Modulated kicks approximation in non-integrable classical and quantum systems

Abstract

Time-dependent harmonic perturbation in dynamical systems is frequently approximated by a sequence of infinitely short pulses (so-called kicks) of alternating sign. In order to improve this approximation we increase the number of kicks per perturbation period. The validity of this method is tested numerically on an exemplary classical system: a particle in the Rosen-Morse potential well, driven by an external harmonic perturbation. Dynamical systems with an unrealistic perturbation, in the form of a sequence of infinitely short pulses, have recently drawn a lot of attention. Such an approach greatly simplifies the analysis of a system, since the time evolution can be described in terms of the appropriate classical (or quantum) map. The well known kicked rotator model was investigated in the study of classical (l, 21 and quantum (3-51 chaos. In the dynamical models of interaction of atoms with strong electromagnetic fields (6-91 the sinusoidal changes of field were simulated by a train of kicks of alternating sign. Not much is yet known about the validity of such an approximation. Leopold and Richards (6), however, reported significant differences between results obtained with impulsive and continuous perturbation. In order to improve the alternated kicks approximation (AKA) and simulate the continuous perturbation in a better way we suggest increasing the number of pulses per wave period. In this letter the modulated kicks approximation (MKA) is defined and tested on an exemplary dynamical model. A correspondence is shown between the approximation of a continuous perturbation in a physical system by a train of infinitely short pulses and the numerical methods of solving the non-linear differential equations by the discretisation of time. Consider an arbitrary smooth function f( t) with period T. Let us define a sequence of distributions jN