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Physical Review A

Quantum theory of multiwave mixing. I. General formalism

Murray Sargent, David A. Holm, Muhammad Suhail Zubairy

Published May 1, 1985• 92 citations
Publisher

Abstract

We present a theory that describes how one strong classical wave and one or two weak quantum-mechanical waves interact in a nonlinear two-level medium. The analysis is applicable to several popular problems with and without cavities. In particular, the theory treats laser and optical bistability instabilities, predicting when the instabilities grow from spontaneous emission. The theory is a multimode extension of Scully-Lamb theory that derives the equations that describe population pulsations, combination tones, mode locking, resonance fluorescence, Rayleigh scattering, and phase conjugation with quantum-mechanical fields. Hence the theory both presents new results on instabilities and phase conjugation, and also unifies the treatment of a variety of phenomena in the context of Lamb theory. The present paper (first of the series) presents the basic formalism, leaving most applications to subsequent papers in this series. The following paper presents equivalent derivations based on a purely operator formalism.

DOI
10.1103/physreva.31.3112

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Physical Review A•Aug 1, 1985•26 citations
We present a theory that describes how one strong classical wave and one or two weak quantum-mechanical waves interact in a nonlinear two-level medium. The analysis is applicable to several popular problems with and without cavities. In particular, the theory treats laser and optical bistability instabilities, predicting when the instabilities grow from spontaneous emission. The theory is a multimode extension of Scully-Lamb theory that derives the equations that describe population pulsations, combination tones, mode locking, resonance fluorescence, Rayleigh scattering, and phase conjugation with quantum-mechanical fields. Hence the theory both presents new results on instabilities and phase conjugation, and also unifies the treatment of a variety of phenomena in the context of Lamb theory. The present paper (first of the series) presents the basic formalism, leaving most applications to subsequent papers in this series. The following paper presents equivalent derivations based on a purely operator formalism.
We present the first reported fully quantum-mechanical theory of laser/optical-bistability instabilities that shows how cavity side modes grow from spontaneous emission. Because of three-wave mixing, a two-peaked spontaneous-emission spectrum is obtained in contrast to the three-peaked spectrum of resonance fluorescence. The theory is a multimode extension of Scully-Lamb theory that shows how population pulsations, combination tones, mode locking, and phase conjugation occur with quantum-mechanical fields.
This paper treats the interaction of one strong classical field mode and one or two weak quantal modes with two-level systems. The analysis is carried out in a purely operator formalism that complements the method presented in the preceding paper and produces the same results. The present approach automatically collects all contributions to second order in the weak-field interactions, and can be easier to extend to more complicated cases. The analysis allows one to treat problems such as predicting the growth of sidemodes in laser- and optical-bistability instabilities, the resonance fluorescence spectrum, Rayleigh scattering, and quantum noise in small-signal phase conjugation.
This dissertation formulates and applies a theory describing how one or two strong classical waves and one or two weak quantum mechanical waves interact in a two-level medium. The theory unifies many topics in quantum optics, such as resonance fluorescence, saturation spectroscopy, modulation spectroscopy, the build up of laser and optical bistability instabilities, and phase conjugation. The theory is based on a quantum population pulsation approach that resembles the semiclassical theories, but is substantially more detailed. Calculations are performed to include the effects of inhomogeneous broadening, spatial hole burning, and Gaussian transverse variations. The resonance fluorescence spectrum in a high finesse optical cavity is analyzed in detail, demonstrating how stimulated emission and multiwave processes alter the spectrum from the usual three peaks. The effects of quantum noise during the propagation of weak signal and conjugate fields in phase conjugation and modulation spectroscopy are studied. Our analysis demonstrates that quantum noise affects not only the intensities of the signal and conjugate, but also their relative phase, and in particular we determine a quantum limit to the semiclassical theory of FM modulation spectroscopy. Finally, we derive the corresponding theory for the two-photon, two-level medium. This yields the first calculation of the two-photon resonance fluorescence spectrum. Because of the greater number of possible interactions in the two-photon two-level model, the theoretical formalism is considerably more complex, and many effects arise that are absent in the one-photon problem. We discuss the role of the Stark shifts on the emission spectrum and show how the Rayleigh scattering is markedly different.
In a recent paper1, a quantum theory of multiwave interactions with a two-level medium is presented. In this paper we apply this theory to sidemode instabilities pertinent to cavity con figurations in optical bistability and ring lasers. We assume the cavity to oscillate on a single mode, and we evaluate the gain and spontaneous emission at other frequencies. Because of the saturating presence of the strong, laser mode, two symmetrically situated sidemodes can grow together, inducing populations pulsations that influence the other sidemode. The instability must therefore be treated as a double sidemode problem.
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 effects of inhomogeneous broadening, spatial hole burning, and Gaussian beams are calculated for the four coefficients derived in previous papers in this series [Phys. Rev. A 31, 3112 (1985); 31, 3124 (1985)] on the quantum theory of multiwave mixing. These coefficients uniformly describe how one or two strong classical-field modes and one or two weak quantum-field modes interact in a two-level medium. The phenomena for which this theory is applicable includes resonance fluorescence, Rayleigh scattering, laser and/or optical bistability instabilities, and phase conjugation. The averaged formulas are illustrated by resonance-fluorescence spectra. We find the sidepeaks can be ``washed out'' and that asymmetries can occur.