Hamiltonian chaos with a cold atom in an optical lattice

TL;DR

This paper explores how a cold atom in an optical lattice can exhibit Hamiltonian chaos, analyzing both classical and quantum dynamics, and identifying conditions for chaotic transport and fractal structures.

Contribution

It develops a semiclassical theory linking atomic chaos to a stochastic map and analyzes quantum effects like nonadiabatic transitions and wave packet splitting.

Findings

Chaotic atomic transport occurs under specific detuning and recoil conditions.

Fractal properties of deterministic atomic transport are analytically derived.

Quantum analysis shows nonadiabatic transitions and wave packet proliferation at nodes.

Abstract

We consider a basic model of the lossless interaction between a moving two-level atom and a standing-wave single-mode laser field. Classical treatment of the translational atomic motion provides the semiclassical Hamilton-Schrodinger equations which are a 5D nonlinear dynamical system with two integrals of motion. The atomic dynamics can be regular or chaotic in dependence on values of the control parameters, the atom-field detuning and recoil frequency. We develop a semiclassical theory of the chaotic atomic transport in terms of a random walk of the atomic electric dipole moment $u$. Based on a jump-like behavior of this variable for atoms crossing nodes of the standing wave, we construct a stochastic map that specifies the center-of-mass motion. We find the relations between the detuning, recoil frequency and the atomic energy, under which atoms may move in a optical lattice in a chaotic way. We obtain the analytical conditions under which deterministic atomic transport has fractal properties and explain a hierarchical structure of the dynamical fractals. Quantum treatment of the atomic motion in a standing wave is studied in the dressed state picture where the atom moves in two optical potentials simultaneously. If the values of the detuning and a characteristic atomic frequency are of the same order, than there is a probability of nonadiabatic transitions of the atom upon crossing nodes of the standing wave. At the same condition exactly, we observe sudden jumps in the atomic dipole moment $u$ when the atom crosses the nodes. Those jumps are accompanied by splitting of atomic wave packets at the nodes. Such a proliferation of wave packets at the nodes of a standing wave is a manifestation of classical atomic chaotic transport. At large values of the detuning, the quantum evolution is shown to be adiabatic in accordance with a regular character of the classical atomic motion.