Geometrical dependence in Casimir-Polder repulsion: Anisotropically polarizable atom and anisotropically polarizable annular dielectric

Casimir-Polder interaction energies between a point anisotropically polarizable atom and an annular dielectric are shown to exhibit localized repulsive long-range forces in specific configurations. We show that when the atom is positioned at the center of the annular dielectric, it is energetically favorable for the atom to align its polarizability with respect to that of the dielectric. As the atom moves away from, but along the symmetry axis of the annular dielectric, it encounters a point where the polarizable atom experiences no torque and the energy is free of orientation dependence. At this height, abruptly, the atom prefers to orient its polarizability perpendicular to that of the dielectric. For certain configurations, it encounters another torsion-free point a larger distance away, beyond which it prefers to again point its polarizability with respect to that of the dielectric. We find when the atom is close enough, and oriented such that the energy is close to maximum, the atom could be repelled. For certain annular polarizations, repulsion can happen below or above the torsion-free height. Qualitative features differ when the atom is interacting with a ring, versus a plate of infinite extent with a hole. In particular, the atom can prefer to orient perpendicular to the polarizability of the plate at large distances, in striking contrast to the expectation that it will orient parallel. To gain insight of this discrepancy, we investigate an annular disc, which captures the results of both geometries in limiting cases. These energies are too weak for immediate applications, nevertheless, we elaborate an interesting application on a prototype of a Casimir machine using these configurations.