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Dynamical tunnelling of ultracold atoms


The divergence of quantum and classical descriptions of particle motion is clearly apparent in quantum tunnelling1,2 between two regions of classically stable motion. An archetype of such non-classical motion is tunnelling through an energy barrier. In the 1980s, a new process, ‘dynamical’ tunnelling1,2,3, was predicted, involving no potential energy barrier; however, a constant of the motion (other than energy) still forbids classically the quantum-allowed motion. This process should occur, for example, in periodically driven, nonlinear hamiltonian systems with one degree of freedom4,5,6. Such systems may be chaotic, consisting of regions in phase space of stable, regular motion embedded in a sea of chaos. Previous studies predicted4 dynamical tunnelling between these stable regions. Here we observe dynamical tunnelling of ultracold atoms from a Bose–Einstein condensate in an amplitude-modulated optical standing wave. Atoms coherently tunnel back and forth between their initial state of oscillatory motion (corresponding to an island of regular motion) and the state oscillating 180° out of phase with the initial state.

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Figure 1: Poincaré section for position q and momentum p of a classical particle in an amplitude-modulated optical lattice.
Figure 2: Diagram of period-one resonances of an atom in an amplitude-modulated sinusoidal potential.
Figure 3: Atomic momentum distributions after n modulation periods, showing dynamical tunnelling.
Figure 4: Mean momentum as a function of the number of modulation periods, n.
Figure 5: Momentum distributions as a function of the number of modulation periods, showing the tunnelling oscillation between negative and positive momenta.

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  1. Tomsovic, S. Tunneling and chaos. Physica Scripta T 90, 162–165 (2001).

    Article  ADS  Google Scholar 

  2. Caldeira, A. O. & Leggett, A. J. Quantum tunneling in a dissipative system. Ann. Phys. 149, 374–456 (1983).

    Article  ADS  Google Scholar 

  3. Davis, M. J. & Heller, E. J. Quantum dynamical tunneling in bound states. J. Chem Phys. 75, 246–254 (1981).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

  5. Haake, F., Kus, M. & Scharf, R. Classical and quantum chaos for a kicked top. Z. Phys. B 65, 381–395 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  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).

    Article  ADS  Google Scholar 

  7. Habib, S., Shizume, K. & Zurek, W. H. Decoherence, chaos, and the correspondence principle. Phys. Rev. Lett. 80, 4361–4365 (1998).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

  9. Moore, F. L., Robinson, J. C., Bharucha, C. F., Sundaram, B. & Raizen, M. G. Atom optics realization of the quantum δ-kicked rotor. Phys. Rev. Lett. 75, 4598–4601 (1995).

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

  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).

    Google Scholar 

  13. Kozuma, M. et al. Coherent splitting of Bose-Einstein condensed atoms with optically induced Bragg diffraction. Phys. Rev. Lett. 82, 871–875 (1999).

    Article  ADS  CAS  Google Scholar 

  14. Steck, D. A., Oskay, W. H. & Raizen, M. G. Observation of chaos-assisted tunneling between islands of stability. Science (in the press).

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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.

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Correspondence to W. K. Hensinger.

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Hensinger, W., Häffner, H., Browaeys, A. et al. Dynamical tunnelling of ultracold atoms. Nature 412, 52–55 (2001).

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