quantum chaos with cesium atoms: pushing the boundaries · quantum chaos with cesium atoms: pushing...

Physica D 131 (1999) 78–89

Quantum chaos with cesium atoms: pushing the boundaries

B.G. Klappauf, W.H. Oskay, D.A. Steck, M.G. Raizen∗Department of Physics, The University of Texas at Austin, Austin, TX 78712-1081, USA

Abstract

Atomic motion in pulsed, periodic optical potentials provides a unique experimental testing ground for quantum chaos. Inthe first generation of experiments with sodium atoms we observed dynamical localization, a quantum suppression of chaoticdiffusion. To go beyond this work we have constructed an experiment with cold cesium atoms, and report our first resultsfrom this system. The larger mass and longer wavelength push out the momentum boundary in phase space that arises fromthe nonzero duration of the pulses. This feature should enable the study of noise effects and dimensionality on dynamicallocalization. We propose a new method of quantum state preparation based on stimulated Raman transitions for studies ofmixed phase space dynamics.c©1999 Elsevier Science B.V. All rights reserved.

PACS:05.45.+b, 32.80.Pj, 42.50.Vk, 72.15.RnKeywords:Kicked rotor; Quantum chaos; Dynamical localization; Atom optics

1. Introduction

The interface between nonlinear dynamics andquantum mechanics has become an active area ofresearch in recent years. One of the key results inthis field, due to Boris Chirikov and co-workers, isdynamical localization. This striking effect is a quan-tum suppression of classical (chaotic) diffusion, andhas stimulated a great deal of interest and discus-sion since it was first predicted almost 20 years ago[1–9].

Rydberg atoms in strong microwave fields pro-vided the first experimental system that addressedthis problem, and the observed suppression of ioniza-tion was attributed to dynamical localization [10–13].To go beyond these results, it is important to findnew experimental systems to investigate dynamical

∗ Corresponding author. E-mail: [email protected].

localization as well as other problems in quantumchaos.

In general, the observation of dynamical localiza-tion requires a predominantly chaotic (classical) phasespace, because diffusion can be restricted by residualstable islands and by classical boundaries accordingto the Kolmogorov, Arnol’d, Moser (KAM) theorem[14]. In addition, the duration of the experiment mustexceed the “quantum break time”, when quantum ef-fects manifest themselves. Finally, the system must besufficiently isolated from the environment that quan-tum interference effects can persist. All of these con-ditions can be satisfied by a system of cold atomsthat are exposed to time-dependent standing waves oflight [15,16]. The typical potentials in this system arehighly nonlinear, resulting in chaotic classical dynam-ics. Since dissipation can be made negligibly small inthis system, quantum effects can become important.These features led to a series of experiments in our

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B.G. Klappauf et al. / Physica D 131 (1999) 78–89 79

group on dynamical localization and quantum chaoswith cold sodium atoms, establishing atom optics as anew experimental testing ground for the field [17–21].The simplest experimental configuration in that workwas a standing wave of light that was pulsed on peri-odically. This system was an experimental realizationof the kicked rotor, which has been a paradigm forclassical and quantum chaos for many years.

To go beyond this first generation of experiments,we wish to address several key topics. The first is tostudy the effects of noise and dissipation on dynamicallocalization. The signature in this case is the destruc-tion of localization, with subsequent diffusive growthin momentum [22–24]. A second direction is to studythe role of dimensionality on dynamical localization.While exponential localization occurs in one dimen-sion, a sharp increase in the width of the exponentiallylocalized momentum distributions is predicted in twodimensions, and a transition to a delocalized state ispredicted in three dimensions [25].

Early efforts to observe these effects in sodiumwere hampered by the presence of a boundary inmomentum space that arises from the nonzero pulseduration of the interaction Hamiltonian [19]. To over-come this problem, we have constructed a new ex-periment with laser-cooled cesium atoms. The largeratomic mass and the longer wavelength of the atomictransition push out the boundary to much highermomenta.

This paper describes our new cesium experiment,which is a realization of the quantum kicked rotor,and presents a detailed study of boundary effects ondynamical localization. In Section 2 we give a theo-retical background on atomic motion in a far-detuneddipole potential, and provide a classical analysis ofthe boundary in phase space. In Section 3 we describethe general experimental approach. In Section 4 wediscuss the new experimental results with cesium. Fi-nally, in Section 5 we describe some directions for fu-ture work including a new method of quantum statepreparation based on stimulated Raman transitions.This method will prepare an initial condition that isa minimum uncertainty state in phase space, openingthe way for detailed studies of quantum transport inmixed phase space.

