Casimir-Polder interaction at finite temperature

An analytic expression valid at any temperature T is derived for the intera ction energy V(R) of two polarizable electric dipoles separated by distance R immersed in the thermal quantum field. The expression is a sum that redu ces to the Casimir-Polder vacuum interaction integral for KR much less than 1 and to the classical limit for KR much greater than 1, where K equivalen t to 2 pi k(B)T/hc. For atomic dipoles with dominant absorption frequency o mega(0) = k(0)c, this V(R) and its Logarithmic derivative are evaluated num erically and plotted vs R for several values of omega(0) and T. Also, simpl e approximate expressions for V(R) and their ranges of validity are derived . Both the numerical results and the simple approximate expressions scale w ith KR for R greater than or similar to R-2 = 110/k(0) and with k(0)R for R less than or similar to R-3 = 0.74/K. They also exhibit the London limit p roportional to R-6 for R less than or similar to 0.023/k(0), the Casimir-Po lder limit proportional to R-7 for R-2 less than or similar to R less than or similar to R-3, if this range exists, and the classical limit proportion al to R-6 for R greater than or similar to 7.4/K. If k(B)T/h omega(0)greate r than or similar to 10(3), then R(2)greater than or similar to R-3, and tr ue R-7 behavior does not occur. A physical analysis of these results and su ggestions for crucial experiments are presented. [S1050-2947(99)07509-5].