Fluctuating surface currents: An algorithm for efficient prediction of Casimir interactions among arbitrary materials in arbitrary geometries

Abstract

This paper presents a new method for the efficient numerical computation of Casimir interactions between objects of arbitrary geometries, composed of materials with arbitrary frequency-dependent electrical properties. Our method formulates the Casimir effect as an interaction between effective electric and magnetic current distributions on the surfaces of material bodies, and obtains Casimir energies, forces, and torques from the spectral properties of a matrix that quantifies the interactions of these surface currents. The method can be formulated and understood in two distinct ways: (1) as a consequence of the familiar stress-tensor approach to Casimir physics, or, alternatively, (2) as a particular case of the path-integral approach to Casimir physics, and we present both formulations in full detail. In addition to providing an algorithm for computing Casimir interactions in geometries that could not be efficiently handled by any other method, the framework proposed here thus achieves an explicit unification of two seemingly disparate approaches to computational Casimir physics.

Casimir forces on a silicon micromechanical chip

“…3a represents the theoretical values of the Casimir force calculated for silicon structures of such geometry, with no fitting parameters. The theoretical calculation involves a boundary-element method (BEM) discretization of the beam and substrate surfaces, combined with a recent fluctuating-surfacecurrent formulation of the Casimir force between dielectric bodies that writes the full Casimir-energy path integral as a simple expression in the classical BEM interaction matrix 14,25 . It includes the contributions of the finite conductivity of silicon and the imperfect etching profiles on the sidewalls of the beam and the electrode (B88°from the substrate surface, see Supplementary Methods).…”

“…3a) neglects the finite length of the beams. The geometry reduces to a two-dimensional problem in the cross-sections (integrated over the longitudinal wavevector), and each object's surface is discretized into set of line segments described by 'rooftop' basis functions 25 . We found that a discretization of approximately 3,200 total points for all surfaces and a substrate truncated to 1 mm was sufficient to obtain convergence to 1% accuracy.…”

Large optical field enhancement for nanotips with large opening angles

“…To rule out systematical errors from the space discretization and time integration in the FDTD simulations, we double-checked the reliability of our simulations by also numerically solving Maxwell's equations with the BEM as implemented in the public-domain SCUFF-EM package [48][49][50]. Being a frequency-domain method, the BEM is free from time integration errors that contribute to the errors in FDTD.…”

Impact of nuclear vibrations on van der Waals and Casimir interactions at zero and finite temperature

“…The individual electronic response of each atom arises from the coupling of valence electronic and phononic excitations via shortrange interactions, represented schematically: for every atom p, a nuclear oscillator of mass mIp with dissipation bIp is connected to nuclear oscillators of other atoms q via anisotropic spring constants Kpq, and to an electronic oscillator of mass mep with dissipation bep and isotropic spring constant kep; only the electrons couple directly to long-range EM fields with effective charge qep. a composite susceptibility V env is given by [24,27,28],…”