This paper develops a consistent formalism for describing nonlinear optical mixing and multiphoton processes of any arbitrary order. The theory uses the time-evolution operators of quantum mechanics, and the related Feynman diagrams.
With the goal of achieving all-optical quantum information processing, singlephoton engineering has become one of the hottest research topics in physics. Rapid progress has been made in generating and detecting single photons, and also in information processing based on linear optics. Furthermore, the discovery of optical nonlinearity that is sensitive to individual photons has raised the possibility of using one photon to control another photon, increasing the need to develop a quantitative theory of nonlinear quantum optics.The simplest method to analyze the nonlinear dynamics of photons is to introduce an effective nonlinearinteraction Hamiltonian based on the singlemode approximation. For example, the following time evolution operator has conventionally been used for the self-Kerr interaction: ˆ
A treatment of nonlinear optical phenomena beginning with a microscopically correct Hamiltonian containing the Bose operators of the light field and the Fermi operators for the optically active electrons in the medium is presented. Using techniques recently developed, a nonperturbative solution to the Schrodinger equation is possible. No linearization or semiclassical approximations are introduced. Two photon emission and absorption including both the degenerate and nondegenerate cases are considered. Parametric amplification and frequency conversion are investigated as examples of multiphoton processes. Atomic cooperation is shown to enhance the above phenomena. A formal treatment of a model illustrating super and subradiance is included.