Fig. 1. Classical calculation of the effective stochasticity parameterKeff as a function of momentum for square pulses of varioustemporal widths. The horizontal line represents theδ-kick case.The other curves represent square pulses with widthsα = 0.014(heavy solid line),α = 0.024 (dashed),α = 0.049 (dash–dot), andα = 0.099 (dotted). The well depth is adjusted in each case to givethe same maximum value ofKeff = 10.5 atp/2~kL = 0. The pointwhereKeff drops to∼ 1 in each case is the classical momentumboundary. The typical limit of our momentum measurements is|p/2~kL | ≈ 80. Note that we have suppressed the curves aftertheir first zero-crossing.

2. Theoretical background

To describe our system we begin with a two-level atom with transition frequencyω0 interactingwith a pulsed standing wave of linearly polarized,near-resonant light of frequencyωL. For sufficientlylarge detuningδL = ω0 − ωL (relative to the nat-ural linewidth), the excited state amplitude can beadiabatically eliminated [16]. The atom can then betreated as a point particle. This approximation leadsto the following kicked-rotor Hamiltonian for thecenter-of-mass motion of the atom

H = p2/2M + V0 cos(2kLx)

∞∑

n=−∞F(t − nT ). (1)

HereV0 = ~Ω2/8δL, kL is the wave number of thelight, Ω = 2µE0/~ the resonant Rabi frequency,µ

the atomic dipole moment,E0 the electric field ofa single traveling wave component of the standing

80 B.G. Klappauf et al. / Physica D 131 (1999) 78–89

Fig. 2. Classical phase portraits for the kicked rotor withK = 10.5, comparingδ-kicks (a) to square pulses of widthsα = 0.014 (b) andα = 0.049 (c). Case (b) is typical for our localization experiments. This case mimics theδ-kicked rotor for a momentum region muchlarger than that used in our experiments (|p/2~kL | . 80).

wave,F(t) a pulse centered att = 0 with durationtp, andT is the period of the standing wave pulses.It is convenient to rescale our coordinates and use thedimensionless Hamiltonian

H = ρ2/2 + k cos(φ)

∞∑

n=−∞f (τ − n), (2)

whereφ = 2kLx, ρ = (2kLT/M)p, τ = t/T , f (τ)

is a pulse of unit amplitude and scaled durationα =tp/T , k = (8V0/~)ωrT

2 is the scaled kick ampli-tude, ωr = ~k2

L/2M is the recoil frequency,H ′ =(4k2

LT 2/M)H , and we have dropped the prime onH .In the quantized model,φ andρ are conjugate vari-ables that satisfy the commutation relation [φ, ρ] =i–k, where–k = 8ωrT is a scaled Planck constant. In thelimit where f (τ) becomes aδ-function, the productηk reduces to the classical stochasticity parameterK

for the δ-kicked rotor, whereη = ∫ ∞−∞f (τ)dτ . The

stochasticity parameter completely specifies the clas-sical δ-kicked rotor dynamics. ForK & 1 the clas-

sical δ-kicked rotor dynamics are globally chaotic, inthe sense that there are no invariant tori that preventtrajectories in the main chaotic region from attainingarbitrarily large momenta. ForK > 4 the primary res-onances become unstable, and the phase space is pre-dominantly chaotic.

The effects of a nonzero pulse width can be seen byrewriting the sequence of kicks as a discrete Fourierseries and analyzing the amplitudes of the primaryresonances of the system. Applying this procedure tothe Hamiltonian (2)

H = ρ2/2 + k

∞∑

m=−∞f (2πm) cos(φ − 2πmτ), (3)

wheref (2πm) is the Fourier transform of the pulsefunction f (τ). This Hamiltonian has primary reso-nances located atρ = dφ/dt = 2πm. Therefore, theFourier transform evaluated atm = ρ/2π modifies theeffective stochasticity parameterKeff as a function of


momentum. For a square pulse, which approximatesthe pulse used in our experiments, this factor can bewritten as

Keff = αksin(αρ/2)

