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Journal of the Optical Society of Korea

Quantum theory of multiwave mixing with a local field correction

Sunghyuck An, B. K. Rhee

Published Sep 1, 1997• 2 citations
Publisher

Abstract

In this paper, we calculate the four coefficients for the quantum theory of multiwave mixing including a local-field correction resulting from dipole-dipole interactions. We make contact with the semiclassical calculations of probe absorption and four-wave-mixing coupling coefficients, and illustrate the effects of local field corrections on resonance-fluorescence and coupled-mode-fluorescence spectra. The method uses the hybrid quantum-Langevin-equation master-equation approach of An and Sargent.

DOI
10.3807/josk.1997.1.2.094

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We calculate the four coefficients for the quantum theory of multiwave mixing1 including a local field correction resulting from dipole–dipole interactions.2 We make contact with the semiclassical calculations of probe absorption and four-wave mixing coupling coefficients, and illustrate the effects of local field corrections on resonance fluorescence and coupled-mode flouorescence spectra. The method uses the hybrid quantum-Langevin-equation master-equation approach of An and Sargent.3,4 We find shifts and changes in the spectral shapes, which can be used to measure local-field corrections.

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In this paper, we calculate the four coefficients for the quantum theory of multiwave mixing including a local-field correction resulting from dipole-dipole interactions. We make contact with the semiclassical calculations of probe absorption and four-wave-mixing coupling coefficients, and illustrate the effects of local field corrections on resonance-fluorescence and coupled-mode-fluorescence spectra. The method uses the hybrid quantum-Langevin-equation master-equation approach of An and Sargent.
We calculate the four coefficients for the quantum theory of multiwave mixing1 including a local field correction resulting from dipole–dipole interactions.2 We make contact with the semiclassical calculations of probe absorption and four-wave mixing coupling coefficients, and illustrate the effects of local field corrections on resonance fluorescence and coupled-mode flouorescence spectra. The method uses the hybrid quantum-Langevin-equation master-equation approach of An and Sargent.3,4 We find shifts and changes in the spectral shapes, which can be used to measure local-field corrections.
We extend the quantum theory of multiwave mixing developed in previous papers in this series to include side-mode saturation. Specifically, the earlier papers calculate the absorption and resonance-fluorescence coefficients to second order in the weak-field interaction for an arbitrarily intense pump field. This paper extends those calculations to fourth order in the weak-field interaction. We find significant differences in the amplified resonance fluorescence spectra given by the second- and fourth-order treatments, particularly as the side mode approaches the laser threshold. In particular, while the second-order spectra diverge at this threshold, the fourth-order spectra remain finite. Although fourth- and higher-order theories are very desirable for applications to larger fields and optical instabilities, the present analysis reveals that their calculation is very complex. This fact poses an important limitation to the generalization of the theory to still higher-order accuracies.
We extend our theoretical formalism describing how one strong classical wave and one or two weak quantum-mechanical waves interact in a nonlinear medium to the ``two-photon two-level'' model. In this model, field transitions between two levels not connected by an electric dipole occur by means of a two-photon transition using off-resonant intermediate levels. This model has many similarities with, as well as significant differences from, the corresponding one-photon two-level model that has been treated in the previous papers in this series. The theory yields the first detailed calculation of the two-photon resonance fluorescence spectrum, as well as the complete set of four coefficients that describe the quantum theory of two-, three-, and four-wave mixing.
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.
This dissertation presents theories of multiwave mixing in two- and three-level media. The first part of the dissertation treats the semiclassical theories in two-level media. Chapter 2 gives the simple semiclassical theory of four-wave mixing when the two pump frequences differ by more than the reciprocal of the population-difference lifetime. This difference washes out the pump spatial holes as well as one of the two reflection gratings. We compare the results to the degenerate treatment of Abrams and Lind and find significant differences in the reflection coefficient spectra. Chapter 3 presents the semiclassical theory of multiwave in a squeezed vacuum characterized by unequal in-phase and in-quadrature dipole decay times. For a highly squeezed vacuum, we find sharp resonances in both probe absorption and reflection coefficients, which provide sensitive ways to measure the amount of squeezing in the vacuum. The second part of the dissertation treats the quantum theories in two- and three-level media. Chapter 4 develops the fourth-order quantum theory of multiwave mixing to describe the effects of sidemode saturation in two-level media. We derive explicit formulas for the fourth-order quantum coefficients and show that the fourth-order quantum theory reproduces the third-order semiclassical coefficient obtained by truncating a continued fraction. We apply the results to cavity problems and find significant differences in the sideband spectra given by the second- and fourth-order treatments, particularly as the sidemode approaches the laser threshold. The final chapter presents a quantum theory of multiwave mixing in three-level cascades with a two-photon pump. The explicit formulas for the resonance fluorescence spectrum and the quantum combination-tone source term are derived. The theory is applied to the generation of squeezed states of light. We find almost perfect squeezing for some strong pump intensities and good broad-band squeezing for low pump intensities.
In the presence of a high-finesse cavity, the resonance fluorescence spectrum can be altered by stimulated emission and multiwave mixing processes. We show that under some conditions it is possible for spontaneous emission to induce conjugate fields similar to semiclassical phase conjugation, requiring the use of the multiwave quantum theory described in earlier papers in this series. The theory is applicable to laser spectroscopy and optical bistability. We present explicit spectra and suggest possible experimental tests. In particular we find a quantum limit for FM spectroscopy that shows that even if the medium under study leaves the FM phase and amplitude relationships intact according to semiclassical theory, spontaneous emission can create an AM component detectable in a square-law detector.