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Physical Review Letters

Wigner Distribution of the Quantized Lorenz Model

R. Graham

Published Nov 19, 1984• 21 citations
Publisher

Abstract

The quantum-mechanical master equation of a homogeneously broadened single-mode laser is solved in the regime of the Lorenz strange attractor. The quantum Wigner distribution is obtained, replacing the classical invariant measure on the attractor, and reducing to it in the classical limit.
DOI
10.1103/physrevlett.53.2020

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The steady-state intensity distribution for a laser operating somewhat above threshold is determined by solving a master equation. The master equation describes the interaction between a single radiation mode and NNN two-level atoms, it contains the effects of pumping and radiation loss from the cavity, and it is applicable to both homogeneous and Doppler broadening. Forms of the master equation applicable to both three- and four-level lasers are considered. Solutions are obtained under the assumption that the distribution of atoms and radiation intensity is a bivariate Gaussian. Fluctuations in the output of giant-pulse lasers are also considered.
The fully quantum-mechanical model of a single-mode laser is examined. A previously introduced method of solution where both the pump and loss reservoirs are treated simultaneously as a weak perturbation of the atom-light system is employed. This method is similar to perturbation theory for degenerate systems in that the reservoirs remove the nonuniqueness (degeneracy) of the steady state of the unperturbed atom-light system. In the limit of weak reservoir coupling, the exact steady-state solution of the master equation is derived. This solution still contains the atomic variables explicitly and therefore remains valid even in situations where the conventional adiabatic elimination of the atomic variables is no longer possible. A compact discrete representation of the master equation in terms of atomic occupation numbers and photon numbers is developed which allows one to perform all calculations in an elementary manner, requiring only appropriate summations over photon and occupation numbers. General exact relations are obtained for the steady-state probability distribution function p(n,m)p(n,m)p(n,m) to find nnn photons in the light field and mmm atoms in the excited state. Our solution constitutes the first steady-state laser distribution function that is valid for an arbitrary number of laser atoms and in the complete region from far below to far above threshold. It provides a quantitative criterion for the range of validity of approximate methods in laser theory.
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.