αρ/2, (4)

where Keff(ρ = 0) = αk plays the role of theδ-kicked rotor stochasticity parameterK. This depen-dence ofKeff on momentum is displayed in Fig. 1,which compares several square-pulse cases to theδ-kick case. The dynamics of the system undergo a tran-sition asKeff drops to∼ 1, because of the presence ofKAM surfaces that span the phase space and act as abarrier against momentum diffusion. Thismomentumboundaryis illustrated in Fig. 2, which shows classicalphase portraits for theδ-kicked rotor (Fig. 2(a)) andtwo square-pulse cases (Figs. 2(b) and (c)). The phaseportrait of Fig. 2(b) is typical of our current cesiumexperiment, and the boundaries are well outside therange of detectable atomic momenta,|p/2~kL | . 80.Our previous experimental work in sodium was lim-ited to a bounded chaotic region similar to that in Fig.2(c). While this situation enabled the observation ofthe transition from the short time classical diffusionto the exponential localization of the momentum dis-tribution after the quantum break time, it could not beused to explore any continued momentum growth dueto delocalization beyond that point.

Fig. 3 shows classical Monte-Carlo simulations(with 2 × 105 particles) of the same systems as inFig. 2. The momentum distributions after 68 kicksare plotted here. The distributions are similar in thecentral region, but the boundary suppresses diffusionin the wings of the distributions in the square-pulsecases, especially in the case with the widest pulses(α = 0.049). The nearly Gaussian initial momentumdistribution is also plotted, and it corresponds to theinitial distribution in the experiments.

A simple classical model of the boundary providesus with relevant scaling parameters and makes clearthe advantage of using cesium rather than sodium.Consider an atom with a momentum such that it travelsone period of the standing wave during a kick. Thenthe momentum transferred to the atom by the potentialaverages to zero, and the particle no longer diffuses.

Fig. 3. Comparison of momentum distributions from classical sim-ulations for different pulse widths after 68 kicks, withK = 10.5.The nearly Gaussian initial condition is shown as a dotted line.Notice that even forα = 0.014 (dashed), the classical growth isslowed. Forα = 0.049 (light solid) the initial condition is unaf-fected outside the boundary. The heavy line is a classical simula-tion for the δ-kicked rotor. The vertical scale is logarithmic andin arbitrary units.

This situation occurs whenvtp = 12λ, and hence an

estimate for the boundary location is

p/2~kL = Mλ2/8π~tp. (5)

Notice that this expression corresponds to the firstzero-crossing of (4). The larger mass and longer op-tical wavelength of cesium provide a 12-fold increaseover sodium in the momentum boundary location fora given pulse width.

3. Experiment

The experimental setup (Fig. 4) is similar to that ofour earlier sodium-based quantum chaos experiments[19]. The experiments are performed on laser-cooledcesium atoms in a magneto-optic trap (MOT) [26,27].

The atoms are trapped in a stainless steel UHVchamber, in contrast to the earlier experiments in ourgroup that used a quartz interaction chamber. All op-tical viewports are anti-reflection coated on both sides


Fig. 4. Schematic diagram of the experimental setup. Two diode lasers provide the light for the MOT, and a Ti : sapphire laser providesthe far-detuned standing wave.

to reduce intensity fringes on the laser beams. An am-pule containing cesium metal is attached to the cham-ber through a valve that is opened occasionally to leakcesium vapor into the chamber. The valve is necessarybecause of the rather high room temperature vaporpressure of cesium. A pair of anti-Helmholtz coils sur-rounding the chamber provides a magnetic field gra-dient of 11 G/cm for the MOT. When the current tothe anti-Helmholtz coils is switched off, the magneticfields decay exponentially after a brief transient, witha 1/e time of about 3 ms. This decay time is longerthan in the earlier sodium-based experiments becauseof induced currents in the metal chamber.

Two single-mode diode lasers (L1, L2) at 852 nmprovide the light for cooling, trapping, and detectionof the cesium atoms. L1 is a 100 mW distributed Braggreflector (DBR) diode laser locked via frequency-modulation (FM) saturated-absorption spectroscopyto the(6S1/2, F = 4) → (6P3/2, F

′ = 4, 5) crossoverresonance. The main beam from L1 is double-passedthrough a tunable acousto-optic modulator (AOM1)centered at 80 MHz that provides fast control overthe intensity and detuning of the beam. During thetrapping stage of the experiment, the light from L1is tuned 15 MHz to the red of the(6S1/2, F = 4) →(6P3/2, F

′ = 5) cycling transition. The light fromL1 passes through a spatial filter (SF), is collimatedwith a waist of 1.4 cm, and has a typical power of

23 mW at the chamber. The beam is divided into threebeams of equal intensity by two beamsplitters withtransmissions of 66% and 50%. The three beams areretroreflected through the center of the chamber in astandard six-beam MOT configuration.

