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Physical Review A

First-order-like transition for colored saturation models of dye lasers: Effects of quantum noise

Cao Li, Wu Da-Jin, Ling Lin

Published Jan 1, 1994• 8 citations

Abstract

A multidimensional form of the unified colored-noise approximation is applied to obtain the Fokker-Planck equation for intensity distributions of dye-laser models simultaneously driven by colored pump noise and quantum noise. The stationary intensity distributions (SID's) of these models are calculated. The first-order-like transitions (FOLT's) of the models are studied and compared. With the help of the combination of Sturm's theorem and Descarte's rule of signs, the following conclusions on the FOLT are obtained. (1) In the limit of white pump noise (the correlation time of the pump noise \ensuremath{\tau}=0), the FOLT disappears for the colored loss-noise model with saturation whereas it does not disappear for the colored gain-noise model. (2) The dependence of the FOLT on \ensuremath{\tau} is intensely affected by the quantum noise. If quantum noise is neglected, the first critical curve which separates the regions of single extremum and double extrema of the SID does not change with \ensuremath{\tau}. However, when the quantum noise is considered, the first critical curve does change with \ensuremath{\tau}. (3) When \ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\infty}, the area of region II in which the SID varies monotonically contracts and tends to zero. Therefore the FOLT disappears in the models. (4) It is found that when the strength of the quantum noise P varies (\ensuremath{\tau} is fixed), the critical curves also change.

DOI: 10.1103/physreva.49.506

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A multidimensional form of the unified colored-noise approximation is applied to obtain the Fokker-Planck equation for intensity distributions of dye-laser models simultaneously driven by colored pump noise and quantum noise. The stationary intensity distributions (SID's) of these models are calculated. The first-order-like transitions (FOLT's) of the models are studied and compared. With the help of the combination of Sturm's theorem and Descarte's rule of signs, the following conclusions on the FOLT are obtained. (1) In the limit of white pump noise (the correlation time of the pump noise \ensuremath{\tau}=0), the FOLT disappears for the colored loss-noise model with saturation whereas it does not disappear for the colored gain-noise model. (2) The dependence of the FOLT on \ensuremath{\tau} is intensely affected by the quantum noise. If quantum noise is neglected, the first critical curve which separates the regions of single extremum and double extrema of the SID does not change with \ensuremath{\tau}. However, when the quantum noise is considered, the first critical curve does change with \ensuremath{\tau}. (3) When \ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\infty}, the area of region II in which the SID varies monotonically contracts and tends to zero. Therefore the FOLT disappears in the models. (4) It is found that when the strength of the quantum noise P varies (\ensuremath{\tau} is fixed), the critical curves also change.

A unified colored-noise approximation is applied to calculate the stationary intensity probabilities of colored-gain-noise and colored-loss-noise models for dye lasers with saturation effects. We compare the results of the unified colored-noise approximation with the predictions of the best Fokker-Planck equation for the stationary intensity probability of a colored-gain-noise model, and find that the results of the unified colored-noise approximation agree better with simulation results than the results of the best Fokker-Planck equation. The first-order-like transition of colored-gain-noise and colored-loss-noise models with saturation effects is discussed and compared. It is found that the emergence of the first-order-like transition is related to the saturation element in the fluctuation term of the dye-laser model for white pump noise, and the occurrence of the first-order-like transition is only due to the existence of colored pump noise in the loss-noise models for dye lasers with saturation effects.

We have derived the approximate formulas of the mean, variance, and skewness of the first-passage-time distribution for the colored-loss-noise model and the colored-gain-noise model of the dye laser by virtue of the best Fokker-Planck-equation approximation. Comparing the colored-loss-noise model with the colored cubic model, it is shown that our analytic results are in good agreement with the numerical simulation given by Zhu, Yu, and Roy [Phys. Rev. A 34, 4333 (1986)]. We find that the prefactor of the pump fluctuation term can curb the variance and skewness of the first-passage-time distribution. In the regime 0.8${\mathit{I}}_{0}$1, where

{\mathit{I}}_{0}

denotes the relative laser intensity, the best Fokker-Planck equation predicts that as the laser intensity is increased, the increases of the variance and skewness of the first-passage-time distribution for the colored saturation models are slower than those of corresponding white saturation models.

A single-mode dye-laser model incorporating pump white noise through gain-parameter fluctuations is analyzed. It includes a fluctuating saturation term. Intensity fluctuations, a first-order-like transition, and intensity correlation functions are calculated and discussed. It is found that effects previously claimed to be a consequence of the presence of colored noise in the context of loss-noise models can be also explained by white gain noise, except for the existence of two time regimes in the early decay of intensity correlation functions.

The intensity correlation functions C(t) for the colored-gain-noise model of dye lasers are analyzed and compared with those for the loss-noise model. For correlation times \ensuremath{\tau} larger than the deterministic relaxation time

{\mathit{t}}_{\mathit{d}}

, we show with the use of the adiabatic approximation that C(t) values coincide for both models. For small correlation times we use a method that provides explicit expressions of non-Markovian correlation functions, approximating simultaneously short- and long-time behaviors. Comparison with numerical simulations shows excellent results simultaneously for short- and long-time regimes. It is found that, when the correlation time of the noise increases, differences between the gain- and loss-noise models tend to disappear. The decay of C(t) for both models can be described by a time scale that approaches the deterministic relaxation time. However, in contrast with the loss-noise model, a secondary time scale remains for large times for the gain-noise model, which could allow one to distinguish between both models.

In this paper, a unified expansion theory that can be simultaneously applied to both large and small correlation times developed by Gang HU is established and applied to the systems driven by multiplicative colored noise. The stationary intensity probabilities are calculated for colored gain-noise and colored-loss-noise models. Comparing with the predictions of the best Fokker–Planck equation and the unified colored-noise approximation for the stationary intensity probability of the two models, it is found that the results of the unified expansion theory are in better agreement with simulations and experimental results than those of the best Fokker–Planck equation approximation and the unified colored-noise approximation.