Quantum theory of nonlinear mixing in multimode fields: general theory

Abstract

A general quantum theory of nonlinear mixing in multimode fields is formulated. The theory is valid for arbitrary media and thus includes various diverse cases such as those corresponding to nonlinear mixing in multilevel systems, optical fibers, etc. Thus Zeeman coherence effects are automatically included. It is applicable to both degenerate and nondegenerate mixing experiments. The theory is in terms of two distinct sets of correlation functions of the polarization operators. One set of correlations is related to the nonlinear susceptibilities. The other set has no counterpart in semiclassical theory and is related to the quantum-mechanical fluctuations of the polarization operator. The general structure of the evolution of the density matrix of the generated fields is discussed. The quantum statistics of the generated fields can be studied in terms of the Wigner distribution function which is explicitly evaluated. Higher-order squeezing characteristics of the field are discussed. Conditions under which the quantization of the semiclassical equations is adequate are examined. The application of the quantum theory to nonlinear mixing in two-photon media is also presented. The theory is also capable of accounting for the interparticle correlations.