Abstract
Unexpected accelerator modes were recently observed experimentally for cold cesium atoms when driven in the presence of gravity. A detailed theoretical explanation of this quantum effect is presented here. The theory makes use of invariance properties of the system, that are similar to the ones of solids, leading to a separation into independent kicked rotor problems. The analytical solution makes use of an asymptotic approximation very similar to the semiclassical one, except that the small parameter is not Planck's constant, but rather the detuning from the frequency that is resonant in absence of gravity.
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REFERENCES
For reviews, see, e.g., F. M. Izrailev, Simple models of quantum chaos: Spectrum and eigenfunctions, Phys. Rep. 196:299–393 (1990).
S. Fishman, in Proceedings of the International School of Physics Enrico Fermi: Varenna Course CXIX, G. Casati, I. Guarneri, and U. Smilansky, eds. (North-Holland, 1993), p. 187.
B. V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep. 52:263(1979).
A. J. Lichtenterg and M. A. Liberman, Regular and Chaotic Dynamics (Springer-Verlag, NY, 1992).
G. M. Zaslavsky, M. Edelman, and B. A. Niyazov, Self-similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics, Chaos 7:159–181 (1997).
G. M. Zaslavsky and M. Edelman, Hierarchical structures in the phase space and fractional kinetics: I. Classical systems, Chaos 10:135–146 (2000).
S. Fishman, D. R. Grempel, and R. E. Prange, Chaos, quantum recurrences, and Anderson localization, Phys. Rev. Lett. 49:509–512 (1982).
D. R. Grempel, R. E. Prange, and S. Fishman, Quantum dynamics of a nonintegrable system, Phys. Rev. A 29:1639–1647 (1984).
J. D. Hanson, E. Ott, and T. M. Antonsen, Influence of finite wavelength on the quantum kicked rotator in the semiclassical regime, Phys. Rev. A 29:819–825 (1984).
A. Iomin, S. Fishman, and G. M. Zaslavsky, Quantum localization for a kicked rotor with accelerator mode islands, Phys. Rev. E 65:036215(2002) (9 pages).
A. Iomin and G. M. Zaslavsky, Hierarchical structures in the phase space and fractional kinetics: II. Immense delocalization in quantized systems, Chaos 10:147–152 (2000).
B. Sundaram and G. M. Zaslavsky, Anomalous transport and quantum-classical correspondence, Phys. Rev. E 59:7231–7234 (1999).
F. M. Izrailev and D. L. Shepelyansky, Quantum resonance for rotator in nonlinear periodic field, Dok. Akad. Nauk SSSR 249:1103–1107 (1979).
G. Casati and I. Guarneri, Non recurrent behaviour in quantum mechanics, Comm. Math. Phys. 95:121(1984).
G. Casati, B. V. Chirikov, D. L. Shepelyansky, and I. Guarneri, Relevance of classical chaos in quantum mechanics in the hydrogen atom in a monochromatic field, Phys. Rep. 154:2(1987).
G. Casati, I. Guarneri, and D. L. Shepelyansky, Hydrogen atom in monochromatic field: Chaos and dynamical photonic localization, IEEE J. Quantum. Electron. 24: 1420–1444 (1988), and references therein.
E. J. Galvez, J. E. Sauer, L. Moorman, P. M. Koch, and D. Richards, Microwave ionization of H atoms: Breakdown of classical dynamics for high frequencies, Phys. Rev. Lett. 61:2011–2014 (1988).
J. E. Bayfield, G. Casati, I. Guarneri, and D. W. Sokol, Localization of classically chaotic diffusion for hydrogen atoms in microwave fields, Phys. Rev. Lett. 63:364–367 (1989).
M. Arndt, A. Buchleitner, R. N. Mantegna, and H. Walther, Experimental study of quantum and classical limits in microwave ionization of rubidium Rydberg atoms, Phys. Rev. Lett. 67:2435–2438 (1991).
D. A. Steck, V. Milner, W. H. Oskay, and M. G. Raizen, Quantitative study of amplitude noise effects on dynamical localization, Phys. Rev. E 62:3461–3475 (2000).
F. L. Moore, J. C. Robinson, C. F. Bharucha, Bala Sundaram, and M. G. Raizen, Atom optics realization of the quantum delta-kicked rotor, Phys. Rev. Lett. 75:4598–4601 (1995).
C. F. Bharucha, J. C. Robinson, F. L. Moore, Qian Niu, Bala Sundaram, and M. G. Raizen, Dynamical localization of ultracold sodium atoms, Phys. Rev. E 60:3881–3895 (1999).
B. G. Klappauf, W. H. Oskay, D. A. Steck, and M. G. Raizen, Quantum chaos with cesium atoms: Pushing the boundaries, Phys. (Amsterdam) 131 D:78–89 (1999).
