Electromagnetic semitransparentδ-function plate: Casimir interaction energy between parallel infinitesimally thin plates

Abstract

We derive boundary conditions for electromagnetic fields on a -function plate. The optical properties of such a plate are shown to necessarily be anisotropic in that they only depend on the transverse properties of the plate. We unambiguously obtain the boundary conditions for a perfectly conducting -function plate in the limit of infinite dielectric response. We show that a material does not ''optically vanish'' in the thin-plate limit. The thin-plate limit of a plasma slab of thickness d with plasma frequency ! 2 p ¼ p =d reduces to a -function plate for frequencies (( 1. We show that the Casimir interaction energy between two parallel perfectly conducting -function plates is the same as that for parallel perfectly conducting slabs. Similarly, we show that the interaction energy between an atom and a perfect electrically conducting -function plate is the usual Casimir-Polder energy, which is verified by considering the thin-plate limit of dielectric slabs. The ''thick'' and ''thin'' boundary conditions considered by Bordag are found to be identical in the sense that they lead to the same electromagnetic fields.

Field theoretic description of electromagnetic boundaries

“…A similar kind of δ-function plates was analyzed in [20]. Also notice that the derivatives in the last term in (1) are taken only in the parallel space to the surface because of the fixed index in the Levi-Civita tensor:…”

“…The results of these references are apparently disparate but, as a matter of fact, since those works are dealing with different kinds of dielectrics, they are not supposed to match in general. In [20] the authors address a broader class of dielectrics, starting off by defining their dielectric permittivity and magnetic permeability that are both directly proportional to a δ-function, in contrast to the relations (2). On the other hand, although they do not identify which kind of material correspond to their boundary conditions, in [22] the authors couple an external potential to the Maxwell action similar to the the one we used in Eq.…”

“…The Casimir energy for soft boundaries has recently been studied in [20,22], by means of different methods. The results of these references are apparently disparate but, as a matter of fact, since those works are dealing with different kinds of dielectrics, they are not supposed to match in general.…”

Dynamical Casimir effect withδ−δ′mirrors

“…(1), have been done by Barone and Barone who proposed an enlarged gauge-invariant Maxwell Lagrangian describing the field in the presence of a δ mirror, investigating the interaction between a static charge and a δ mirror [24], and also the static Casimir effect between two δ mirrors [25]. Parashar, Milton, Shajesh and Schaden [26], in a different approach, considered the electric and magnetic properties of an infinitesimally thin mirror by means of the electric permittivity and magnetic permeability described in terms of δ functions, deriving the boundary conditions for the electromagnetic field on the mirror and investigating the Casimir-Polder interaction between an atom and the mirror. The use of δ − δ ′ potentials (δ ′ is the derivative of the Dirac δ) for simulating partially reflecting mirrors, in the context of the static Casimir effect, was considered by Muñoz-Castañeda and Guilarte [27], resulting in a generalization of (1), done by adding a δ ′ term in the potential, namely…”

Nonanalytic behavior of the Casimir force across a Lifshitz transition in a spin-orbit-coupled material

“…Others have considered the Casimir effect with twodimensional plates [23], however our particular model requires a more material-centered approach (see the supplement [24]). We consider the Casimir force at zero temperature between two parallel plates where at least one is modeled as a two-band spin-orbit coupled material (sufficiently thin to be considered quasi-two dimensional) with a fixed chemical potential and tunable Zeeman splitting due to an external magnetic field.…”