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Physical Review Letters

Dynamical Localization: Classical vs Quantum Oscillations in Momentum Spread of Cold Atoms

P. J. Bardroff, Iwo Białynicki‐Birula, D. S. Krähmer, Gershon Kurizki, E. Mayr, P. Stifter, Wolfgang P. Schleich

Published May 15, 1995• 54 citations
Publisher

Abstract

We investigate the classical and quantum dynamics of atoms moving in a phase-modulated standing light field. In both cases the width of the momentum distribution exhibits characteristic oscillations as a function of the modulation amplitude. We argue that at the maxima of these oscillations the system is chaotic, whereas in the valleys it is almost regular. Quantum localization appears only in the chaotic regime. We connect our analysis with a recent experiment [F. Moore et al., Phys. Rev. Lett. 73, 2974 (1994)].

DOI
10.1103/physrevlett.74.3959

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We investigate the classical and quantum dynamics of atoms moving in a phase-modulated standing light field. In both cases the width of the momentum distribution exhibits characteristic oscillations as a function of the modulation amplitude. We argue that at the maxima of these oscillations the system is chaotic, whereas in the valleys it is almost regular. Quantum localization appears only in the chaotic regime. We connect our analysis with a recent experiment [F. Moore et al., Phys. Rev. Lett. 73, 2974 (1994)].
In this paper the model of two level atomic momentum spread in amplitude and phase modulated standing light wave was investigated. this is a nonlinear quantum pendulum driven by a time dependent perterbation with two frequencies. This system shows chaotic behaviour in the classical limit. The system exists the characteristic of dynamical localization for the same parameters as that in the classical model correspoinding to it. Localization length of the system with two incommensurate perturbing frequency is much larger than that of the system with one perturbing frequency.
Dynamics of the quantum system that consists of a cold atom in a modulated standing wave of light is analyzed using the time-dependent variational principle formulation based on squeezed coherent states. A group of ordinary differential equations describing evolution of two pairs of canonically conjugate variables (q(t), p(t); G(t), Π(t)) are derived where G(t) and Π(t) describe the quantum fluctuations of the system. It has been shown that a transition from the regular motion to the chaotic motion in G(t), Π(t) phase space. Quantum system seems to be capable to show the classical-like chaotic structure.
The recent pioneering experiments of the [Nature 412, 52 (2001)] and [Science, 293, 274 (2001)] groups have demonstrated the dynamical tunneling of cold atoms interacting with standing electromagnetic waves. It has been shown [Phys. Rev. Lett. 89, 253201 (2002)], that the tunneling oscillations observed in these experiments correspondingly stems from two- and three-Floquet quantum state mechanism and can be controlled by varying the experimental parameters. The question where are the fingerprints of the classical chaotic dynamics in a quantum dynamical process which is controlled by 2 or 3 quantum states remains open. Our calculations show that although the effective $\ensuremath{\hbar}$ associated with the two experiments is large, and the quantum system is far from its semiclassical limit, the quantum Floquet-Bloch quasienergy states still can be classified as regular and chaotic states. In both experiments the quantum and the classical phase-space entropies are quite similar, although the classical phase space is a mixed regular-chaotic space. It is also shown that as the wave packet which is initially localized at one of the two inner regular islands oscillates between them through the chaotic sea, it accumulates a random phase which causes the decay of the amplitude of the oscillating mean momentum, ⟨P(t)⟩⟨P(t)⟩⟨P(t)⟩, as measured in both experiments. The extremely high sensitivity of the rate of decay of the oscillations of ⟨P(t)⟩⟨P(t)⟩⟨P(t)⟩ to the very small changes in the population of different Floquet-Bloch states, is another type of fingerprint of chaos in quantum dynamics that presumably was measured in the NIST and AUSTIN experiments for the first time.