General framework for interatomic interaction energy of two ground-state atoms in a thermal bath

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We give a general derivation of the fourth-order DDC formalism (a formalism proposed by Dalibard, Dupont-Roc, and Cohen-Tannoudji) for calculating the interaction energy between two ground-state multilevel atoms which are coupled to electromagnetic fields in a thermal bath at temperature $T$. Both the contributions of the thermal field fluctuations and the radiation reaction of atoms are separately identified. As an application of the formalism, we revisit the interaction energy of two static ground-state two-level atoms in a free space and discover new behaviors such as the $\ensuremath{\sim}T{L}^{\ensuremath{-}2}$ behavior of the van der Waals interaction energy in the region where ${\ensuremath{\lambda}}^{3/4}{T}^{\ensuremath{-}1/4}\ensuremath{\ll}L\ensuremath{\ll}\ensuremath{\lambda}$, with $\ensuremath{\lambda}$ the transition wavelength of the atoms and $L$ the interatomic separation, and an oscillatory behavior of the Casimir-Polder interaction energy as $L$ varies which is superimposed on the monotonic $\ensuremath{\sim}T{L}^{\ensuremath{-}6}$ dependence when $L\ensuremath{\gg}\ensuremath{\lambda}\ensuremath{\gg}{T}^{\ensuremath{-}1}$.