Casimir interaction between a plate and a cylinder

We find the exact Casimir force between a plate and a cylinder, a geometry intermediate between parallel plates, where the force is known exactly, and the plate--sphere, where it is known at large separations. The force has an unexpectedly weak decay \sim L/(H^3 \ln(H/R)) at large plate--cylinder separations H (L and R are the cylinder length and radius), due to transverse magnetic modes. Path integral quantization with a partial wave expansion additionally gives a qualitative difference for the density of states of electric and magnetic modes, and corrections at finite temperatures.Comment: 4 pages, 3 figure

On the Casimir effect for parallel plates in the spacetime with one extra compactified dimension

In this paper, the Casimir effect for parallel plates in the presence of one compactified universal extra dimension is reexamined in detail. Having regularized the expressions of Casimir force, we show that the nature of Casimir force is repulsive if the distance between the plates is large enough, which is disagree with the experimental phenomena.Comment: 7 pages, 3 figure

Casimir Forces between Compact Objects: I. The Scalar Case

We have developed an exact, general method to compute Casimir interactions between a finite number of compact objects of arbitrary shape and separation. Here, we present details of the method for a scalar field to illustrate our approach in its most simple form; the generalization to electromagnetic fields is outlined in Ref. [1]. The interaction between the objects is attributed to quantum fluctuations of source distributions on their surfaces, which we decompose in terms of multipoles. A functional integral over the effective action of multipoles gives the resulting interaction. Each object's shape and boundary conditions enter the effective action only through its scattering matrix. Their relative positions enter through universal translation matrices that depend only on field type and spatial dimension. The distinction of our method from the pairwise summation of two-body potentials is elucidated in terms of the scattering processes between three objects. To illustrate the power of the technique, we consider Robin boundary conditions $\phi -\lambda \partial_n \phi=0$, which interpolate between Dirichlet and Neumann cases as $\lambda$ is varied. We obtain the interaction between two such spheres analytically in a large separation expansion, and numerically for all separations. The cases of unequal radii and unequal $\lambda$ are studied. We find sign changes in the force as a function of separation in certain ranges of $\lambda$ and see deviations from the proximity force approximation even at short separations, most notably for Neumann boundary conditions.Comment: 27 pages, 9 figure

Mode summation approach to Casimir effect between two objects

In this paper, we explore the TGTG formula from the perspective of mode summation approach. Both scalar fields and electromagnetic fields are considered. In this approach, one has to first solve the equation of motion to find a wave basis for each object. The two T's in the TGTG formula are T-matrices representing the Lippmann-Schwinger T-operators, one for each of the objects. The two G's in the TGTG formula are the translation matrices, relating the wave basis of an object to the wave basis of the other object. After discussing the general theory, we apply the prescription to derive the explicit formulas for the Casimir energies for the sphere-sphere, sphere-plane, cylinder-cylinder and cylinder-plane interactions. First the T-matrices for a plane, a sphere and a cylinder are derived for the following cases: the object is imposed with general Robin boundary conditions; the object is semitransparent; and the object is magnetodielectric. Then the operator approach is used to derive the translation matrices. From these, the explicit TGTG formula for each of the scenarios can be written down. Besides summarizing all the TGTG formulas that have been derived so far, we also provide the TGTG formulas for some scenarios that have not been considered before.Comment: 42 page

Casimir Forces: An Exact Approach for Periodically Deformed Objects

A novel approach for calculating Casimir forces between periodically deformed objects is developed. This approach allows, for the first time, a rigorous non-perturbative treatment of the Casimir effect for disconnected objects beyond Casimir's original two-plate configuration. The approach takes into account the collective nature of fluctuation induced forces, going beyond the commonly used pairwise summation of two-body van der Waals forces. As an application of the method, we exactly calculate the Casimir force due to scalar field fluctuations between a flat and a rectangular corrugated plate. In the latter case, the force is found to be always attractive.Comment: 4 pages, 3 figure

