Nonlinear coherent dynamics of an atom in an optical lattice

We consider a simple model of lossless interaction between a two-level single atom and a standing-wave single-mode laser field which creates a one-dimensional optical lattice. Internal dynamics of the atom is governed by the laser field which is treated to be classical with a large number of photons. Center-of-mass classical atomic motion is governed by the optical potential and the internal atomic degree of freedom. The resulting Hamilton-Schrödinger equations of motion are a five-dimensional nonlinear dynamical system with two integrals of motion, the total atomic energy and the length of the Bloch vector are conserved during the interaction. In our previous papers the motion of the atom has been shown to be regular or chaotic (in the sense of exponential sensitivity to small variations in initial conditions and/or the system's control parameters) in dependence on values of the control parameters, the atom-field detuning and recoil frequency. At exact atom-field resonance, exact solutions for both the external and internal atomic degrees of freedom can be derived. The center-of-mass motion does not depend in this case on the internal variables, whereas the Rabi oscillations of the atomic inversion is a frequency modulated signal with the frequency to be defined by the atomic position in the optical lattice. We study analytically correlations between the Rabi oscillations and the center-of-mass motion in two limiting cases of a regular motion out off the resonance: (1) far-detuned atoms and (2) fastly moving atoms. The main focus of the paper is chaotic atomic motion that may be quantified strictly by positive values of the maximal Lyapunov exponent. It is shown that atom, depending on the value of its total energy, can either oscillate chaotically in a well of the optical potential or fly ballistically with weak chaotic oscillations of its momentum or wander in the optical lattice changing the direction of motion in a chaotic way. In the regime of chaotic wandering atomic motion is shown to have fractal properties. We find a useful tool to visualize complicated atomic motion — Poincaré mapping of atomic trajectories in an effective three-dimensional phase space onto planes of atomic internal variables and momentum. The Poincaré mappings are constructing using a translational invariance of the standing laser wave. We find common features with typical non-hyperbolic Hamiltonian systems — chains of resonant islands of different sizes imbedded in a stochastic sea, stochastic layers, bifurcations, and so on. The phenomenon of sticking of …