The repump laser, L2, is a 150 mW Littrow cavitygrating-stabilized diode laser. This beam is used toprevent optical pumping into theF = 3 ground stateduring the trapping and detection stages. This laser iselectronically locked to the center of the(6S1/2, F =3) → (6P3/2, F

′ = 4) saturated-absorption reso-nance. The beam has a typical power of 27 mW atthe chamber and a waist of 7.5 mm. The intensityis controlled by a mechanical shutter (SH) with arise/fall time of 1 ms. The beam is combined with thevertical arm of the light from L1 via a polarizing cubebeamsplitter and retro-reflected through the chamber.Optical isolators (ISO1, ISO2) are used to minimizeoptical feedback to L1 and L2.

The entire experimental timing and data acquisitionsequences are computer controlled. The timing dia-gram for the experiment is given in Fig. 5. After trap-ping and initial cooling, the intensity of L1 is reducedfor 1 ms and the detuning is increased to 39 MHz tofurther cool the sample. Typically, we trap 106 atomsfrom the background vapor. The momentum distri-bution of the atoms is nearly Gaussian, although thetails of the distribution are more populated than in a


Fig. 5. An idealized timing diagram for the experiment.

purely Gaussian distribution. The center of the dis-tribution typically fits well to a Gaussian distributionwith σp/2~kL = 4.4, and 96% of the atoms are con-tained in this Gaussian. The position distribution ofthe atoms is also Gaussian, withσx = 0.1 mm.

After the final cooling, the trapping fields are turnedoff, leaving the momentum distribution unchanged,and the interaction potential pulses are then turnedon. A stabilized single-mode Ti : sapphire laser (L3)pumped by an argon-ion laser provides the pulsedstanding wave. The light from L3 passes througha fixed-frequency 80 MHz acousto-optic modulator(AOM2) that controls the pulse sequence. The beamis spatially filtered, centered on the atoms, and retro-reflected through the chamber to form a standingwave. The beam has a typical power of 290 mWat the chamber and a waist of 1.44 mm. The abso-lute wavelength of L3 is measured with a scanningMichelson interferometer wavemeter. A scanning con-focal Fabry–Perot cavity with a 1.5 GHz free spectralrange is used to monitor long-term drift of L3 duringmeasurements.

For all the experiments described here we detunedthis beam 6.1 GHz to the red of the cycling transi-tion, with typical fluctuations of about 100 MHz. The

pulse sequence consisted of a series of fixed-lengthpulses with a rise/fall time of 75 ns and less than 3 nsvariation in the pulse duration. Sequences using pulsewidths between 283 and 1975 ns (full width at halfmaximum) were used in these experiments to demon-strate the role of the momentum boundary. The periodwas 20µs with less than 4 ns variation per pulse pe-riod as measured with a fast photodiode. The proba-bility of spontaneous scattering was less than 0.5% perkick period for all of the parameters used. The phasenoise of the standing wave due to vibrations in theoptical system resulted in less than 8% of a standingwave period phase drift after 200 kicks, with a typicalfluctuation timescale of 0.5 ms.

The detection of momentum is accomplished bya time-of-flight method. The atoms drift in the darkfor a controlled duration, typically 15 ms. The trap-ping beams are then turned on in zero magnetic field,forming an optical molasses [26] that freezes the po-sition of the atoms. The final spatial distribution isrecorded via fluorescence imaging in a short (10 ms)exposure on a cooled charge-coupled device (CCD)camera. The final distribution and the free-drift timeenable the determination of the momentum distribu-tion. The maximum momentum that we can measure is


Fig. 6. Time evolution of a typical kicked-rotor experiment forK = 11.5 ± 10%, T = 20µs, 0.283µs pulse width, and–k = 2.08. Theinitial distribution is nearly Gaussian withσp/2~kL = 4.4. The final distribution is exponential with a localization length of 13. Note thatthe vertical axis is linear and in arbitrary units, and the time increment between distributions is 2 kicks.

limited by the size of the CCD. For a 15 ms drift time,we can detect momenta within|p/2~kL | ≈ 80. Usingshorter drift times, we could measure larger momentaat the expense of resolution. The initial spatial distri-bution is not deconvolved from the final momentumdistributions, because the effect on the RMS width ofour final distributions is on the order of 1%.

