Casimir Interaction Energy Research Articles

Casimir Interactions from Infinite Range and Dilation Symmetry

The Casimir interaction energy for a class of discrete self-similar configuration of parallel plates is evaluated using existing methods. The similarities to characteristics of an attractive Casimir force is deduced only at infinite range of configuration. Further, the emergence of Casimir-like energy is qualitatively described for a Gaussian model of Landau-Ginzburg scalar field. Its relevance to self-similarity in the statistical field is shown at infinite range of fluctuations.

Casimir Interaction between a Plane and a Sphere: Correction to the Proximity-Force Approximation at Intermediate Temperatures

We consider the Casimir interaction energy between a plane and a sphere of radius R at finite temperature T as a function of the distance of closest approach L. Typical experimental conditions are such that the thermal wavelength λT=ℏc/kBT satisfies the condition L≪λT≪R. We derive the leading correction to the proximity-force approximation valid for such intermediate temperatures by developing the scattering formula in the plane-wave basis. Our analytical result captures the joint effect of the spherical geometry and temperature and is written as a sum of temperature-dependent logarithmic terms. Surprisingly, two of the logarithmic terms arise from the Matsubara zero-frequency contribution.

Casimir energy due to inhomogeneous thin plates

We study the Casimir energy due to a quantum real scalar field coupled to two planar, infinite, zero-width, parallel mirrors with non-homogeneous properties. These properties are represented, in the model we use, by scalar functions defined on each mirror's plane. Using the Gelfand-Yaglom's theorem, we construct a Lifshitz-like formula for the Casimir energy of such a system. Then we use it to evaluate the energy perturbatively, for the case of almost constant scalar functions, and also implementing a Derivative Expansion, under the assumption that the spatial dependence of the properties is sufficiently smooth. We point out that, in some particular cases, the Casimir interaction energy for non-planar perfect mirrors can be reproduced by inhomogeneities on planar mirrors.

Scattering theory of the screened Casimir interaction in electrolytes

We apply the scattering approach to the Casimir interaction between two dielectric half-spaces separated by an electrolyte solution. We take the nonlocal electromagnetic response of the intervening medium into account, which results from the presence of movable ions in solution. In addition to the usual transverse modes, we consider longitudinal channels and their coupling by reflection at the surface of the local dielectric. The Casimir interaction energy is calculated from the matrix describing a round-trip of coupled transverse and longitudinal waves between the interacting surfaces. The nonzero-frequency contributions are approximately unaffected by the presence of ions. We find, at zero frequency, a contribution from longitudinal channels, which is screened over a distance of the order of the Debye length, alongside an unscreened term arising from transverse-magnetic modes. The latter defines the long-distance asymptotic limit for the interaction.

Electromagnetic Casimir effect of concentric cylinders in (D + 1)-dimensional space–time

We study the electromagnetic Casimir interaction energy between two parallel concentric cylinders in [Formula: see text]-dimensional Minkowski space–time for different combinations of perfectly conducting boundary condition and infinitely permeable boundary condition. We consider two cases where one cylinder is outside each other and where one is inside the other. By solving the equation of motion and computing the TGTG formulas, explicit formulas for the Casimir interaction energy can be derived and asymptotic behavior of the Casimir interaction energy in the nanoregime is calculated by using perturbation technique. We computed the interaction energy analytically up to next-to-leading order term.

Electromagnetic δ -function sphere

We develop a formalism to extend our previous work on the electromagnetic $\delta$-function plates to a spherical surface. The electric ($\lambda_e$) and magnetic ($\lambda_g$) couplings to the surface are through $\delta$-function potentials defining the dielectric permittivity and the diamagnetic permeability, with two anisotropic coupling tensors. The formalism incorporates dispersion. The electromagnetic Green's dyadic breaks up into transverse electric and transverse magnetic parts. We derive the Casimir interaction energy between two concentric $\delta$-function spheres in this formalism and show that it has the correct asymptotic flat plate limit. We systematically derive expressions for the Casimir self-energy and the total stress on a spherical shell using a $\delta$-function potential, properly regulated by temporal and spatial point-splitting, which are different from the conventional temporal point-splitting. In strong coupling, we recover the usual result for the perfectly conducting spherical shell but in addition, there is an integrated curvature-squared divergent contribution. For finite coupling, there are additional divergent contributions; in particular, there is a familiar logarithmic divergence occurring in the third order of the uniform asymptotic expansion that renders it impossible to extract a unique finite energy except in the case of an isorefractive sphere, which translates into $\lambda_g=-\lambda_e$.

