Cascading and Local-Field Effects in Non-Linear Optics Revisited; A Quantum-Field Picture Based on Exchange of Photons

Abstract: The semi-classical theory of radiation-matter coupling misses local-field effects that may alter the pulse time-ordering and cascading that leads to the generation of new signals. These are then introduced macroscopically by solving Maxwell's equations. This procedure is convenient and intuitive but ad hoc. We show that both effects emerge naturally by including coupling to quantum modes of the radiation field in the vacuum state to second order. This approach is systematic and suggests a more general class of corrections that only arise in a QED framework. In the semi-classical theory, which only includes classical field modes, the susceptibility of a collection of $N$ non-interacting molecules is additive and scales as $N$. Second-order coupling to a vacuum mode generates an effective retarded interaction that leads to cascading and local field effects both of which scale as $N^2$.