We reconsider the problem of the adiabatic elimination of selected dynamical variables in the description of nonlinear systems, with a special emphasis on the identification of suitable criteria for the global validity of this procedure. The problem is analyzed in detail using as a guideline the one-mode homogeneously broadened laser model, with an injected signal and an arbitrary population difference for added flexibility. We propose five conditions for the global validity of the adiabatic limit, after consideration not only of the relative size of the time scales involved, but also of the magnitude of all parameters, of the physical variables, and of their fluctuations. The adiabatic elimination is formulated in the context of the dressed-mode description, leading to a generalization of the procedure adopted in earlier studies of multimode absorptive bistability. In addition, we establish a rigorous link between the adiabatic-elimination procedure and the multiple-time-scale approach to dynamic systems. The scaling behavior of the variables, of the eigenvalues of the linearized equations, and of the dressed-mode amplitudes and eigenvectors with respect to the smallness parameter of the problem is studied in detail. The relevance of what we shall define as "normal" and "anomalous" scaling in the text to the applicability of the adiabatic-elimination process is clarified. We also develop an alternative adiabatic-elimination scheme for the special case where all the rate constants of the system have the same order of magnitude, but another smallness parameter can be identified. From our analysis, it should be clear that our main conclusions are model independent, and not at all restricted to the specific features of the dynamical system selected as a test case for our discussion.
We reconsider the problem of the adiabatic elimination of selected dynamical variables in the description of nonlinear systems, with a special emphasis on the identification of suitable criteria for the global validity of this procedure. The problem is analyzed in detail using as a guideline the one-mode homogeneously broadened laser model, with an injected signal and an arbitrary population difference for added flexibility. We propose five conditions for the global validity of the adiabatic limit, after consideration not only of the relative size of the time scales involved, but also of the magnitude of all parameters, of the physical variables, and of their fluctuations. The adiabatic elimination is formulated in the context of the dressed-mode description, leading to a generalization of the procedure adopted in earlier studies of multimode absorptive bistability. In addition, we establish a rigorous link between the adiabatic-elimination procedure and the multiple-time-scale approach to dynamic systems. The scaling behavior of the variables, of the eigenvalues of the linearized equations, and of the dressed-mode amplitudes and eigenvectors with respect to the smallness parameter of the problem is studied in detail. The relevance of what we shall define as "normal" and "anomalous" scaling in the text to the applicability of the adiabatic-elimination process is clarified. We also develop an alternative adiabatic-elimination scheme for the special case where all the rate constants of the system have the same order of magnitude, but another smallness parameter can be identified. From our analysis, it should be clear that our main conclusions are model independent, and not at all restricted to the specific features of the dynamical system selected as a test case for our discussion.
The adiabatic elimination principle [91, 92] is universal in the area of nonlinear dissipative dynamical systems, and its importance arises from the fact that it allows one to reduce the number of the equations which govern the dynamics of the system. We introduce it at this stage in the book in order to be able to describe the optical pumping mechanisms which can be exerted to attain population inversion. However, this principle will be applied repeatedly in the following chapters for other purposes. In the first subsection we describe the principle in general, while the following subsections concern the issue of optical pumping. We show first that it is not possible to obtain population inversion between the two levels of the lasing transition if the pump involves only these two levels. Next we illustrate the three-level pumping scheme and the four-level pumping scheme. We demonstrate that, by adiabatically eliminating the probabilities of the additional energy levels, one arrives at an equation identical to Eq. (4.12), which, for appropriate ranges of values of the parameters involved in the pumping, allows one to obtain population inversion. General formulation of the principle In a dissipative dynamical system the dynamical variables undergo relaxation processes associated with suitable rate constants such as, for example, the rate γ ⊥ for the normalized atomic polarization P and the rate γ ∥ for the normalized population difference D . The inverses of these rates provide the time scales which characterize the evolution of each variable. Therefore we can distinguish a set of fast variables , which evolve over short time scales and are therefore characterized by large relaxation rates, and a set of slow variables , which display small relaxation rates. A remarkable simplification in the analysis of dynamical systems is obtained by using the technique of adiabatic elimination of fast variables, which Haken considered as a basis for the discipline which he formulated and called Synergetics [91, 92].
