedom only. The dynamics perpendicular to thestanding wave are not significant, therefore we are led to aneffectively one-dimensional system. The one-dimensionalsystem can be described in the corresponding two-dimensional phase space which is spanned by momentumand position coordinates along the standing wave. Single fre-quency modulation of the intensity of the standing waveleads to an effective Hamiltonian for the center-of-mass mo-tion given by

H =px

2

2m+

"Veff

4f1 − 2« sinsvt + fdgsin2skxd, s1d

where the effective Rabi frequency isVeff=V2/d, V

=GÎI / Isat is the resonant Rabi frequency,« is the modulationparameter,v is the modulation angular frequency,G is theinverse spontaneous lifetime,d is the detuning of the stand-ing wave,t is the time,px is the momentum component ofthe atom along the standing wave, andk is the wave number.HereI is the spatial-mean of the intensity of the unmodulatedstanding wave(which is half of the peak intensity) so V

*Electronic address: [email protected]; Present address: De-partment of Physics, University of Michigan, 2477 Randall Labo-ratory, 500 East University Ave., Ann Arbor, MI 48109-1120, USA.

†Electronic address: [email protected]

PHYSICAL REVIEW A 70, 013408(2004)

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=GÎIpeak/2Isat whereIsat=hcG /l3 is the saturation intensity.l is the wavelength of the standing wave.f determines thestart phase of the amplitude modulation. Using scaled vari-ables[7] the Hamiltonian is given by

H = p2/2 + 2kf1 − 2« sinst + fdgsin2sq/2d, s2dwhereH=s4k2/mv2dH, q=2kx, andp=s2k/mvdpx.

The driving amplitude is given by

k = vrVeff/v2 =

"k2Veff2v2m

=4U0vrv2"

, s3d

wherevr ="k2/2m is the recoil frequency,t= tv is the scaled

time variable, andU0 is the well depth. The commutator ofscaled position and momentum is given by

fp,qg = i k–, s4d

where the scaled Planck’s constant isk–=8vr /v. For k=1.2and «=0.20 the classical Poincaré surface of section isshown in Fig. 1. Two symmetric regular regions can be ob-served aboutsq=0,p=1d and sq=0,p=−1d. These regionscorrespond to oscillatory motion in phase with the amplitudemodulation in each well of the standing wave. In the experi-ment [1,2] atoms are loaded in a period-1 region of regularmotion by controling their initial position and momentumand by choosing the starting phase of the amplitude modula-tion appropriately. Classically atoms should retain their mo-mentum state when observed stroboscopically(time step isone modulation period). A distinct signature of dynamical

tunneling is a coherent oscillation of the stroboscopicallyobserved mean momentum as shown in Fig. 2 and reportedin Ref. [1].

In Sec. II we introduce the theoretical tools to analyzedynamical tunneling by discussing Gross-Pitaevskii simula-tions and the appropriate Floquet theory. We present a thor-ough analysis of the experiments from Ref.[1] in Sec. III.After showing some theoretical results for the experimentalparameters we give a small overview of what to expect whensome of the system parameters in the experiments are variedin Sec. IV and give some initial analysis. In Sec. V we pointto pathways to analyze the quantum-classical transition forour experimental system and give conclusions in Sec. VI.

II. THEORETICAL ANALYSIS OF THE DYNAMICEVOLUTION OF A BOSE-EINSTEIN CONDENSATE

A. Dynamics using the Gross-Pitaevskii equation

The dynamics of a Bose-Einstein condensate in a time-dependent potential in the mean-field limit are described bythe Gross-Pitaevskii equation[8,9]

i"] Csr ,td

] t= F− "

2m¹2 + Vtrapsr ,td + Vsr ,td

+ N4p "2a

muCsr ,tdu2GCsr ,td, s5d

whereN is the mean number of atoms in the condensate, anda is the scattering length witha=2.8 nm for sodium.Vtrapsr dis the trapping potential which is turned off during the inter-action with the standing wave andVsr ,td is the time-

FIG. 1. Poincaré section for a classical particle in an amplitude-modulated optical standing wave. Momentum and position(onewell of the standing wave) of the particle along the standing waveare plotted stroboscopically with the stroboscopic period beingequal to the modulation period. The central region consists of smallamplitude motion. Chaos(dotted region) separates this region fromtwo period-1 regions of regular motion(represented in the Poincarésection as sets of closed curves) located left and right of the centeralong momentump=0. Further out in momentum are two stableregions of motion known as librations. At the edges are bands ofregular motion corresponding to above barrier motion. It is plottedfor modulation parameter«=0.20 and scaled well depthk=1.20.

FIG. 2. Stroboscopic mean momentum as a function of the in-teraction time with the modulated standing wave measured inmodulation periods,n, for modulation parameter«=0.29, scaledwell depthk=1.66, modulation frequencyv /2p=250 kHz, and aphase shiftw=0.2132p. (a) and(b) correspond to the two differentinteraction timesn+0.25 andn+0.75 modulation periods, respec-tively. Results from the dynamic evolution of the Gross-Pitaevskiiequation are plotted as a solid line(circles) and the experimentaldata are plotted as a dashed line(diamonds).

HENSINGERet al. PHYSICAL REVIEW A 70, 013408(2004)

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dependent optical potential induced by the optical standingwave. The Gross-Pitaevskii equation is propagated in timeusing a standard numerical split-operator, fast Fourier trans-form method. The size of the spatial grid of the numericalsimulation is chosen to contain the full spatial extent of theinitial condensate(therefore all the populated wells of thestanding wave), and the grid has periodic boundary condi-tions at each side(a few unpopulated wells are also includedon each side).

