Cumulant Expansions and Pressure Broadening as an Example of Relaxation

Abstract

Pressure broadening by neutral atoms is treated in a time-dependent formalism making use of generalized cumulants due to Kubo. A thermal-equilibrium initial density matrix is assumed, unlike in previous theories by Baranger and Fano who neglect initial correlations between atoms; it is pointed out that the wings of the spectrum depend in an essential manner on these initial correlations. The treatment centers on a time-evolution operator $U(t)={({\mathrm{Tr}}_{B}\ensuremath{\rho})}^{\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}{\mathrm{Tr}}_{B}\ensuremath{\rho}{e}^{\mathrm{it}{H}^{\mathrm{x}}}$ operating in the Liouville space of the radiating atom and governing its motion under the influence of the perturbing gas or bath {${\mathrm{Tr}}_{B}$ is the trace over bath coordinates, $\ensuremath{\rho}\ensuremath{\sim}{e}^{\ensuremath{-}\ensuremath{\beta}H}$ is the density matrix, and ${H}^{\mathrm{x}}$ is the quantum-mechanical Liouvillian: ${H}^{\mathrm{x}}( )=[H, ( )]$; $H$ is the total Hamiltonian for the system radiator plus bath}; $U(t)$ is written as $U(t)={T}_{\ensuremath{\rightarrow}}\mathrm{exp}[i\ensuremath{\int}{0}^{t}d{t}^{\ensuremath{'}}L({t}^{\ensuremath{'}})]$, $L(t)={H}_{s}^{\mathrm{x}}+R(t)$ (${T}_{\ensuremath{\rightarrow}}$ is a time-ordering operator), where the effect of the bath is contained in the time-dependent non-Hermitian perturbation $R(t)$ added to the Liouvillian ${H}_{s}^{\mathrm{x}}$ of the unperturbed radiator. The operator $R(t)$ is expanded in powers of a "reduced" density equal to the perturbing-gas density multiplied by the ratio of the fugacities corresponding to mutually interacting and noninteracting perturbing atoms, respectively; the terms of the expansion are expressed by means of generalized cumulants, and describe interactions of the radiator with clusters of perturbers. By setting $R(t)=\overline{R}+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{R}(t)$, where $\overline{R}\ensuremath{\equiv}{\mathrm{lim}}_{T\ensuremath{\rightarrow}\ensuremath{\infty}}[(\frac{1}{T})\ensuremath{\int}{0}^{T}\mathrm{dtR}(t)]$, the spectrum is written as the sum of its impact approximation, determined by $\overline{R}$, plus a correction expanded in powers of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{R}(t)$, which to first order in the gas density equals the one-perturber spectrum minus its singularities at the resonance frequencies.