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Quantum theory of nondegenerate multiwave mixing

Seng‐Tiong Ho, Prem Kumar, Jeffrey H. Shapiro

Published Jun 9, 1986• 0 citations
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Abstract

Recently there has been much interest in the quantum theory of multiwave mixing in atomic media, partly due to the promise of generating squeezed states of light using four-wave mixing (FWM) schemes, which need a quantized electro-magnetic-field theory to assess the effect of propagation loss and spontaneous emission on squeezing. In fact, squeezed states have recently been observed in an intracavity nondegenerate backward FWM experiment in an atomic beam of sodium.1

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Quantum theory of nondegenerate multiwave mixing

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•Jun 9, 1986•0 citations

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Quantum theory of nondegenerate multiwave mixing: General formulation

Seng‐Tiong Ho, Prem Kumar, Jeffrey H. Shapiro

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.

Physical Review A•Mar 1, 1988•8 citations
Recently there has been much interest in the quantum theory of multiwave mixing in atomic media, partly due to the promise of generating squeezed states of light using four-wave mixing (FWM) schemes, which need a quantized electro-magnetic-field theory to assess the effect of propagation loss and spontaneous emission on squeezing. In fact, squeezed states have recently been observed in an intracavity nondegenerate backward FWM experiment in an atomic beam of sodium.1
A quantum theory of nondegenerate multiwave mixing in a two-level atomic medium is formulated, with emphasis on traveling-wave interaction geometries. Two equivalent alternative methods are developed to treat, with some rigor, the spatial propagation of the interacting multimode traveling-wave field. The theory is applied to the case of a single beam propagating through the medium to obtain the spectrum of squeezing. The effects of spontaneous emission, propagation loss, atomic collisions, and pump-probe phase mismatch are comprehensively taken into account for the first time.
Multiphoton transitions are of great interest in laser spectroscopy because they permit the study of new physical phenomena and because of their potential applications. Malcuit et al.1 studied two-photon transitions in atomic sodium and showed that for appropriate phase-matching four-wave mixing can overcome the usual spontaneous decay processes. We have recently developed a quantum theory2 to treat such multiwave mixing processes in two-photon media. Because of the greater complexity of two-photon transitions, many effects arise that are absent in the one-photon case, some of which have important consequences in the generation of squeezed states. For the two-photon two-level model, the field modes have frequencies approximately one-half the frequency difference between the levels. We find that our coefficients differ from the corresponding one-photon theory in that dynamic Stark shifts have a major effect and that the pump-induced coherence between the two levels has a new term in the conjugate coupling terms, which leads to improved squeezing.
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.