Diagrammatic schemes for nonlinear optical interactions

Nonlinear optical interactions are usually understood diagrammatically to sift only those contributions to the dielectric susceptibility that matter most to the signal being measured. Here, we review the main techniques of diagrammatic perturbation theory (i.e., double-sided Feynman diagrams, Liouville pathways, and Albrecht notation) that make use of the semiclassical approximation. A brief mention of other recent diagrammatic proposals is also made. The limitations of such approaches are discussed in view of a fully quantum-mechanical treatment of optical interactions. It is then suggested that the limitations of the conventional approaches can be overcome by a field-type modification of the Albrecht notation that combines the simplicity of the energy-level representation with the wealth of information provided by more elaborated diagrams (namely, double-sided Feynman diagrams and Liouville pathways). Rules to replace the Albrecht notation with the matter field representation are given, and instructive examples are illustrated for Raman spectroscopy.