Methods of Adiabatic Elimination of Variables in Simple Laser Models

Abstract

In the last years, the study of low dimensional systems of nonlinear differential equations has shown the richness of phenomena associated with simple mathematical models [1]. Self-oscillating behavior and deterministic chaos are two widely studied aspects of nonlinear dynamics. However, while a qualitative agreement between theory and experiment is often found due to the’ structural stability’ of such phenomena, a quantitative agreement is very rarely obtained. In fact, the starting models, yielding a realistic description of physical systems, often invoke partial differential equations or, more generally, high-dimensional sets of ordinary differential equations. As a consequence, even a numerical analysis of such models represents a hard task. However, in many cases, the asymptotic motion, i. e. the most interesting one from an experimental point of view, involves only a few degrees of freedom, despite the complexity of the model. A striking example is given by periodic convection in Rayleigh-Benard experiments of hydrodynamics. The behavior of single mode homogeneously broadened lasers, is also known to be described by a few macroscopic variables, because of cooperative effects. In this last case, a further reduction of degrees of freedom can be accomplished when a large separation exists among the relaxation time scales of electric field, polarization and population inversion. All these examples indicate the existence of simple sets of differential equations, describing only the behavior of the ‘relevant’ variables. The procedure to reduce the number of degrees of freedom, is the Adiabatic Elimination (AE) also called Slaving Principle by Haken, who was the first to introduce such a method [2]. The idea is to divide the variables into two groups (relevant, irrelevant) depending on their damping rates (small, large, respectively). The second group can be eliminated since the associated variables describe a fast relaxation towards an instantaneous equilibrium position, whereas they are slaved by the motion of the slow ones.