To obtain the initial wave function a Gaussian test func-tion is evolved by imaginary time evolution to converge tothe ground state of the stationary Gross-Pitaevskii equation.Then the standing wave is turned on adiabatically withVsr ,td approximately having the form of a linear ramp. Afterthe adiabatic turn-on, the condensate wave function is foundto be localized at the bottom of each well of the standingwave. The standing wave is shifted and the time-dependentpotential now has the form

Vsr ,td ="Veff

4f1 − 2« sinsvt + fdgsin2skx+ wd, s6d

wherew is the phase shift which is applied to selectively loadone region of regular motion[1]. The position representationof the atomic wave functionuCsr du2 just before the modula-tion starts(and after the phase shift) is shown in Fig. 3st=0Td.

The Gross-Pitaevskii equation is used to model the ex-perimental details of a Bose-Einstein condensate in an opti-cal one-dimensional lattice. The experiment effectively con-sists of many coherent single-atom experiments. Thecoherence is reflected in the occurrence of diffraction peaksin the atomic momentum distribution(see Fig. 4). Utilizing

the Gross-Pitaevskii equation, the interaction between thesesingle-atom experiments is modeled by a classical field. Al-though ignoring quantum phase fluctuations in the conden-sate, the wave nature of atoms is still contained in the Gross-Pitaevskii equation and dynamical tunneling is a quantumeffect that results from the wave nature of the atoms. Theassumption for a common phase for the whole condensate iswell justified for the experimental conditions as the timescales of the experiment and the lattice well depth are suffi-ciently small. It will be shown in the following(see Fig. 8)that the kinetic energy is typically of the order of 105 Hzwhich is much larger than the nonlinear term in the Gross-Pitaevskii equation(5) which is on the order of 400 Hz. Theexperimental results, in particular dynamical tunneling,could therefore be modeled by a single particle Schrödingerequation in a one-dimensional single well with periodicboundary conditions. Nevertheless the Gross-Pitaevskiiequation is used to model all the experimental details of aBose-Einstein condensate in an optical lattice to guaranteemaximum accuracy. We will discuss and compare the Gross-Pitaevskii and the Floquet approaches below(last paragraphof Sec. II).

Theoretical analysis of the dynamical tunneling experi-ments will be presented in this paper utilizing numerical so-lutions of the Gross-Pitaevskii equation. Furthermore we willanalyze the system parameter space which is spanned by thescaled well depthk, the modulation parameter«, and the

scaled Planck’s constantk–. In fact variation of the scaledPlanck’s constant in the simulations allows one to move thequantum system towards the classical limit.

B. Floquet analysis

The quantum dynamics of a periodically driven Hamil-tonian system can be described in terms of the eigenstates of

FIG. 3. Position representationof the atomic wave function as afunction of the number of modu-lation periods calculated using theGross-Pitaevskii equation. Thewave packet is plotted strobo-scopically at a phase so it is lo-cated classically approximately atits highest point in the potentialwell. The position of the standingwave wells is also shown(dottedline). Dynamical tunneling can beobserved. At t=5T most atomshave tunneled into the otherperiod-1 region of regular motion.The position axis is given inFermi units (scaled by the meanThomas-Fermi diameter).

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the Floquet operatorF, which evolves the system in time byone modulation period. In the semiclassical regime, the Flo-quet eigenstates can be associated with regions of regularand irregular motion of the classical map. However, when"is not sufficiently small compared to a typical classical actionthe phase-space representation of the Floquet eigenstates donot necessarily match with some classical(regular or irregu-lar) structures[10,11]. However, initial states localized at thestable region around a fixed point in the Poincaré section canbe associated with superpositions of a small number of Flo-quet eigenstates. Using this state basis, one can reveal theanalogy of the dynamical tunneling experiments and conven-tional tunneling in a double well system. Two states of op-posite parity which can be responsible for the observed dy-namical tunneling phenomenon are identified. Floquet statesare stroboscopic eigenstates of the system. Their phase spacerepresentation therefore provides a quantum analog to theclassical stroboscopic phase space representation, thePoincaré map.

Only very few states are needed to describe the evolutionof a wave packet that is initially strongly localized on a re-gion of regular motion. A strongly localized wave packet isused in the experiments(strongly localized in each well ofthe standing wave) making the Floquet basis very useful. Incontrast, describing the dynamics in momentum or positionrepresentation requires a large number of states so that someof the intuitive understanding which one can obtain in theFloquet basis is impossible to gain. For example, the tunnel-ing period can be derived from the quasi-eigenenergies of therelevant Floquet states, as will be shown below.

For an appropriate choice of parameters the phase spaceexhibits two period-1 fixed points, which for a suitablePoincaré section lie on the momentum axis at ±p0, as in Ref.

[1]. For certain values of the scaled well depthk and modu-lation parameter« there are two dominant Floquet statesuf±lthat are localized on both fixed points but are distinguishedby being even or odd eigenstates of the parity operator thatchanges the sign of momentum. A state localized on just onefixed point is therefore likely to have dominant support on aneven or odd superposition of these two Floquet states:

ucs±p0dl < suf+l ± uf−ld/Î2. s7dThe stroboscopic evolution is described by repeated ap-

plication of the Floquet operator. As this is a unitary opera-tor,

Fuf±l = es−i2pf±/k–duf±l. s8d

f± are the Floquet quasienergies. Thus at a time which isntimes the period of modulation, the state initially localizedon +p0 evolves to

ucsndl < ses−i2pnf+/k–duf+l + es−i2pnf−/k

–duf−ld/Î2. s9dIgnoring an overall phase and defining the separation be-tween Floquet quasienergies as

Df = f− − f+, s10d

one obtains

ucsndl < suf+l + es−i2pnDf/k–duf−ld/Î2. s11d

At

n = k–/s2Dfd s12d

periods, the state will form the antisymmetric superpositionof Floquet states and thus is localized on the other fixed point

FIG. 4. Momentum distribu-tions as a function of the interac-tion time with the modulatedstanding wave calculated usingnumerical solutions of the Gross-Pitaevskii equation for the sameparameters as Fig. 3. Initially at-oms have mostly negative mo-mentumst=0.25Td. After approxi-mately five modulation periodsmost atoms populate a state withpositive momentum, thereforehaving undergone dynamical tun-neling. Corresponding experimen-tal data from Ref.[1] is also in-cluded as insets.


