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.
We theoretically investigate the response of a coherently-driven nonlinear optical cavity to an additional incident single photon. Using a quantum description of the nonlinear dynamics that fully accounts for the quantum fluctuations of the cavity field and for the discrete nature of the incident photon, we characterize the quantum dynamics of single-photon-stimulated jumps from the low-photon-number to the high-photon-number state of the optical bistability loop. We find that the system can exhibit a giant response to this single quantum of excitation, rooted in the phase-transition picture of optical bistability. In addition to shedding light on the role of quantum fluctuations in the non-equilibrium dynamics of a nonlinear optical cavity, our results suggest a strategy for an all-optical single-photon avalanche detector.
We specialize the quantum-statistical theory of optical bistability developed in "Optical Bistability. II" to a low-
Q cavity. From a linearized Fokker-Planck equation derived in the positive
P 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.