pressure affects the phonon spectrum essen-tially by increasing all phonon frequencies,we can explain the decrease in Tc observed inmany superconductors on compression, ashas been done recently for MgB2 (10, 29).

That Tc in boron increases substantiallywith pressure may be due to one of the fol-lowing: (i) The factor h 5 N(0)^I2& mayincrease with pressure in B, thus suppressingthe effect of increasing ^v2&. The Hopfieldparameter, h, may also contribute to the in-crease in Tc if the character of the conductionelectrons also changes under pressure, as ins-d transfer. (ii) The ^v2& factor may actuallydecrease with pressure if the phonon modesresponsible for electron-phonon couplingsoften under pressure. (iii) The parameter m*decreases on compression, which would berelated to pressure-induced additional screen-ing of the electron-electron interaction. Ofthese, the first and third options are possibil-ities: Both may be effective if B is approach-ing a covalent instability (with h increasing),as discussed by Allen and Dynes (30); or ittransforms to a compensated metal, as in thecases discussed by Richardson and Ashcroft(31). A similar increase in critical tempera-ture (dTc/dP ' 0.05 K/GPa) is observed inmetallic S after transforming to the b-Postructure at 160 GPa (32), suggesting that themechanism could be related.

We have found superconductivity in B atpressures above 160 GPa. The pressure ofmetallization is in the general range of (butsomewhat lower than) theory, which predict-ed that the transition would be accompaniedby the loss of covalent bonding to form adense nonicosahedral structure (12). Themagnitude of Tc appears to be consistent withsuch a transition and with an electron-cou-pling origin for the superconductivity. Thiswork extends the range of electrical conduc-tivity measurements to a record value of 250GPa. These observations should stimulatetheoretical calculations of superconductivityin elemental B and related low-Z substances.

References and Notes1. J. Bardeen, L. N. Cooper, J. R. Schrieffer, Phys. Rev.

108, 1175 (1957).2. N. W. Ashcroft, Phys. Rev. Lett. 21, 1748 (1968); C. F.

Richardson, N. W. Ashcroft, Phys. Rev. Lett. 78, 118(1997).

3. D. A. Papaconstantopoulos et al., Phys. Rev. B 15,4221 (1977).

4. K. A. Johnson, N. W. Ashcroft, Nature 403, 632(2000); M. Stadele, R. M. Martin, Phys. Rev. Lett. 84,6070 (2000).

5. K. M. Lang et al., J. Low Temp. Phys. 114, 445 (1999).6. N. E. Christensen, D. L. Novikov, Phys. Rev. Lett. 86,

1861 (2001); J. B. Neaton, N. W. Ashcroft, Nature400, 141 (1999).

7. K. Takei, K. Nakamura, Y. Maeda, J. Appl. Phys. 57,5093 (1985); V. I. Tutov, E. E. Semenenko, Sov. J. LowTemp. Phys. 16, 22 (1990).

8. R. J. Cava et al., Nature 367, 146 (1994); Nature 367,252 (1994).

9. J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani,J. Akimitsu, Nature 410, 63 (2001).

10. M. Monteverde et al., Science 292, 75 (2001); T. Tomita,

J. J. Hamlin, J. S. Schilling, D. G. Hinks, J. D. Jorgensen,preprint available at http://xxx.lanl.gov/abs/cond-mat/0103538; B. Lorenz, R. L. Meng, C. W. Chu, Phys. Rev. B64, 12507 (2001); V. G. Tissen et al., preprint availableat http://xxx.lanl.gov/abs/cond-mat/0105475; V. V.Struzhkin et al., prerpint available at http://xxx.lanl.gov/abs/cond-mat/0106576, and references therein.

