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

This paper presents a 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.

Figure 1

Enforcing boundary conditions via surface-current Lagrange multipliers. (a) Consider a single point $\mathbf{x}$ on the surface of an object in a Casimir geometry. The boundary conditions at $\mathbf{x}$, which must be satisfied in the constrained path integral (19), are that the tangential components of the $\mathbf{E}$ and $\mathbf{H}$ fields be continuous as we pass from inside to outside the object [Eq. (20)]; here $\{{\mathbf{t}}^1,{\mathbf{t}}^2\}$ are vectors tangent to the surface at $\mathbf{x}$. (b) We rewrite the boundary conditions in terms of differential operators ${\mathbf{L}}^{\mathrm{E},\mathrm{M}}$ operating on the $\mathcal{A}$ field, and we introduce Lagrange multipliers $\{K^1,K^2,N^1,N^2\}$ to enforce the boundary conditions at $\mathbf{x}$; specifically, $K^1,K^2$ enforce the tangential $\mathbf{E}$-field continuity at $\mathbf{x}$, while $N^1,N^2$ enforce the tangential $\mathbf{H}$-field continuity [Eq. (23)]. (c) Repeating this procedure for all points on the object surface, we obtain Lagrange multiplier fields $\mathbf{K}\left(\mathbf{x}\right),\mathbf{N}\left(\mathbf{x}\right)$, which have an obvious interpretation as the electric and magnetic surface currents of Fig. 4. Integrating the photon field out of the path integral then yields an effective action describing the interactions of these surface currents [Eqs. (28), (41), and (42)], leading ultimately to our FSC formulas for the Casimir energy.

Figure 2

Casimir force between PEC spheres and between PEC cubes. The solid green circles indicate sphere-sphere data computed using the FSC method described in this paper, while the open blue circles indicate sphere-sphere data computed using a numerical implementation of the scattering-matrix method of Ref. [44]. (The lower dashed-line curve indicates the first four terms of the asymptotic series for the sphere-sphere Casimir force reported in Ref. [44].) The solid red squares indicate cube-cube data computed using the FSC method described in this paper; for this geometry, scattering-matrix methods and indeed almost all existing Casimir methods would be unwieldy or impossible to apply. The upper dashed-line curve indicates the proximity-force approximation (PFA) to the cube-cube force, $F_{\mathrm{PFA}}=4R^2\frac{{\pi }^2\hslash c}{240L^4}$ for cubes of face area $4R^2$.

Figure 3

Casimir force on an elongated cylindrical nanoparticle above a square aperture in a thin plate. The upper inset (not to scale) shows the particle-plate geometry, while the lower inset is a close-up view of the surface mesh used to represent the particle. The axis of the nanoparticle (the $z$ axis) is perpendicular to the plane of the plate; the center of the nanoparticle lies a distance $Z$ above the center of the plate. In the regime $0\le Z\le 330$ nm, the $z$-directed force on the nanoparticle is positive; i.e., the particle is repelled from the plate. The shaded portion of the graph indicates this repulsive-force regime. For $Z>185$ the nanoparticle lies entirely above the plate and the force is thus unambiguously repulsive. Both the nanoparticle and the plate are made of finite-conductivity gold (see text).

Figure 4

Schematic depiction of a scattering geometry in the SIE picture. A collection of arbitrarily shaped homogeneous bodies, with frequency-dependent relative electrical properties $\{{\epsilon }^r,{\mu }^r\}$, is embedded in a homogeneous medium with electrical properties $\{{\epsilon }^e,{\mu }^e\}$. Incident radiation, characterized by electric and magnetic fields ${\mathbf{E}}^{\mathrm{inc}},{\mathbf{H}}^{\mathrm{inc}}$, impinges on the objects to induce surface currents; for perfectly conducting objects we have only electric surface currents ($\mathbf{K}$), while for general objects we have equivalent electric and magnetic ($\mathbf{N}$) surface currents. The goal of SIE methods is to solve for the surface-current distributions in terms of the incident fields, after which we can compute the scattered fields anywhere in space from the surface currents. [The dotted line indicates a fictitious bounding contour $\mathcal{C}$ surrounding one of the objects over which we integrate the Maxwell stress tensor to compute the Casimir force on that object (Sec. 2).]

Figure 5

One possible choice of expansion functions for tangential currents on object surfaces is obtained by discretizing object boundaries into unions of small flat panels and introducing localized basis functions describing elemental currents sourced and sunk at panel vertices: These are “RWG'' basis functions, associated with each edge (pair of triangles) in the mesh [64].

Figure 6

Schematic summary of the integral identity (C5) proved in Appendix  . We consider a closed bounding surface $\mathcal{C}$ and choose two points $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$, which may lie both inside (left panel), both outside (center panel), or on opposite sides (right panel) of $\mathcal{C}.$ We write an expression [Eq. (C4)] that involves products of DGFs, one connecting $\mathbf{r}$ to a point $\mathbf{x}$ on $\mathcal{C}$ and a second connecting $\mathbf{x}$ to ${\mathbf{r}}^{\prime }.$ Then we evaluate the surface integral of this expression as $\mathbf{x}$ ranges over all of $\mathcal{C}$. The answer we obtain depends on the relative positioning of $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$ with respect to $\mathcal{C}$: If the two points lie both inside or both outside $\mathcal{C}$, the surface integral vanishes, while if the two points lie on opposite sides of $\mathcal{C}$ then the surface integral yields the derivative of a DGF connecting $\mathbf{r}$ to ${\mathbf{r}}^{\prime }$.