letters to nature 5.Haake,E,Kus,M.Scharf,R.Classical and quantum chaos for a kicked top.Z Phys B 65,381-395 (19871. 6.Sanders,B.C.Milburn,G.I.The effect of measurement on the quantum features of a chaotic 3tem.Z.PhxB77,497-510(1989). WA 7.Habib,S..Shizume.K.Zurek.W.H.Decoherence,chaos.and the correspondence principle Ph3.ReK.Left.80,4361-43651998). 8.Graham.R..Schlautmann.M.Zoller,P.Dynamical localization of atomic-beam deflection by a modulated standing light wave.Phys Rev.A 45,R19-R22(1992). 9.Moore,F.L.Robinson,LC..Bharucha,C.F,Sundaram,B.&Raizen,M.G.Atom optics realization of the quantum 8-kicked rotor.Plrys.Rev.Lett.75,4598-4601(1995). 10.Hensinger,W.K,Truscott,A.G..Upcroft,B.Heckenberg N.R&Rubinsztein-Dunlop,H.Atoms in 8 an amplitude-modulated standing wave-dynamics and pathways to quantum chaos.IOpt.B2,659- 667(20001. 0 10 20 30 40 11.Hensinger,W.K.et al Experimental study of the quantum driven pendulum and its cassical analog in Modulation periods atom optics.Phrys.Rev.A (in the press). 12.Arnold,V.1.Mathematical Methads of Classical Mechanics (Springer,New York,1979). 13.Kozuma,M.etal Coherent splitting of Bose-Finstein condensed atoms with optically induced Bragg Figure 5 Momentum distributions as a function of the number of modulation periods, diffraction.Phry's.Rev.Lett.82,871-875 (1999). showing the tunnelling oscillation between negative and positive momenta.We note that 14.Steck,D.A..Oskay,W.H.&Raizen,M.G.Observation of chaos-assisted tunneling between islands of the zero-momentum state remains mostly unpopulated,even when the mean momentum stabilitv.Science (in the press). is zero.The colour coding ranges from blue to red for atomic populations ranging from small to large Acknowledgements We thank C.Holmes for discussions.The NIST group was supported by the ONR,NASA and ARDA,and the University of Qucensland group was supported by the ARC.A.B. was partially supported by DGA(France).and H.H.was partially supported by the A.v.Humboldt Foundation.W.K.H.and B.U.thank NIST for hospitality during the ity is that the tails of the oscillating quantum wave packets may extend outside the region of regular motion,allowing the atoms to 'leak out'into the classically chaotic region.This possibility is Correspondence and requests for materials should be addressed to W.K.H. (e-mail:hensinge@physics.uq.cdu.au). currently under investigation.The contribution of multiple Floquet states could lead to complicated multi-frequency oscillations,and an envelope for the tunnel oscillations appearing as decay,as observed for some parameters in our simulations.The effects of spontaneous emission and atom-atom interactions should be small. Quantum theory predicts dynamical tunnelling to occur for Quantum interference various values of the scaled well depth K,the modulation parameter e and modulation frequency and also predicts a strong sensitivity of superfluid 3He of the tunnelling period and amplitude on these parameters.For e=0.23,k=1.75 and a/2=250kHz we measured a tunnelling R.W.Simmonds,A.Marchenkov,E.Hoskinson,J.C.Davis period of approximately 13 modulation periods.As shown in R.E.Packard Fig.4b,for e=0.30,K=1.82 and a/2=222kHz we find a tunnelling period of 6 modulation periods with a significantly Physics Department,University of California,Berkeley,California 94720,USA longer decay time than in Fig.4a.We have also experimentally 444 observed an increase in the tunnelling period when k is decreased Celebrated interference experiments have demonstrated the and when all other parameters are held constant.This is the wave nature of light'and electrons,quantum interference opposite of what one would expect for spatial barrier tunnelling. being the manifestation of wave-particle duality.More recently, Our observation of dynamical tunnelling of atoms in a modu- double-path interference experiments have also demonstrated lated standing wave opens the door to further studies in quantum the quantum-wave nature of beams of neutrons',atoms'and nonlinear dynamics.By varying the hamiltonian parameters and the Bose-Einstein condensates'.In condensed matter systems, initial conditions,we observe dynamical tunnelling for a variety of double-path quantum interference is observed in the d.c.super- mixed phase space configurations.This may be 'chaos-assisted conducting quantum interference device(d.c.SQUID).Here we tunnelling,and a tunnelling rate that varies wildly as system report a double-path quantum interference experiment involv- parameters are changed would be a signature of such tunnelling'. ing a liquid:superfluid 