11. J. Kortus, I. I. Mazin, K. D. Belashchenko, V. P.Antropov, L. L. Boyer, Phys. Rev. Lett. 86, 4656(2001).

12. C. Mailhiot, J. B. Grant, A. K. McMahan, Phys. Rev. B42, 9033 (1990).

13. D. A. Young, Phase Diagrams of the Elements (Univ. ofCalifornia Press, Berkeley, CA, 1991).

14. N. Vast et al., Phys. Rev. Lett. 78, 693 (1997).15. M. Fujimori et al., Phys. Rev. Lett. 82, 4452 (1999).16. Y. Ma et al., Trans. Am. Geophys. Union 81, S39

(2000).17. M. I. Eremets, K. Shimizu, T. C. Kobayashi, K. Amaya,

Science 281, 1333 (1998).18. M. I. Eremets et al., Phys. Rev. Lett. 85, 2797 (2000).19. H. K. Mao, J. Xu, P. M. Bell, J. Geophys. Res. 91, 4673

(1986).20. O. Madelung, M. Schultz, H. Weiss, Eds., Landolt-

Bornstein. New Series, vol. 17e (Springer-Verlag, Ber-lin, 1983).

21. M. I. Eremets, High Pressure Experimental Methods(Oxford Univ. Press, Oxford, 1996).

22. The superconducting step cannot be attributed to Pt orPd electrical wires in series with the sample, because thesame superconducting transitions were observed forboth electrodes. In addition, we proved that the Ptelectrodes did not reveal a superconducting transitionon cooling down to 50 mK in previous experiments onCsI up to 220 GPa (17). In those experiments, the onlysuperconducting transition was that of the CsI sample,which appeared at 1.6 K and 203 GPa and shifted tolower temperatures with pressure. We also did notobserve any superconductivity in experiments on Xe to155 GPa and 27 mK, using the same arrangement of Ptelectrodes (18). Only recently has superconductivitybeen found in compacted Pt powder, but at very lowtemperatures; i.e., Tc ^ 1.38 mK [R. Konig, A. Schindler, T.Herrmannsdorfer, Phys. Rev. Lett. 82, 4528 (1999)]. Wenote that the BN/epoxy layer is also in contact with the

sample and electrodes. However, this mixture remainsinsulating to at least 240 GPa, as demonstrated byrecent experiments on N [M. I. Eremets et al., Nature411, 170 (2001)], and consistent with the experimentsdescribed above. Thus, although the Meissner effect wasnot examined (17), the resistance steps measured herecan be attributed to superconductivity in the B samples.

23. The initial pressure derivative is very large (;0.2K/GPa). Extrapolation of this line to lower pressuregives Tc 5 0 K at ;130 GPa, which correlates with akink in the room-temperature R(P) curve. This sug-gests the possibility that the transition to the metal-lic state occurs at ;130 GPa, whereas the changenear 180 GPa arises from a phase transition.

24. N. F. Mott, Metal-Insulator Transitions ( Taylor &Francis, London, ed. 2, 1990).

25. We note that bct and face-centered tetragonal (fct)share the same Bravais lattice. Hence, a transitionfrom bct to fcc is structurally similar to moving fromIn to Al within the group IIIA family. The latter isaccompanied by an increase in zero-pressure Tc by afactor of three [N. W. Ashcroft, N. D. Mermin, SolidState Physics (Harcourt, New York, 1976)].

26. The modes in a-B extend to 1300 cm21 at highpressure (13). The frequency of diatomic B is ;1000cm21 [K. P. Huber, G. Herzberg, Constants of Diatom-ic Molecules (Van Nostrand, New York, 1979)].

27. D. U. Gubser, A. W. Webb, Phys. Rev. Lett. 35, 104(1975); M. M. Dacorogna, M. L. Cohen, P. K. Lam, Phys.Rev. B 34, 4865 (1986).

28. W. L. McMillan, Phys. Rev. 167, 331 (1968).29. I. Loa, K. Syassen, Solid State Commun. 118, 279

(2001); T. Vogt, G. Schneider, J. A. Hriljac, G. Yang,J. S. Abell, Phys. Rev. B. 63, 220505 (2001); A. F.Goncharov et al., preprint available at http://xxx.lanl.gov/abs/cond-mat/0104042.

30. P. B. Allen, R. C. Dynes, Phys. Rev. B 12 (1975).31. C. F. Richardson, N. W. Ashcroft, Phys. Rev. B 55,

15130 (1997).32. V. V. Struzhkin, R. J. Hemley, H. K. Mao, Y. A.

