The quantum theory of optical bistability in nonlinear systems
Permanent link to Research Commons versionhttps://hdl.handle.net/10289/16060
This thesis is concerned with the investigation of the behaviour of various nonlinear systems when excited by a coherent light-field. Of particular interest to us is the possibility of observing optical bistability in these systems. In Chapter One we introduce the general theoretical methods we employ to perform all investigations. In Chapters Two and Three we develop a quantum theory of the interaction of light with a variety of different systems, i.e. Raman active media, the parametric oscillator, the two-photon absorber and a system of three-level atoms. In each case, a master equation, containing all statistical information about the system, is derived. This enables the systematic inclusion of a damping mechanism into each model. Discussion is limited to the steady state behaviour of the systems, and in general we assume the deterministic limit in which we ignore quantum fluctuations in the field variables. It is then possible to factorise the steady state expectation values of these variables. Steady state calculations indicate that each system may exhibit optical bistability in output field/intensity dependent on input field/intensity. To determine whether a system will display optical bistability, it is necessary to perform a stability analysis. Where possible, such analytical calculations are performed. However, in certain cases the complexity of the highly nonlinear systems results in these calculations becoming extremely difficult. In such cases, conclusions relating to the stability of the system are drawn from graphical plots of its steady state behaviour. Chapters Four to Eight are devoted to a study of the intracavity interaction of coherent radiation with semiconductors. As explained in Chapter Four (of an introductory nature), this work was prompted by two recent experiments which indicated the existence of optical bistability in semiconductors. Although both experiments used semiconductors as the nonlinear material, the mechanism proposed to produce the observed bistability in each case was vastly different. In Chapter Eight we discuss one of these mechanisms – interband excitation. We present a very simple theory of this effect and show that in essence it is equivalent to the theory of optical bistability in two-level systems. However, we stress that our theory is only approximate and relies on the validity of several simplifying assumptions. Chapters Five and Six concern the other form of semiconductor optical bistability - arising from the interaction between a light field and excitons comprising the semiconductor. In order to develop a quantum theory of this system using master equation techniques, it is necessary to transform the fermion system to a boson system. Bosonisation transformations required to effect this are developed in Chapter Five. In Chapter Six we present a fully quantum mechanical theory of optical bistability in excitonic systems. Two types of bistability are found: bistability in output intensity and also exciton number, dependent on input intensity. In Chapter Seven we investigate the effects of quantum fluctuations in a low density exciton system, by considering its Fokker-Planck equation. We discuss in detail the adiabatic elimination of stochastic variables with regard to the system’s Fokker-Planck equation.
The University of Waikato
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