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arXiv:quant-ph/0102085 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Atomic motion in magneto-optical double-well potentials: A testing ground for quantum chaos

Shohini Ghose, Paul M. Alsing, Ivan Deutsch

Published Oct 24, 2001• 17 citations
arXivHTMLPDF

Abstract

We have identified ultracold atoms in magneto-optical double-well potentials as a very clean setting in which to study the quantum and classical dynamics of a nonlinear system with multiple degrees of freedom. In this system, entanglement at the quantum level and chaos at the classical level arise from nonseparable couplings between the atomic spin and its center of mass motion. The main features of the chaotic dynamics are analyzed using action-angle variables and Poincaré surfaces of section. We show that for the initial state prepared in current experiments [D. J. Haycock et al., Phys. Rev. Lett. 85, 3365 (2000)], classical and quantum expectation values diverge after a finite time, and the observed experimental dynamics is consistent with quantum-mechanical predictions. Furthermore, the motion corresponds to tunneling through a dynamical potential barrier. The coupling between the spin and the motional subsystems, which are very different in nature from one another, leads to interesting questions regarding the transition from regular quantum dynamics to chaotic classical motion.

DOI
10.1103/physreve.64.056119

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We have identified ultracold atoms in magneto-optical double-well potentials as a very clean setting in which to study the quantum and classical dynamics of a nonlinear system with multiple degrees of freedom. In this system, entanglement at the quantum level and chaos at the classical level arise from nonseparable couplings between the atomic spin and its center of mass motion. The main features of the chaotic dynamics are analyzed using action-angle variables and Poincaré surfaces of section. We show that for the initial state prepared in current experiments [D. J. Haycock et al., Phys. Rev. Lett. 85, 3365 (2000)], classical and quantum expectation values diverge after a finite time, and the observed experimental dynamics is consistent with quantum-mechanical predictions. Furthermore, the motion corresponds to tunneling through a dynamical potential barrier. The coupling between the spin and the motional subsystems, which are very different in nature from one another, leads to interesting questions regarding the transition from regular quantum dynamics to chaotic classical motion.
The transport of ultra‐cold atoms in magneto‐optical potentials provides a clean setting in which to investigate the distinct predictions of classical versus quantum dynamics for a system with coupled degrees of freedom. In this system, entanglement at the quantum level and chaos at the classical level arise from the coupling between the atomic spin and its center‐of‐mass motion. Experiments, performed deep in the quantum regime, correspond to dynamic quantum tunneling. This nonclassical behavior is contrasted with the predictions for an initial phase space distribution produced in the experiment, but undergoing classical Hamiltonian flow. We study conditions under which the trapped atoms can be made to exhibit classical dynamics through the process of continuous measurement, which localizes the probability distribution to phase space trajectories, consistent with the uncertainty principle and quantum “back‐action” noise. This method allows us to analytically and numerically identify the quantum‐classical boundary.
Cold atoms in optical potentials provide an ideal test bed to explore quantum nonlinear dynamics. Atoms are prepared in a magneto-optic trap or as a dilute Bose–Einstein condensate and subjected to a far detuned optical standing wave that is modulated. They exhibit a wide range of dynamics, some of which can be explained by classical theory while other aspects show the underlying quantum nature of the system. The atoms have a mixed phase space containing regions of regular motion which appear as distinct peaks in the atomic momentum distribution embedded in a sea of chaos. The action of the atoms is of the order of Planck’s constant, making quantum effects significant. This tutorial presents a detailed description of experiments measuring the evolution of atoms in time-dependent optical potentials. Experimental methods are developed providing means for the observation and selective loading of regions of regular motion. The dependence of the atomic dynamics on the system parameters is explored and distinct changes in the atomic momentum distribution are observed which are explained by the applicable quantum and classical theory. The observation of a bifurcation sequence is reported and explained using classical perturbation theory. Experimental methods for the accurate control of the momentum of an ensemble of atoms are developed. They use phase space resonances and chaotic transients providing novel ensemble atomic beamsplitters. The divergence between quantum and classical nonlinear dynamics is manifest in the experimental observation of dynamical tunnelling. It involves no potential barrier. However a constant of motion other than energy still forbids classically this quantum allowed motion. Atoms coherently tunnel back and forth between their initial state of oscillatory motion and the state 180 ° out of phase with the initial state.
Summary form only given. The recovery of the classical limit in quantum systems that have underlying chaotic dynamics is a key question in current studies of the quantum to classical transition. Interaction with an environment results in decoherence that can cause the quantum Wigner function to approach the corresponding classical phase space distribution by suppressing quantum interference effects, but it fails to extract localized classical trajectories from the quantum dynamics. Such trajectories are crucial for quantifying chaos both theoretically and in experiments through the measure of the Lyapunov exponents. One can recover trajectories from the quantum dynamics through the process of continuous measurements when the record is retained. Ehrenfest s theorem guarantees that quantum systems that are well localized by the measurement effectively obey classical mechanics. We have studied the role of continuous measurement in recovering the classical chaotic dynamics of laser cooled atoms in a magneto-optical lattice of double wells.
Summary form only given. We represent the first experiment evidence of dynamic tunneling of ultra-cold rubidium atoms. Rather then tunneling through a potential well, an atom tunnels from its initial motion into a different and oppositely directed motion. This process is classically forbidden. To observe this effect we have implemented an experiment that probes the dynamics of cold rubidium atoms in a far detuned optical standing wave. Rubidium atoms are prepared in a standard magneto-optic trap. The light intensity of the standing wave is modulated leading to a quantum chaotic mixed phase space for the atomic dynamics, consisting of regions of regular motion encircled by a sea of chaos. Phase space resonances can be observed as distinct peaks (with non-zero mean momentum) in the atomic momentum distribution.
We review the quantum transport of ultra-cold alkali atoms trapped in a one-dimensional optical lattice of double-potential wells, engineered through a combination of ac-Stark shifts and Zeeman interactions. The system is modelled numerically through analysis of the bandstructure and integration of the time-dependent Schrödinger equation. By these means we simulate coherent control of the atomic wavepackets. We present results from ongoing experiments on laser-cooled caesium, including the demonstration of quantum state preparation and preliminary evidence for coherent tunnelling. Entanglement between the internal and motional degrees of freedom allows us to access the tunnelling dynamics by Stern-Gerlach measurements of the ground state magnetic populations. A scheme to extend this into a full reconstruction of the density matrix for the ground state angular momentum is presented. We further consider the classical dynamics of our system, which displays deterministic chaos. This has important implications for the distinction between classical and quantum mechanical transport.
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.