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Physical Review A

Heuristic stochastic model of mirrorless optical bistability

F. A. Hopf, C. M. Bowden

Published Jul 1, 1985• 36 citations
Publisher

Abstract

A heuristic microscopic two-level-atom model with fluctuations is developed to explore, theoretically, the effect of fluctuations on mirrorless optical bistability. In stochastic models, bistability appears as a bimodal distribution function. We show that the predictions of bimodality and bistability are the same in the heuristic and semiclassical (nonstochastic) microscopic models.

DOI
10.1103/physreva.32.268

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A heuristic microscopic two-level-atom model with fluctuations is developed to explore, theoretically, the effect of fluctuations on mirrorless optical bistability. In stochastic models, bistability appears as a bimodal distribution function. We show that the predictions of bimodality and bistability are the same in the heuristic and semiclassical (nonstochastic) microscopic models.
Recent interest has focussed on the subject of mirrorless optical bistability, the simplest system showing bistability theoretically is a system of N ≥ 2 atoms. Microscopic theory is derived from quantum theory by a factorization of binary products of operators. There are strong inter atomic interactions in the bistable regime, it is unlikely that these factorizations are valid. Quantum calculations that avoid factorization imply that bistability does not exist. Hence in the previous bistability conference there was skepticism that bistability could really occur in a mirrorless system with only a few atoms.
We consider a gas of homogeneously broadened two-level atoms in a Fabry-Perot cavity injected with a coherent resonant electric field. Adopting a macroscopic birth-death description, we construct a stochastic master equation for the output intensity distribution. From this starting point we find the steady-state distribution and show it to be bimodal in the region of the optical bistability. We compare our results for the mean and variance with the microscopic results of Bonifacio, Gronchi, and Lugiato as a test of the stochastic-master-equation approach.
Absorptive optical bistability is shown to exist for a single two-level atom inside a resonant optical cavity. Solutions for the quantum-mechanical density operator are obtained numerically for a parameter regime at the interface between the quantum limit, in which quantum-mechanical noise invalidates the semiclassical prediction of bistability, and the classical limit, in which quantum noise is a negligible perturbation on semiclassical results. Bimodal photon number distributions and Q functions are obtained, and two-state transition rates are calculated.>
In the limit of large field-relaxation rates as compared to the rates of collective and atomic relaxation, we model the stochastic evolution of a bistable system with the aid of the Langevin equations for atomic variables. This semiclassical approach leads to a Fokker-Planck equation for the joint probability distribution of the atomic polarization and inversion. The Monte Carlo integration scheme, consisting in construction of approximate random trajectories, enables us to evaluate numerically the adiabatic values of the output field, intensity, and intensity fluctuations as functions of time. The final model used in numerical work is equivalent to ignoring microscopic fluctuations and considering delta-correlated external fluctuations impressed on the driving field.