'He.Using a geometry analogous to the By introducing noise or spontaneous emission in a controlled superconducting d.c.SQUID,we control a quantum phase shift manner,we could systematically investigate the role of decoherence by using the rotation of the Earth,and find the classic inter- in tunnelling and explore the classical limit of chaotic systems.By ference pattern with periodicity determined by the He quantum carefully following the evolution of wave packets loaded into the of circulation. chaotic region from a Bose-Einstein condensate,we could probe Our basic interferometer topology is shown in Fig.la.Schemati- quantum chaos'with the unprecedented resolution afforded by cally,the device is a circular loop of radius R which includes two minimum uncertainty wave packets. superfluid He Josephson weak links's.These weak links each During the preparation of this Letter,we learned of an consist of a 65 x 65 array of 100-nm apertures etched in a 60-nm- experiment4 reporting tunnelling of a different motional state in thick silicon nitride membrane.Similar arrays have previously been a similar system. □ shown to be characterized by a current-phase relation givenby the Josephson formula: Received 10 May:accepted 12 June 2001. I=I sino (1) 1.Tomsowvic,S.Tunneling and chaos.Physica Scripta T90,162-165 (2001) 2.Caldeira,A.O.Leggett,A.J.Quantum tunneling in a dissipative system.Arn.Phrys.149,374-456 Here I is the mass current flowing through the array,is the (1983). quantum phase difference across the array,and I.is the critical 3.Davis,M.J.Heller,E.J.Quantum dynamical tunneling in bound states.J.Chem Phys.75,246-254 current characterizing the array. (1981). 4.Dyrting.S..Milburn,G.J.Holmes,C.A.Nonlinear quantum dynamics at a classical second order The interferometer is predicted to behave as a single weak link resonance.P%y.RaE48,969-978(1993). with an effective critical current,(see Methods).If the inter- NATURE|VOL 4125 JULY 2001 www.nature.com 然©2001 Macmillan Magazines Ltd 55letters to nature NATURE | VOL 412 | 5 JULY 2001 | www.nature.com 55 ity is that the tails of the oscillating quantum wave packets may extend outside the region of regular motion, allowing the atoms to `leak out' into the classically chaotic region. This possibility is currently under investigation. The contribution of multiple Floquet states could lead to complicated multi-frequency oscillations, and an envelope for the tunnel oscillations appearing as decay, as observed for some parameters in our simulations. The effects of spontaneous emission and atom±atom interactions should be small. Quantum theory predicts dynamical tunnelling to occur for various values of the scaled well depth k, the modulation parameter e and modulation frequency q, and also predicts a strong sensitivity of the tunnelling period and amplitude on these parameters. For e ˆ 0:23, k ˆ 1:75 and q=2p ˆ 250 kHz we measured a tunnelling period of approximately 13 modulation periods. As shown in Fig. 4b, for e ˆ 0:30, k ˆ 1:82 and q=2p ˆ 222 kHz we ®nd a tunnelling period of 6 modulation periods with a signi®cantly longer decay time than in Fig. 4a. We have also experimentally observed an increase in the tunnelling period when k is decreased and when all other parameters are held constant. This is the opposite of what one would expect for spatial barrier tunnelling. Our observation of dynamical tunnelling of atoms in a modu￾lated standing wave opens the door to further studies in quantum nonlinear dynamics. By varying the hamiltonian parameters and the initial conditions, we observe dynamical tunnelling for a variety of mixed phase space con®gurations. This may be `chaos-assisted tunnelling', and a tunnelling rate that varies wildly as system parameters are changed would be a signature of such tunnelling1 . By introducing noise or spontaneous emission in a controlled manner, we could systematically investigate the role of decoherence in tunnelling and explore the classical limit of chaotic systems. By carefully following the evolution of wave packets loaded into the chaotic region from a Bose±Einstein condensate, we could probe `quantum chaos' with the unprecedented resolution afforded by minimum uncertainty wave packets. During the preparation of this Letter, we learned of an experiment14 reporting tunnelling of a different motional state in a similar system. M Received 10 May; accepted 12 June 2001. 1. Tomsovic, S. Tunneling and chaos. Physica Scripta T 90, 162±165 (2001). 2. Caldeira, A. O. & Leggett, A. J. Quantum tunneling in a dissipative system. Ann. Phys. 149, 374±456 (1983). 