In a recent experiment [ R. G. Devoe and R. G. Brewer, Phys. Rev. Lett. 50, 1269 ( 1983)], it was found that the optical Bloch equations could not satisfactorily explain the signal that was observed for free-induction decay in the impurity ion cyrstal Pr3+:LaF3. Several theories have been proposed to explain this failure of the Bloch equations. In this paper, the general validity conditions for the optical Bloch equations are examined within the limits of a Markovian relaxation model. The specific problem to be considered is the interaction of an optical field with two-level atoms. The atoms undergo relaxation as a result of coupling to a perturber bath that, itself, is negligibly affected by the relaxation process. First, a simple decay-parameter model is assumed for the relaxation of atomic density-matrix elements. Such a model is found to lead to a set of generalized Bloch equations of which the conventional Bloch equations form a subset. Subsequently, more-realistic models for relaxation in both vapors and solids are considered within the limits of the impact approximation (i.e., the duration of a fluctuation can be viewed as instantaneous with respect to all relevant time scales in the problem). It is found that, even in the impact (Markovian) approximation, the generalized Bloch equations cannot be expected to provide an adequate description of relaxation, owing to effects in which a fluctuation-induced change in the atomic transition frequency persists between fluctuations. In vapors this frequency shift is produced by velocity-changing collisions that change the atomic resonance frequency (as seen in the laboratory frame), whereas in solids it is produced by local-field fluctuations. The limiting conditions under which one can expect both the generalized and the conventional Bloch equations to retain their validity are explored.
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