Timofeev, Nature 390, 382 (1997).33. We are grateful to N. W. Ashcroft for useful discus-

sions and S. Gramsch for comments on the manu-script. Supported by NSF.

4 May 2001; accepted 30 May 2001

Observation of Chaos-AssistedTunneling Between Islands of

StabilityDaniel A. Steck, Windell H. Oskay, Mark G. Raizen*

We report the direct observation of quantum dynamical tunneling of atomsbetween separated momentum regions in phase space. We study how thetunneling oscillations are affected as a quantum symmetry is broken and as theinitial atomic state is changed. We also provide evidence that the tunneling rateis greatly enhanced by the presence of chaos in the classical dynamics. Thistunneling phenomenon represents a dramatic manifestation of underlying clas-sical chaos in a quantum system.

Quantum-mechanical systems can displayvery different behavior from their classicalcounterparts. In particular, quantum effectssuppress classical chaotic behavior, wheresimple deterministic systems exhibit compli-cated and seemingly random dynamics (1).

Nevertheless, aspects of quantum behaviorcan often be understood in terms of the pres-ence or absence of chaos in the classical limit.In this report, we focus on quantum transportin a mixed system, where the classical dy-namics are complicated by the coexistence ofchaotic and stable behavior. We study quan-tum tunneling between two stable regions(referred to as nonlinear resonances or islandsof stability) in the classical phase space. Theclassical transport between these islands is

Department of Physics, The University of Texas atAustin, Austin, TX 78712–1081, USA.

*To whom correspondence should be addressed. E-mail: raizen@physics.utexas.edu

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forbidden by dynamical “barriers” in phasespace. In contrast, quantum tunneling cancouple the two islands so that a wave packetoscillates coherently between the two sym-metry-related stable regions (2–14).

“Dynamical tunneling,” where the classi-cal transport is forbidden because of the sys-tem dynamics and not a potential barrier, wasoriginally introduced in the context of a two-dimensional, time-independent potential (2).Subsequently, it was found that the presenceof chaos could markedly enhance the tunnel-ing rate in a driven, double-well potential (3),and the role of a discrete symmetry in thissystem was highlighted in the tunneling pro-cess (4). In addition to symmetry, the pres-ence of regular islands is important for pro-ducing coherent tunneling, because the is-lands cause localization of the Floquet states(5), which are the analogs of energy eigen-states in time-periodic quantum systems (1).Thus, dynamical tunneling between islandsof stability is analogous to tunneling in thesimple double-well potential, where the lo-calized eigenstates split into a symmetric/antisymmetric pair, and the tunneling can beunderstood in terms of the dephasing of thisnearly degenerate Floquet-state doublet. Itwas found that the tunneling rate is correlatedwith the degree of overlap of the tunnelingstates with the chaotic region, again pointingto the role of the chaotic sea in assisting thetunneling transport (6). The possible en-hancement of the tunneling rate because ofthe presence of the chaotic region was under-stood in terms of a three-level process, wherethe tunneling doublet interacts with a thirdstate associated with the chaotic region. Theterm “chaos-assisted tunneling” was intro-duced (7, 8) to distinguish this process fromordinary dynamical tunneling, which is atwo-state process. Chaos-assisted tunnelinghas also been explained in terms of indirectpaths, which are multiple-step transitions thattraverse the chaotic region, as opposed todirect paths, which tunnel in a single step andare responsible for regular dynamical tunnel-ing (9). Because of these coexisting directand indirect mechanisms, the presence of thechaotic region produces large fluctuations inthe tunneling rate as the system parametersvary, sometimes increasing the tunneling rateby orders of magnitude.

Previous experimental work on dynamicaland chaos-assisted tunneling has mainly fo-cused on wave analogies to these effects.Chaos-assisted tunneling has been studied inmicrowave billiards, where the enhancementof mode doublet splittings due to classicalchaos has been detected spectroscopically(15). The Shnirelman peak in the level spac-ing distribution is a similar statistical signa-ture of dynamical tunneling (16) and hasbeen observed in acoustic resonator (17) andmicrowave cavity experiments (18). Finally,

another recent atom-optics experiment hasexamined coherent tunneling in a double-welloptical lattice potential (19, 20).

