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Physical Review A

Relaxation from a marginal state in optical bistability

Pere Colet, Maxi San Miguel, Jaume Casademunt, José María Sancho

Published Jan 1, 1989• 50 citations
Publisher

Abstract

Characteristic decay times for relaxation close to the marginal point of optical bistability are studied. A model-independent formula for the decay time is given which interpolates between Kramers time for activated decay and a deterministic relaxation time. This formula gives the decay time as a universal scaling function of the parameter which measures deviation from marginality. The standard deviation of the first-passage-time distribution is found to vary linearly with the decay time, close to marginality, with a slope independent of the noise intensity. Our results are substantiated by numerical simulations and their experimental relevance is pointed out.

DOI
10.1103/physreva.39.149

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Characteristic decay times for relaxation close to the marginal point of optical bistability are studied. A model-independent formula for the decay time is given which interpolates between Kramers time for activated decay and a deterministic relaxation time. This formula gives the decay time as a universal scaling function of the parameter which measures deviation from marginality. The standard deviation of the first-passage-time distribution is found to vary linearly with the decay time, close to marginality, with a slope independent of the noise intensity. Our results are substantiated by numerical simulations and their experimental relevance is pointed out.
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.
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 investigate the scaling behavior of the relaxation process for an unstable state near a subcritical Hopf bifurcation point. When the parametric modulation is applied to a magneto-optical trap, the atomic cloud becomes unstable and decays to the dynamic bistable states. Near the subcritical Hopf bifurcation point, we experimentally show that the relaxation process exhibits the scaling behavior; the relaxation time shows a scaling exponent of -1.002 (±0.024). We also present the passage time distribution for the statistical interpretation of the escape process associated with the relaxation of the unstable state. We compare the experimental results to the numerical and analytic results, demonstrating the good agreement between them.
As a result of quantum fluctuations, a bistable system will undergo spontaneous transitions from one state to the other. For a bistable two-mode laser we present a theoretical analysis of the switching statistics by treating it as a one-dimensional first-passage---time (FPT) problem for the intensity of one of the modes. The methodology is also applicable to other bistable systems, such as in optical bistability. Several new asymptotic results for the small- and large-time behavior of various FPT distributions are given. The switching-time distribution is introduced as a weighted integral over FPT distributions and asymptotic expressions for this distribution are given. Several results of Monte Carlo computer simulations of the FPT problem are presented.