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Physical Review A

Quantum theory of multiwave mixing. VII. Connection to quantum Langevin theory

David A. Holm, Murray Sargent

Published Jun 1, 1986• 20 citations
Publisher

Abstract

We transform our quantum theory of multiwave mixing, which is based on a reduced density operator, to the corresponding nondegenerate quantum-noise Langevin equations. We derive the diffusion coefficients from the drift coefficients using the generalized Einstein relation and illustrate the resulting quantum Langevin theory of multiwave mixing by specializing to the Reid and Walls degenerate theory of squeezed states of light.

DOI
10.1103/physreva.33.4001

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We transform our quantum theory of multiwave mixing, which is based on a reduced density operator, to the corresponding nondegenerate quantum-noise Langevin equations. We derive the diffusion coefficients from the drift coefficients using the generalized Einstein relation and illustrate the resulting quantum Langevin theory of multiwave mixing by specializing to the Reid and Walls degenerate theory of squeezed states of light.
The present paper combines a Langevin quantum-regression method with a density-operator approach to derive the master equation for the quantum theory of multiwave mixing in a very efficient way. The approach is quite general and is particularly valuable for analyzing complicated media such as semiconductors. It is used in the present paper to derive the quantum multiwave-mixing equations in a squeezed vacuum. We find improved formulas for resonance fluorescence in a squeezed vacuum as well as the squeezing coefficients in a squeezed vacuum. Comparing squeezing spectra in squeezed and ordinary vacua, we find significantly enhanced squeezing for the appropriate pump-vacuum relative phase.
We apply our quantum theory of multiwave mixing to the generation of squeezed states of light via three- and four-wave mixing in cavities and in propagation. We compare the cavity predictions to the experimental results of Slusher et al. [Phys. Rev. Lett. 55, 2409 (1985)], finding reasonably good agreement. The squeezing found there is due to the fact that the number operators are driven by resonance fluorescence, which nearly vanishes for the large detunings chosen, while the combination tone operators are driven by relatively large source terms. We give a physical discussion as to why the spectral quantities outside the cavity are given by those inside multiplied by the cavity linewidth. We have found analytic formulas for the variances for arbitrary propagation distances, detunings, and pump intensities. We derive these formulas both from the density operator and from the Langevin methods. To obtain significant squeezing, the propagation distances must be large compared with the resonant Beer's law length.
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.