Photon statistics of a bistable absorber

Abstract

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 ${E}_{T}$ is a discontinuous function of the incident light ${E}_{I}$ 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 ${E}_{T}$ 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 ${E}_{I}$. 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 ${E}_{I}$. 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 ${E}_{T}$, 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 ${E}_{T}$ vs ${E}_{I}$ (hysteresis cycle). On the contrary, the mean value $〈{E}_{T}〉$ is a continuous single-valued function of ${E}_{I}$, which behaves similarly to the Maxwell construction of a first-order phase transition: $〈{E}_{T}〉$ jumps from low to high values in a very sharp transition region of ${E}_{I}$, 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 ${E}_{T}$ 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.