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Physical Review A

Activation rates in dispersive optical bistability with amplitude and phase fluctuations: A case without detailed balance

Peter Talkner A, Peter Hänggi A

Published Feb 1, 1984• 27 citations
Publisher

Abstract

For the two-dimensional model of dispersive optical bistability put forward by Graham and Schenzle [Phys. Rev. A 23, 1302 (1981)] the activation rates of the metastable states at low noise are evaluated explicitly. The rates are calculated in terms of the mean first-passage time of a two-variable Fokker-Planck equation which does not obey detailed balance and which has a drift field not expressable as the gradient of the corresponding nonequilibrium potential. The forward rate, describing the transition from the state with low transmission to the state with high transmission, is exponentially decreased with increasing detuning ${\ensuremath{\delta}}^{2}$, whereas the backward rate is exponentially enhanced with increasing ${\ensuremath{\delta}}^{2}$. The prefactor of the rate exhibits a complicated dependence on the detuning parameter which for the absorptive case with zero detuning, $\ensuremath{\delta}=0$, reduces to the familiar Kramers result of a Fokker-Planck system with a drift field derivable from a potential.

DOI
10.1103/physreva.29.768

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For the two-dimensional model of dispersive optical bistability put forward by Graham and Schenzle [Phys. Rev. A 23, 1302 (1981)] the activation rates of the metastable states at low noise are evaluated explicitly. The rates are calculated in terms of the mean first-passage time of a two-variable Fokker-Planck equation which does not obey detailed balance and which has a drift field not expressable as the gradient of the corresponding nonequilibrium potential. The forward rate, describing the transition from the state with low transmission to the state with high transmission, is exponentially decreased with increasing detuning \ensuremathδ2{\ensuremath{\delta}}^{2}\ensuremathδ2, whereas the backward rate is exponentially enhanced with increasing \ensuremathδ2{\ensuremath{\delta}}^{2}\ensuremathδ2. The prefactor of the rate exhibits a complicated dependence on the detuning parameter which for the absorptive case with zero detuning, \ensuremathδ=0\ensuremath{\delta}=0\ensuremathδ=0, reduces to the familiar Kramers result of a Fokker-Planck system with a drift field derivable from a potential.
Systems exhibiting dispersive optical bistability are considered as examples of nonequilibrium steady states lacking detailed balance. Methods for describing such states are developed. The Fokker-Planck equation for the steady-state distribution function of the transmitted electromagnetic field is derived. In the limit of small fluctuations the Fokker-Planck equation is reduced to the form of a Hamilton-Jacobi equation for a "nonequilibrium thermodynamic potential." This equation is solved in various approximations. The nonequilibrium thermodynamic potential acts like a free-energy for the first-order-type transition far from thermodynamic equilibrium lacking detailed balance. The potential entirely determines the steady-state distribution, the coexistence curve in the bistable domain, a Lyapunoff function of the deterministic equations of motion, and allows us to cast the deterministic equations of motion into the standard form of nonequilibrium thermodynamics.
Using the Bonifacio-Lugiato mean field model of an absorptive bistable device, we discuss in some detail the long time dynamics and the influence of quantum noise on the metastable states of a bi-stable system. We develop our analysis along two seemingly different lines: first, by means of a suitable generalization and reinterpretation of a method due to Kramers, we obtain an explicit analytic expression for the escape time out of the metastable state, which is valid not only near steady state, but through the long time evolution of the system as well; next, we calculate the same escape time by the so-called mean first passage time method. The two results are found to differ by a numerical factor of two. In our discussion we have made an effort to explore details of both the original Kramers' scheme and of the first passage time method that needed clarification. We have also related the global long time evolution to the escape processes out of the metastable states.
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 generalized Fokker-Planck equation is derived for a system undergoing optical bistability when the dominant source of fluctuations is due to fluctuations in the incident field rather than spontaneous emission. The fluctuations are treated as a time-dependent Gaussian process whose properties are determined by the individual laser's characteristics. Both amplitude and phase fluctuations are included on the incident laser field. The effect of fluctuations of the incident laser on optical bistability are different for amplitude and phase fluctuations and are different for absorptive and dispersive optical bistability. The combination of incident laser phase fluctuations and dispersive optical bistability leads to large amplitude fluctuations near the turning points of the optical bistability curve and causes the system to make a transition from one branch of the optical bistability curve to the other before reaching the mean-field turning points. We take the high-QQQ cavity limit of our generalized Fokker-Planck equation, solve the linearized equations, and provide criteria for the magnitude of fluctuations in optical bistability in terms of laser and cavity parameters. The magnitude of incident laser fluctuations are typically orders of magnitude larger than fluctuations due to spontaneous emission and are therefore important for practical applications of optical bistability devices.
In a recent paper (1) we formulated a running wave, singlemode model for purely dispersive optical bistability, which incorporates amplitude and frequency fluctuations in the incident field, cavity length fluctuations, thermal noise in the radiation field and in the material. In the bad-cavity case of miniaturized bistable devices which utilize semiconductor dispersive media, our model reduces to a single stochastic differential equation for the material variable, which contains several terms of additive and multiplicative noise. In the white-noise case, it is equivalent to a Fokker-Planck equation in one variable, whereas in the case of colored noise it is equivalent to a multidimensional Fokker-Planck equation. On the basis of this model, we examined both problem of practical (P) and of fundamental (F) interest.