Our experiment studies the motion of coldcesium atoms in an amplitude-modulatedstanding wave of light. Because the light isdetuned far from the D2 line (50 GHz, or 104

natural linewidths, to the red of the F 5 33F9 transition, where F is the atomic hyperfinequantum number), the internal dynamics ofthe atom can be adiabatically eliminated (21,22). The atomic center-of-mass Hamiltoniancan then be written in scaled units as

H 5 p2/2 2 2acos2(pt)cos(x) (1)

where x and p are the canonical position andmomentum coordinates, respectively, t istime, and a is given by (8vrT

2/\) V0 [V0 isthe amplitude of the ac Stark shift corre-sponding to the time-averaged laser intensity,T is the period of the temporal modulation, \is the reduced Planck constant, and vr is therecoil frequency, which has the numericalvalue 2p 3 2.07 kHz for this experiment);more details on the unit scaling can be foundin (22). The quantum description of this sys-tem is governed by one additional parameter,the effective Planck constant \vk 5 8vrT, sothat the scaled coordinate operators satisfy [x,p] 5 i\vk (note, however, that for the experi-mental data we report momentum in units ofdouble photon recoils, 2\kL, which is equiv-alent to the scaled momentum expressed inmultiples of \vk). The time-dependent potentialin this system can be decomposed into a sumof three unmodulated cosine terms (23). Oneof these terms is stationary, whereas the othertwo move with velocity 6 2p, so that in thelimit of vanishing a, the phase space of thissystem has three primary resonances, two ofwhich are symmetric partners about the p 5 0axis. The value of a used in the experimentwas 10.5 6 5%. At this large value of a, thecentral island has mostly vanished, leaving alarge chaotic region surrounding the twosymmetry-related islands (Fig. 1, A and B).To study chaos-assisted tunneling, we pre-pared the atoms in one of the resonances andobserved the atoms coherently oscillate be-tween the two islands by monitoring the evo-lution of the atomic momentum distribution.The possibility of experimentally observingchaos-assisted tunneling in this system hasbeen a subject of recent discussion (10–12),and the tunneling and band structure in thissystem were recently treated in an extensivenumerical study (10).

The basic experimental apparatus hasbeen described in detail in (22), although wehave made several major improvements, aswe now describe. To prepare the initial atom-ic state, we first cooled and trapped 106 ce-sium atoms from the background vapor in astandard six-beam magneto-optic trap(MOT), at a temperature of 10 mK (corre-

sponding to a Gaussian momentum distribu-tion with sp/\kL 5 4). The atoms are thenfurther cooled and stored for 300 ms in athree-dimensional, far detuned, linearly po-larized optical lattice similar to that of (24).After adiabatic release from the lattice, theatoms achieve a temperature of 400 nK (sp/\kL 5 1.4). The atoms are then opticallypumped to the F 5 4, m 5 0 magneticsublevel, resulting in a temperature of 3 mK(sp/\kL 5 4). The atomic orientation is main-tained with a 1.5-G bias field. A velocity-selective, stimulated Raman pulse on the 9.2-GHz clock transition (which is insensitive toZeeman shifts to first order) “tags” a narrowvelocity slice (of less than 1% of the atoms)into the F 5 3, m 5 0 sublevel near p 5 0.The Raman fields are generated with a setupsimilar to that in (25), and the 800-ms squaretemporal pulse yields a momentum slice witha half-width at half-maximum of 0.03 32\kL. The remaining atoms are then removedby applying low-intensity, circularly polar-ized light resonant on the F 5 4 3 F9 5 5cycling transition for 800 ms.