The pulse period wasT = 20µs, correspondingto –k = 2.08. The kick strengthk was chosen to pro-vide the best exponentially localized momentum dis-tributions. For the shortest pulses, we usedV0/h =3.55 MHz, yielding a classical stochasticity parameterof K = 13.1. For the longest pulses we usedV0/h =0.94 MHz, corresponding toK = 24. The absoluteuncertainty inK is ±10%, and the largest contribu-tions are due to the measurement of beam profile andthe absolute laser power calibration.

The momentum boundary due to the nonzero pulsewidth is |p/2~kL | = 213 for our typical operatingparameters. This value is a factor of four larger thanin our earlier sodium experiments. The correspondingreduction in the effective value ofK is only 6% out to|p/2~kL | = 40 and 25% at our maximum detectablemomentum of|p/2~kL | ≈ 80.

We can also estimate the effects of collisionsfrom the experimental work of Dalibard and cowork-

ers [28], who measured a collision cross-section of5 × 10−11 cm2 for cesium atoms prepared in theF = 4, mF = 4 ground sublevel with a tempera-ture of 5µK. In our initial distribution, the densityis about 1011 cm−3, and the mean velocity is 5 cm/s.These figures lead to a collision probability of only2.5% in 1 ms, or 0.05% per kick period. This resultactually overestimates the collision probability be-cause our atoms are distributed among the variousmF

sublevels, and so the actual collision cross-section issmaller than the figure used here.

4. Data and results

The results presented here illustrate the capabilitiesof the cesium setup. We observe dynamical localiza-tion and study experimentally the effects of nonzeropulse width on the kicked-rotor system. The resultingmomentum distributions are compared with classicalsimulations. We will present a detailed study of the de-pendence of quantum localization on the stochasticityparameterK in a future publication.

Fig. 6 shows the time evolution for a typicalkicked-rotor experiment, from the initial distributionthrough 68 kicks. The momentum distribution starts


Fig. 7. Comparison of momentum distribution evolution for four different kick pulse widths from 0.3 to 2µs. Distributions for five differenttimes are shown on each graph: 0 kicks (heavy solid line), 17 kicks (dotted), 34 kicks (dashed), 51 kicks (dash–dot), and 62 kicks (solid).These data show the effect of the boundary on the evolution of the momentum distribution. The vertical scales are logarithmic and inarbitrary units.


Fig. 8. Comparison of experimental momentum distributions after51 kicks for two different pulse widths,α = 0.014 (heavy line)and α = 0.049 (thin line). In theα = 0.014 case, also shown inFig. 7(a), the boundary is at|p/2~kL | = 213. This boundary widthis much larger than the width of the distribution, and hence theboundary does not significantly affect the exponential distribution.In the α = 0.049 case, also shown in Fig. 7(c), the boundary islocated at|p/2~kL | = 61. This boundary width is clearly insidethe base of the exponential distribution; the momentum distributionis now distorted in its wings, while the shape at the center isnearly identical to the first case. Note that the vertical scale islogarithmic and in arbitrary units.

out with a Gaussian profile, which makes a rapidtransition to a broader, exponential distribution. Weobserve a continued slow growth in the momentumdistribution until the end of the experiment, with thedistributions remaining exponential. Fig. 7(a) showsfive momentum distributions for an experiment withsimilar parameters, taken at increments of 17 kicks.Note that the distribution remains exponential overthe nearly three orders of magnitude in intensity thatare resolvable in our experiment. The slow growthof the localized distribution is also evident in thisfigure.

Fig. 7 compares experiments using four differ-ent pulse widths with the corresponding momentumboundaries as predicted by Eq. (5). In the first case,Fig. 7(a), the boundary of|p/2~kL | = 213 is threetimes larger than the|p/2~kL | = 70 resolvable widthof our distribution. This situation is ideal for exper-

Fig. 9. Growth of energy with time for the cases in Fig. 8. Themomentum boundaries are|p/2~kL | = 213 for the caseα = 0.014(circles) and|p/2~kL | = 61 for the caseα = 0.049 (triangles).Note that the initial growth rate is the same until the distributionsdiffuse out to larger momenta whereKeff begins to drop sharply.