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at −p0. In other words the atoms have tunneled from one ofthe fixed points to the other. This is reminiscent of barriertunneling between two wells, where a particle in one well, ina superposition of symmetric and antisymmetric energyeigenstates, oscillates between wells with a frequency givenby the energy difference between the eigenstates.

Tunneling can also occur when the initial state has signifi-cant overlap with two nonsymmetric states. For example, ifthe initial state is localized on two Floquet states, one local-ized inside the classical chaotic region and one inside theregion of regular motion, a distinct oscillation in the strobo-scopic evolution of the mean momentum may be visible. Thefrequency of this tunneling oscillation depends on the spac-ing of the corresponding quasi-eigenenergies in the Floquetspectrum. In many cases multiple tunneling frequencies oc-cur in the stroboscopic evolution of the mean momentumsome of which are due to tunneling between nonsymmetricstates.

Quantum dynamical tunneling may be defined in that aparticle can access a region of phase space in a way that isforbidden by the classical dynamics. This implies that itcrosses a Kolmogorov-Arnold-Moser(KAM ) surface[12,13]. The clearest evidence of dynamical tunneling can beobtained by choosing the scaled Planck’s constantk– suffi-ciently small so that the atomic wave function is muchsmaller than the region of regular motion(the size of thewave function is given byk–). Furthermore, it should be cen-tered inside the region of regular motion. However, even ifthe wave packet is larger than the region of regular motionand also populates the classical chaotic region of phase spaceone can still analyze quantum-classical correspondence andtunneling. One assumes a classical probability distribution ofpoint particles with the same size as the quantum wave func-tion and compares the classical evolution of this point par-ticle probability distribution with the quantum evolution ofthe wave packet. A distinct difference between the two evo-lutions results from the occurrence of tunneling assumingthat the quantum evolution penetrates a KAM surface vis-ibly.

C. Comparison

Using the Gross-Pitaevskii equation one can exactlysimulate the experiment because the momentum or positionrepresentation is used. Therefore the theoretical simulationcan be directly compared with the experimental result. Incontrast Floquet states do not have a straightforward experi-mental intuitive analog. In the Floquet states analysis onecan compare the quasienergy splitting between the tunnelingFloquet state with the experimentally measured tunneling pe-riod. The occurrence of multiple frequencies in the experi-mentally observed tunneling oscillations might also be ex-plained with the presence of more than two dominantFloquet states, the tunneling frequencies being the energysplitting between different participating Floquet states. Usingthe Gross-Pitaevskii approach one can simulate the experi-ment with high precision. The same number of populatedwells as in the experiment can be used and the turn-on of thestanding wave can be simulated using an appropriate turn-on

Hamiltonian. Therefore one can obtain the correct initialstate with high precision. Furthermore, the mean-field inter-action is simulated which is not contained in the Floquetapproach.

III. NUMERICAL SIMULATIONS OF THE EXPERIMENTS

A. Gross-Pitaevskii simulations

In this section experimental results that were discussed ina previous paper[1] are compared with numerical simula-tions of the Gross-Pitaevskii(GP) equation.

In the experiment[1] the atomic wave function was pre-pared initially to be localized around a period-1 region ofregular motion. Figure 2 shows the stroboscopically mea-sured mean momentum as a function of the interaction timewith the standing wave for modulation parameter«=0.29,scaled well depthk=1.66, modulation frequencyv /2p=250 kHz, and a phase shift ofw=0.2132p. (a) and (b)correspond to the two different interaction timesn+0.25(standing wave modulation ends at a maximum of the ampli-tude modulation) and n+0.75 modulation periods(standingwave modulation ends at a minimum of the amplitude modu-lation), respectively. Results from the simulations(solid line,circles) are compared with the experimental data(dashedline, diamonds). Dynamical tunneling manifests itself as acoherent oscillation of the stroboscopically observed meanmomentum. This occurs in contrast to the classical predictionin which atoms should retain their momentum state whenobserved stroboscopically(time step is one modulation pe-riod). There is good agreement between experiment andtheory as far as the tunneling period is concerned. However,the experimentally measured tunneling amplitude is smallerthan the theoretical prediction.

It should be noted that the theoretical simulations do nottake account of any possible spatial and temporal variationsof the scaled well depth(e.g., light intensity), which couldpossibly lead to the observed discrepancy. It was decided toproduce simulations without using any free parameters.However, the uncertainty associated with the experimentwould allow small variations of the modulation parameter«and the scaled well depthk. There was a 10% uncertainty inthe value of the scaled well depthk and a 5% uncertainty inthe modulation parameter« (all reported uncertainties are 1standard deviation combined systematic and statistical uncer-tainties). Both temporal and spatial uncertainty during onerun of the experiment are contained in these values as well asthe systematic total measurement uncertainty. It was verifiedthat there are no important qualitative changes when varyingthe parameters in the uncertainty regime for the simulationspresented here. However, the agreement between experimentand theory often can be optimized(not always non-ambiguously, meaning that it is sometimes hard to decidewhich set of parameters produces the best fit). Although thetheoretical simulation presented does not show any decay inthe mean momentum curve, a slight change of parametersinside the experimental uncertainty can lead to decay whichis most likely caused by another dominant Floquet statewhose presence leads to the occurrence of a beating of thetunneling oscillations which appears as decay(and revival on


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longer time scales). A detailed analysis of the correspondingFloquet states and their meaning will be presented in Sec.III C. [An example of how a change of parameters inside theexperimental uncertainty in the simulations can optimize theagreement between theory and experiment is shown in Figs.6 and 7. Both figures are described in more detail later in thissection.] Another reason for the observed discrepancy couldbe the evolution of noncondensed atoms that is not containedin the GP approach and the interaction of noncondensed at-oms with the condensate. With a sufficiently long adiabaticturn-on time of the far detuned standing wave the productionof noncondensed atoms should be negligible. The interactionof the condensate with noncondensed atoms should also benegligible due to the low atomic density(note that in theexperiments the condensate is expanded before the standingwave is turned on). However, further studies are needed togive an exact estimate of these effects.

