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

Test of Replica Theory: Thermodynamics of 2D Model Systems with Quenched Disorder

We study the statistics of thermodynamic quantities in two related systems with quenched disorder: A (1+1)-dimensional planar lattice of elastic lines in a random potential and the 2-dimensional random bond dimer model. The first system is examined by a replica-symmetric Bethe ansatz (RBA) while the latter is studied numerically by a polynomial algorithm which circumvents slow glassy dynamics. We establish a mapping of the two models which allows for a detailed comparison of RBA predictions and simulations. Over a wide range of disorder strength, the effective lattice stiffness and cumulants of various thermodynamic quantities in both approaches are found to agree excellently. Our comparison provides, for the first time, a detailed quantitative confirmation of the replica approach and renders the planar line lattice a unique testing ground for concepts in random systems.Comment: 16 pages, 14 figure

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

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

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 cylinders and plates

We study collective interaction effects that result from the change of free quantum electrodynamic field fluctuations by one- and two-dimensional perfect metal structures. The Casimir interactions in geometries containing plates and cylinders is explicitly computed using partial wave expansions of constrained path integrals. We generalize previously obtained results and provide a more detailed description of the technical aspects of the approach \cite{Emig06}. We find that the interactions involving cylinders have a weak logarithmic dependence on the cylinder radius, reflecting that one-dimensional perturbations are marginally relevant in 4D space-time. For geometries containing two cylinders and one or two plates, we confirm a previously found non-monotonic dependence of the interaction on the object's separations which does not follow from pair-wise summation of two-body forces. Qualitatively, this effect is explained in terms of fluctuating charges and currents and their mirror images

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

Geometry and material effects in Casimir physics - Scattering theory

We give a comprehensive presentation of methods for calculating the Casimir force to arbitrary accuracy, for any number of objects, arbitrary shapes, susceptibility functions, and separations. The technique is applicable to objects immersed in media other than vacuum, to nonzero temperatures, and to spatial arrangements in which one object is enclosed in another. Our method combines each object's classical electromagnetic scattering amplitude with universal translation matrices, which convert between the bases used to calculate scattering for each object, but are otherwise independent of the details of the individual objects. This approach, which combines methods of statistical physics and scattering theory, is well suited to analyze many diverse phenomena. We illustrate its power and versatility by a number of examples, which show how the interplay of geometry and material properties helps to understand and control Casimir forces. We also examine whether electrodynamic Casimir forces can lead to stable levitation. Neglecting permeabilities, we prove that any equilibrium position of objects subject to such forces is unstable if the permittivities of all objects are higher or lower than that of the enveloping medium; the former being the generic case for ordinary materials in vacuum.Comment: 44 pages, 11 figures, to appear in upcoming Lecture Notes in Physics volume in Casimir physic

Casimir effect in the boundary state formalism

Casimir effect in the planar setting is described using the boundary state formalism, for general partially reflecting boundaries. It is expressed in terms of the low-energy degrees of freedom, which provides a large distance expansion valid for general interacting field theories provided there is a non-vanishing mass gap. The expansion is written in terms of the scattering amplitudes, and needs no ultraviolet renormalization. We also discuss the case when the quantum field has a nontrivial vacuum configuration.Comment: 11 pages. Proceedings contribution of talk given at the Workshop on Quantum Field Theory under the Influence of External Conditions (QFEXT07), University of Leipzig, September 16-21, 2007. To appear in J. Phys.

Parity violating cylindrical shell in the framework of QED

We present calculations of Casimir energy (CE) in a system of quantized electromagnetic (EM) field interacting with an infinite circular cylindrical shell (which we call `the defect'). Interaction is described in the only QFT-consistent way by Chern-Simon action concentrated on the defect, with a single coupling constant $a$. For regularization of UV divergencies of the theory we use % physically motivated Pauli-Villars regularization of the free EM action. The divergencies are extracted as a polynomial in regularization mass $M$, and they renormalize classical part of the surface action. We reveal the dependence of CE on the coupling constant $a$. Corresponding Casimir force is attractive for all values of $a$. For $a\to\infty$ we reproduce the known results for CE for perfectly conducting cylindrical shell first obtained by DeRaad and Milton.Comment: Typos corrected. Some references adde