imental studies of the quantum kicked rotor, sincethe momentum distribution remains well within theboundary. In Fig. 7(b), we see that although the dis-tance between the boundaries at|p/2~kL | = 125 isless than twice the width of the final distribution, thereis little effect on the shapes of the distributions. InFig. 7(c), the boundary is located at|p/2~kL | = 61,and it has a clear effect on the wings of the distribu-tions. Notice, however, that after 17 kicks the distri-bution still looks exponential. Fig. 7(d) shows a casewhere the boundary at|p/2~kL | = 30 is well withinthe exponential distributions shown in Fig. 7(a). Inthis case, the distribution quickly reaches the bound-ary, and the diffusion process halts before quantumlocalization sets in. It is also interesting to note thatthe portions of the initial distributions that are outsidethe boundary in Figs. 7(c) and (d) remain stationary.This behavior is a result of the presence of the invari-ant curves past the boundary, as shown in Fig. 2, thatstrongly inhibit momentum transport. These observa-tions are consistent with a previous theoretical studyof the kicked rotor with finite pulses [29].

The dynamics near the center of the distribution ap-pear to be relatively insensitive to the location of theboundary. Fig. 8 compares the distributions after 51


Fig. 10. Comparison of experimental momentum distributions after68 kicks to the classical simulations of Fig. 3. Both square-pulsecases, the experiment (light solid line) and the simulation (dashedline), have the same pulse widthα = 0.049 and hence the sameboundary, althoughK is slightly different between the two cases(K = 11.1 and 10.5, respectively). The experimental data clearlymanifests its quantum nature through a characteristic exponentialprofile in the central region. The classical square-pulse simulationinstead matches the classicalδ-kicked rotor simulation (heavy solidline) in the central region. The initial distribution (dotted line)remains unchanged outside the boundary. Note that the verticalscale is logarithmic and in arbitrary units.

kicks from Figs. 7(a) and (c). It is clear that the por-tion of the distribution in the central region remainsunaffected by the boundary, but there is substantial de-viation between the two cases in the wings of the dis-tributions. Fig. 9 shows that the initial energy growthrate is the same for the two boundary locations in Fig.8. It is only as the distribution nears the boundary thatfurther growth is inhibited. This point is significant be-cause much of the theoretical analysis of this systemhas been done using the long-term diffusion in energy,which is especially sensitive to the high-momentumtails of the distribution.

In Fig. 10 we compare the intermediate case of Fig.7(c) (α = 0.049) to the classical simulation shown inFig. 3. The distributions shown correspond to 68 kicks.There are several important features in this compari-

son. First, the classical square-pulse case follows theδ-kick case out to the boundary. Second, the experimen-tal distribution is characteristically exponential nearits center, and then it drops off at the boundary as inthe classical simulation. Finally, the initial conditionsin both the classical simulation and the experiment re-main unchanged outside the boundary. Therefore, inthe central region, both the classical simulation andthe experiment behave as they would in the limit ofδ-kicks, displaying classical diffusion and quantum lo-calization, respectively. Their behavior in the bound-ary region, however, is quite similar.

5. Future directions

So far, we have considered the observation of dy-namical localization and the role of the boundary inour experiment. We will now discuss experiments thatare currently under development in our laboratory thatextend these results. Specifically, we will consider twoclasses of experiments: noise and dissipation in thequantum kicked rotor and the probing of local phasespace structure using localized initial states.

The study of noise and dissipation is of great impor-tance in many areas of physics. In particular, the roleof noise and dissipation in quantum mechanics hasgenerated much theoretical interest in recent years, es-pecially in the context of classically chaotic systems.In the case of the quantum kicked rotor, noise and dis-sipation are expected to lead to destruction of quan-tum localization. Thisdelocalizationphenomenon isthe prototype for demonstrating the important role ofnoise in recovering the (chaotic) classical limit fromits quantum description [22–24,30]. The kicked rotoris also an interesting setting for the study of noise anddissipation, because of its fundamental importance inquantum chaos.

The noise-induced delocalization experiments arerelatively straightforward generalizations of the dy-namical localization experiments. To introduce noisein the experiment, one simply adds a random, time-dependent perturbation to one of the system parame-ters. Suitable parameters include the kick strength, thetime between kicks, or the phase of the standing-wave


potential. Alternatively, one may introduce dissipationby coupling the atoms to a heat reservoir; this cou-pling can be accomplished by allowing the atoms toundergo spontaneous emission events, by either tun-ing the standing-wave light close to resonance or in-troducing weak, near-resonant light from the trappingbeams. The predicted behavior of the quantum kickedrotor in the presence of noise [8,22,24] is that insteadof localizing after the quantum break time, the mo-mentum distribution will continue to diffuse at a re-duced rateDq, where the magnitude ofDq increaseswith the noise amplitude. Delocalization due to am-plitude noise was seen in the previous sodium-basedexperiments, but the momentum boundary preventeda quantitative study of the effect [31].

