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Physical Review A

Fluctuation theory in quantum-optical systems. II. Rigorous adiabatic elimination by the hierarchy method

Federico Casagrande, Elia Eschenazi, L. A. Lugiato

Published Jan 1, 1984• 14 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.29.239

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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.
An exact ccc-number generalized Fokker-Planck equation is derived for six noncommuting variables: the field variables, atomic populations, and atomic polarization of a laser model treated extensively by Lax. Langevin procedures yield the correct Fokker-Planck terms. Adiabatic eliminations are performed on the classical Markoff process given by the Fokker-Planck equation to compare with previous results. It is found that higher-order diffusion coefficients influence the second-order ones slightly during adiabatic elimination.
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.
Quantum fluctuations for multiphoton transitions in driven optical systems are treated on the basis of a Fokker-Planck equation for the Wigner function. The equation is considered without adiabatic elimination of the atoms or the field. The linear stability is analyzed and time correlations are derived. A spectrum of transmitted light associated with vacuum Rabi splitting is obtained. The squeezing and antibunching effects in the atomic system and the field are discussed. This theory coincides with the adiabatic elimination theory for the case of both good- and bad-cavity limits.
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.
In the general framework of the system size expansion of Van Kampen and Kubo, we consider the Fokker-Planck equation for a model of absorptive bistability in the bad-cavity limit. The physical system is described by the reduced atomic density operators after adiabatic elimination of the cavity field variables. Mapping of the master equation into ccc-number form according to the normal-ordering mapping scheme yields known results for the atomic fluctuations and correlation functions; however, it also leads to a Fokker-Planck equation with a non-positive-definite diffusion matrix. The symmetrical-order-mapping scheme eliminates this difficulty. The leading contribution to the system size expansion yields a Fokker-Planck equation for the symmetrical-ordered density function having a positive-definite diffusion matrix. The atomic expectation values and fluctuations previously derived from the quantum Langevin equations emerge naturally from this Fokker-Planck equation.
A quantum treatment of a coherently driven cavity with a medium of two-level atoms is given. The system displays both absorptive and dispersive bistability. Quantum fluctuations are included via a Fokker-Planck equation in a generalized PPP representation, after adiabatic elimination of the atomic variables. The transmitted spectrum and correlation function is calculated in a linearized (Gaussian) approximation. In the limit of large detuning and radiative damping, the quantum fluctuations are identical to those of a simple nonlinear polarizability theory. An analysis is given of the nonclassical statistical properties of the intensity-correlation spectrum.