3. Davis, M. J. & Heller, E. J. Quantum dynamical tunneling in bound states. J. Chem Phys. 75, 246±254 (1981). 4. Dyrting, S., Milburn, G. J. & Holmes, C. A. Nonlinear quantum dynamics at a classical second order resonance. Phys. Rev. E 48, 969±978 (1993). 5. Haake, F., Kus, M. & Scharf, R. Classical and quantum chaos for a kicked top. Z. Phys. B 65, 381±395 (1987). 6. Sanders, B. C. & Milburn, G. J. The effect of measurement on the quantum features of a chaotic system. Z. Phys. B 77, 497±510 (1989). 7. Habib, S., Shizume, K. & Zurek, W. H. Decoherence, chaos, and the correspondence principle. Phys. Rev. Lett. 80, 4361±4365 (1998). 8. Graham, R., Schlautmann, M. & Zoller, P. Dynamical localization of atomic-beam de¯ection by a modulated standing light wave. Phys. Rev. A 45, R19±R22 (1992). 9. Moore, F. L., Robinson, J. C., Bharucha, C. F., Sundaram, B. & Raizen, M. G. Atom optics realization of the quantum d-kicked rotor. Phys. Rev. Lett. 75, 4598±4601 (1995). 10. Hensinger, W. K., Truscott, A. G., Upcroft, B., Heckenberg, N. R. & Rubinsztein-Dunlop, H. Atoms in an amplitude-modulated standing wave±dynamics and pathways to quantum chaos. J. Opt. B 2, 659± 667 (2000). 11. Hensinger, W. K.et al. Experimental study of the quantum driven pendulum and its classical analog in atom optics. Phys. Rev. A (in the press). 12. Arnold, V. I. Mathematical Methods of Classical Mechanics (Springer, New York, 1979). 13. Kozuma, M. et al. Coherent splitting of Bose-Einstein condensed atoms with optically induced Bragg diffraction. Phys. Rev. Lett. 82, 871±875 (1999). 14. Steck, D. A., Oskay, W. H. & Raizen, M. G. Observation of chaos-assisted tunneling between islands of stability. Science (in the press). Acknowledgements We thank C. Holmes for discussions. The NIST group was supported by the ONR, NASA and ARDA, and the University of Queensland group was supported by the ARC. A.B. was partially supported by DGA (France), and H. H. was partially supported by the A. v. Humboldt Foundation. W.K.H. and B.U. thank NIST for hospitality during the experiments. Correspondence and requests for materials should be addressed to W.K.H. (e-mail: hensinge@physics.uq.edu.au). 20 Modulation periods Momentum (ùk) 10 30 40 4 0 –4 –8 8 0 Figure 5 Momentum distributions as a function of the number of modulation periods, showing the tunnelling oscillation between negative and positive momenta. We note that the zero-momentum state remains mostly unpopulated, even when the mean momentum is zero. The colour coding ranges from blue to red for atomic populations ranging from small to large. ................................................................. Quantum interference of super¯uid 3 He R. W. Simmonds, A. Marchenkov, E. Hoskinson, J. C. Davis & R. E. Packard Physics Department, University of California, Berkeley, California 94720, USA .............................................................................................................................................. Celebrated interference experiments have demonstrated the wave nature of light1 and electrons2 , quantum interference being the manifestation of wave±particle duality. More recently, double-path interference experiments have also demonstrated the quantum-wave nature of beams of neutrons3 , atoms4 and Bose±Einstein condensates5 . In condensed matter systems, double-path quantum interference is observed in the d.c. super￾conducting quantum interference device6 (d.c. SQUID). Here we report a double-path quantum interference experiment involv￾ing a liquid: super¯uid 3 He. Using a geometry analogous to the superconducting d.c. SQUID, we control a quantum phase shift by using the rotation of the Earth, and ®nd the classic inter￾ference pattern with periodicity determined by the 3 He quantum of circulation. Our basic interferometer topology is shown in Fig. 1a. Schemati￾cally, the device is a circular loop of radius R which includes two super¯uid 3 He Josephson weak links7,8. These weak links each consist of a 65 ´ 65 array of 100-nm apertures etched in a 60-nm￾thick silicon nitride membrane. Similar arrays have previously been shown to be characterized by a current±phase relation given9 by the Josephson formula: I ˆ Ic sinf …1† Here I is the mass current ¯owing through the array, f is the quantum phase difference across the array, and Ic is the critical current characterizing the array. The interferometer is predicted to behave as a single weak link with an effective critical current, I p c (see Methods). If the inter- © 2001 Macmillan Magazines Ltd