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Письма в Журнал экспериментальной и теоретической физики

ATOM-fiELD CORRELATIONS IN THE WEAK-EXCITATION LIMIT OF ABSORPTIVE OPTICAL BISTABILITY

Th. K. Mavrogordatos

Published Jan 1, 2020• 0 citations
Publisher

Abstract

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.

DOI
10.31857/s1234567820170036

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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.
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.
We determine the incoherent spectrum, squeezing properties, and second-order correlation function of the collective atomic degrees of freedom in absorptive optical bistability. This is accomplished via a linearized Fokker-Planck equation in the positive PPP representation, guided by the analysis of H. J. Carmichael [Phys. Rev. A 33, 3262 (1986)] which does not resort to adiabatic elimination. We focus on the regimes of weak and strong intracavity excitation, addressing the good-cavity and bad-cavity limits as well as the limit of collective strong coupling. Adiabatic elimination of the intracavity field sustained by an auxiliary resonator coupled to the ensemble is used to probe the atomic correlations via the formation of a collective emission channel. We compare to the corresponding expressions for the forward-scattered light with reference to experimental results, discussing key differences between the lower and upper branch of the steady-state semiclassical bistability curve. Our analysis is carried out around the stable states situated far away from the turning points, where analytical expressions can be obtained self-consistently, demonstrating a clear departure from classical behavior.
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.
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.
A linearized theory of fluctuations for absorptive bistability based on the positive P representation is developed without adiabatic elimination of the atoms or the field. An analytic expression for the steady-state covariance matrix is derived from which the size of quantum-statistical effects can be estimated without restriction to the good- or the bad-cavity limit. When the atom and field relaxation rates are similar the intensity correlation function of the transmitted light exhibits an oscillatory relaxation associated with vacuum Rabi splitting.
In the general framework of the system size expansion of Van Kampen and Kubo, we consider the Fokker-Planck equation for a model of absorptive bistability in the bad-cavity limit. The physical system is described by the reduced atomic density operators after adiabatic elimination of the cavity field variables. Mapping of the master equation into ccc-number form according to the normal-ordering mapping scheme yields known results for the atomic fluctuations and correlation functions; however, it also leads to a Fokker-Planck equation with a non-positive-definite diffusion matrix. The symmetrical-order-mapping scheme eliminates this difficulty. The leading contribution to the system size expansion yields a Fokker-Planck equation for the symmetrical-ordered density function having a positive-definite diffusion matrix. The atomic expectation values and fluctuations previously derived from the quantum Langevin equations emerge naturally from this Fokker-Planck equation.