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Annual Meeting Optical Society of America

Quantum theory of optical bistability with a spatially varying field mode in the good cavity limit

Xiao Min, H. J. Kimble, H. J. Carmichael

Published Jan 1, 1985• 0 citations
Publisher

Abstract

A straightforward but general extension of the quantum theory of optical bistability is made to include spatial variations of the field mode in the good cavity limit. The analysis proceeds by dividing the field mode into small sections which are each microscopically large in terms of the atomic number to allow truncation of the generalized Fokker-Planck equation but which are macroscopically small to justify the assumption of constant field amplitude. In a linearized approximation, analytic expressions are obtained for the ratio of incoherent to coherent intensity and for the intensity correlation function of the transmitted field for the two particular examples of a Gaussian-mode field in a ring cavity and a plane-wave field in a standing-wave cavity. In the weak field limit the results of the plane-wave ring cavity are recovered independent of the form of the spatial dependence of the cavity mode. However, more generally a nonuniform distribution tends to suppress certain quantum features such as photon antibunching.

DOI
10.1364/oam.1985.fg7

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A straightforward but general extension of the quantum theory of optical bistability is made to include spatial variations of the field mode in the good cavity limit. The analysis proceeds by dividing the field mode into small sections which are each microscopically large in terms of the atomic number to allow truncation of the generalized Fokker-Planck equation but which are macroscopically small to justify the assumption of constant field amplitude. In a linearized approximation, analytic expressions are obtained for the ratio of incoherent to coherent intensity and for the intensity correlation function of the transmitted field for the two particular examples of a Gaussian-mode field in a ring cavity and a plane-wave field in a standing-wave cavity. In the weak field limit the results of the plane-wave ring cavity are recovered independent of the form of the spatial dependence of the cavity mode. However, more generally a nonuniform distribution tends to suppress certain quantum features such as photon antibunching.
An extension of the quantum theory of the atom-field interaction within a high-finesse resonator is made to include spatial variations of the field mode in the limit of small cavity decay rate. The two particular examples of a Gaussian mode field in a ring cavity and a plane-wave field in a standing-wave interferometer are presented to illustrate the method. Analytic expressions are obtained for the incoherent intensity and photon correlations of the transmitted field. In qualitative terms, effects such as sub-Poissonian photon statistics predicted by the plane-wave theory in a ring cavity are preserved. In either the weak-field or dispersive limit the results of the plane-wave theory in a ring cavity are recovered independent of the form of the spatial dependence of the cavity mode.
A generalized Fokker-Planck equation is used to study the problem of single-atom optical bistability in the good-cavity limit. The effects of quantum fluctuations are investigated by linearizing fluctuations around the steady-state value. It is shown that in the good-cavity limit, quantum fluctuations are relatively small, and hence there exist both absorptive and dispersive single-atom bistability. By comparing our single-atom Fokker-Planck equation with the N-atom Fokker-Planck equation, we find that in the good-cavity limit, these two different Fokker-Planck equations reduce to identical equations. We therefore come to an important conclusion that the results obtained from the many-atom Fokker-Planck equation in the good-cavity limit are also valid for a single atom. \textcopyright{} 1996 The American Physical Society.
We calculate the steady-state and first-order time varying atom—field correlation functions in the weak-excitation limit of absorptive optical bistability from a linearized theory of quantum fluctuations. We formulate a Fokker—Planck equation in the positive P representation following the phase-space analysis in [H. J. Carmichael, Phys. Rev. A 33 , 3262 (1986)], which is suitable for the determination of cross-correlations as it does not resort to adiabatic elimination. Special emphasis is placed on the limit of collective strong coupling as attained from a vanishing photon-loss rate. We compare to the cavity-transmission spectrum with reference to experimental results obtained for macroscopic dissipative systems, discussing the role of anomalous correlations arising as distinct nonclassical features.
A quantum treatment of a coherently driven cavity with a medium of two-level atoms is given. The system displays both absorptive and dispersive bistability. Quantum fluctuations are included via a Fokker-Planck equation in a generalized PPP representation, after adiabatic elimination of the atomic variables. The transmitted spectrum and correlation function is calculated in a linearized (Gaussian) approximation. In the limit of large detuning and radiative damping, the quantum fluctuations are identical to those of a simple nonlinear polarizability theory. An analysis is given of the nonclassical statistical properties of the intensity-correlation spectrum.
We specialize the quantum-statistical theory of optical bistability developed in "Optical Bistability. II" to a low-QQQ cavity. From a linearized Fokker-Planck equation derived in the positive PPP representation, the steady-state correlation matrix and spectrum of fluctuations are calculated. Expressions which normally assume a positive-definite diffusion matrix are shown to be valid for nonpositive-definite diffusion. For absorptive bistability we find agreement with results derived previously by quite different methods. More generally we extend previously published results by including atomic and cavity detunings. We compare the transmitted spectrum for dispersive bistability with that for absorptive bistability, and photon antibunching in dispersive bistability is discussed.
We calculate the steady-state and first-order time varying atom-field correlation functions in the weak-excitation limit of absorptive optical bistability from a linearized theory of quantum fluctuations.We formulate a Fokker-Planck equation in the positive P representation following the phase-space analysis of [H.J. Carmichael, Phys.Rev. A 33, 3262 (1986)] which is suitable for the determination of cross-correlations as it does not resort to adiabatic elimination.Special emphasis is placed on the limit of collective strong coupling as attained from a vanishing photon-loss rate.We compare to the cavity-transmission spectrum with reference to experimental results obtained for macroscopic dissipative systems, discussing the role of anomalous correlations arising as distinct nonclassical features.