The position representation of the atomic wave functionuCsxdu2 is plotted stroboscopically after multiples of onemodulation period in Fig. 3 for the same parameters as Fig.2. The position of the standing wave wells is also shown(dotted line), their amplitude is given in arbitrary units. Theposition axis is scaled with the mean Thomas-Fermi diam-eter. The initial modulation phase is chosen so that the wavepacket should be located classically approximately at itshighest point of the potential well[2]. Choosing this strobo-scopic phase the two regions of regular motion are alwaysmaximally separated in position space. Using this phase forthe stroboscopic plots enables the observation of dynamicaltunneling in position space as the two regions of regularmotion are located to the left and to the right of the minimumof the potential well, being maximally spatially separated. Incontrast, in the experiments, tunneling is always observed inmomentum space(the standing wave is turned off when theregions of regular motion are at the bottom of the well, over-lapping spatially but having oppositely directed momenta) asit is difficult to optically resolve individual wells of thestanding wave. The first picture in Fig. 3st=0Td shows theinitial wave packet before the modulation is turned on. Sub-sequent pictures exemplify the dynamical tunneling process.At t=2T, half the atoms have tunneled; most of the atoms arein the other region of regular motion att=5T. The atomshave returned to their initial position at aboutt=9T. Thedouble peak structure att=5T and the small central peak att=2T could indicate that Floquet states other than the twodominant ones are also loaded which is likely as a relativelylarge Planck’s constant was used enabling the initial wavepacket to cover a substantial phase space area.

Figure 4 shows simulations of the stroboscopically mea-sured momentum distributionsuCspdu2 as bar graphs for thesame parameters as Fig. 3. Corresponding experimental datafrom Ref. [1] are also included as insets. The momentumdistributions are plotted atn+0.25 modulation periods,wheren is an integer. At this modulation phase the amplitudemodulation is at its maximum and atoms in a period-1 regionof regular motion are classically at the bottom of the wellhaving maximum momentum. The two period-1 regions ofregular motion can be distinguished in their momentum rep-resentation at this phase as they have opposite momenta. Att=0.25T the atoms which were located initially halfway up

the potential well as shown in Fig. 3 att=0T have “rolled”down the well having acquired negative momentum. Mo-mentum distributions for subsequent times illustrate the dy-namical tunneling process. Att=5.25 modulation periodsmost atoms have reversed their momentum and they return totheir initial momentum state at approximatelyt=9.25T. Thesimulations are in reasonable agreement with the experimen-tally measured data.

Dynamical tunneling is sensitive to the modulation pa-rameter, the scaled well depth, and the scaled Planck’s con-stant. To illustrate this, tunneling data along with the appro-priate evolution of the Gross-Pitaevskii equation will beshown for another two parameter sets. Even though this rep-resents only a small overview of the parameter dependencyof the tunneling oscillations it may help to appreciate thevariety of features in the atomic dynamics. Figure 5 showsthe theoretical simulation and the experimental data for«=0.28, k=1.49, v /2p=250 kHz, andw=0.2232p. Themean momentum is plotted stroboscopically with the inten-sity modulation at maximum(n+0.25 modulation periods,nbeing an integer). The solid line(circles) is produced by aGross-Pitaevskii simulation and the dashed line(diamonds)consists of experimental data. There are approximately 3.5tunneling periods in 40 modulation periods in the theoreticalcurve. Figure 6 shows the mean momentum as a function ofthe interaction time with the standing wave for modulationparameter«=0.30, scaled well depthk=1.82, modulationfrequencyv /2p=222 kHz, and phase shiftw=0.2132p.For these parameters the tunneling frequency is larger thanfor the parameters shown in Fig. 5.

The simulation shows good agreement with the experi-ment. However, the theoretical mean momentum tunnelingamplitude is larger than the one measured in the experimentand the theoretical tunneling frequency for this set of param-

FIG. 5. Mean momentum as a function of the interaction timewith the modulated standing wave measured in modulation periods,n, for «=0.28, k=1.49, v /2p=250 kHz, andw=0.2232p. Thepoints are plotted stroboscopically with an interaction time ofn+0.25 modulation periods which corresponds to turning off thestanding wave at maximum. Results from the dynamic evolution ofthe Gross-Pitaevskii equation are plotted as a solid line(circles) andthe experimental data is plotted as a dashed line(diamonds).


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eters is slightly larger than the experimentally measured one.Figure 7 illustrates that one can achieve much better agree-ment between experiment and simulation if one of the pa-rameters is varied inside the experimental regime of uncer-tainty. The theoretical curve in this figure is obtained usingthe same parameters as in Fig. 6, but the modulation param-eter« is reduced from 0.30 to 0.28. The tunneling amplitudeand frequency is now very similar to the experimental re-

sults. Note that the experimental data is not centered at zeromomentum. This is also the case in the theoretical simula-tions for both«=0.28 and«=0.30. The mean momentumcurve appears much more sinusoidal than the one for«=0.29,k=1.66,v /2p=250 kHz andw=0.2132p (Fig. 2).This could imply that the initial wave function has supporton fewer Floquet states.

