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Journal of the Optical Society of America B

Quantum theory of multiwave mixing VI Effects of quantum noise on modulation spectroscopy

David A. Holm, Murray Sargent

Published May 1, 1986• 6 citations
Publisher

Abstract

Treating the weak-sideband fields quantum mechanically, we study the effects of quantum noise on amplitude-modulation and frequency-modulation spectroscopy of two-level media. Our analysis demonstrates that quantum noise affects not only the intensities of the modulation sidebands but also their relative phase, which leads in particular to a quantum limit for frequency-modulation spectroscopy.

DOI
10.1364/josab.3.000732

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Treating the weak-sideband fields quantum mechanically, we study the effects of quantum noise on amplitude-modulation and frequency-modulation spectroscopy of two-level media. Our analysis demonstrates that quantum noise affects not only the intensities of the modulation sidebands but also their relative phase, which leads in particular to a quantum limit for frequency-modulation spectroscopy.
AM and FM modulation spectroscopies1,2 have been used to study the properties of matter as well as for applications such as reading out optical memories.3 Advances in the accuracy of these methods have turned what used to be an academic subject, quantum noise, into a practical consideration. For example, memory readout fields must be very weak to keep the medium from heating up too much and quantum noise puts a lower limit on the readout field strength. Similarly if weak modulation sidebands are used to study coherent dips via AM spectroscopy, quantum noise becomes relevant. We use our quantum theory of multiwave mixing4 to predict the effects of quantum noise on these problems. We find that, for modulation frequencies about equal to the Rabi frequency for the strong wave, spontaneous emission contributes a pronounced AM component that might be confused with the appearance of a hole in reading out optical memories. In AM spectroscopy we find that the coherent dips are partly filled in by spontaneous emission.
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.
Over the past three years we have developed a quantum theory of multiwave mixing designed to determine effects of quantum noise on laser and optical bistability instabilities, pump/probe saturation spectroscopy, AM and FM modulation spectroscopy, and phase conjugation. The theory treats a classical pump field and one or two quantum-mechanical sidemode fields interacting with one- or two-photon, two-level media. The theory1 produces a master equation for the reduced sidemode density operator able to describe all of the above phenomena in a uniform way closely related to the corresponding semiclassical theories. It has led to a number of significant predictions about the effects of spontaneous emission on laser spectroscopy and optical instabilities.
Due to improvements in detector sensitivity, quantum noise is known to be the limiting factor of many lasers and other optical devices. In this work we present a fully quantum mechanical theory of multiwave mixing to study the effects of spontaneous emission noise on three- and four- wave mixing. The foundations of this theory have been presented in a series of papers1-3 demonstrating how such diverse topics as resonance fluorescence, saturation spectroscopy, phase conjugation, and squeezed states of light share a common origin.
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.
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.