On the static Casimir effect with parity-breaking mirrors

We study the Casimir interaction energy due to the vacuum fluctuations of the electromagnetic (EM) field in the presence of two mirrors, described by 2+1-dimensional, generally nonlocal actions, which may contain both parity-conserving and parity-breaking terms. We compare the results with the ones corresponding to Chern–Simons boundary conditions and evaluate the interaction energy for several particular situations.

Electromagnetic Casimir energy of a disk opposite a plane

Building on work of Meixner [J. Meixner, Z. Naturforschung 3a, 506 (1948)], we show how to compute the exact scattering amplitude (or $T$-matrix) for electromagnetic scattering from a perfectly conducting disk. This calculation is a rare example of a non-diagonal $T$-matrix that can nonetheless be obtained in a semi-analytic form. We then use this result to compute the electromagnetic Casimir interaction energy for a disk opposite a plane, for arbitrary orientation angle of the disk, for separations greater than the disk radius. We find that the proximity force approximation (PFA) significantly overestimates the Casimir energy, both in the case of the ordinary PFA, which applies when the disk is parallel to the plane, and the "edge PFA," which applies when the disk is perpendicular to the plane.

Casimir energies of self-similar plate configurations

We construct various self-similar configurations using parallel $\delta$-function plates and show that it is possible to evaluate the Casimir interaction energy of these configurations using the idea of self-similarity alone. We restrict our analysis to interactions mediated by a scalar field, but the extension to electromagnetic field is immediate. Our work unveils an easy and powerful method that can be easily employed to calculate the Casimir energies of a class of self-similar configurations. As a highlight, in an example, we determine the Casimir interaction energy of a stack of parallel plates constructed by positioning $\delta$-function plates at the points constituting the Cantor set, a prototype of a fractal. This, to our knowledge, is the first time that the Casimir energy of a fractal configuration has been reported. Remarkably, the Casimir energy of some of the configurations we consider turn out to be positive, and a few even have zero Casimir energy. For the case of positive Casimir energy that is monotonically decreasing as the stacking parameter increases the interpretation is that the pressure of vacuum tends to inflate the infinite stack of plates. We further support our results, derived using the idea of self-similarity alone, by rederiving them using the Green's function formalism. These expositions gives us insight into the connections between the regularization methods used in quantum field theories and regularized sums of divergent series in number theory.

Domain wall interactions due to vacuum Dirac field fluctuations in2+1dimensions

We evaluate quantum effects due to a $2$-component Dirac field in $2+1$ space-time dimensions, coupled to domain-wall like defects with a smooth shape. We show that those effects induce non trivial contributions to the (shape-dependent) energy of the domain walls. For a single defect, we study the divergences in the corresponding self-energy, and also consider the role of the massless zero mode, corresponding to the Callan-Harvey mechanism, by coupling the Dirac field to an external gauge field. For two defects, we show that the Dirac field induces a non trivial, Casimir-like effect between them, and provide an exact expression for that interaction in the case of two straight-line parallel defects. As is the case for the Casimir interaction energy, the result is finite and unambiguous.

Casimir energy between a sinusoidally corrugated sphere and a plate using proximity force approximation

The aim of this paper is to obtain the Casimir energy between a sinusoidally corrugated sphere and a plate. We first present a derivation of the sphere-plate Casimir force obtained by applying proximity force approximation, and afterwards, we obtain the effect of deformation on the Casimir interaction energy by considering the sphere to have different kinds of sinusoidal corrugations. We suppose a/r ≤ 0.00755 for the sake of proximity force approximation validity and an experimental accuracy goal of 1 % in the case of a sphere with the radius r at a minimal distance a from a plate. The effect of finite conductivity is taken into account for such short distances. Furthermore, we consider that mirrors are not perfect reflectors indeed which concludes in exact theoretical prediction. Thus, we consider the correction due to finite conductivity. We note that for open geometries, temperature dependence of the Casimir force is larger than that of closed geometries. We also investigate thermal corrections of the Casimir force.