B. Stroboscopic evolution of the system energies

Calculating the expectation values of the relevant systemenergies can give important information about the relevantenergy scales and it might also help to obtain a deeper in-sight into the stroboscopic evolution of a Bose-Einstein con-densate in a periodically modulated potential. Figure 8 showsthe energy, expectation values for the mean-field energy, thepotential energy, and the kinetic energy given in Hz(scaledby Planck’s constant). For comparison the stroboscopic meanmomentum expectation values are also shown. The energyand momentum expectation values are plotted stroboscopi-cally and the solid(circles) and dashed(diamonds) curvescorrespond to the two different interaction timesn+0.25 andn+0.75 modulation periods, respectively. Figure 8 is plottedfor modulation parameter«=0.29, scaled well depthk=1.66, modulation frequencyv /2p=250 kHz, and a phaseshift of w=0.2132p which corresponds to Fig. 2. Themean-field energy is three orders of magnitude smaller thanthe potential or the kinetic energy. While the kinetic energydoes not show a distinct oscillation, an oscillation is clearlyvisible in the stroboscopic potential energy evolution. Thisoscillation frequency is not equal to the tunneling frequencybut it is smaller as can be seen in Fig. 8. One obtains a period

FIG. 6. Mean momentum as a function of the interaction timewith the modulated standing wave measured in modulation periods,n, for modulation parameter«=0.30, scaled well depthk=1.82,modulation frequencyv /2p=222 kHz, and phase shiftw=0.2132p. The points are plotted stroboscopically with an interactiontime of n+0.25 modulation periods which corresponds to turningoff the standing wave at maximum. Results from the dynamic evo-lution of the Gross-Pitaevskii equation are plotted as a solid line(circles) and the experimental data is plotted as a dashed line(diamonds).

FIG. 7. Mean momentum as a function of the interaction timewith the modulated standing wave measured in modulation periods,n, using the same parameters as Fig. 6 but the modulation parameter« is reduced from 0.30 to 0.28 for the theoretical simulation. Amuch better fit is obtained.

FIG. 8. Stroboscopic evolution of mean-field energy, the kineticenergy, and the potential energy. The energies are given in Hz(en-ergy is scaled with Planck’s constant). For comparison the meanmomentum is also shown as a function of the interaction time withthe standing wave(in units of modulation periods). The solid(circles) and dashed(diamonds) curves correspond to the two dif-ferent interaction timesn+0.25 andn+0.75 modulation periods,respectively. The evolution is plotted for modulation parameter«=0.29, scaled well depthk=1.66, modulation frequencyv /2p=250 kHz, and a phase shiftw=0.2132p.


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of approximately 8.9 modulation periods compared to a tun-neling period of approximately 10.0 modulation periods.Considering the energy scales one should note that this os-cillation is also clearly visible in the stroboscopic evolutionof the total energy of the atoms. The origin of this oscillationis not known yet and will be the subject of future investiga-tion.

C. Floquet analysis for some experimental parameters

Time and spatial periodicity of the Hamiltonian allow uti-lization of the Bloch and Floquet theorems[11,14]. Becauseof the time periodicity, there still exist eigenstates of theevolution operator over one period(Floquet theorem). Itseigenvalues can be written in the forme−i2pfnt/k

–wherefn is

called the quasienergy of the states.n is a discrete quantumnumber. Due to the spatiall /2-periodicity, in addition ton,these states are labeled by a continuous quantum number, theso-called quasi-momentumqP f−2p /l ,2p /lg (Bloch theo-rem). The quasienergy spectrumfnsqd is therefore made ofbands labeled byn (see for instance Fig. 10). More precisely,the states can be written as

ufn,qstdl = ef−i2pfnsqdt/k–ges−iqqducn,qstdl s13d

wherehucn,qstdlj is now strictly periodic in space and time(i.e., not up to a phase).

The evolution of the initial atomic wave function can beeasily computed from its expansion on theufn,qs0dl once theFloquet operator has been diagonalized. Theucn,qstdl, arethe eigenstates of the modified Floquet-Bloch Hamiltonian

H = sp + k–qd2/2 + 2ks1 − 2« sin tdsin2sq/2d s14dsubjected to strictly periodic space-time boundary condi-tions.

Dominant Floquet states may be determined by calculat-ing the inner product of the Floquet states with the initialatomic wave function. To obtain a phase space representationof Floquet states in momentum and position space one cancalculate the Husimi orQ-function. It is defined as

Qsq,p,td =1

2pk–ukq + ipuflu2, s15d

whereuq+ ipl is the coherent state of a simple harmonic os-cillator with frequencyv0 chosen asÎk in scaled units. Theposition representationkq8 uq+ ipl of the coherent state[15,16] is given by

kq8uq + ipl = S v0pk–D1/4expH− Fq8 − q2D G2 + ipqk–J s16d

up to an overall phase factor and whereD=Îk–/2v0. Floquetanalysis will be shown for some of the experimental param-eters which were presented in the previous section. Figure 9shows contour plots of the Husimi functions of two Floquetstates with opposite parity[(a) and(b)] for these parameterswhose presence allows dynamical tunneling to occur. Both ofthem are approximately localized on the classical period-1

regions of regular motion and they were selected so that theinitial atomic wave function has significant overlap withthem (26% and 44%, respectively). The initial experimentalstate has also significant overlaps22%d with a third statethat is shown in Fig. 9(c). The overlap is calculated using a

FIG. 9. Phase space representations of two Floquet states forq=0 that are involved in the dynamical tunneling are shown in(a)and (b). The Floquet states correspond to the experimental param-eters: modulation parameter«=0.29, scaled well depthk=1.66, andmodulation frequencyv /2p=250 kHz, which were utilized to ob-tain the experimental results shown in Fig. 2. A third state, shown in(c) also has significant overlap with the initial experimental state.


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coherent state that is centered on the periodic region of regu-lar motion, which is a good approximation of the initial ex-perimental state. The quasi-eigenenergy spectrum for this setof parameters is shown in Fig. 10. The quasi-eigenenergiesare plotted as a function of the quasi-momentum(Blochangle). Each line corresponds to one Floquet state labeled byn [see Eq.(13)]. The arrows point to the two Floquet statesshown in Figs. 9(a) and 9(b) that correspond to the tunnelingsplitting in Eq. (10). Most of the other lines correspond toFloquet states which lie in the classical chaotic phase spaceregion. Examples of phase space representations of such Flo-quet states can be found in Ref.[11]. Note that the spectrumis k– periodic in quasienergies. For quasi-momentumq=0 thesplitting between the two states that are shown in Fig. 9 isapproximately 0.08 in reduced units(energy in frequencyunits, fenergy/s2pk–dg). With a modulation frequencyv /2p=250 kHz one obtains a scaled Planck’s constantk–, thereforethe tunneling periodTtun which follows from Eq.(12) is 10which is in good agreement with the experiment.

