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.
For a quantum system describable in a Markoffian way, via a set of variables $\mathrm{a}={{a}_{1}, {a}_{2}, \ensuremath{\cdots}}$, we show that the Langevin noise sources ${F}_{\ensuremath{\mu}}$ in the operator equations of motion $\frac{d{a}_{\ensuremath{\mu}}}{\mathrm{dt}}={A}_{\ensuremath{\mu}}(\mathrm{a})+{F}_{\ensuremath{\mu}}$ possess second moments $〈{F}_{\ensuremath{\mu}}(t){F}_{\ensuremath{\nu}}(u)〉=2〈{D}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathrm{a}, t)〉\ensuremath{\delta}(t\ensuremath{-}u)$. The diffusion coefficients ${D}_{\ensuremath{\mu}\ensuremath{\nu}}$ can be determined from a knowledge of the mean equations of motion via the (exact) time-dependent Einstein relation $2〈{D}_{\ensuremath{\mu}\ensuremath{\nu}}〉=\ensuremath{-}〈{A}_{\ensuremath{\mu}}{a}_{\ensuremath{\nu}}〉\ensuremath{-}〈{a}_{\ensuremath{\mu}}{A}_{\ensuremath{\nu}}〉+\frac{d〈{a}_{\ensuremath{\mu}}(t){a}_{\ensuremath{\nu}}(t)〉}{\mathrm{dt}}$, where
〈〉 represents a reservoir average. The sources ${F}_{\ensuremath{\mu}}$, ${F}_{\ensuremath{\nu}}$ do not commute with one another, and as a result the commutation rules of the ${a}_{\ensuremath{\mu}}$ are shown to be preserved in time. The mean motion and diffusion coefficients are calculated for a harmonic oscillator, and for a set of atomic levels. We prove that two dynamically coupled systems (e.g., field and atoms) have uncorrelated Langevin forces if they are coupled to independent reservoirs. Radiation-field-atom coupling adds no new noise sources. We thus obtain simply the maser model including noise sources used in Quantum Noise V. Direct calculations of the mean motion and fluctuations in a system coupled to a reservoir yield relationships in agreement with the Einstein relation. For reservoirs violating time reversal, anomalous frequency shifts are found possible that violate the Ritz combination principle since $\ensuremath{\Delta}{\ensuremath{\omega}}_{12}+\ensuremath{\Delta}{\ensuremath{\omega}}_{23}+\ensuremath{\Delta}{\ensuremath{\omega}}_{31}$ need not vanish.
Using the recently developed phase-space techniques for the treatment of quantum-mechanical problems, we set up a procedure for calculating multitime-correlation functions in terms of the joint distribution functions. The correlation functions are then expressed simply as integrals over the associated phase space. Explicit expressions are given for these joint distribution functions, in terms of Green's functions of the
c-number equations of motion for the phase-space equivalent of the density operator. Using these joint distribution functions, an exact regression theorem is rederived, and the connection with the multitime correspondence between classical and quantum stochastic processes is discussed.
Quantum coherence and its destruction (dephasing) by coupling to a dissipative environment plays an important role in time-resolved nonlinear optical response as well as nonadiabatic transitions and tunneling processes in condensed phases. Generating functions of density-matrix elements and multitime coordinate and momentum correlation functions related to these phenomena are calculated using a pathintegral approach by performing functional integration. The dissipative environment is assumed to be an ensemble of harmonic oscillators and is taken into account by using Feynman-Vernon influence functional. Closed-form expressions for generating functions in terms of the bath spectral density are derived. The present theory generalizes earlier calculations of these quantities to arbitrary temperatures, any dependence of the transition coupling on coordinates (non-Condon effects), and arbitrary order in the interstate coupling. Conditions for factorizing the Liouville-space generating functions that allow a reduced description based on the classical Langevin equation are established. Possible applications to four-wave-mixing spectroscopy and nonadiabatic rate processes are discussed.