Cumulant Expansions and Pressure Broadening as an Example of Relaxation

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 $U\left(t\right)={\left({\mathrm{Tr}}_B\rho \right)}^{-1\mathrm{}}\times {\mathrm{Tr}}_B\rho e^{\mathrm{it}{H}^{\mathrm{x}}}$ 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, $\rho \sim e^{-\beta H}$ is the density matrix, and $H^{\mathrm{x}}$ is the quantum-mechanical Liouvillian: $H^{\mathrm{x}}\left( \right)=[H, ( \left)\right]$; $H$ is the total Hamiltonian for the system radiator plus bath}; $U\left(t\right)\mathrm{}$ is written as $U\left(t\right)=T_{\to }\exp \left[i\int 0^{t}d{t}^{\prime }L\left(t^{\prime }\right)\right]$, $L\left(t\right)=H_s^{\mathrm{x}}+R\left(t\right)\mathrm{}$ ($T_{\to }$ is a time-ordering operator), where the effect of the bath is contained in the time-dependent non-Hermitian perturbation $R\left(t\right)\mathrm{}$ added to the Liouvillian $H_s^{\mathrm{x}}$ of the unperturbed radiator. The operator $R\left(t\right)\mathrm{}$ 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 $R\left(t\right)=\overline{R}+\tilde{R}\mathrm{}\left(t\right)\mathrm{}$, where $\overline{R}\equiv {\lim }_{T\to \infty }\left[\right(\frac{1}{T}\left)\int 0^T\mathrm{dtR}\left(t\right)\mathrm{}\right]$, the spectrum is written as the sum of its impact approximation, determined by $\overline{R}$, plus a correction expanded in powers of $\tilde{R}\mathrm{}\left(t\right)\mathrm{}$, which to first order in the gas density equals the one-perturber spectrum minus its singularities at the resonance frequencies.