The quasi-momentum plays a significant role in the ex-periments. The quasi-momentumq is approximately equal tothe relative velocityv between the wave packet(before the

lattice is turned on) and the lattice,v=q /m [17] if the stand-ing wave is adiabatically turned on. It was found in[11] thatit is of importance to populate a state whose quasi-momentum average is equal to zero. Moreover, quasi-momentum spread is also of importance. It has been shownin [18] that if the thermal velocity distribution is too broad,then the tunneling oscillation disappears. As can be seen inFig. 10 the quasi-eigenenergies of the two contributing Flo-quet states depend on the quasi-momentum(or Bloch angle).The tunneling period(which is inversely proportional to theseparation between these two quasi-eigenenergies) dependson the quasi-momentum. Using a thermal atomic cloud oneobtains a statistical ensemble of many quasi-momenta asthey initially move in random directions with respect to theoptical lattice. Atoms localized in individual wells can bedescribed by a wave packet in the plane wave basis andtherefore they are characterized by a superposition of manyquasi-momenta. The resulting quasi-momentum widthwashes out the tunneling oscillations(see[18], Fig. 3). Infact in another experiment Stecket al. [3] found that theamplitude of the mean momentum oscillations resulting froma tunneling process between two librational islands of stabil-ity decreased when the initial momentum width of the atomiccloud was increased.

Figure 11 shows contour plots of Husimi functions for thetwo dominant Floquet states for the modulation parameter«=0.30, scaled well depthk=1.82, and modulation fre-quencyv /2p=222 kHz, which corresponds to experimentalresults shown in Fig. 6. The states are selected to have maxi-mum overlap with the initial wave packet(38% and 44%). Incontrast to the experimental results shown in Fig. 2 there areonly two dominant Floquet states. The quasi-eigenenergyspectrum for this set of parameters(not shown) reveals alevel splitting of approximately 0.15 in reduced units, thecalculated tunneling period is 6 modulation periods which isin good agreement with the experiment.

When comparing the stroboscopic evolution of the meanmomentum shown in Fig. 2(modulation parameter«=0.29,scaled well depthk=1.66, and modulation frequencyv /2p=250 kHz) with the one shown in Fig. 6(modulation param-eter«=0.30, scaled well depthk=1.82, and modulation fre-quencyv /2p=222 kHz), one finds that it is less sinusoidal.This can be explained in the Floquet picture. While there arethree dominant Floquet states for the first case(Figs. 2 and 9)(three Floquet states with significant overlap with the initialexperimental state), there are only two dominant Floquetstates for the second case(Figs. 6 and 11) resulting in a moresinusoidal tunneling oscillation.

D. Loading analysis of the Floquet superposition state

The initial atomic wave packet is localized around theclassical period-1 region of regular motion by inducing asudden phase shift to the standing wave. This enables theobservation of dynamical tunneling. In the Floquet picturethe observation of dynamical tunneling requires that the ini-tial state has support on only a few dominant Floquet states,preferably populating only two with a phase space structureas shown in Fig. 9. Optimizing the overlap of the initial state

FIG. 10. Quasi-eigenenergy spectrum for parameters: modula-tion parameter«=0.29, scaled well depthk=1.66, and modulationfrequencyv /2p=250 kHz. The quasi-eigenenergy is measured inreduced energy units and it is a function of the Bloch angle. TheBloch angle is equal to the quasi-momentum multiplied with thespatial periodl /2 of the lattice. Each line corresponds to one Flo-quet state labeled byn [see Eq.(13)]. The quasi-eigenenergy ofmost Floquet states is strongly dependent on the quasi-momentumbut it is not the case for the states of the tunneling doublet. Recallthat the quasi-momentumq is proportional to the phase taken by astate when spatially translated byl /2. In other words, fixingqimposes on the states some conditions on the boundary of one el-ementary cellfq,q+l /2g. If a state is localized deep inside a cell,changing the boundary conditions(i.e., q) will not affect the stateso much and the corresponding quasienergies appear as curves thatare approximately parallel to theq axis. However, one should ex-pect a strongq-dependence for a state that spreads over at leastDq,l /2.


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with these Floquet “tunneling” states should maximize theobserved tunneling amplitude. Here we carry out analysisconfirming this prediction. This corresponds to optimizingthe overlap of the initial experimental state with the period-1regions of regular motion. Figure 12 shows the mean atomicmomentum as a function of the interaction time with thestanding wave which is plotted for a range of the initialphase shiftw of the standing wave. The solid line(circles)corresponds to an interaction time with the standing wave ofn+0.25 modulation periods and the dashed line(diamonds)corresponds to an interaction time ofn+0.75 modulation pe-riods. The simulations are made using the Gross-Pitaevskiiequation for modulation parameter«=0.29, scaled welldepth k=1.66, and modulation frequencyv /2p=250 kHzwhich corresponds to Fig. 2. A phase shiftw=0 correspondsto localizing the wave packet exactly at the bottom of thewell and w=p corresponds to localizing it exactly at themaximum of the standing wave well. Symmetry dictates thatno tunneling oscillation can occur for these two loadingphases. This is also shown in Fig. 12(the mean momentum

curve forw=p is not shown but it is the same as forw=0).The amplitude of the tunneling oscillations changes stronglywhen the initial phase shiftw of the standing wave ischanged. The best overlap with the tunneling Floquet statesis obtained for the phase shiftw somewhere in between0.2532p and 0.3032p which corresponds to placing thewave packet halfway up the standing wave well. This resultis in good agreement with the structure of the tunneling Flo-quet states as shown in Fig. 9. When changingw there is nochange in the observed tunneling period. This is to be ex-pected as mainly the loading efficiency for the “tunneling”Floquet states varies whenw is varied. It should be noted thatthe simulations are carried out for a relatively large scaled

Planck’s constantk– which means that the wave packet size israther large compared to characteristic classical phase spacefeatures like the period-1 regions of regular motion. Thisanalysis shows that the tunneling amplitude sensitively de-pends on the loading efficiency of the tunneling Floquetstates and that there is a smooth dependency of the tunnelingamplitude on this loading parameter.

