Semiclassical and Quantum Statistical Dressed Mode Description of Optical Bistability

Abstract

Since the self-pulsing is a many-mode problem, it cannot be described by the mean field theory of OB, even for αL≪l, T≪l. Hence it seems at first hopeless to obtain a simple description of this phenomenon. In fact, the direct numerical solution of the Maxwell-Bloch equations (MBE) amounts to little more than a crude registration of data, without any predictive power. In order to get an insight into the behavior of these self-pulsing instabilities it is necessary to give an analytical or quasi-analytical description. We have achieved this goal by elaborating a formalism that we have called “dressed mode theory of OB.” This formalism is based on Haken’s formulation of generalized Ginzburg-Landau equations for Phase Transitions in systems far from thermal equilibrium. This method translates the MBE into an infinite set of equations in time only for the mode variables. We call them “dressed modes” because, even if each mode has a dominant field or atomic character, they incorporate in part the atom-field interaction. By selecting the dressed modes that play the dominant role and using the adiabatic elimination principle, we reduce the problem to a pair of coupled equations in time only, that describe self-pulsing, fully including both nonlinearity and propagation. The results obtained from these two equations agree very satisfactorily with the data obtained from the numerical solutions of the MBE. The picture of self-pulsing that arises from this bidimensional phase space is quite appealing and leads to new predictions. In particular, we find new types of hysteresis cycles,that involve both cw and pulsing solutions. In this picture the system appears as multistable rather than bistable, in the sense that some of the stable states are cw, others are pulsing. This dressedmode formalism has also been extended to the quantum statistical theory. Here the starting point is the many-mode master equation for absorptive and dispersive OB that has been recently derived by one of us (LAL). This equation holds for a ring cavity provided αL≪1, T≪l. Using the quantum statistical dressed mode treatment, we have studied the spectrum of transmitted light by considering all the modes of the cavity. Thanks to the flexibility of the formalism, we can obtain a quite general expression of the spectrum without adiabatically eliminating either the atomic or the field variables. We find that when the self-pulsing instability is approached the spectrum develops sidebands in correspondence to the modes that are going unstable. Thus once again the rise of an instability is heralded by the fluctuations of the system; crossing the instability threshold, the self-pulsing becomes manifest at a macroscopic level.