Starting with the quantum-noise-source model of a maser developed in the fourth paper in this series and specializing to the case in which all atomic parameters except the population difference vary rapidly compared with the field, an equation is developed for the density operator $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, D, t)$ that depends on the electromagnetic field variables
b, ${b}^{\ifmmode\dagger\else\textdagger\fi{}}$ and the population difference
D. The corresponding equation obeyed by the density matrix ${\ensuremath{\rho}}_{\mathrm{mn}}(D, t)$ in the "
n representation" is obtained. The antinormal ordering correspondence $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, D, t)=\mathcal{a}{\overline{\ensuremath{\rho}}}^{(a)}(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, D, t)$ [in which $\ensuremath{\beta}$ is replaced by
b and ${\ensuremath{\beta}}^{*}$ by ${b}^{\ifmmode\dagger\else\textdagger\fi{}}$ with all ${b}^{\ifmmode\dagger\else\textdagger\fi{}}'\mathrm{s}$ to the right of all
b′s] defines an associated classical function ${\overline{\ensuremath{\rho}}}^{(a)}$. The equation for $\frac{\ensuremath{\partial}\ensuremath{\rho}}{\ensuremath{\partial}t}$ corresponds to a Fokker-Planck equation for $\frac{\ensuremath{\partial}{\overline{\ensuremath{\rho}}}^{(a)}}{\ensuremath{\partial}t}$. Under the Markoffian assumption, an associated classical random process is uniquely defined. The quantum-mechanical average of normal-ordered time-ordered operators is given by simple integrals over the corresponding classical distribution functions. For one-time operators this result is well known. For two-time operators, a proof is given using the quantum-regression theorem of the second paper in this series. Amplitude and phase fluctuations are discussed by rewriting the Fokker-Planck equation in the variables
I, $\ensuremath{\varphi}$,
D, where $\ensuremath{\beta}={I}^{\frac{1}{2}}\mathrm{exp}(\ensuremath{-}i\ensuremath{\varphi})$. The Langevin description of the classical random process associated with ${\overline{\ensuremath{\rho}}}^{(a)}(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, D, t)$ is used to make a classical adiabatic elimination of
D. The resulting equations for ${\overline{\ensuremath{\rho}}}^{(a)}(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, t)$ and $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, t)$ are shown to agree with the corresponding results of the ninth paper in this series, obtained by a more difficult quantum adiabatic elimination of
D. The equation for $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, t)$, rewritten in the
n representation agrees with corresponding results of Scully and Lamb.
We generalize the method of deriving a kinetic equation for a gas laser developed in the first paper of this series to include radiation-matter correlations. As a consequence, we are able to show that the first Born approximation with asymptotic conditions which contain radiation-matter correlations is sufficient to explain the nonthermal photon distribution observed in photon-counting experiments. Our derivation includes the spatial and velocity effects of the motion of the gas atoms. After adiabatically eliminating the matter variables, we obtain a master equation for the radiation density matrix
R with nonlinear coefficients. Our radiation master equation for stationary atoms without spatial effects is the same as the generalized Fokker-Planck equation of the Langevin-noise-equation treatment of lasers. We show that the generalized Fokker-Planck equation is the result of the first-Born-approximation treatment of the j\ifmmode\cdot\else\textperiodcentered\fi{}A interaction and is not dependent upon the microscopic structure of the reservoirs. In the first Born approximation, the diagonal and off-diagonal matrix elements of
R do not interact with each other. Consequently, an initially diagonal radiation density matrix remains diagonal. However, we show that even though the stationary state is diagonal it is necessary to know the propagator of the off-diagonal part of
R to answer time-dependent questions about quantities, such as the line shape, that depend on phase.
The evolution of the radiation field in a model for interacting radiation and matter in cooperative-atom states is studied via the density matrix. The exact treatment possible for small numbers of cooperating atoms reveals effects not apparent from approximation treatments for long interaction times and/or strong interactions. The superradiance concept needs modification under these conditions, and the photon statistics can be greatly modified by the interaction. The representation for the density matrix uses eigensolutions of the model obtained earlier.
We present a derivation of a Fokker - Planck equation for the micromaser based on the Wigner function and valid in the semiclassical limit. The resulting equation is shown to correspond to the conventionally used approximate Fokker - Planck equations obtained for a continuous approximation to the discrete Fock representation of the density matrix of the light field. We thus establish a connection between the continuous approximation and the Wigner function. Then expressing expectation values as properly symmetrized products, we can extend in an exact way the validity of the Fokker - Planck approach to the micromaser down to the zero pump off-state of few photons and purely thermal statistics.
Abstract We report a comparative analysis of the limits of laser cooling of atoms in light fields both on the basis of a numerical solution of the quantum kinetic equation for the atomic density matrix, which makes it possible to accurately take into account quantum recoil effects during the interaction of atoms with field photons, and within the framework of the semiclassical approach using various forms of the Fokker – Plank equation. The analysis allows one to outline the limits of applicability of the semiclassical approach, as well as to choose an unambiguous form for introducing the diffusion contribution to the Fokker – Planck equation in the semiclassical description of atomic kinetics.