Virtual photons in imaginary time: Computing exact Casimir forces via standard numerical-electromagnetism techniques

TL;DR

This paper introduces a numerical method leveraging classical electromagnetism techniques to accurately compute Casimir forces in complex geometries and materials, enabling new insights into force behaviors.

Contribution

It presents a novel approach that directly uses Green's functions at imaginary frequencies for precise Casimir force calculations in arbitrary setups.

Findings

Validated method on known geometries like cylinders and plates.

Discovered non-monotonic lateral forces in piston-like geometries.

Applicable to both perfect and realistic metallic materials.

Abstract

We describe a numerical method to compute Casimir forces in arbitrary geometries, for arbitrary dielectric and metallic materials, with arbitrary accuracy (given sufficient computational resources). Our approach, based on well-established integration of the mean stress tensor evaluated via the fluctuation-dissipation theorem, is designed to directly exploit fast methods developed for classical computational electromagnetism, since it only involves repeated evaluation of the Green's function for imaginary frequencies (equivalently, real frequencies in imaginary time). We develop the approach by systematically examining various formulations of Casimir forces from the previous decades and evaluating them according to their suitability for numerical computation. We illustrate our approach with a simple finite-difference frequency-domain implementation, test it for known geometries such as a cylinder and a plate, and apply it to new geometries. In particular, we show that a piston-like geometry of two squares sliding between metal walls, in both two and three dimensions with both perfect and realistic metallic materials, exhibits a surprising non-monotonic ``lateral'' force from the walls.