In this paper we discuss the influence of laser fluctuations on the hysteresis cycle in Optical Bistability. For the simplest case of an Absorptive Optical Bistability in a good cavity driven by an external laser light with non-white Gaussian amplitude fluctuations we derive a Fokker-Planck equation in the limit of large laser linewidth and good amplitude stabilization. This Fokker-Planck equation has a nonconstant diffusion term which shifts the most probable values of the steady state probability distribution from its deterministic positions. From the form of the steady state solution of the Fokker-Planck equation with non-white amplitude fluctuations we discuss the relative stability of the bistable branches. As in the case of quantum fluctuations the random amplitude of the laser field leads to a small range of values of the injected laser intensity in which the two bistable peaks have a comparable area. The transmitted field averaged with respect to the stationary distribution function coincides with one of the two deterministic branches except on the narrow transition region where we have large fluctuations.
On the basis of the absorptive bistability model, including the fluctuations, we rederive in the good-cavity limit the Fokker-Planck equation describing the time evolution of the quasiprobability distribution for the transmitted field amplitude. It is shown that on a semiclassical level different variants of the Fokker-Planck equation are possible. The equation of Bonifacio, Gronchi, and Lugiato is solved in terms of a path integral which involves a generalized Wiener measure. In order to calculate the averages of transmitted field moments as functions of time, a Monte Carlo integration scheme is used. This scheme consists in constructing random trajectories which result from the Langevin equation statistically equivalent to the derived Fokker-Planck equation. Numerical results in the bistable domain are presented for large values of the fluctuation parameter.
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.
Stochastic differential equations for absorptive optical bistability derived using the positive P representation are solved numerically. The stochastic model is formulated in ten dimensions, twice the dimensions of the space of physical observables. Extra dimensions are needed to include the quantum character of the noise. Ensemble averages are computed for the field amplitude, the incoherent intensity, and the second-order correlation function of the transmitted light. Comparisons with linearized theory are made and detailed results are given in the vicinity of the critical point where nonlinear effects are paramount. The scaling of fluctuations with system size at the critical point is investigated. Transitions between deterministic steady states are studied in a system just beyond the threshold for bistability. A potential difficulty with the use of the positive P representation for modeling nonlinear quantum-statistical systems is identified.
We discuss the quantum-statistical properties of a system of absorbing atoms in a ring cavity under the action of a resonant classical field. At a semiclassical level, this system exhibits a bistable behavior in the sense that the transmitted light
ET is a discontinuous function of the incident light
EI with a hysteresis cycle, which resembles a first-order phase transition. Here we specify this analogy stressing the nonthermodynamic properties of this coherently driven system. The quantized field
ET is described by a Glauber
P function, which obeys to a Fokker-Planck equation. This equation is derived from a very general master equation in the case of a good quality cavity by neglecting derivatives of order higher than second. A remarkable feature of this equation is that the diffusion coefficient is a nonlinear function of the field intensity. The steady-state solution of the Fokker-Planck equation has the following features. The
P function can be single or double peaked according to the value of
EI. In fact, our Fokker-Planck equation is like the one of a Brownian particle moving in a potential which presents one minimum or minima separated by a barrier, depending on the value of
EI. The peaks of the
P function correspond to the minima of the potential. In the double-peaked case, the peaks have different widths. The maxima of the
p function occur at values of
ET, which coincide with the classical stationary solutions that are stable according to the linear stability analysis. These semiclassical solutions give a discontinuous two-valued function of
ET vs
EI (hysteresis cycle). On the contrary, the mean value
〈ET〉 is a continuous single-valued function of
EI, which behaves similarly to the Maxwell construction of a first-order phase transition:
〈ET〉 jumps from low to high values in a very sharp transition region of
EI, which lies between the discontinuity points of the hysteresis cycle. However, the Maxwell rule that we obtain is quite different from that of equilibrium thermodynamics. The mean-square fluctuations of
ET are always very small except in the transition region where they are quite remarkable. We also present an exact expression of the stationary
P function, which takes into account all higher-order derivatives, and we show that the Fokker-Planck approximation is very good for calculating mean values and mean-square fluctuations. Finally, we discuss the connections which link bistable absorption to the laser with injected signal and to the laser with a saturable absorber.