The relaxation rates κ , γ ⊥ and γ ∥ which appear in the single-mode model are irrelevant for the stationary state. Instead, their values become critical when we consider dynamical, i.e. time-dependent, phenomena. In particular, they control the possibility of eliminating adiabatically some of the variables in play, thus reducing the complexity of the problem. In this chapter we first illustrate, in Section 19.1, the useful laser classification introduced by Arecchi. Next, in Section 19.2, we discuss the dynamical behavior of the solutions of a model obtained from the single-mode model (derived in Chapter 12) by performing the adiabatic elimination of the atomic variables, which procedure is called the good-cavity limit in the literature. Section 19.2.1 is devoted to the case of the free-running laser ( class-A laser ) and discusses in particular, at the end, the cubic single-mode laser model which is the simplest model capable of describing a laser. On the other hand, in Section 19.2.2 we discuss the case of optical bistability in the good-cavity limit. If we eliminate adiabatically the atomic polarization only, we obtain, instead, the single- mode rate equations which are discussed in Section 19.3. In particular, Section 19.3.1 concerns the case of class-B lasers and focusses especially on the phenomenon of relaxation oscillations. Special attention is paid to demonstrating the equivalence of the model which describes class-B lasers to the rate-equation laser model derived in Chapter 1 and the semiconductor laser model obtained in Chapter 16. In Section 19.3.2 we describe the generation of a giant pulse in class-B lasers, a phenomenon which is called active Q- switching . The final section, Section 19.4, deals with the model obtained by adiabatically eliminating the field variable in the so-called bad-cavity limit . Classification of lasers Arecchi [243] identifies three especially important kinds of laser, which he calls class-A, class-B and class-C lasers, respectively. (1) Class-A lasers are characterized by the condition κ ≪ γ ⊥ , γ ∥ . In this case the atomic variables can be adiabatically eliminated. The He–Ne laser, argon laser and quantum-cascade laser belong to this class.
We present a regorous adiabatic elimination of the atomic variables in the framework of the quantum-statistical single-mode laser model. This procedure, which is systematic and similar to the one used in the semiclassical theory, is based on the hierarchy method proposed years ago by one of us. Using a rigorous definition of the adiabatic limit, we show that the hierarchy is exactly truncated by the limit itself. Thus we recover the Fokker-Planck equation previously derived by us using a nonsystematic procedure. We discuss the advantages of the present adiabatic elimination method over the ones previously used by various authors.
We carry out a systematic adiabatic elimination of the atomic degrees of freedom from the quantum-mechanical master equation for the single-mode laser. We represent the reduced density operator of the field mode by various quasiprobabilities and construct the respective equations of motion. The generators of infinitesimal time translations are obtained as series in powers of two parameters, the smallness of which defines the adiabatic limit. Our "adiabatic" expansion treats that part of the atom-field interaction as a zeroth-order effect which describes the action of the field on the atoms, while the reaction of the atoms is treated perturbatively. The adiabatic equilibrium thus assigned to the atomic variables at all times is a conditional one, contingent on the current state of the field mode. As a result, saturation effects in the atoms are fully accounted for in low orders of our expansion. In second order, especially, we obtain a Fokker-Planck equation for the Wigner function of the field mode which is valid for arbitrary pump strengths below, near, and above threshold. We compare our results with those of previous theories.
We discuss the steady-state and the linearized response of a multimode laser with an injected signal. The steady-state analysis is carried out for arbitrary values of the laser gain and resonator parameters, while the linearized stability calculation is limited to parameter ranges where the mean-field assumption is a reasonable approximation. For suitably large values of the polarization and population decay rates, we review the problem of the adiabatic elimination of the atomic variables and demonstrate explicitly the existence of possible pitfalls and how to avoid them.
Introduction The transient behavior of nonlinear optical resonators which display bistability and its reaction to fluctuations of system parameters are of extreme importance in all possible applications of these devices. Moreover, however, studies on these processes may give insight into some more general aspects of the behavior of nonlinear dynamical systems. In fact it has been emphasized in the early stages of the development of the field that optically bistable systems, in a suitable model, may exhibit a first-order phase transition (Bonifacio and Lugiato, 1978). Since the threshold behavior of a laser can be described as an example of a second-order phase transition (Haken, 1964,1975; see also Haken, 1977), optical bistability in (passive) nonlinear resonators complements nicely lasers as nonlinear dissipative systems in optics. Both types of optical systems are easily accessible to measurements, and the time scales coming into play are convenient for detailed experimental studies of the dynamics. It is worth mentioning that there is a close connection between laser operation and optical bistability in nonlinear (passive) resonators (see Lugiato, 1984). The standard single-mode laser model with distributed losses (Haken, 1970; Sargent, Scully and Lamb, 1974) can be generalized to include the effect of an external field that continuously drives the atoms within the optical resonator. The standard model of absorptive bistability, the Bonifacio–Lugiato model, is a special case of this generalized model. It represents the situation of a pumping mechanism being absent.