We will also implement a method for preparing lo-calized initial conditions, so that we can study lo-cal structures in the classical phase space. Our cur-rent initial conditions are completely delocalized inposition over the scale of one period of the standingwave. These initial conditions are a serious limita-tion for studies of mixed phase space, because the re-sulting dynamics are an average over many structurescontained within the initial distribution. If we wereable to produce initial states that were narrow in bothposition and momentum, we could study many inter-esting effects, including local fluctuations in the lo-calization length, phase-space scarring, tunneling be-tween islands of stability, and transport across KAMtori.

In order to prepare localized initial states, we mustfirst prepare a distribution that is further localized inmomentum. We will accomplish this task via a stimu-lated Raman “tagging” approach [32]. This approachtakes advantage of the fact that the cesiumD2 linehas a ground state that is split into two hyperfine lev-els,F = 3 andF = 4, separated by∼ 9.2 GHz. It ispossible to induce two-photon stimulated Raman tran-sitions between the two ground levels by illuminatingthe atoms with two laser fields that are each detunedseveral GHz from theD2 line, but have a relative fre-quency offset that is near the ground state splitting.Because the fields are far-detuned, this process is co-herent. If the two fields are counterpropagating, theinteraction will be sensitive to Doppler shifts, and the

resonance condition is

ν2 − ν1 = 9.2 GHz− v

c(ν1 + ν2), (6)

whereν1 andν2 are the frequencies of the two fields,v

the velocity of the atom, andc is the speed of light. Theexperimental sequence begins by cooling and trappingatoms in the MOT. The repump light is then turned off,and the atoms are optically pumped into theF = 3ground level. Then the trapping beams are turned off,and the stimulated Raman fields are pulsed on. Therelative detunings of the Raman beams are chosen sothat the atoms promoted to theF = 4 ground state arenearp = 0 after the Raman pulse. The width of themomentum distribution of these “tagged” atoms is setmainly by the duration of the pulse. Once the coldestatoms have been tagged, the experiment can proceedas usual. At the end of the experiment, we turn on thefreezing molasses without the repump light in orderto illuminate and image only those atoms that weretagged. The untagged atoms effectively drop out of theexperiment, leaving behind a smaller atomic samplewith a narrower initial momentum distribution.

We now consider localization in position. We firstprepare the atoms using the stimulated Raman tech-nique such that the momentum distribution is withinthe interval(−~kL , +~kL). The standing-wave poten-tial is then turned on adiabatically. Under these condi-tions, all the atoms are loaded into the lowest energyband of the periodic potential [33]. As the depth ofthe potential is increased, the atoms become localizednear the bottoms of the potential wells, at the expenseof increased temperature. In the limit of deep poten-tial wells, the lowest energy band approximates theground state of the harmonic oscillator, since the wellbottoms are locally parabolic; hence, in this limit, theprepared states are in principle minimum uncertaintyGaussian wave packets, modulo 2π in position. Theaspect ratio of the distribution in phase space can becontrolled via the final well depth.

The prepared initial distribution that we have de-scribed so far is localized aboutx = p = 0. Itis useful to have techniques for centering the distri-bution at other points of interest in phase space. Ashift in the position of the distribution can be accom-


plished by suddenly shifting the phase of the standingwave at the beginning of the experiment. Such a phaseshift can, for example, be accomplished by placing anelectro-optic modulator in the path of one of the twobeams that form the standing-wave potential. A shiftin the momentum of the distribution can be achievedby introducing a relative frequency shift between thetwo standing-wave beams, producing a standing wavemoving at a velocity proportional to the shift.

6. Summary

We have constructed a quantum chaos experiment incesium that models theδ-kicked rotor. This apparatusis capable of a much wider range of experiments thanthe sodium-based experiments. The primary advantageof this new experiment lies in the physical character-istics of cesium, which reduce the effects of the mo-mentum boundary and bring the experiment closer tothe δ-kick limit. We will soon extend the capabilitiesof this experiment to study the effects of noise anddecoherence on localization and to study structures inmixed phase space.

Acknowledgements

This work was supported by the Robert A. WelchFoundation and the National Science Foundation. DASacknowledges support from a National Science Foun-dation Graduate Research Fellowship.

