Quantum Noise. XII. Density-Operator Treatment of Field and Population Fluctuations

Abstract

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 The field and population reservoirs have been replaced by dissipation coefficients (or transition probabilities) and noncommuting noise sources.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 ${N}_{1}$ and ${N}_{2}$ 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 $\ensuremath{\beta}$, ${\ensuremath{\beta}}^{*}$, ${N}_{1}$, and ${N}_{2}$ 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 $N=\frac{1}{2}(m+n)$, $k=m\ensuremath{-}n$, 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 $\ensuremath{\rho}(N)$.