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Physical Review A

First-passage—time distributions and switching statistics in a bistable two-mode laser

Daan Lenstra, Surendra Singh

Published Oct 1, 1983• 23 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.28.2318

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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.
A two-mode laser operating in the bistable region switches spontaneously from one mode to the other under the influence of quantum fluctuations. We present here a Monte Carlo simulation of the coupled Langevin equations describing the dynamics of a two-mode laser. We present results on the variation of average switching time with pump parameter. We show that if fluctuations in the pump parameter are incorporated, the Monte Carlo estimates agree well with measured switching times. To further check this observation we reanalyze the results of a recent analytical study and show that by including the same pump parameter fluctuations, the agreement with measurements becomes much better.
The general time-dependent problem associated with a two-mode-laser Fokker-Planck equation has been considered. Analytic expressions for the eigenvalues and the eigenfunctions have been derived for the below- and above-threshold operation of the system. It is shown that depending on the values of the pump parameters a1,a2{a}_{1},{a}_{2}a1​,a2​ and the coupling constant $\ensuremath{\xi}$, a wide variety of dynamical behavior may be encountered. Thus for $\ensuremath{\xi}<1$ interesting threshold effects may appear in the behavior of correlation times and linewidths as a function of excitation which may change their behavior whenever a mode passes through its threshold of oscillation. For $\ensuremath{\xi}=1$ the correlation time of the stronger mode may pass through a maximum before it starts to decrease with increasing excitation. The weaker mode may show particularly interesting behavior for $\ensuremath{\xi}=1$. Both the correlation time and the linewidth may approach a constant value independent of the excitation. For $\ensuremath{\xi}>1$ the system may show bistability. Our analysis provides a deeper mathematical understanding of the onset of bistability and the analogy to the thermodynamic phase transition. Expressions for the switching time have been derived using several different procedures. We have also been able to derive closed-form expressions for the two-time joint probability densities. It should be possible to test some of these predictions in photon-counting experiments. Finally, we mention that our results and method should also be applicable in other similar problems, e.g., to the classical theory of coupled fields.