References

[1] G. Casati, B.V. Chirikov, J. Ford, F.M. Izrailev, in: G. Casati,J. Ford (Eds.), Stochastic Behaviour in Classical and QuantumHamiltonian Systems, Lecture Notes in Physics, vol. 93,Springer, Berlin, 1979.

[2] B. Chirikov, F.M. Izrailev, D.L. Shepelyansky, Sov. Sci. Rev.C 2 (1981) 209.

[3] S. Fishman, D.R. Grempel, R.E. Prange, Phys. Rev. Lett. 49(1982) 509.

[4] D.R. Grempel, R.E. Prange, S. Fishman, Phys. Rev. A 29(1984) 1639.

[5] D.L. Shepelyansky, Physica D 8 (1982) 208.[6] D.L. Shepelyansky, Phys. Rev. Lett. 56 (1986) 677.[7] D.L. Shepelyansky, Physica D 28 (1987) 103.[8] D. Cohen, Phys. Rev. A 44 (1991) 2292.[9] R. Blumel, W.P. Reinhardt, Chaos in Atomic Physics,

Cambridge, 1997.[10] J.E. Bayfield, P.M. Koch, Phys. Rev. Lett. 33 (1974) 258.[11] E.J. Galvez, B.E. Sauer, L. Moorman, P.M. Koch, D. Richards,

Phys. Rev. Lett. 61 (1988) 2011.[12] J.E. Bayfield, G. Casati, I. Guarneri, D.W. Sokol, Phys. Rev.

Lett. 63 (1989) 364.[13] R. Blumel, R. Graham, L. Sirko, U. Smilansky, H. Walther,

K. Yamada, Phys. Rev. Lett. 62 (1989) 341.[14] L.E. Reichl, The Transition to Chaos in Conservative Classical

Systems: Quantum Manifestations, Springer-Verlag, Berlin,1992.

[15] R. Graham, M. Schlautmann, D.L. Shepelyansky, Phys. Rev.Lett. 67 (1991) 255.

[16] R. Graham, M. Schlautmann, P. Zoller, Phys. Rev. A 45(1992) R19.

[17] F.L. Moore, J.C. Robinson, C. Bharucha, P.E. Williams, M.G.Raizen, Phys. Rev. Lett. 73 (1994) 2974.

[18] J.C. Robinson, C. Bharucha, F.L. Moore, R. Jahnke, G.A.Georgakis, Q. Niu, M.G. Raizen, B. Sundaram, Phys. Rev.Lett. 74 (1995) 3963.

[19] F.L. Moore, J.C. Robinson, C.F. Bharucha, B. Sundaram,M.G. Raizen, Phys. Rev. Lett. 75 (1995) 4598.

[20] J.C. Robinson, C.F. Bharucha, K.W. Madison, F.L. Moore,B. Sundaram, S.R. Wilkinson, M.G. Raizen, Phys. Rev. Lett.76 (1996) 3304.

[21] G.P. Collins, Phys. Today 48 (1995) 18.[22] E. Ott, T.M. Antonsen Jr., J.D. Hanson, Phys. Rev. Lett. 53

(1984) 2187.[23] T. Dittrich, R. Graham, Europhys. Lett. 4 (1987) 263.[24] S. Fishman, D.L. Shepelyansky, Europhys. Lett. 16 (1991)

643.[25] G. Casati, I. Guarneri, D.L. Shepelyansky, Phys. Rev. Lett.

62 (1989) 345.[26] S. Chu, Science 253 (1991) 861.[27] C. Cohen-Tannoudji, in: J. Dalibard, J.M. Raimond, J. Zinn-

Justin (Eds.), Fundamental Systems in Quantum Optics, LesHouches, 1990, Elsevier, London, 1992.

[28] M. Arndt, M. Ben Dahan, D. Guery-Odelin, M.W. Reynolds,J. Dalibard, Phys. Rev. Lett. 79 (1997) 625.

[29] R. Blumel, S. Fishman, U. Smilansky, J. Chem. Phys. 84(1986) 2604.

[30] W.H. Zurek, J.P. Paz, Phys. Rev. Lett. 72 (1994) 2508.[31] J.C. Robinson, Ph.D. Dissertation, The University of Texas

at Austin, 1995.[32] M. Kasevich, D. Weiss, E. Riis, K. Moler, S. Kasapi, S. Chu,

Phys. Rev. Lett. 66 (1991) 2297.[33] M. Raizen, C. Salomon, Q. Niu, Phys. Today (1997) 30.

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