Density Expansion of the Memory Operator in Pressure-Broadening Theory

Abstract

The motion of a radiating atom immersed in a gas of other atoms, the bath, may be described by means of a nonunitary time-evolution operator $U\left(t\right)={\left({\mathrm{Tr}}_B\rho \right)}^{-1\mathrm{}}{\mathrm{Tr}}_B\rho e^{\mathrm{it}{H}^{\times }}$ acting in its Liouville space (${\mathrm{Tr}}_B$ is the trace over bath variables, $\rho$ is the density matrix, and $H^{\times }$ is the quantum-mechanical Liouvillian). In a previous paper, $U\left(t\right)\mathrm{}$ was written in the form $U\left(t\right)={\exp }_{\to }\left[i\int 0^t{\mathrm{dt}}^{\prime }{L}_1\left(t^{\prime }\right)\right]$ (${\exp }_{\to }$ denotes a time-ordered exponential), and the time-dependent effective Liouvillian $L_1\mathrm{}\left(t\right)\mathrm{}$ was expanded in powers of a "reduced density," or activity. In this paper the Fourier transform $U\left(\omega \right)=\int 0^{\infty }\mathrm{dt}{e}^{-i\omega t}U\left(t\right)\mathrm{}$ is written in the form $iU\left(\omega \right)={[\omega -L_2(\omega \left)\right]}^{-1\mathrm{}}$, and the frequency-dependent effective Liouvillian $L_2\left(\omega \right)$, or "memory operator," is expanded in powers of the reduced density. It is argued that in treating $L_2\left(\omega \right)$ to first order in the reduced density, one effectively allows the radiator to interact with only one perturber at a time, thus neglecting all multiple-collision effects. This is in contrast to performing the same-order approximation on $L_1\mathrm{}\left(t\right)\mathrm{}$, which is equivalent to treating different perturbers as uncorrelated, but still allows for multiple-collision effects. By adding the terms of higher order in the expansion of $L_2\left(\omega \right)$, one allows the radiator to interact simultaneously with two, three,... perturbers.