Casimir-Polder interaction at finite temperature

An analytic expression valid at any temperature T is derived for the interaction energy V(R) of two polarizable electric dipoles separated by distance R immersed in the thermal quantum field. The expression is a sum that reduces to the Casimir-Polder vacuum interaction integral for KR≪1 and to the classical limit for KR≫1, where K≡2πk_BT/ħc. For atomic dipoles with dominant absorption frequency ω₀=k₀c, this V(R) and its logarithmic derivative are evaluated numerically and plotted vs R for several values of ω₀ and T. Also, simple approximate expressions for V(R) and their ranges of validity are derived. Both the numerical results and the simple approximate expressions scale with KR for R≳R₂=110/k₀ and with k₀R for R≲R₃=0.74/K. They also exhibit the London limit ∝R^-6 for R≲0.023/k₀, the Casimir-Polder limit ∝R^-7 for R₂≲R≲R₃, if this range exists, and the classical limit ∝R^-6 for R≳7.4/K. If k_BT/ħω₀≳10³, then R₂≳R₃, and true R^-7 behavior does not occur. A physical analysis of these results and suggestions for crucial experiments are presented.