We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with
N stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with
N stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.
A fully quantum-mechanical treatment is given of field and population fluctuations in a homogeneously broadened laser. The field and population reservoirs have been replaced by dissipation coefficients (or transition probabilities) and noncommuting noise sources. The atomic polarization variables are adiabatically eliminated. The mean motion of an arbitrary operator function $M(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, {N}_{1}, {N}_{2})$ of the field variables
b, ${b}^{\ifmmode\dagger\else\textdagger\fi{}}$ and the population variables
N1 and
N2 is calculated. The equation of motion for the density matrix $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, {N}_{1}, {N}_{2}, t)$ is deduced, as well as the Fokker-Planck equation for the associated classical distribution function $P(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, {N}_{1}, {N}_{2}, t)$. The diffusion coefficients in the latter equation demonstrate the shot-noise origin of the fluctuations. The Fokker-Planck equation for
P is replaced by Langevin equations for $\ensuremath{\beta}$, ${\ensuremath{\beta}}^{*}$,
N1, and
N2 to facilitate adiabatic elimination of the population variables (when valid). The resulting Langevin equations for $\ensuremath{\beta}$, ${\ensuremath{\beta}}^{*}$ are converted to a Fokker-Planck equation for $P(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, t)$, to an operator equation for $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, t)$, and to an equation for ${\ensuremath{\rho}}_{\mathrm{mn}}$, the field density matrix in the photon-number representation. The transformation $\ensuremath{\beta}={I}^{\frac{1}{2}}{e}^{\ensuremath{-}i\ensuremath{\phi}}$ to an ($I, \ensuremath{\phi}$) representation leads (a) to the steady-state solution
P(I) valid at all operating levels, and (b) to the phase linewidth. The transformation to new variables
N=21(m+n), $k=m\ensuremath{-}n$, permits the equation for ${\ensuremath{\rho}}_{\mathrm{mn}}$ to be replaced by a Fokker-Planck equation for $\ensuremath{\rho}(N, k, t)$. The latter permits a verification of the previously obtained phase linewidth and a solution for the steady-state $\ensuremath{\rho}(N)$.
We derive the equation of motion for the quantum-mechanical radiation density matrix of a gas laser to lowest order in the dimensionless coupling constant. Our derivation is fully quantum mechanical and we can calculate the coherence properties from the radiation density matrix. Our model consists of
N two-level systems interacting with radiation in a cavity in the presence of dissipation, pumping, and collisions. The method we use is a generalization of the Bogoliubov derivation of the kinetic equation for a small parameter. Our derivation holds for any physically realizable pump power, and near threshold reduces to Lamb's near-threshold theory. With the equation of motion for the radiation density matrix, we obtain solutions both for when the average field is nonzero and for when it is zero. The steady-state electromagnetic density is the same in both cases except for a small spontaneous-emission term. We show that the reason gas lasers do not satisfy rate equations is the existence of zeroth-order correlations between the internal atomic variables and atomic center-of-mass variables. It is these same zeroth-order correlations which are responsible for the Lamb dip. Our derivation includes collisions and reduces the calculations of their effect to quadrature.
In a previous paper we showed how time-dependent projection operators may be employed to enable the Nakajima-Zwanzig projection-operator technique for deriving exact master equations to deal efficiently with two or more coupled classical or quantum systems, neither of which is reservoir like. We considered in detail the case where the relevant part of the classical-system probability-density function (PDF) or quantumsystem density operator (DO) is a product of the PDF's or DO's for the separate subsystems, and we applied the techniques developed to problems in quantum optics and in the kinetic theory of dilute nonideal gases. In this paper we make the time-dependent projection-operator approach useful for a greater variety of systems by allowing the relevant part of the PDF or DO to include correlations between two of the interacting subsystems. This extension allows us to describe well the dynamics of strongly interacting systems in a low degree of approximation while avoiding the use of infinite resummations. We derive exact generalized master equations in this manner for the same two cases as in our earlier work, namely a classical gas of
N molecules interacting via a two-body potential and a quantum-optical system of
N two-level atoms interacting with an electromagnetic field. In the former case, the relevant part of the PDF contains two-body correlations, and we obtain two exact coupled master equations for the singlet and doublet PDF's
F1(t) and
F2(t). In the latter case, the relevant part of the DO contains atom-field correlations, and the result is three exact coupled equations for the single-atom DO ${\ensuremath{\rho}}_{1}(t)$, the field DO
R(t), and the DO for one atom plus the field $\ensuremath{\chi}(t)$. From the exact equations we derive approximate equations by making simplifying assumptions. In the case of the gas we carry out a straightforward density expansion of the ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{F}}_{2}$ master equation and obtain a set of kinetic equations for dense gases, which have been derived previously by Frieman, Goldman, and Dorfman and which describe well effects due to the finite mean free path. In the quantum-optical case we derive kinetic equations for a single-mode laser by treating the $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\chi}}$ equation in the Born-Markoff approximation. The equations describe gain saturation and other effects on the field dynamics which are of infinite order in the atom-field coupling constant. It is shown that these equations reduce to previously derived laser master equations if atom-atom correlations can be neglected. Finally, we mention possible generalizations of the time-dependent projection-operator approach used in this paper and briefly discuss its application to other problems, including the kinetics of liquids, collisional line broadening, superradiance, and amplified spontaneous emission.
We derive quantum-mechanical kinetic equations for the matter density $\ensuremath{\rho}$ and radiation density matrix
R, which describe optical pumping phenomena. The resultant kinetic equations are a set of coupled nonlinear equations for $\ensuremath{\rho}$ and
R. With appropriate linearizations, we can obtain the present theories of optical pumping. The nonlinear equations describe multiple scattering and line narrowing due to imprisonment of resonant radiation. We show that the coupled equations for $\ensuremath{\rho}$ and
R are equivalent to coupled equations for $\ensuremath{\rho}$ and a generalized polarization matrix II, whose matrix elements are the second moments of
R. The polarization matrix II constitutes a complete description of linear phenomena in the same manner as present theories describe optical pumping phenomena by using the matter density matrix $\ensuremath{\rho}$ alone. As a consequence of our nonphenomenological treatment of radiation, we can provide a completely microscopic treatment of the externally modulated light-beam experiment of Bell and Bloom. We show that the atom absorbs the modulation envelope directly from the external thermal light beam in the same way that the atom absorbs transverse polarization directly from the light beam in optical pumping experiments.