At this point, the atoms have been pre-pared in a very narrow distribution about p 50, but they are not localized in position on thescale of the standing-wave period. A one-dimensional optical lattice is ramped on adi-abatically so that the atoms localize in thepotential wells. The lattice is then suddenlyspatially shifted by 1/4 of the lattice period(in several hundred ns) with an electroopticmodulator placed before the standing-waveretroreflector. After 6 ms of evolution in thelattice, the atoms return to the centers of thepotential wells, acquiring kinetic energy inthe meantime. The resulting Gaussian mo-mentum profile is peaked at 4.1 3 2\kL, witha width sp 5 1.1 3 2\kL. This state prepa-ration procedure produces a localized atomicwave packet centered on one of the islands ofstability (Fig. 1, A and B). The three redellipses are the 50% contours of a classicaldistribution with the same position and mo-mentum marginal distributions as the Wignerfunction. (The Wigner function has addition-al structures that reflect the coherences of theinitial state.) The initial condition shown doesnot reflect a slight distortion due to anhar-monic motion in the lattice. The importanceof the extremely narrow velocity selection istwofold. First, the atomic distribution must beselected to be well within one photon recoilof zero momentum, so that all the atoms loadinto the lowest energy band of the lattice.Then, in the deep-well limit, the atomic dis-tribution becomes minimum-uncertaintyGaussian (modulo the standing-wave period).Second, only atoms whose momenta arenearly a multiple of \kL will tunnel, as wediscuss further below. As the lattice onlyimparts momentum in multiples of 2\kL, theramping and shifting of the lattice result in a

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distribution with an overall Gaussian enve-lope but concentrated in narrow momentumslices around n(2\kL) for integer n. Thisstructure indicates coherence of the wavepacket over multiple periods of the opticallattice.

After the state preparation, the atoms aresubjected to the time-dependent interactiondescribed by Eq. 1, with a modulation periodof T 5 20 ms (\vk 5 2.08). The atoms are thenallowed to expand freely for 20 ms, afterwhich the optical molasses is turned on,freezing the atoms in place. The fluorescenceof the atoms is collected on a charge-coupleddevice camera. As a result of the long free-drift time, this process yields a measurementof the atomic momentum distribution. Be-cause of the relatively large size (sx 5 0.15mm) of the initial atomic cloud, the individ-ual momentum slices are not resolved in themeasured distributions. To compensate forthe loss in signal that results from discardingmost of the atoms, we averaged the datapresented in this report over 20 iterations,except for the distributions in Fig. 1D (100iterations) and Fig. 5A (19 iterations). Themomentum distributions are sampled everytwo modulation periods (40 ms) for all thedata presented here except for the high tem-poral resolution data (Fig. 5A).

The measured evolution of the momentumdistribution (Fig. 1C) shows clear tunneling

oscillations between the initial momentumpeak and its symmetric partner, which is lo-cated 8 3 2\kL away in momentum. Fourdamped oscillations are apparent in this mea-surement out to 80 modulation periods (1600ms), and after this time the oscillations havecompletely damped away. Four of the mo-mentum distributions near the beginning ofthe evolution are shown in more detail in Fig.1D. During the first oscillation, nearly half ofthe atoms appear in the secondary peak (26).

As mentioned above, only atoms with mo-mentum of approximately a multiple of aphoton recoil momentum (or scaled momen-tum of nearly a multiple of \vk/2) will tunnel.This is clear from the requirement of symme-try for tunneling to occur, because only statesthat involve these special velocity classes arecoupled to their symmetric reflections (aboutthe p 5 0 axis) by two-photon transitions.This is essentially the same condition re-quired for Bragg scattering (27–29). The bro-ken symmetry resulting from selecting othervelocity classes is formally equivalent to abroken time-reversal symmetry (30) and sup-presses the formation of symmetric/antisym-metric doublets (30, 31). We can study thisbroken symmetry directly by varying the Ra-man detuning of the velocity-selection pulsefrom the optimum value and monitoring theeffect on the evolution of the average mo-mentum ^ p& (Fig. 2). The case with the stron-

gest momentum oscillations corresponds tothe data shown in Fig. 1, C and D. Alsoshown are measurements with Raman detun-ings corresponding to momentum offsets of0.05 3 2\kL and 0.12 3 2\kL. In the formercase, the oscillations are partially suppressed,and for the larger detuning, the tunnelingoscillations have mostly disappeared. Be-cause of this sensitivity to the initial momen-tum, the tunneling oscillations are not visiblewithout subrecoil velocity selection, as wehave experimentally verified (32). Addition-ally, this effect is largely responsible for thedamping of the tunneling oscillations that weobserve, because the states near the edge ofthe Raman velocity selection profile will nottunnel as efficiently as the “resonant” statesat the profile center. The various states willalso tunnel at slightly different rates, leadingto dephasing of the oscillations, similar tobroadened excitation of a two-level system.Hence, narrower velocity selection shouldlead to longer damping times, although noiseand decoherence sources may also limit thecoherence of the oscillations.

