The Dynamics of Periodically Forced Systems

Abstract

The classical and quantum dynamics of a one-dimensional atomic system perturbed by a periodic electric field of frequency, Ω, in the regimes of high and low field frequency is studied. At high frequencies various ionisation mechanisms are considered in both dynamics. We show that for systems having analytic potentials, and for sufficiently high frequencies, the classical system can ionize through regular orbits, in contradistinction to the driven Coulomb system. An area-preserving map is constructed which approximates the classical motion well at high frequencies; explicit quantization of this map, in terms of the Fourier components of the classical motion, provides a very efficient means of obtaining approximate solutions to the one-dimensional, time-dependent Schrödinger equation. The Morse oscillator is considered in detail: the classical map is found to agree well with the numerical solution of Hamilton’s equations. Classical and quantal ionization probabilities are compared and circumstances delineated where they agree. Comparisons of various theoretical models with experimental data for the ionization of excited hydrogen atoms in low frequency microwave fields arc used to distinguish between tunnelling through and classical escape over the slowly oscillating barrier and between one- and many-state dynamical processes. Formulae used to interpret low frequency laser multi-photon ionization data are found not to describe the experimental data which are best reproduced by the new semiclassical model presented here. Ranges of validity of other models are delineated. A new analytic approximation for the solutions of the two-state equations of motion is obtained and used to predict the positions and widths of each member of the infinite set of resonances between any finite value of Ω and 0. This analysis shows why recent experiments on the microwave ionization of hydrogen atoms by low frequency fields failed to observe any resonances