Transition Operators in Radiative Damping Theory

The application of transition operators ${\mathcal{P}}_{m,m^{\prime }}\mathrm{}\left(t\right)=\exp \mathrm{}\left(\mathrm{iHt}\right)\left|m〉〈m^{\prime }\right|\exp \mathrm{}(-iHt)\mathrm{}$ is studied for the problem of an atomic system $S$ with eigenstates $\mathrm{}\left\{m\right\}\mathrm{}$ interacting with one or more damping reservoirs $R$. The average value of these operators gives the reduced density matrix ${{\rho }_{m^{\prime },m}}^{\mathrm{}\left(S\right)\mathrm{}}\mathrm{}\left(t\right)\mathrm{}$ for $S$. If $R$ consists of broad-band distributions of harmonic oscillators, (e.g., radiative damping), then damped equations of motion can be derived for all ${\mathcal{P}}_{m,m^{\prime }}\mathrm{}\left(t\right)\mathrm{}$, even if $S$ is a multilevel system. One need not specify the initial states of $R$, nor restrict the treatment to second order in the S-R coupling. The formalism is illustrated for the case where $S$ consists of (i) a four-level atom in a resonant cavity (with broad-band modes also present), and (ii) a collection of atoms that can be treated as a multilevel spin system. Density-matrix equations are obtained for the case where no damping radiation is present initially. In (ii), the formalism is used to derive a two-time corrleation function without the aid of the fluctuation-regression theorem.