We also verified that the tunneling isstrongest if the wave packet is initially cen-tered on the island of stability. As the initialwave packet is moving, we can displace theinitial condition in the x direction in phasespace simply by inserting a short delay timewhere the standing wave is off before begin-ning the driven pendulum interaction. Theoscillations in ^p& were compared for delaytimes of 0, 3.8, 7.6, and 15.1 ms, correspond-ing to displacements of 0, 1/4, 1/2, and 1periods of the optical potential away from theisland center (Fig. 3). For the 1/4-period dis-placement, the oscillations are suppressed,but still present. The initial wave packet inthis case only weakly excites the tunnelingFloquet states and mostly populates the statesin the chaotic sea (resulting in diffusionthroughout the sea) and states in the outerstability band (resulting in trapping of the

Fig. 1. Experimental observation of tunneling oscillations. (A) The classical phase space for theexperimental parameters. The islands of stability involved in the tunneling appear as two blueregions inside the green chaotic region and are symmetric reflections about the p 5 0 axis. Aschematic of the initial atomic state is superimposed on the upper island in red, appearing as threenarrow ellipses. (B) Magnified view of the upper stability island and the initial condition. (C) Themeasured evolution of the momentum distribution in time, showing several coherent oscillationsbetween the two islands, which are separated in momentum by 8 3 2\kL. In this plot, thedistribution is sampled every 40 ms (every two modulation periods). (D) Detailed view of the firstfour highlighted distributions in (C), where it is clear that a substantial fraction of the atoms tunnelto the other island (2).

Fig. 2. Comparison of tunneling oscillations fordifferent Raman detunings. The strongest oscil-lations observed (v) correspond to Raman ve-locity selection at p 5 0. The other two casesare for velocity selection at p 5 0.05 3 2\kL(f), where the oscillations are partially sup-pressed, and p 5 0.12 3 2\kL (Œ), where theoscillations are almost completely suppressed.

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wave packet at the high momenta). For the1/2-period displacement, the oscillations arecompletely suppressed, because the wavepacket is almost entirely trapped in the outerstable region. For the longest delay, the wavepacket returns to the island center, and thetunneling oscillations are once again present.The amplitude of the oscillations is somewhatsuppressed in this case, however, because thefree evolution stretches the wave packet, andhence the tunneling states are not as efficient-ly populated.

So far, we have discussed the tunnelingoscillations and how they are affected by abroken quantum symmetry and the initial lo-cation in phase space, which are importantcharacteristics of dynamical tunneling. Todemonstrate chaos-assisted tunneling re-quires further evidence, and an important testis to compare the tunneling in the time-de-pendent potential with tunneling in the ab-

sence of chaos. A sensible integrable coun-terpart of the amplitude-modulated standing-wave system arises by simply considering thetime-averaged potential, resulting in thequantum pendulum. Because the initial dis-tribution is centered outside the separatrix,classical transport across the p 5 0 axis isalso forbidden in this system. However, thereis a well-known dynamical tunneling mech-anism in the pendulum, high-order Braggscattering (27–29), which is a manifestationof quantum above-barrier reflection (33). Asthe wave packet is initially peaked near 4 32\kL, the dominant transport process iseighth-order Bragg scattering. For the param-eters in the experiment, the calculated eighth-order Bragg period is around 1 s, which ismuch longer than the 400-ms tunneling peri-od in the (chaotic) driven pendulum. Wecompared the evolution of ^p& for the drivenpendulum to the transport in the undrivenpendulum (Fig. 4), and indeed no coherentoscillations are observed in the undriven caseduring the interaction times measured in theexperiment. Hence, we observe that the clas-sical chaos enhances the tunneling rate forthese experimental parameters, in the sensethat the tunneling in the presence of classicalchaos occurs at a substantially greater ratethan the tunneling in the integrable case.

