Symmetrical Formulation of the Quantum Theory of Three-Wave Optical Parametric Interactions

The quantum theory of three-wave optical parametric interactions in a crystal such as LiNb${\mathrm{O}}_3$ is formulated in a symmetrical manner with respect to interaction time and distance. The formalism leads to a unique treatment of parametric interactions of both cavity type and propagation type, in that the resemblance between time in the cavity type and distance in the propagation type of interaction in the classical description is preserved. Three-wave effects of harmonic generation, parametric amplification (including spontaneous emission), and frequency conversion are treated in a unique and symmetric manner. Using Feynman's operator techniques, quantum states resulting from parametric interactions, previously unavailable, are derived, from which characteristics of the processes are deduced.