K. V. Shajesh

The Casimir-Polder interaction energy between a unipolarizable point atom and a unipolarizable dielectric ring has been limited, until now, to the case when the atom is confined on the axis of symmetry of the ring. We find the generalized analytical expression for any position of the atom relative to the ring in terms of complete elliptic integrals. This is aided by the construction of a class of integrals of a Jacobian elliptic function as a linear combination of complete elliptic integrals. Our expression for the interaction energy allows us to investigate the instability of the atom even for the equilibrium points that exist off the axis of symmetry.

The Casimir-Polder interaction energy between a unipolarizable point atom and a unipolarizable dielectric ring has been limited, until now, to the case when the atom is confined on the axis of symmetry of the ring. We find the generalized analytical expression for any position of the atom relative to the ring in terms of complete elliptic integrals. This is aided by the construction of a class of integrals of a Jacobian elliptic function as a linear combination of complete elliptic integrals. Our expression for the interaction energy allows us to investigate the instability of the atom even for the equilibrium points which exists off the axis of symmetry.

We investigate the implications of a fundamental length scale on the centripetal force on a rotating Casimir apparatus in $κ$-space-time. We model the Casimir apparatus rotating with constant angular speed using appropriate $κ$-deformed coordinates. We find the $κ$-deformed centripetal force on a single plate, as well as for parallel plates. We show that the Casimir energy, including the divergent part (self energies of the plates) experiences centripetal forces like a conventional mass. We also find centripetal force on oriented parallel plates rotating with constant angular speed in $κ$-space-time. Results show that the mass-energy equivalence principle holds in the $κ$-space-time.

We investigate the response of Casimir energies to fluctuations in a scalar field in a weak gravitational field in the $κ$-deformed space-time. We model the Casimir plates in a gravitational field by $κ$-deformed Rindler coordinates and calculate the Casimir energy using the $κ$-deformed scalar field. We show that the Casimir energy accelerates in a weak gravitational field like a mass. Thus, our calculations show that the mass-energy equivalence principle holds in $κ$-deformed space-time even though a length scale is introduced through space-time non-commutativity.

Repulsion, induced from quantum vacuum fluctuations, for an anisotropically polarizable atom on the symmetry axis of an anisotropically polarizable annular disk is studied. There exist two torsion-free points on each side of the annular disk, where the interaction energy is orientation independent. The position of the second of the two torsion-free points, on either side of the disk, is shown to determine the orientation dependence of the atom at distances far from the plate, revealing a geometrical dependence. In the ring limit of the annular disk, an additional region of repulsion emerges.

We offer in this review a description of the vacuum energy of self-similar systems. We describe two views of setting self-similar structures and point out the main differences. A review of the authors’ work on the subject is presented, where they treat the self-similar system as a many-object problem embedded in a regular smooth manifold. Focused on Dirichlet boundary conditions, we report a systematic way of calculating the Casimir energy of self-similar bodies where the knowledge of the quantum vacuum energy of the single building block element is assumed and in fact already known. A fundamental property that allows us to proceed with our method is the dependence of the energy on a geometrical parameter that makes it possible to establish the scaling property of self-similar systems. Several examples are given. We also describe the situation, shown by other authors, where the embedded space is a fractal space itself, having fractal dimension. A fractal space does not hold properties that are rather common in regular spaces like the tangent space. We refer to other authors who explain how some self-similar configurations “do not have any smooth structures and one cannot define differential operators on them directly”. This gives rise to important differences in the behavior of the vacuum.

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.

Repulsion, induced from quantum vacuum fluctuations, for an anisotropically polarizable atom on the symmetry axis of an anisotropically polarizable annular disc is studied. There exists two torsion free points on each side of the annular disc, where the interaction energy is orientation independent. The position of second of the two torsion free points, on either side of the disc, is shown to determine the orientation dependence of the atom at distances far from the plate, revealing a geometrical dependence. In the ring limit of the annular disc, new repulsion emerges.