Casimir Repulsion between Metallic Objects in Vacuum

We give an example of a geometry in which two metallic objects in vacuum experience a repulsive Casimir force. The geometry consists of an elongated metal particle centered above a metal plate with a hole. We prove that this geometry has a repulsive regime using a symmetry argument and confirm it with numerical calculations for both perfect and realistic metals. The system does not support stable levitation, as the particle is unstable to displacements away from the symmetry axis.

Figure 1

(a) Schematic geometry achieving Casimir repulsion: an elongated metal particle above a thin metal plate with a hole. The idealized version is the limit of an infinitesimal particle polarizable only in the $z$ direction. (b) At $z=0$, vacuum-dipole field lines are perpendicular to the plate by symmetry, and so dipole fluctuations are unaffected by the plate (for any $\omega$, shown here for $\omega =0$). (c) Schematic particle-plate interaction energy $U(z)-U(\infty )$: zero at $z=0$ and at $z\to \infty$, and attractive for $z\gg W$, so there must be Casimir repulsion (negative slope) close to the plate.

Figure 2

(a) Schematic electrostatic interaction of a dipole with a neutral perforated plate (side view), depicting the charge density $\sigma$ induced on the plate. Since $\sigma$ is positive for small $r$ and negative for large $r$, $\sigma$ can be constructed out of a superposition of dipoles in the $z=0$ plane, oriented radially inward about the $z$ axis. A simple calculation then shows that the interaction is repulsive for small $z$. (b) In contrast, a dipole oriented parallel to the plate is always attracted, as one can see from the induced charge density shown above.

Figure 3

Exact Casimir force for cylinder-plate geometry (inset) for perfect metals (computed with BEM) and gold (computed with FDTD); positive (shaded) force is repulsive. For $d\lesssim 300\mathrm{nm}$ and $d>170\mathrm{nm}$ (vertical dashed line), the force is unambiguously repulsive as the cylinder is entirely above the plate. In contrast, a perfect-metal sphere (diameter 60 nm) is always attracted to the plate.

Figure 4

Exact (BEM) $2d$ Casimir force for ellipse-line geometry (lower inset) with perfect metals and TE polarization (in-plane electric field); positive force (shaded) is repulsive. The effects of both particle width $a_y$ and line thickness $t$ are shown, for fixed $a_z=0.002W$. As the ellipticity $\gamma =a_z/a_y$ decreases or $t$ increases, the repulsive force diminishes. Upper-left inset: frequency-resolved force $F(\mathrm{Im}\omega )$ at fixed separation $d=0.2W$ and $t=0^{+}$: as $\gamma$ decreases, attractive contributions arise from small $\mathrm{Im}\omega$. Upper-right inset: $F(\mathrm{Im}\omega )$ at fixed $d=0.2W$ and $\gamma =4$: as $t$ increases, attractive contributions arise from large $\mathrm{Im}\omega$.