Material dependence of Casimir interaction between a sphere and a plate: First analytic correction beyond proximity force approximation

TL;DR

This paper analytically derives the first correction beyond the proximity force approximation for the Casimir interaction between a sphere and a plate, revealing dependence on dielectric properties and providing numerical estimates for gold models.

Contribution

It provides the first analytic correction beyond PFA for sphere-plate Casimir interaction, including dielectric dependence and numerical validation.

Findings

Next-to-leading order term depends on dielectric functions.

Error in plasma model approximation is within 2%.

The ratio of correction terms can reach up to 4.5%.

Abstract

We derive analytically the asymptotic behavior of the Casimir interaction between a sphere and a plate when the distance between them, $d$, is much smaller than the radius of the sphere, $R$. The leading order and next-to-leading order terms are derived from the exact formula for the Casimir interaction energy. They are found to depend nontrivially on the dielectric functions of the objects. As expected, the leading order term coincides with that derived using the proximity force approximation. The result on the next-to-leading order term complements that found by Bimonte, Emig and Kardar [Appl. Phys. Lett. \textbf{100}, 074110 (2012)] using derivative expansion. Numerical results are presented when the dielectric functions are given by the plasma model or the Drude model, with the plasma frequency (for plasma and Drude models) and relaxation frequency (for Drude model) given respectively by 9eV and 0.035eV, the conventional values used for gold metal. It is found that if plasma model is used instead of Drude model, the error in the sum of the first two leading terms is at most 2%, while the error in $\theta_1$, the ratio of the next-to-leading order term divided by $d/R$ to the leading order term, can go up to 4.5%.