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arXiv:quant-ph/0701038 · Physical Review A

Theory of chaotic atomic transport in an optical lattice

Victor Yu. Argonov, Sergey V. Prants

Published Jun 29, 2007• 42 citations
arXivHTMLPDF

Abstract

A semiclassical theory of chaotic atomic transport in a one-dimensional nondissipative optical lattice, when the standing-wave field and center-of-mass motion are treated classically, is developed. Using the basic equations of motion for the Bloch and translational atomic variables, we derive a stochastic map for the synchronized component of the atomic dipole moment that determines the center-of-mass motion. We find the analytical relations between the atomic and lattice parameters under which atoms typically alternate between flying through the lattice and being trapped in the wells of the optical potential. We use the stochastic map to derive formulas for the probability density functions (PDFs) for the flight and trapping events. Statistical properties of chaotic atomic transport strongly depend on the relations between the atomic and lattice parameters. We show that there is a good quantitative agreement between the analytical PDFs and those computed with the stochastic map and the basic equations of motion for different ranges of the parameters. Typical flight and trapping PDFs are shown to be broad distributions with power law segments with the slope $\ensuremath{-}1.5$ and exponential ``tails.'' The lengths of the power law and exponential segments of the PDFs depend on the values of the parameters. We find analytical conditions under which deterministic atomic transport has fractal properties and explain a hierarchical structure of the dynamical fractals.