The role of Surface Plasmon modes in the Casimir Effect

In this paper we study the role of surface plasmon modes in the Casimir effect. First we write the Casimir energy as a sum over the modes of a real cavity. We may identify two sorts of modes, two evanescent surface plasmon modes and propagative modes. As one of the surface plasmon modes becomes propagative for some choice of parameters we adopt an adiabatic mode definition where we follow this mode into the propagative sector and count it together with the surface plasmon contribution, calling this contribution "plasmonic". The remaining modes are propagative cavity modes, which we call "photonic". The Casimir energy contains two main contributions, one coming from the plasmonic, the other from the photonic modes. Surprisingly we find that the plasmonic contribution to the Casimir energy becomes repulsive for intermediate and large mirror separations. Alternatively, we discuss the common surface plasmon defintion, which includes only evanescent waves, where this effect is not found. We show that, in contrast to an intuitive expectation, for both definitions the Casimir energy is the sum of two very large contributions which nearly cancel each other. The contribution of surface plasmons to the Casimir energy plays a fundamental role not only at short but also at large distances.Comment: 10 pages, 3 figures. TQMFA200

Modification of energy shifts of atoms by the presence of a boundary in a thermal bath and the Casimir-Polder force

We study the modification by the presence of a plane wall of energy level shifts of two-level atoms which are in multipolar coupling with quantized electromagnetic fields in a thermal bath in a formalism which separates the contributions of thermal fluctuations and radiation reaction and allows a distinct treatment to atoms in the ground and excited states. The position dependent energy shifts give rise to an induced force acting on the atoms. We are able to identify three different regimes where the force shows distinct features and examine, in all regimes, the behaviors of this force in both the low temperature limit and the high temperature limit for both the ground state and excited state atoms, thus providing some physical insights into the atom-wall interaction at finite temperature. In particular, we show that both the magnitude and the direction of the force acting on an atom may have a clear dependence on atomic the polarization directions. In certain cases, a change of relative ratio of polarizations in different directions may result in a change of direction of the force.Comment: 29 pages, 3 figure

Fluctuation induced quantum interactions between compact objects and a plane mirror

The interaction of compact objects with an infinitely extended mirror plane due to quantum fluctuations of a scalar or electromagnetic field that scatters off the objects is studied. The mirror plane is assumed to obey either Dirichlet or Neumann boundary conditions or to be perfectly reflecting. Using the method of images, we generalize a recently developed approach for compact objects in unbounded space [1,2] to show that the Casimir interaction between the objects and the mirror plane can be accurately obtained over a wide range of separations in terms of charge and current fluctuations of the objects and their images. Our general result for the interaction depends only on the scattering matrices of the compact objects. It applies to scalar fields with arbitrary boundary conditions and to the electromagnetic field coupled to dielectric objects. For the experimentally important electromagnetic Casimir interaction between a perfectly conducting sphere and a plane mirror we present the first results that apply at all separations. We obtain both an asymptotic large distance expansion and the two lowest order correction terms to the proximity force approximation. The asymptotic Casimir-Polder potential for an atom and a mirror is generalized to describe the interaction between a dielectric sphere and a mirror, involving higher order multipole polarizabilities that are important at sub-asymptotic distances.Comment: 19 pages, 7 figure

Casimir Energy of a Spherical Shell

The Casimir energy for a conducting spherical shell of radius $a$ is computed using a direct mode summation approach. An essential ingredient is the implementation of a recently proposed method based on Cauchy's theorem for an evaluation of the eigenfrequencies of the system. It is shown, however, that this earlier calculation uses an improper set of modes to describe the waves exterior to the sphere. Upon making the necessary corrections and taking care to ensure that no mathematically ill-defined expressions occur, the technique is shown to leave numerical results unaltered while avoiding a longstanding criticism raised against earlier calculations of the Casimir energy.Comment: LaTeX, 14 pages, 1 figur

Casimir forces between arbitrary compact objects

We develop an exact method for computing the Casimir energy between arbitrary compact objects, either dielectrics or perfect conductors. The energy is obtained as an interaction between multipoles, generated by quantum current fluctuations. The objects' shape and composition enter only through their scattering matrices. The result is exact when all multipoles are included, and converges rapidly. A low frequency expansion yields the energy as a series in the ratio of the objects' size to their separation. As an example, we obtain this series for two dielectric spheres and the full interaction at all separations for perfectly conducting spheres.Comment: 4 pages, 1 figur