A fully quantum-mechanical treatment of resonant light scattering is presented. The incident field is assumed to be described by a coherent state, and is allowed to be intense enough to cause saturation. Complete solutions are obtained for the correlated atom-field pure state vector, including multiphoton contributions of arbitrary order. The frequency spectrum of the scattered field is evaluated and is found to agree exactly with the result previously obtained by means of the quantum fluctuation-regression theorem. A derivation of the fluctuation-regression theorem and of the optical Bloch equations is given which is fully quantum mechanical and which relies upon no assumption of statistical factorization of atom and field states. The accuracy of the result found for the scattered - field spectrum is thus shown to be limited only by the assumption of the smallness of the saturated linewidth compared to the (optical) atomic resonance frequency. The one-photon approximation is analyzed in some detail. The method of adding an imaginary term to the upper-atomic-state energy is clarified, and it is shown how the vacuum and one-photon amplitudes thereby obtained may be used, within a simple and plausible iteration scheme, to construct the complete multiphoton spectrum. A variety of commonly used injection schemes and methods of representing atomic relaxation are discussed, and comparisons are made with results found by other authors. The entire analysis is performed with the aid of a canonical transformation which replaces the applied field by a $c$ number. It is thus proved quite rigorously and generally that the use of a $c$-number applied field is a fully quantum-mechanical procedure, provided only that radiation-reaction terms are retained.

• Received 29 November 1973

DOI:https://doi.org/10.1103/PhysRevA.12.1919