Although it is customary to study time-periodic systems in a stroboscopic sense, sam-pling only at a particular phase of the modula-tion as we have done up to this point, it is alsointeresting to study the continuous tunnelingdynamics in our system. We studied the evolu-tion of the momentum distribution during thefirst half of the first tunneling period, sampling

the system at 1-ms intervals, or 20 times permodulation period (Fig. 5A). The most obviousaspect of this data is that the initial and second-ary (tunneled) peaks exhibit complementary butopposite momentum oscillations at the modu-lation frequency. These oscillations can be ex-plained in terms of the continuous motion of thecorresponding islands in phase space (Fig. 5B).As the two islands have opposite momentum,they move in opposite directions but oscillate inmomentum because of repulsion by the rem-nants of the center island (34 ). In thispicture, the islands constitute a pair of non-intersecting “flux tubes” (14 ) that remainconfined in separated momentum intervals.The tunneling atoms can be viewed as arealization of a dynamical Schrodinger cat,because they represent a coherent superpo-sition of two states separated in momentumspace, each one corresponding to motion ina classical island of stability.

The evolution in Fig. 5A also showsother interesting transport behavior. Thereis another oscillation that proceeds morequickly than the tunneling, which appearsas population oscillating between the initialpeak and the chaotic region near p 5 0.This can be seen most clearly as an en-hanced population near zero momentumduring the third, fifth, and seventh modu-lation periods. This process also points tochaos-assisted tunneling, because it sug-gests that a third (chaotic) state is involvedin the transport between the two islands.

Note added in proof: After the submissionof this paper, we became aware of an exper-iment reporting dynamical tunneling in a sim-ilar setting (35).

Fig. 3. Comparison of chaos-assisted tunnelingfor different free-drift times before the stand-ing-wave interaction. The strongest oscillationsoccur for zero drift time (v), where the initialwave packet is centered on the island of sta-bility as in Fig. 1A. The oscillations are substan-tially suppressed for a 3.8-ms drift time (f),which displaces the initial wave packet centerby 1/4 of a period of the standing wave. Tun-neling oscillations are completely suppressedfor a 7.6-ms drift time (Œ), corresponding to a1/2-period offset of the initial wave packet. Fora 15.1-ms drift time (V), the wave packet isagain centered on the island, and coherentoscillations are restored.

Fig. 4. Comparison of chaos-assisted tunnel-ing oscillations (v) to transport in the cor-responding quantum pendulum (f). No tun-neling oscillations are observed in the pen-dulum case over the interaction times stud-ied in the experiment.

Fig. 5. High temporal resolution tunneling measurement. (A) Evolution of the momentumdistribution during the first tunneling oscillation, sampled 20 times per modulation period. The twopeaks show complementary oscillations at the modulation frequency in addition to the slowertunneling oscillation. Some population also appears in the chaotic region between the islands,especially during the third, fifth, and seventh modulation periods (34). (B) Phase space plots (axesas in Fig. 1A) at four different phases of the lattice modulation, showing the classical origin of thefast oscillations in (A). At the start of the modulation period, the islands of stability are maximallyseparated but move inward as they drift away from x 5 0 and return to their initial momenta bythe end of the modulation period. The two islands always remain separated in momentum and donot cross the p 5 0 axis (34).

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Online at www.sciencemag.org/cgi/content/full/1061569.

37. This work was supported by the NSF, the R. A. WelchFoundation, the Sid W. Richardson Foundation, and aFannie and John Hertz Foundation Fellowship(D.A.S.). D.A.S. acknowledges fruitful discussions withK. W. Madison.

12 April 2001; accepted 1 June 2001Published online 5 July 2001;10.1126/science.1061569Include this information when citing this paper.

