Casimir-Polder energy for axially symmetric systems

TL;DR

This paper develops a formalism for calculating Casimir-Polder energies in axially symmetric systems, deriving explicit interaction energies for delta-function plates and spheres with anisotropic molecules, and highlighting mode contributions.

Contribution

It introduces a new formalism for axially symmetric media and derives explicit Casimir-Polder energies for delta-function geometries with anisotropic molecules.

Findings

Interaction energy is insensitive to molecular orientation in the perfect conductor limit.

Only the m=0 azimuth mode contributes to the Casimir-Polder energy in axially symmetric systems.

The formalism simplifies calculations by separating TE and TM modes for m=0.

Abstract

We develop a formalism suitable for studying Maxwell's equations in the presence of a medium that is axially symmetric, in particular with respect to Casimir-Polder interaction energies. As an application, we derive the Casimir-Polder interaction energy between an electric $\delta$-function plate and an anisotropically polarizable molecule for arbitrary orientations of the principal axes of polarizabilities of the molecule. We show that in the perfect conductor limit for the plate the interaction is insensitive to the orientation of the polarizabilities of the molecule. We obtain the Casimir-Polder energy between an electric $\delta$-function sphere and an anisotropically polarizable molecule, again for arbitrary orientations of the principal axes of polarizabilities of the molecule. We derive results when the polarizable molecule is either outside the sphere, or inside the sphere. We present the perfectly conducting limit for the $\delta$-function sphere, and also the interaction energy for the special case when the molecule is at the center of the sphere. Our general proposition is that the Casimir-Polder energy between a dielectric body with axial symmetry and an unidirectionally polarizable molecule placed on the axis with its polarizability parallel to the axis gets non-zero contribution only from the $m=0$ azimuth mode. This feature in conjunction with the property that the $m=0$ mode separates into transverse electric and transverse magnetic modes allows the evaluation of Casimir-Polder energies for axially symmetric systems in a relatively easy manner.