Quantum Noise. X. Density-Matrix Treatment of Field and Population-Difference Fluctuations

Abstract

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 $\rho (b, b^{†}, D, t)$ that depends on the electromagnetic field variables $b$, $b^{†}$ and the population difference $D$. The corresponding equation obeyed by the density matrix ${\rho }_{\mathrm{mn}}(D, t)$ in the "$n$ representation" is obtained. The antinormal ordering correspondence $\rho (b, b^{†}, D, t)=\mathcal{a}{\overline{\rho }}^{\mathrm{}\left(a\right)\mathrm{}}(\beta , {\beta }^{*}, D, t)$ [in which $\beta$ is replaced by $b$ and ${\beta }^{*}$ by $b^{†}$ with all $b^{†}\text{'}\mathrm{s}$ to the right of all $b\text{'}\mathrm{s}$] defines an associated classical function ${\overline{\rho }}^{\mathrm{}\left(a\right)\mathrm{}}$. The equation for $\frac{\partial \rho }{\partial t}$ corresponds to a Fokker-Planck equation for $\frac{\partial {\overline{\rho }}^{\mathrm{}\left(a\right)\mathrm{}}}{\partial t}$. 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 $I$, $\phi$, $D$, where $\beta =I^{\frac{1}{2}}\exp (-i\phi )$. The Langevin description of the classical random process associated with ${\overline{\rho }}^{\mathrm{}\left(a\right)\mathrm{}}(\beta , {\beta }^{*}, D, t)$ is used to make a classical adiabatic elimination of $D$. The resulting equations for ${\overline{\rho }}^{\mathrm{}\left(a\right)\mathrm{}}(\beta , {\beta }^{*}, t)$ and $\rho (b, b^{†}, t)$ 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 $\rho (b, b^{†}, t)$, rewritten in the $n$ representation agrees with corresponding results of Scully and Lamb.