Entropy, stability and fluctuations in lasing systems

Abstract

We shall be concerned in this work, with the fundamental questions of entropy and stability in lasing systems, these being necessarily far from equilibrium. It is well known that a stable steady state near thermal equilibrium has associated with it a minimum entropy production. For an open system, the flow of entropy into the system (and out of it) must be considered as well. Further, the entropy production in a steady state near equilibrium, serves as a Liapounov function i.e. it is of definite sign (positive in this case) and its first time derivative has the opposite sign. Indeed, such a function exists for all dynamic states that are asymptotically stable. Recently, a generalized N-body form of the entropy was derived by Prigogine and co-workers in Brussels and Austin. This entropy which is obtained via a non-canonical transformation has the advantage of including both diagonal and off-diagonal elements of the density operator in contrast to the conventional Boltzmann form which holds only for dilute gases and contains only diagonal elements of the density matrix. In chapter two we present the theory underlying this generalized entropy and by means of simple stochastic examples, demonstrate how it overcomes the shortcomings of the Boltzmann form. We then extend our definition of the entropy to open systems by introducing an entropy flow. Finally, we outline the general criteria for dynamic stability and show how these lead to a modification of the celebrated Einstein formula for calculating fluctuations around non-equilibrium steady states. In particular, the theory predicts the existence of an N-body Liapounov function which is related to the generalized entropy, and which we use in our subsequent analysis. Chapter three is devoted to the semi-classical laser model. The stability of this model is demonstrated both near and far from equilibrium and the near-equilibrium results allow us to define an entropy production and derive the results of linear irreversible thermodynamics. In chapter four we consider a Liapounov function for the fully quantum-mechanical model below and above threshold. In particular, we show how the transition to the semi-classical theory is made in the limit of large photon numbers, which shows that our choice of Liapounov function is the same in a semi-classical theory as in a fully quantum-mechanical approach. Chapter five is devoted exclusively to the study of fluctuations. In addition to using the modified Einstein formula to derive the semi-classical first order fluctuations, we write a macroscopic 'birth-and-death' master equation for the laser which is readily soluble in the steady state. The fluctuations for high photon numbers are derived and found to agree with the strong-signal Scully-Lamb results. Further, it is possible to derive all the higher order moments in closed form, and a consistent truncation of the BBGKY hierarchy is obtained. Finally, we consider the stochastic model of Haken in chapter six in which we represent the laser as a rotating wave van der Pol oscillator with an external driving force. Our Liapounov function approach yields a very simple derivation of the Spiking condition.