Casimir-Polder repulsion near edges: Wedge apex and a screen with an aperture

Although repulsive effects have been predicted for quantum vacuum forces between bodies with nontrivial electromagnetic properties, such as between a perfect electric conductor and a perfect magnetic conductor, realistic repulsion seems difficult to achieve. Repulsion is possible if the medium between the bodies has a permittivity in value intermediate to those of the two bodies, but this may not be a useful configuration. Here, inspired by recent numerical work, we initiate analytic calculations of the Casimir-Polder interaction between an atom with anisotropic polarizability and a plate with an aperture. In particular, for a semi-infinite plate, and, more generally, for a wedge, the problem is exactly solvable, and for sufficiently large anisotropy, Casimir-Polder repulsion is indeed possible, in agreement with the previous numerical studies. In order to achieve repulsion, what is needed is a sufficiently sharp edge (not so very sharp, in fact) so that the directions of polarizability of the conductor and the atom are roughly normal to each other. The machinery for carrying out the calculation with a finite aperture is presented. As a motivation for the quantum calculation, we carry out the corresponding classical analysis for the force between a dipole and a metallic sheet with a circular aperture, when the dipole is on the symmetry axis and oriented in the same direction.

Figure 1

Two-dimensional geometry of a needle of length $L$ a distance $Z$ above a line with a gap of width $a$.

Figure 2

Three-dimensional geometry of a polarizable atom a distance $Z$ above a dielectric slab with a circular aperture of radius $a$.

Figure 3

Configuration of three dipoles, two of which are antiparallel, and one perpendicular to the other two.

Figure 4

Polarizable atom, located at polar coordinates $\rho$, $\theta$, within a conducting wedge with dihedral angle $\Omega$.

Figure 5

Polarizable atom, above a half conducting plane, free to move on a line perpendicular to the plane but a distance $X$ to the left of the plane.

Figure 6

The $z$ component of the force between an anisotropic atom (with ratio of transverse to longitudinal polarizabilities $\gamma$) and a semi-infinite perfectly conducting plane, $z=0$, $x>0$. $F_z=-{\alpha }_{\mathit{zz}}/(32\pi X^5)f(u)$ in terms of the variable $u=X/\rho =-\cos \theta$. Here the atom lies on the line $y=0$, $x=-X$, and $\rho$ is the distance from the edge of the plane and the atom. Here, $f>0$ corresponds to an attractive force on the $z$ direction, and $f<0$ corresponds to a repulsive force. The different curves correspond to different values of $\gamma$, $\gamma =0$ to 1 by steps of 0.1, from bottom to top. For $\gamma <1/4$ a repulsive regime always occurs when the atom is sufficiently close to the plane of the conductor.

Figure 7

Same as Fig. 6. The region close to the plane, $1⩾u⩾0.99$, with $\gamma$ near the critical value of 1/4. Here from bottom to top are shown the results for values of $\gamma$ from 0.245 to 0.255 by steps of 0.001.

Figure 8

A polarizable atom outside a perfectly conducting wedge of interior angle $\beta$. The atom is located at polar angles $\rho$, $\varphi$ relative to the symmetry plane of the wedge.

Figure 9

The $z$ component of the force on a completely anisotropic atom moving on a line perpendicular to a wedge. The different curves are for various values of $\beta$ from 0 to $\pi$ by steps of $\pi /20$, from bottom up. The last few values of $\beta$ have a markedly different character from the others.