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Physical Review A

Comparison of a semiclassical stochastic-master-equation approach to laser fluctuations with the Scully-Lamb theory

Adi R. Bulsara, William C. Schieve

Published Nov 1, 1978• 5 citations
Publisher

Abstract

The analogy of a laser with an autocatalytic chemical reaction is used to write down a macroscopic stochastic master equation of the birth-and-death type for the photon probability function in a semiclassical laser formalism. This equation may be solved exactly in the steady state using a generating function. This leads to a prescription for calculating all the higher-order moments of the photon number as well as providing a unique truncation of the hierarchy of moment equations at any given order. Further, the variance is given by a Poisson distribution, in agreement with the strong-signal Scully-Lamb theory. This master equation is also shown to lead to an exact Fokker-Planck equation by using a modified Kramers-Moyal expansion. A comparison between our results and the (microscopic) Scully-Lamb theory shows that the two approaches give the same results at large photon number.

DOI
10.1103/physreva.18.2253

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Physical Review A•Jun 1, 1975•5 citations
The analogy of a laser with an autocatalytic chemical reaction is used to write down a macroscopic stochastic master equation of the birth-and-death type for the photon probability function in a semiclassical laser formalism. This equation may be solved exactly in the steady state using a generating function. This leads to a prescription for calculating all the higher-order moments of the photon number as well as providing a unique truncation of the hierarchy of moment equations at any given order. Further, the variance is given by a Poisson distribution, in agreement with the strong-signal Scully-Lamb theory. This master equation is also shown to lead to an exact Fokker-Planck equation by using a modified Kramers-Moyal expansion. A comparison between our results and the (microscopic) Scully-Lamb theory shows that the two approaches give the same results at large photon number.
We consider the problem of two-photon transitions between atomic levels having the same parity. The semiclassical theory is developed from Maxwell's equations and the resulting transcendental equations for the pulse envelope solved exactly. They lead to an unstable pulse envelope if the higher-order processes are included. We then treat the laser as a birth/death process, deriving a Markovian (macroscopic) master equation for the probability function, which may be solved exactly in the steady state. By a suitable truncation procedure, this equation predicts a stable steady-state envelope, and is consistent with the microscopic quantum theory when virtual processes are neglected. A comparison is made between the macroscopic (master equation) approach and the microscopic quantum theory at the level of the moment equations. We find that, in the high-gain limit, the fluctuations predicted by the two theories are the same.
We discuss a generalized version of the Scully-Lamb laser master equation for a collection of driven multilevel atoms in a lambda-type configuration. We are especially interested in the effect of the external driving field on the laser process with or without a population inversion between the lasing levels. We provide an exact solution of the photon probability distribution in steady state and discuss regimes of operation that can lead to sub-Poissonian statistics, as measured by the ratio between the variance and the average photon number. We solve the field master equation in the transient regime and analyze the effects of the external driving field on the linewidth of the emitted radiation. The largest reduction of the photon variance relative to the mean number corresponds to the injection into the cavity of atoms prepared in the nonlasing level. Injection of atoms in an incoherent mixture of the three energy states normally leads to super-Poissonian statistics.
We arrive at transition probabilities for a laserlike system by neglecting wave properties of matter and radiation, and treating the problem, in an analogy to chemical systems, entirely in a number representation. Thus, for example, stimulated emission is idealized as a collision between a photon and an excited atom and the transition rate is proportional to the amount of each species present in the cavity. In all, transition probabilities for stimulated emission, absorption of radiation, cavity loss, atomic deexcitation (other than stimulated emission), and pumping of atoms are included in the model. The resulting stochastic master equation analogous to that employed for the discussion of fluctuations in chemical systems is then used to obtain the deterministic behavior of atoms and field. This agrees with that of the quantum mechanical laser model of Casagrande and Lugiato; thus the rate parameters of the stochastic model may be related to the parameters of their model. Returning to the master equation, we project out the atomic variable and use adiabatic elimination to arrive at a closed, single-variable equation for P(n), the probability of having n photons in the cavity. The stationary solution of this equation is obtained and graphs of (n) and log[(n2)/(n)2] versus the pumping parameter show close agreement with the microscopic results of the Casa-grande-Lugiato model. A similar discussion of the fluctuations can also be given for a laser with saturable absorption. It shows a first-order phase transition. A comparison with the microscopic results of Lugiato et al. is outlined. Again a remarkable agreement of the stochastic approach via the master equation and the microscopic theory is obtained.
The single-mode laser model as formulated by Haken and co-workers is considered. Within the approximation which takes into account only the interaction (at all orders in the coupling constant) between the mode and the single atoms, this model leads to a linear time-evolution equation for the Glauber quasiprobability distribution $P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)$ of the field mode, containing derivatives of all orders in $\ensuremath{\beta}$ and ${\ensuremath{\beta}}^{*}$. This equation is solved exactly in the stationary situation. Thus the quantum effects due to the terms with derivatives of order higher than the second one are fully taken into account. It is shown that these terms have a shrinking effect on the stationary distribution above threshold, counteracting the broadening effect due to the second-derivative term. The photon and photoelectron distributions are given. Exact relations are deduced, which allow the calculation of all the moments of the steady-state distribution from the first one. This distribution is compared with the stationary solutions of Risken's equation, of P. Mandel's equation, and with the stationary distribution deduced by Weidlich, Risken, and Haken. The connection with the single-mode laser theory of Scully and Lamb is analyzed in the high-intensity region. The complete Scully-Lamb master equation is translated into the Glauber representation, yielding a linear equation for $P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)$ containing derivatives of all orders in $\ensuremath{\beta}$ and ${\ensuremath{\beta}}^{*}$. Suitably connecting the parameters AAA, BBB, and CCC of the Scully-Lamb equation with the parameters of the model of Haken and co-workers, it is shown that there is perfect agreement between the strong-signal Scully-Lamb steady-state distribution and that of Weidlich, Risken, and Haken for all values of the pump parameter.
The photon-number distributions for a laser with a saturable-absorber and a dye laser are obtained by extensions of the Scully and Lamb optical-maser theory. The results are shown to be in correspondence with those of Lugiato et al. for the saturable-absorber laser and with those of Schaefer and Willis for the dye laser. Moreover, they are in a form which clearly shows the relation between the saturable-absorber laser and the dye laser. The thermodynamic potentials for the first-order phase-transition analogy are calculated and are found to be of the form predicted by Scott, Sargent, and Cantrell using semiclassical equations.
Intensity fluctuations for a single-mode laser near threshold are calculated from the Scully-Lamb equation of motion for the reduced density matrix of the radiation field. The magnitude and correlation time of the intensity fluctuations are contrasted with the results obtained from the usual Fokker-Planck analysis of the rotating-wave, Van-der-Pol (RWVP) model. When the photon number at threshold is not large, these calculations predict measurable differences from the RWVP Fokker-Planck results. Two methods are used to calculate the dynamics. One is a direct eigenfunction expansion of the Green's function of the equation of motion. The other is based on the calculation of the initial derivatives of the correlation function.