The application of transition operators ${\mathcal{P}}_{m,{m}^{\ensuremath{'}}}(t)=\mathrm{exp}(\mathrm{iHt})|m〉〈{m}^{\ensuremath{'}}|\mathrm{exp}(\ensuremath{-}iHt)$ is studied for the problem of an atomic system $S$ with eigenstates ${m}$ interacting with one or more damping reservoirs $R$ . The average value of these operators gives the reduced density matrix ${{\ensuremath{\rho}}_{{m}^{\ensuremath{'}},m}}^{(S)}(t)$ 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 ${\mathcal{P}}_{m,{m}^{\ensuremath{'}}}(t)$, 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.The application of transition operators ${\mathcal{P}}_{m,{m}^{\ensuremath{'}}}(t)=\mathrm{exp}(\mathrm{iHt})|m〉〈{m}^{\ensuremath{'}}|\mathrm{exp}(\ensuremath{-}iHt)$ is studied for the problem of an atomic system $S$ with eigenstates ${m}$ interacting with one or more damping reservoirs $R$. The average value of these operators gives the reduced density matrix ${{\ensuremath{\rho}}_{{m}^{\ensuremath{'}},m}}^{(S)}(t)$ 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 ${\mathcal{P}}_{m,{m}^{\ensuremath{'}}}(t)$, 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.

Physical ReviewMay 5, 196930 citations

S

with eigenstates

{m}

interacting with one or more damping reservoirs

R

. The average value of these operators gives the reduced density matrix ${{\ensuremath{\rho}}_{{m}^{\ensuremath{'}},m}}^{(S)}(t)$ 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 ${\mathcal{P}}_{m,{m}^{\ensuremath{'}}}(t)$, 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.