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 The field and population reservoirs have been replaced by dissipation coefficients (or transition probabilities) and noncommuting noise sources.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 present a regorous adiabatic elimination of the atomic variables in the framework of the quantum-statistical single-mode laser model. This procedure, which is systematic and similar to the one used in the semiclassical theory, is based on the hierarchy method proposed years ago by one of us. Using a rigorous definition of the adiabatic limit, we show that the hierarchy is exactly truncated by the limit itself. Thus we recover the Fokker-Planck equation previously derived by us using a nonsystematic procedure. We discuss the advantages of the present adiabatic elimination method over the ones previously used by various authors.
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.
A dynamic correspondence is set up between the density matrix of the electromagnetic field and an associated classical function. Means of one and two time quantum operators are shown to be obtainable from the corresponding averages of the associated classical random problem. When all atomic response rates are fast compared to photon rates, the equation of motion is obtained for the field density matrix. The associated classical problem is shown to obey a Fokker-Planck equation. Over a broad region that includes the threshold region, the Fokker-Planck equation is reduced to that for the rotating-wave van der Pol oscillator for which exact numerical solutions have been previously obtained by Hempstead and Lax.
We show that the most complete and accurate description of fluctuations in quantum-optical systems is that obtained using the generalized Wigner distribution for the macroscopic observables of the system. The smallness of the inverse of the saturation photon number entitles us to neglect the terms with derivatives of order higher than second order in the time-evolution equation for the quasiprobability distribution. This treatment allows us to also describe correctly nonclassical effects such as photon antibunching or "squeezing," which are maltreated or even destroyed if one neglects the atom-atom correlations. We derive and analyze the Fokker-Planck equation for the Wigner function in the case of the usual laser and compare the results with those following from other approaches. In the region very high above threshold we find a small antibunching effect, which was not discovered in previous treatments due to the neglect of atom-atom correlations.
We summarize the basic problems in laser coherence and the concepts and techniques needed to solve them. Special techniques needed to handle partition noise in semiconductor lasers will be touched on. We address the problem of how a self-sustained oscillator works. Why is the laser line so narrow? Why does a small amount of noise produce a broadening of the line as opposed to additive side bands? What is the role of zero point oscillations and spontaneous emission in determining the line width? In what sense can the transition to a lasing state be described as a phase transition? The concepts used to deal with the response of a laser system involve a quasi-Markoffian description. This in turn validates a quantum regression theorem which relates the spectrum of the noise to the transport response even in a system far from equilibrium. A significant simplification is made by introducing non-commuting (quantum) noise sources. The techniques required in the solution of laser noise problems involve a multi-time correspondence between quantum variables and c-number random variables that reduces a quantum stochastic problem to a classical stochastic problem. Adiabatic elimination is used to eliminate the fast variables to obtain a reduced problem for the fields alone or the fields and some total atomic variables (total inversion and polarization). Reduction to a rotating wave version of the van der Pol oscillator and numerical solution of the Fokker-Planck equation yields (a) the line shape (b) the spectrum of intensity fluctuations and (c) the distribution of photocounts observed over some time interval. Comparison of theoretical results with later experiments will be presented.