Casimir forces beyond the proximity approximation

The proximity force approximation (PFA) relates the interaction between closely spaced, smoothly curved objects to the force between parallel plates. Precision experiments on Casimir forces necessitate, and spur research on, corrections to the PFA. We use a derivative expansion for gently curved surfaces to derive the leading curvature modifications to the PFA. Our methods apply to any homogeneous and isotropic materials; here we present results for Dirichlet and Neumann boundary conditions and for perfect conductors. A Pad\'e extrapolation constrained by a multipole expansion at large distance and our improved expansion at short distances, provides an accurate expression for the sphere-plate Casimir force at all separations.Comment: 4 pages, 1 figur

Photon density of states for deformed surfaces

A new approach to the Helmholtz spectrum for arbitrarily shaped boundaries and a rather general class of boundary conditions is introduced. We derive the boundary induced change of the density of states in terms of the free Green's function from which we obtain both perturbative and non-perturbative results for the Casimir interaction between deformed surfaces. As an example, we compute the lateral electrodynamic Casimir force between two corrugated surfaces over a wide parameter range. Universal behavior, fixed only by the largest wavelength component of the surface shape, is identified at large surface separations. This complements known short distance expansions which are also reproduced.Comment: 8 pages, J Phys A Special Issue QFEXT0

Casimir force between sharp-shaped conductors

Casimir forces between conductors at the sub-micron scale cannot be ignored in the design and operation of micro-electromechanical (MEM) devices. However, these forces depend non-trivially on geometry, and existing formulae and approximations cannot deal with realistic micro-machinery components with sharp edges and tips. Here, we employ a novel approach to electromagnetic scattering, appropriate to perfect conductors with sharp edges and tips, specifically to wedges and cones. The interaction of these objects with a metal plate (and among themselves) is then computed systematically by a multiple-scattering series. For the wedge, we obtain analytical expressions for the interaction with a plate, as functions of opening angle and tilt, which should provide a particularly useful tool for the design of MEMs. Our result for the Casimir interactions between conducting cones and plates applies directly to the force on the tip of a scanning tunneling probe; the unexpectedly large temperature dependence of the force in these configurations should attract immediate experimental interest

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

The Casimir effect as scattering problem

We show that Casimir-force calculations for a finite number of non-overlapping obstacles can be mapped onto quantum-mechanical billiard-type problems which are characterized by the scattering of a fictitious point particle off the very same obstacles. With the help of a modified Krein trace formula the genuine/finite part of the Casimir energy is determined as the energy-weighted integral over the log-determinant of the multi-scattering matrix of the analog billiard problem. The formalism is self-regulating and inherently shows that the Casimir energy is governed by the infrared end of the multi-scattering phase shifts or spectrum of the fluctuating field. The calculation is exact and in principle applicable for any separation(s) between the obstacles. In practice, it is more suited for large- to medium-range separations. We report especially about the Casimir energy of a fluctuating massless scalar field between two spheres or a sphere and a plate under Dirichlet and Neumann boundary conditions. But the formalism can easily be extended to any number of spheres and/or planes in three or arbitrary dimensions, with a variety of boundary conditions or non-overlapping potentials/non-ideal reflectors.Comment: 14 pages, 2 figures, plenary talk at QFEXT07, Leipzig, September 2007, some typos correcte

Casimir forces between arbitrary compact objects: Scalar and electromagnetic field

We develop an exact method for computing the Casimir energy between arbitrary compact objects, both with boundary conditions for a scalar field and dielectrics or perfect conductors for the electromagnetic field. The energy is obtained as an interaction between multipoles, generated by quantum source or 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 examples, we obtain this series for two spheres with Robin boundary conditions for a scalar field and dielectric spheres for the electromagnetic field. The full interaction at all separations is obtained for spheres with Robin boundary conditions and for perfectly conducting spheres.Comment: 24 pages, 3 figures, contribution to QFEXT07 proceeding

Geothermal Casimir Phenomena

We present first worldline analytical and numerical results for the nontrivial interplay between geometry and temperature dependencies of the Casimir effect. We show that the temperature dependence of the Casimir force can be significantly larger for open geometries (e.g., perpendicular plates) than for closed geometries (e.g., parallel plates). For surface separations in the experimentally relevant range, the thermal correction for the perpendicular-plates configuration exhibits a stronger parameter dependence and exceeds that for parallel plates by an order of magnitude at room temperature. This effect can be attributed to the fact that the fluctuation spectrum for closed geometries is gapped, inhibiting the thermal excitation of modes at low temperatures. By contrast, open geometries support a thermal excitation of the low-lying modes in the gapless spectrum already at low temperatures.Comment: 8 pages, 3 figures, contribution to QFEXT07 proceedings, v2: discussion switched from Casimir energy to Casimir force, new analytical results included, matches JPhysA versio

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

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

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