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Journal of Physics A Mathematical and General

Dynamics of atom-cavity system and elimination of momentum recoil

Li-Xiang Cen, Shun-Jin Wang

Published May 5, 2000• 4 citations
Publisher

Abstract

In the framework of algebraic dynamics, we investigate an atom-cavity system with atomic centre-of-mass motion included. Using nonlinear transformations, the Hamiltonian is linearized in terms of Lie algebraic generators so that the algebraic dynamical structure of the system appears quite clear and its solution is easy to handle by an algebraic dynamical method. In more detail, for the three-level system of Ξ configuration, we show that the momentum recoil of the atom can be eliminated under certain conditions.

DOI
10.1088/0305-4470/33/19/305

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In the framework of algebraic dynamics, we investigate an atom-cavity system with atomic centre-of-mass motion included. Using nonlinear transformations, the Hamiltonian is linearized in terms of Lie algebraic generators so that the algebraic dynamical structure of the system appears quite clear and its solution is easy to handle by an algebraic dynamical method. In more detail, for the three-level system of Ξ configuration, we show that the momentum recoil of the atom can be eliminated under certain conditions.
Strong coupling of the internal and external degrees of freedom of a cold atom to each other and to the spatially periodic field of the standing light wave in a high-finesse cavity is responsible for the dynamic instability of the atomic center-of-mass motion. Due to a weak interaction of the internal nonlinear resonances in the standard model of cavity QED, a stochastic layer appears, whose width in the semiclassical approximation is estimated in terms of the main parameters of the system: atomic recoil frequency, mean number of excitations, and detuning from the resonance. As a result, the atomic motion in the absolutely regular potential has the fractal character, with long Lévy flights alternating with small chaotic oscillations in potential wells.
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.