Harmonic oscillator subject to parametric pulses: An amplitude (Milne) oscillator approach

Authors: Thylwe, K.-E., Korsch, H.-J
Document Type: Article
Pubstate: Published
Journal: J. Phys. A: Math. Gen
Volume: 34   3497-3510.
Year: 2001


A harmonic oscillator subject to a parametric pulse is examined. The aim of the paper is to present a new theory for analysing transitions caused by parametric pulses. The new theoretical notions which are introduced relate the pulse parameters in a direct way with transition matrix elements. The harmonic-oscillator transitions are expressed in terms of the asymptotic properties of a companion oscillator, the Milne (amplitude) oscillator. A traditional phase-amplitude decomposition of the harmonic-oscillator solutions results in the so-called Milne’s equation for the amplitude, and the phase is determined by an exact relation to the amplitude. This approach is extended in the present analysis with new relevant concepts and parameters for pulse dynamics of classical and quantal systems. The amplitude oscillator has a particularly nice numerical behaviour. In the case of strong pulses it does not possess any of the fast oscillations induced by the pulse on the original harmonic oscillator. Furthermore, the new dynamical parameters introduced in this approach are closely related to the relevant characteristics of the pulse. The relevance to quantum mechanical problems such as reflection and transmission from a localized well and the mechanical problem of controlling vibrations is illustrated.