Quantum chaos and a periodically perturbed Eberly-Chirikov pendulum

Abstract

A two-level system contained in a single-mode resonant cavity tuned to the energy-level separation of the two-level system is studied. A purely quantum-level description is converted into five coupled ordinary differential equations for certain relevant expectation values. These equations are identical with a system of equations proposed by Belobrov, Zaslavskii, and Tartakovskii on semiclassical grounds, and very closely related to a similar system proposed by Milonni, Ackerhalt, and Galbraith on semiclassical grounds. The level-population expectation value for the two-level system shows a transition to chaos as the coupling strength is increased. This transition is suggested by power spectra and confirmed by calculation of corresponding Liapunov exponents. It is shown that the equations can be transformed so that they exhibit the presence of a periodically perturbed Eberly-Chirikov pendulum as the key dynamical element responsible for the observed behavior. Numerical simulation of this periodically perturbed pendulum is shown to reproduce the peculiar features of the observed spectra obtained for the full, five-variable model. We discuss the relationship of these studies to the issue of bona fide chaos in a purely quantum-mechanical-level description of the system.