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

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 pistonlike geometry of two squares sliding between metal walls, in both two and three dimensions with both perfect and realistic metallic materials, exhibits a surprising nonmonotonic “lateral” force from the walls.

Figure 1

(Color online) Schematic of one-dimensional geometry, showing two 1D metal plates separated by a distance $a$, embedded in a computational cell of length $L=a+4a$, with periodic boundary conditions, and resolution $\Delta x$.

Figure 2

(Color online) Top: Plot of force summand, or spectral density, $\hslash \Delta \omega ∕2\Delta x$ vs $\omega$ for 1D parallel metal plates from Fig. 1. Bottom: Plot of force partial sum ${\sum }^{\omega }\hslash \Delta \omega ∕2\Delta x$ vs $\omega$.

Figure 3

(Color online) Plot of Casimir force integrand $dU∕da$ between two 1D parallel plates separated by a distance $a=1$ vs imaginary frequency $w=\mathrm{Im}\omega$, using the method of Sec. 3.

Figure 4

(Color online) Schematic illustration of a possible contour around a body; the force on the body is given by an integral of the stress tensor around this contour.

Figure 5

(Color online) Plot of Casimir force integrand $∯T\bullet d\mathbf{S}$ between two 1D parallel plates separated by a distance $a=1$ vs imaginary frequency $w=\mathrm{Im}\omega$, using the stress-tensor method of Sec. 4.

Figure 6

(Color online) Casimir force between a 3D radius-$R$ cylinder and a plate (inset), relative to the proximity-force approximation $F_{\mathrm{PFA}}$, vs normalized separation $a∕R$. The solid lines are the Casimir force computed in Ref. 57 for TE (gray) and TM (blue) polarizations, along with results computed by our method with a simple finite-difference discretization (gray squares). Error bars were estimated for some data points by using computations at multiple spatial resolutions. Inset shows interaction stress tensor $\Delta ⟨T_{xx}⟩$ at a typical imaginary frequency $w=2\pi c∕a$, where red indicates attractive stress.

Figure 7

(Color online) Casimir force between 2D ($z$-invariant fields) metal squares $F∕F_{\mathrm{PFA}}$, vs distance from metal plate $h$ (inset), normalized by the total force $(\mathrm{TE}+\mathrm{TM})$ obtained using the PFA, $F_{\mathrm{PFA}}=\hslash c\zeta \left(3\right)s∕8\pi a^3$. The total force is plotted (black squares) along with the TE (red dots) and TM (blue circles) contributions.

Figure 8

(Color online) [(a)–(f)] TM stress map of the geometry in Fig. 7 for various $h$. The interaction stress tensors $⟨T_{xx}⟩$ (left) and $⟨T_{xy}⟩$ (right) for: [(a) and (d)] $h=0.5a$; [(b) and (e)] $h=a$; and [(c) and (f)] $h=2a$, where blue/white/$\text{red}=\text{repulsive}$/zero/attractive.

Figure 9

(Color online) Casimir force between 2D ($z$-invariant fields) dielectric $(\epsilon =4)$ squares $F∕F_{\mathrm{PFA}}$, vs distance from metal plate $h$ (inset), normalized by the total force $(\mathrm{TE}+\mathrm{TM})$ obtained using the PFA. (Here, the PFA force is computed for $x$-infinite slabs of dielectric $\epsilon =4$.) The total force is plotted (black squares) along with the TE (red dots) and TM (blue circles) contributions.

Figure 10

(Color online) Casimir force between 2D ($z$-invariant fields) gold squares $F∕F_{\mathrm{PFA}}$, vs distance from metal plate $h$ (inset), normalized by the total force $(\mathrm{TE}+\mathrm{TM})$ obtained using the PFA. (Here, the PFA force is computed for $x$-infinite gold slabs.) The total force is plotted (black squares) along with the TE (red dots) and TM (blue circles) contributions.

Figure 11

(Color online) Casimir force per unit length between $z$-invariant 3D metal blocks $F∕F_{\mathrm{PFA}}$, vs distance from metal plate $h$ (inset), normalized by the total force $(\mathrm{TE}+\mathrm{TM})$ obtained using the PFA, $F_{\mathrm{PFA}}=\hslash cs{\pi }^2∕480a^4$. The total force is plotted (black squares) along with the TE (red dots) and TM (blue circles) contributions.