An analytic expression valid at any temperature T is derived for the interaction energy $V\left(R\right)\mathrm{}$ 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 $\mathrm{KR}\ll 1$ and to the classical limit for $\mathrm{KR}\gg 1,$ where $K\equiv 2\pi k_{B}T/ħc.\mathrm{}$ For atomic dipoles with dominant absorption frequency ${\omega }_0{=k}_{0}c,$ this $V\left(R\right)\mathrm{}$ and its logarithmic derivative are evaluated numerically and plotted vs R for several values of ${\omega }_0$ and T. Also, simple approximate expressions for $V\left(R\right)\mathrm{}$ and their ranges of validity are derived. Both the numerical results and the simple approximate expressions scale with $\mathrm{KR}$ for $R\gtrsim R_2{=110/k}_0$ and with $k_{0}R$ for $R\lesssim R_3=0.74/K.\mathrm{}$ They also exhibit the London limit $\propto R^{-6}$ for $R\lesssim {0.023/k}_0,$ the Casimir-Polder limit $\propto R^{-7}$ for $R_2\lesssim R\lesssim R_3,$ if this range exists, and the classical limit $\propto R^{-6}$ for $R\gtrsim 7.4/K.$ If $k_{B}T/ħ{\omega }_0\gtrsim {10}^3,$ then $R_2\gtrsim R_3,$ and true $R^{-7}$ behavior does not occur. A physical analysis of these results and suggestions for crucial experiments are presented.

• Received 11 September 1998

DOI:https://doi.org/10.1103/PhysRevA.60.2577