IV. PARAMETER DEPENDENCY OF THE TUNNELINGOSCILLATIONS

The scaled parameter space for the dynamics of the sys-tem is given by the scaled well depthk and the modulationparameter«. Both parameters will significantly change thestructure and number of the contributing Floquet states. Ithas been found that a strong sensitivity of the tunneling fre-quency on the system parameters is a signature of chaos-assisted tunneling[11,18] where a third state associated withthe classical chaotic region interacts with the tunneling Flo-quet states. A Floquet state that is localized inside a region ofregular motion that surrounds another resonance can also in-teract with the tunneling doublet(two tunneling Floquetstates), this phenomenon is known as resonance-assisted tun-neling [19].

Comprehensively exploring the parameter space and itsassociated phenomena is out of the scope of this paper. In-stead an analysis associated with our experiments will bepresented here showing one scan of the scaled well depthkand another scan of the modulation parameter« around theexperimental parameter regime. The results are shown in theform of plots of the mean momentum as a function of theinteraction time with the modulated standing wave. The solidline (circles) corresponds to an interaction time with thestanding wave ofn+0.25 modulation periods and the dashedline (diamonds) corresponds to an interaction time ofn+0.75 modulation periods.

Figure 13 shows the scaled well depthk being variedfrom 1.10 to 1.75. The other parameters are held constant(modulation parameter«=0.29, modulation frequencyv /2p=250 kHz, and phase shiftw=0.2132p). The mo-mentum distribution evolution is shown to illustrate intricatechanges in the tunneling dynamics when the parameters arevaried. Both the tunneling frequency spectrum and the tun-neling amplitude are strongly dependent onk. Often one cansee more than just one dominant tunneling frequency. Forexample, fork=1.35 the two dominant tunneling frequencies

FIG. 11. Phase space representations of two Floquet stateswhose presence can lead to the occurrence of dynamical tunneling.The Floquet states correspond to the experimental parameters:modulation parameter«=0.30, scaled well depthk=1.82, andmodulation frequencyv /2p=222 kHz, which were utilized to ob-tain the experimental results shown in Fig. 6.


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which contribute to the tunneling oscillation have a period ofapproximately 3.9 and 34 modulation periods. In the intervalof approximatelyk=1.5 and 1.8 the tunneling oscillationshave a more sinusoidal shape indicating the presence of onlyapproximately two dominant tunneling states. In this intervalthe tunneling frequency is peaked atk=1.55 with a tunnelingperiod of approximately 8.9 modulation periods. In order toanalyze further the behavior of the tunneling frequency, weshow the low frequency(diamonds) and the high frequencycomponent(squares) of the tunneling frequency as a functionof the scaled well depthk in Fig. 14. The high frequencycomponent is plotted only for values ofk where it is visiblein the stroboscopic momentum evolution shown in Fig. 13.The occurrence of multiple tunneling frequencies resultsfrom the presence of more than two dominant Floquet states.The minimum of the low frequency component of the tun-neling frequency is due to the level crossing of two contrib-uting Floquet states(see squares and filled circles atk=1.25 in Fig. 15). The error bars result from the readoutuncertainty of the tunneling period from the simulations.This analysis reveals some of the intricate features of thetunneling dynamics. Instead of a smooth parameter depen-dency a distinct rise and fall(in the low frequency compo-nent) in the tunneling frequency versus scaled well depthappears. The tunneling frequency minimum is centered atk

smaller values of« the corresponding classical phase space ismainly regular. There is no distinct oscillation at«=0.10which is centered at zero momentum. Distinct tunneling os-cillations start to occur at«=0.14 with a gradually increasingtunneling frequency. While there is a tunneling period ofapproximately 67 modulation periods at«=0.14, the tunnel-ing period is only approximately 7.7 modulation periods at«=0.38. For larger values of the modulation parameter« theoscillations become less sinusoidal indicating the presence ofan increasing number of dominant Floquet states. The errorbars result from the readout uncertainty of the tunneling pe-riod from the simulations.

V. MOVING THE QUANTUM SYSTEM TOWARDS THECLASSICAL LIMIT

A fundamental strength of the experiments which are dis-cussed here is that they are capable of exploring the transi-tion of the quantum system towards the classical limit by

decreasing the scaled Planck’s constantk–.Here a quantum system with mixed phase space exhibit-

ing classically chaotic and regular regions of motion ismoved towards the classical limit. By adjusting the scaled

Planck’s constantk– of the system, the wave and particlecharacter of the atoms can be probed although some experi-

FIG. 13. Mean momentum asa function of the interaction timewith the modulated standing wavefor different values of the scaledwell depthk (modulation param-eter «=0.29, modulation fre-quency v /2p=250 kHz, andphase shift w=0.2132p). Thesolid line (circles) corresponds toan interaction time with the stand-ing wave of n+0.25 modulationperiods and the dashed line(dia-monds) corresponds to an interac-tion time of n+0.75 modulationperiods.