Deterministic Delivery of aSingle Atom

Stefan Kuhr,* Wolfgang Alt, Dominik Schrader, Martin Muller,Victor Gomer, Dieter Meschede

We report the realization of a deterministic source of single atoms. A standing-wave dipole trap is loaded with one or any desired number of cold cesium atomsfrom a magneto-optical trap. By controlling the motion of the standing wave,we adiabatically transport the atom with submicrometer precision over mac-roscopic distances on the order of a centimeter. The displaced atom is observeddirectly in the dipole trap by fluorescence detection. The trapping field can alsobe accelerated to eject a single atom into free flight with well-defined velocities.

The manipulation of individual atomic parti-cles is a key factor in the quantum engineer-ing of microscopic systems. These techniquesrequire full control of all physical degrees offreedom with long coherence times. In com-parison to well-established single-ion trap-ping methods (1–4), a similar level of controlof neutral atoms has yet to be achieved be-cause of their weaker interactions with exter-nal electromagnetic fields.

Thermal sources of neutral atoms, suchas atomic beams, provide a flux of uncor-related atoms with random arrival times.However, there is great interest in a sourcethat would deliver a desired number of coldatoms at a time set by the experimentalist.Micromaser experiments, for example, usea dilute atomic beam, which results in amean number of atoms inside the resonatorthat is much less than 1. Poissonian statis-tics, however, dictate that the probability ofhaving more than one atom inside the res-

onator simultaneously does not vanish; thiscan easily destroy the ideal one-atom-maseroperation (5). Another possible applicationis the controlled generation of single opti-cal photons triggered by atoms entering aresonator with mirrors of ultrahigh reflec-tivity (a “high-finesse” resonator) one byone (6, 7). Other experiments require theplacement of more than one atom into theregion of interest. Quantum logic gates (8)can be implemented by entangling (2, 4, 9,10) neutral atoms through their simulta-neous coupling to the optical field of aresonator (11, 12). This is possible with thecurrent technology, but in recent experi-ments (13, 14) atoms enter the cavity in arandom way, rendering it impossible tohave a certain small number of atoms ondemand.

In comparison, our technique guaranteescontrol of the position of individual neutralatoms with submicrometer precision. Astanding-wave dipole trap is used to storeany desired small number of cold atoms ina laser field interference pattern, localizingthe trapped atoms to better than half of theoptical wavelength. Changing the laser pa-

rameters moves this interference patternalong with the trapped atom in a prescribedway. Whereas the transportation of atomicclouds has recently been realized usingmagnetic potentials (15), here we demon-strate the controlled transport of a singleatom.

Optical dipole traps (16 –21) are basedon the interaction between an electric com-ponent of the light field E and the inducedatomic electric dipole moment d, which isproportional to E. The interaction energyU 5 –^d z E&/2 is proportional to the locallight intensity. If the laser frequency issmaller than the atomic resonance frequen-cy, the atom is attracted to the region ofmaximum intensity. Thus, the simplest op-tical dipole trap is a focused laser beam.Tuning the laser frequency far away fromall atomic resonances substantially reducesthe photon scattering rate, and the atom istrapped in a nearly conservative potential.In contrast, a magneto-optical trap (MOT )(22) provides dissipative forces and servesas a convenient source of single cold atoms(23, 24). Atoms captured from the back-ground gas interact with the near-resonantlight field of the MOT and scatter photonsfrom the laser beams. This fluorescencesignal monitors the number of trapped at-oms in real time (Fig. 1). These atoms canbe transferred into a dipole trap superim-posed on the MOT without any loss, thusallowing us to experiment with a predeter-mined number of atoms (24).

Our dipole trap consists of two counter-propagating laser beams with equal inten-sities and optical frequencies n1 and n2,producing a position-dependent dipole po-tential U(z, t) 5 U0 cos2[p(Dnt 2 2z/l)],where U0 is the local trap depth, z is theposition of the atoms, l 5 1064 nm is theoptical wavelength (l 5 c/n1 ' c/n2, where

Institut fur Angewandte Physik, Universitat Bonn,Wegelerstrasse 8, D-53115 Bonn, Germany.

*To whom correspondence should be addressed. E-mail: kuhr@iap.uni-bonn.de

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