H. Ammann, R. Gray, I. Shvarchuck, and N. Christensen, Quantum delta-kicked rotor: Experimental observation of decoherence, Phys. Rev. Lett. 80:4111–4115 (1998).
H. Ammann, R. Gray, I. Shvarchuck, and N. Christensen, Experimental observation of dynamical localization and decoherence in the atomic δ-kicked rotor, J. Phys. B 31:2449–2455 (1998)
R. Graham, M. Schlautmann, and P. Zoller, Dynamical localization of atomic-beam deflection by a modulated standing light wave, Phys. Rev. A 45:R19-R22 (1992).
C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions: Basic Processes and Applications (Wiley, New-York, 1992).
R. Blumel, S. Fishman, and U. Smilansky, Excitation of molecular rotation by periodic microwave pulses. A testing ground for Anderson localization, J. Chem. Phys. 84: 2604–2614 (1986).
F. M. Izrailev, Limiting quasienergy statistics for simple quantum systems, Phys. Rev. Lett. 56:541–544 (1986).
W. H. Oskay, D. A. Steck, V. Milner, B. G. Klappauf, and M. G. Raizen, Ballistic peaks at quantum resonance, Opt. Comm. 179:137–148 (2000).
S. Wimberger, I. Guarneri and S. Fishman, in preparation.
M. K. Oberthaler, R. M. Godun, M. B. d'Arcy, G. S. Summy, and K. Burnett, Observation of quantum accelerator modes, Phys. Rev. Lett. 83:4447–4451 (1999).
R. M. Godun, M. B. d'Arcy, M. K. Oberthaler, G. S. Summy, and K. Burnett, Quantum accelerator modes: A tool for atom optics, Phys. Rev. A 62:013411(2000) (8 pages).
M. B. d'Arcy, R. M. Godun, M. K. Oberthaler, G. S. Summy, K. Burnett, and S. A. Gardiner, Approaching classicality in quantum accelerator modes through decoherence, Phys. Rev. E 64:056233(2001) (15 pages).
S. Fishman, I. Guarneri, and L. Rebuzzini, Stable quantum resonances in atom optics, Phys. Rev. Lett. 89:084101(2002).
D. L. Shepelyansky, Some statistical properties of simple classically stochastic quantum systems, Phys. D 8:208–222 (1983).
G. Casati, G. Mantica, and D. L. Shepelyansky, Regular and anomalous quantum diffusion in the Fibonacci kicked rotator, Phys. Rev. E 63:066217(2001) (6 pages), and references therein.
L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3d Ed. (Pergamon, Oxford, 1977), p. 76.
T. Geisel, G. Radons, and J. Rubner, Kolmogorov-Arnol'd-Moser barriers in the quantum dynamics of chaotic systems, Phys. Rev. Lett. 57:2883–2886 (1986).
R. S. MacKay and J. D. Meiss, Relation between quantum and classical thresholds for multiphoton ionization of excited atoms, Phys. Rev. A 37:4702–4706 (1988).
J. D. Meiss, Comment on "Microwave ionization of H atoms: Breakdown of classical dynamics for high frequencies," Phys. Rev. Lett. 62:1576(1989).
D. R. Grempel, S. Fishman, and R. E. Prange, Finite-Planck's-constant scaling at stochastic transitions of dynamical systems, Phys. Rev. Lett. 53:1212–1215 (1984).
S. Fishman, D. R. Grempel and R. E. Prange, Temporal crossover from classical to quantal behavior near dynamical critical points, Phys. Rev. A 36:289–305 (1987).
See, e.g., ref. 38, Chap. VII.
V. V. Sokolov, O. V. Zhirov, and G. Casati, Quantum resonances of the kicked rotor and the SU(q) group, Phys. Rev. Lett. 84:3566–3569 (2000).
For a review see, e.g., G. Nenciu, Dynamics of band electrons in electric and magnetic fields: Rigorous justification of the effective Hamiltonians, Rev. Mod. Phys. 63:91–127 (1993) and references therein.
Q. Niu, X-G. Zhao, G. A. Georgakis, and M. G. Raizen, Atomic Landau-Zener tunneling and Wannier-Stark ladders in optical potentials, Phys. Rev. Lett. 76:4504–4507 (1996).
S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, Observation of Atomic Wannier-Stark Ladders in an Accelerating Optical Potential, Phys. Rev. Lett. 76:4512–4515 (1996).
M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Bloch oscillations of atoms in an optical potential, Phys. Rev. Lett. 76:4508–4511 (1996).
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Fishman, S., Guarneri, I. & Rebuzzini, L. A Theory for Quantum Accelerator Modes in Atom Optics. Journal of Statistical Physics 110, 911–943 (2003). https://doi.org/10.1023/A:1022176306198
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DOI: https://doi.org/10.1023/A:1022176306198