Quantum theory of nondegenerate multiwave mixing: General formulation

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Abstract
In this first paper of a series on the quantum theory of nondegenerate multiwave mixing applicable to traveling-wave interaction geometries, we describe our problem formulation. We consider the explicit dynamics of a subset of the field quantization modes interacting with a system of stationary two-level atoms contained in a volume much smaller than the field quantization volume. Because we make the realistic assumption of leaving the remaining infinite set of field modes as a common thermal-field reservoir, the resulting Langevin equations contain extra decay terms due to collective spontaneous emission or super-radiance. We show that all but one of these super-radiance terms are negligible in the following two limits: (a) when the number of atoms in a diffraction volume is small; (b) when the atoms are pumped far off resonance. There is, however, an anomalous decay term which does not appear in a classical model of super-radiance based upon coherently phased atomic dipoles. The magnitude of this anomalous term is neither dependent upon the number of atoms nor on the pump-frequency detuning and may not be negligible at a low pump intensity. Neglecting the super-radiance terms, we then present a general Fourier-expansion solution technique for obtaining the atomic polarization in the presence of any number of field modes. The expansion is shown to be convergent in a commonly occurring situation in which all the strong pump modes are frequency degenerate and the remaining nondegenerate modes are all weak compared to the atomic saturation intensity. In subsequent papers of this series, we will present methods to treat, with some rigor, the spatial propagation of an interacting multimode quantum field and apply these methods to traveling-wave squeezed-state generation experiments.