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Physical Review A

Quantum theory of optical bistability. II. Atomic fluorescence in a high- Q cavity

P. D. Drummond, Daniel F. Walls

Published May 1, 1981• 194 citations
Publisher

Abstract

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 $P$ 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.

DOI
10.1103/physreva.23.2563

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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.

Physical Review A•Nov 1, 1996•21 citations
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.
A quantum treatment of a coherently driven dispersive cavity is given based on a cubic nonlinearity in the polarisability of the internal medium. This system displays bistability and hysteresis in the semiclassical solutions. Quantum fluctuations are included via a Fokker-Planck equation in a generalised P representation. The transmitted light shows a transition from a single-peaked spectrum to a double-peaked spectrum above the threshold of the lower branch. Fluctuations in the field are reduced on the upper branch and both photon bunching and photon antibunching are predicted, for different operating points. An exact solution obtained for the steady-state generalised P function shows decidedly non-equilibrium behaviour, e.g. the lack of a Maxwell construction.
A quantum theory describing N two-level atoms in a coherently driven single-mode optical ring cavity is presented. The usual approximation that one may adiabatically eliminate either the cavity or atomic variables is not made. Hence the results are valid for a range of relative cavity and atomic relaxation rates. The theory is based on the linearized equations derived in the positive P representation. A simple analytical solution is presented, including the effect of detunings, for the steady-state transmitted spectrum. The transmitted intensity and squeezing spectra are discussed.
The topic of this thesis is a theoretical study of nonequilibrium transitions and quantum statistical properties of nonlinear quantum optical systems driven by an external radiation source. In certain limiting cases, a comparison is made between these transitions and the phase transitions found in equilibrium physical systems. In chapters one and two, the mathematical tools are introduced. In operator terms, the time development is described by a Markovian master equation in the interaction picture. This is equivalent to a corresponding time development equation or Fokker-Planck equation in a vector space of c-numbers. In order to deal with the type of Fokker-Planck equation that results, we introduce a quantum classical correspondence resulting in a distribution function over a complex phase-space, which is a generalisation of the Glauber-Sudarshan P-function. In chapter three this is applied to a model of a coherently driven mode with nonlinear dispersion and absorption. We find in the limit of zero temperature, that the spectrum is symmetric relative to the input frequency, and an exact solution is obtained for the distribution function. For a detuned driving field and nonlinear dispersion, optical bistability can occur. In chapter four a model of sub/second harmonic generation is introduced. This has several non-equilibrium transitions, including dispersive optical bistability, and bistable behaviour with coherent phase-locked input to both modes. Exact solutions occur in the limit of zero temperature and adiabatic elimination of one mode. In both chapters three and four, steady-state photon anti-bunching occurs with an absorptive nonlinearity. In chapter five we include interactions between the radiation mode and a fluorescent atomic system. In this case different behaviour occurs depending on the relative decay rates of the individual atoms and of the radiation mode. In the case of a high-Q interferometer, the atomic variables can be adiabatically eliminated. Both dispersive and absorptive bistability can occur. We show by analytic and numerical calculations (in the case of inhomogeneous broadening) that dispersive operation has advantages in requiring a lower atomic density and input field to observe bistability. When the input field has Gaussian (rather than coherent) photon statistics, there is no bistability, but enhanced photon bunching occurs. Finally, there is a different type of behaviour when the field mode decays rapidly and can be adiabatically eliminated. Within the cooperation lifetime the system can be described by a J²-invariant Hamiltonian, giving the special case of only collective damping. The result is a new type of critical point transition in the thermodynamic limit, with the appearance of a family of solutions like Lotka-Volterra cycles for a coherent driving field above threshold.
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 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.