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mental and numerical restrictions limit the extent of thisquantum-classical probe. This should enhance our under-standing of nonlinear dynamical systems and provide insightinto their quantum and classical origin. It is out of the scopeof this paper to present anything more than a short analysisrelevant to the experimental results. The quantum-classicalborderland is analyzed by considering the mean momentumas a function of the interaction time with the modulatedstanding wave for different values of the scaled Planck’sconstantk–. Figure 17 shows results for modulation parameter«=0.29, scaled well depthk=1.66, and phase shiftw=0.2132p which corresponds to experimental results shown inFig. 2. The scaled Planck’s constantk– is varied by adjustingthe modulation frequencyv /2p and leaving the scaled welldepth k, the modulation parameter«, and the initial phaseshift w constant. Results are shown for the scaled Planck’sconstant k– ranging from 0.40sv /2p=500 kHzd to 1.33sv /2p=150 kHzd. The error bars result from the readout un-certainty of the tunneling period from the simulations. Asshown in previous work[21], the momentum of the regionsof regular motion is proportional to the modulation fre-quency. Note that this is a purely classical feature that woulddisappear if the scaled momentum would be plotted instead

FIG. 14. Tunneling frequency as a function of the scaled welldepth k (modulation parameter«=0.29, modulation frequencyv /2p=250 kHz, and phase shiftw=0.2132p) taken from Fig. 13.Diamonds and squares denote low and high frequency components,respectively.

FIG. 15. Floquet spectrum as a function of the scaled well depthk (modulation parameter«=0.29, modulation frequencyv /2p=250 kHz, and phase shiftw=0.2132p. The Floquet states with maximum overlap are marked by black bullets, white bullets are used forsecond most overlap, and white squared bullets show third most overlap. Phase space representations(Husimi functions) of some of theFloquet states for different values ofk are also shown.


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of the real momentum. Intuitively one would expect the tun-neling frequency to decrease as the system becomes moreclassical. However, a different quite surprising phenomenonoccurs. Considering the modulation frequency interval be-tween 225 and 325 kHz the tunneling frequency graduallydecreases as the system becomes more classical. The sameapplies for the interval between 150 and 175 kHz and theinterval from 425 to 475 kHz. Qualitative changes occur atapproximately 200, 350(375), and 500 kHz where the tun-neling oscillation shows multifrequency contributions andthe tunneling frequency then significantly increases as theeffective Planck’s constant is further decreased. A more de-tailed analysis will be the subject of future study. The threeintervals where the tunneling frequency decreases appear asthree arms in the graph. Atv /2p=200, 350, and 375 kHzthe system cannot be well described using one dominant tun-neling frequency, therefore no data is shown at these fre-quencies.

The results of this first analysis indicate that the transitionfrom quantum to classical physics contains many fascinatingdetails to be explored. The results for the driven pendulum inatom optics shows that the transition from quantum to clas-sical dynamics is not smooth.

VI. CONCLUSION

Recent dynamical tunneling experiments[1] were ana-lyzed using simulations of the Gross-Pitaevskii equation andthe corresponding Floquet theory. The main features of theexperiments can be explained by a two or three state frame-work that is provided by Floquet theory. We have identifiedthe relevant Floquet states and shown their Husimi functions.Note that the mean-field interaction was negligible in theexperiments. Tunnelling period and amplitude are in goodagreement with GP simulation and Floquet theory.

An analysis of the parameter space has shown that tunnel-ing frequency is strongly dependent on the system param-eters scaled well depthk, modulation parameter«, and the

scaled Planck’s constantk–. While we find an approximatelylinear dependence of the tunneling frequency on the modu-lation parameter« for the set of experimental parameters,there is a distinct spike in the tunneling frequency as a func-tion of the scaled well depthk.

However, this cannot be interpreted as a signature ofchaos-assisted tunneling essentially because the states cannotbe clearly identified with chaotic or regular regions of theclassical phase space, therefore the notion of chaos-assistedtunneling is difficult to use in this context. We note that it isimportant to decrease the scaled Planck’s constant by at leastan order of magnitude[18] for an observation of chaos-assisted tunneling to be nonambiguous. We have simulatedwhen the system is moved towards the classical limit. Abifurcationlike behavior results which can be understood interms of quantum bifurcations of the contributing Floquetstates.

ACKNOWLEDGMENTS

This work was supported by the Australian ResearchCouncil. The experimental results that are shown here wereaccomplished by W. K. Hensinger, H. Häffner, A. Browaeys,N. R. Heckenberg, K. Helmerson, C. McKenzie, G. J. Mil-burn, W. D. Phillips, S. L. Rolston, H. Rubinsztein-Dunlop,and B. Upcroft in the context of Ref.[1] and we would liketo thank the authors. The corresponding author W. K. H.would like to acknowledge interesting discussions with Wil-liam D. Phillips and Steven Rolston during the preparation of

FIG. 16. Tunneling frequency as a function of the modulationparameter« (scaled well depthk=1.66, modulation frequencyv /2p=250 kHz, and phase shiftw=0.2132p). The error bars re-sult from the readout uncertainty of the tunneling oscillations fromthe simulations. The high frequency component that is present inthe data for 0.00,«,0.18 is not shown.

FIG. 17. Dominant tunneling frequency as a function of themodulation frequencyv /2p for modulation parameter«=0.29,scaled well depthk=1.66, and phase shiftw=0.2132p which cor-responds to experimental results shown in Fig. 2. As expected, theoverall tendency consists of a decreasing of the tunneling frequencyas the modulation frequency increases and therefore as the scaledPlanck’s constant decreases, however, the transition is not assmooth as one might expect.


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theoretical results presented in this paper. W. K. H. wouldlike to thank NIST and Laboratoire Kastler-Brossel for hos-pitality during part of the work for this paper. LaboratoireKastler Brossel de l’Université Pierre et Marie Curie et de

l’Ecole Normale Supérieure is Unité Mixte de Recherche8552 du CNRS. Laboratoire de mathématiques et de phy-sique théorique de l’Université François Rabelais is UnitéMixte de Recherche 6083 du CNRS.

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analysis of dynamical tunneling experiments with a bose ...73356/uq73356.pdftrapsr,td + vsr,td + n...

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