DOI
10.1103/physreva.75.063428

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A semiclassical theory of chaotic atomic transport in a one-dimensional nondissipative optical lattice, when the standing-wave field and center-of-mass motion are treated classically, is developed. Using the basic equations of motion for the Bloch and translational atomic variables, we derive a stochastic map for the synchronized component of the atomic dipole moment that determines the center-of-mass motion. We find the analytical relations between the atomic and lattice parameters under which atoms typically alternate between flying through the lattice and being trapped in the wells of the optical potential. We use the stochastic map to derive formulas for the probability density functions (PDFs) for the flight and trapping events. Statistical properties of chaotic atomic transport strongly depend on the relations between the atomic and lattice parameters. We show that there is a good quantitative agreement between the analytical PDFs and those computed with the stochastic map and the basic equations of motion for different ranges of the parameters. Typical flight and trapping PDFs are shown to be broad distributions with power law segments with the slope \ensuremath−1.5\ensuremath{-}1.5\ensuremath−1.5 and exponential ``tails.'' The lengths of the power law and exponential segments of the PDFs depend on the values of the parameters. We find analytical conditions under which deterministic atomic transport has fractal properties and explain a hierarchical structure of the dynamical fractals.
We study dissipative transport of spontaneously emitting atoms in a one-dimensional (1D) standing-wave laser field in the regimes where the underlying deterministic Hamiltonian dynamics is regular and chaotic. A Monte Carlo stochastic wave-function method is applied to simulate semiclassically the atomic dynamics with coupled internal and translational degrees of freedom. It is shown in numerical experiments and confirmed analytically that chaotic atomic transport can take the form either of ballistic motion or a random walking with specific statistical properties. The character of spatial and momentum diffusion in the ballistic atomic transport is shown to change abruptly in the atom-laser detuning regime where the Hamiltonian dynamics is irregular in the sense of dynamical chaos. We find a clear correlation between the behavior of the momentum diffusion coefficient and Hamiltonian chaos probability which is a manifestation of chaoticity of the fundamental atom-light interaction in the diffusivelike dissipative atomic transport. We propose to measure a linear extent of atomic clouds in a 1D optical lattice and predict that, beginning with those values of the mean cloud's momentum for which the probability of Hamiltonian chaos is close to 1, the linear extent of the corresponding clouds should increase sharply. A sensitive dependence of statistical characteristics of dissipative transport on the values of the detuning allows us to manipulate the atomic transport by changing the laser frequency.
Abstract This paper is a review article on some aspects of quantum–classical correspondence in chaotic dynamics of cold atoms interacting with a standing-wave laser field forming an optical lattice. The problem is treated from both (semi)classical and quantum points of view. In both approaches, the interaction of an atomic electic dipole with the laser field is treated quantum mechanically. Translational motion is described, at first, classically (atoms are considered to be point-like objects) and then quantum mechanically as a propagation of matter waves. Semiclassical equations of motion are shown to be chaotic in the sense of classical dynamical chaos. Point-like atoms in an absolutely deterministic and rigid optical lattice can move in a random-like manner demonstrating a chaotic walking with typical features of classical chaos. This behavior is explained by random-like ‘jumps’ of one of the atomic internal variable when atoms cross nodes of the standing wave and occurs in a specific range of the atom-field detuning. When treating atoms as matter waves, we show that they can make nonadiabatic transitions when crossing the standing-wave nodes. The point is that atomic wave packets split at each node in the same range of the atom-field detuning where the classical chaos occurs. The key point is that the squared amplitude of those semiclassical ‘jumps’ equal to the quantum Landau–Zener parameter which defines the probability of nonadiabatic transitions at the nodes. Nonadiabatic atomic wave packets are much more complicated compared to adiabatic ones and may be called chaotic in this sense. A few possible experiments to observe some manifestations of classical and quantum chaos with cold atoms in horizontal and vertical optical lattices are proposed and discussed.
Centre-of-mass atomic motion in an optical lattice is shown to be near the resonance a chaotic walking due to the interplay between coherent internal atomic dynamics and spontaneous emission. Statistical properties of chaotic atomic motion can be controlled by the single parameter, the detuning between the atomic transition frequency and the laser frequency. We derive a Fokker-Planck equation in the energetic space to describe the atomic transport near the resonance and demonstrate numerically how to manipulate the atomic motion varying the detuning.
Near the resonance, centre-of-mass atomic motion in an optical lattice is shown to be a chaotic walking due to the interplay between coherent internal atomic dynamics and spontaneous emission. Statistical properties of chaotic atomic motion can be controlled by a single parameter, the detuning between the atomic transition frequency and the laser frequency. We derive a Fokker-Planck equation in the energetic space to describe the atomic transport near the resonance and demonstrate numerically how to manipulate the atomic motion varying the detuning.
We study dynamics of the atom-photon interaction in cavity quantum electrodynamics, considering a cold two-level atom in a single-mode high-finesse standing-wave cavity as a nonlinear Hamiltonian system with three coupled degrees of freedom: translational, internal atomic, and the field. The system proves to have different types of motion including Lévy flights and chaotic walkings of an atom in a cavity. The corresponding equations of motion for expectation values of the atom and field variables have two characteristic time scales: fast Rabi oscillations of the internal atomic and field quantities and slow translational oscillations of the center of the atom mass. It is shown that the translational motion, related to the atom recoils, is governed by an equation of a parametric nonlinear pendulum with a frequency modulated by the Rabi oscillations. This type of dynamics is chaotic with some width of the stochastic layer that is estimated analytically. The width is fairly small for realistic values of the control parameters, the normalized detuning delta and atomic recoil frequency alpha. We consider the Poincaré sections of the dynamics, compute the Lyapunov exponents, and find a range of the detuning, |delta| less, similar 3, where chaos is prominent. It is demonstrated how the atom-photon dynamics with a given value of alpha depends on the values of delta and initial conditions. Two types of Lévy flights, one corresponding to the ballistic motion of the atom and the other corresponding to small oscillations in a potential well, are found. These flights influence statistical properties of the atom-photon interaction such as distribution of Poincaré recurrences and moments of the atom position x. The simulation shows different regimes of motion, from slightly abnormal diffusion with <x(2)> approximately tau(1.13) at delta=1.2 to a superdiffusion with <x(2)> approximately tau(2.2) at delta=1.92 that corresponds to a superballistic motion of the atom with an acceleration. The obtained results can be used to find new ways to manipulate atoms, to cool and trap them by adjusting the detuning delta.
Centre-of-mass atomic motion in an optical lattice near the resonance is shown to be a chaotic walking due to the interplay between coherent internal atomic dynamics and spontaneous emission. Statistical properties of chaotic atomic motion can be controlled by the single parameter, the detuning between the atomic transition frequency and the laser frequency. We derive a Fokker-Planck equation in the energetic space to describe the atomic transport near the resonance and demonstrate numerically how to manipulate the atomic motion varying the detuning.