Scalar cylinder-plate and cylinder-cylinder Casimir interaction in higher dimensional spacetime

We study the cylinder-plate and the cylinder-cylinder Casimir interaction in the $(D+1)$-dimensional Minkowski spacetime due to the vacuum fluctuations of massless scalar fields. Different combinations of Dirichlet (D) and Neumann (N) boundary conditions are imposed on the two interacting objects. For the cylinder-cylinder interaction, we consider the case where one cylinder is inside the other, and the case where the two cylinders are outside each other. By computing the transition matrices of the objects and the translation matrices that relate different coordinate systems, the explicit formulas for the Casimir interaction energies are derived. Using perturbation technique, we compute the small separation asymptotic expansions of the Casimir interaction energies up to the next-to-leading order terms. The leading terms coincide with the respective results obtained using proximity force approximation, which is of order $d^{-D+1/2}$, where $d$ is the distance between the two objects. The results on the next-to-leading order terms are more interesting and important. We find some universal behaviors. It is also noticed that for the case of Dirichlet-Dirichlet cylinder-plate interaction, the next-to-leading order term agrees with that obtained using derivative expansion. Hence, based on our results on other boundary conditions and on the cylinder-cylinder interaction, we postulate a formula for the derivative expansion to expand the Casimir interaction energy up to the next-to-leading order terms for DD, DN, ND and NN boundary conditions, for the interaction between two curved surfaces in $(D+1)$-dimensional Minkowski spacetime. It is found that the postulate agrees with our previous results on the sphere-sphere interactions except when $D=4$.

Derivative expansion for the electromagnetic and Neumann-Casimir effects in2+1dimensions with imperfect mirrors

We calculate the Casimir interaction energy in $d=2$ spatial dimensions between two (zero-width) mirrors---one flat and the other slightly curved---upon which imperfect conductor boundary conditions are imposed for an electromagnetic (EM) field. Our main result is a second-order derivative expansion (DE) approximation for the Casimir energy, which is studied in different interesting limits. In particular, we focus on the emergence of a nonanalyticity beyond the leading-order term in the DE, when approaching the limit of perfectly conducting mirrors. We also show that the system considered is equivalent to a dual one, consisting of a massless real scalar field satisfying imperfect Neumann conditions (on the very same boundaries). Therefore, the results obtained for the EM field hold true also for the scalar field model.

Ferminoic Casimir effect between spheres

We consider the Casimir interaction between two spheres corresponding to massless Dirac fields with MIT-bag boundary conditions. Using operator approach, we derive the TGTG-formula for the Casimir interaction energy between the two spheres. A byproduct is the explicit formula for the translation matrix that relates the fermionic spherical waves in different coordinate systems. In the large separation limit, it is found that the order of the Casimir interaction energy is $L^{-5}$, where $L$ is the separation between the centers of the spheres. This order is intermediate between that of two Dirichlet spheres (of order $L^{-3}$) and two Neumann spheres (of order $L^{-7}$). In the small separation limit, we derive analytically the asymptotic expansion of the Casimir interaction energy up to the next-to-leading order term. The leading term agrees with the proximity force approximation. The result for the next-to-leading order term is compared to the corresponding results for scalar fields and electromagnetic fields.

A microscopic approach to Casimir and Casimir–Polder forces between metallic bodies

We consider the Casimir–Polder interaction energy between a metallic nanoparticle and a metallic plate, as well as the Casimir interaction energy between two macroscopic metal plates, in terms of the many-body dispersion interactions between their constituents. Expressions for two- and three-body dispersion interactions between the microscopic parts of a real metal are first obtained, both in the retarded and non-retarded limits. These expressions are then used to evaluate the overall two- and three-body contributions to the macroscopic Casimir–Polder and Casimir force, and to compare them with each other, for the two following geometries: metal nanoparticle/half-space and half-space/half-space, where all the materials are assumed perfect conductors. The above evaluation is obtained by summing up the contributions from the microscopic constituents of the bodies (metal nanoparticles). In the case of nanoparticle/half-space, our results fully agree with those that can be extracted from the corresponding macroscopic results, and explicitly show the non-applicability of the pairwise approximation for the geometry considered. In both cases, we find that, while the overall two-body contribution yields an attractive force, the overall three-body contribution is repulsive. Also, they turn out to be of the same order, consistently with the known non applicability of the pairwise approximation. The issue of the rapidity of convergence of the many-body expansion is also briefly discussed.

Casimir energies of periodic dielectric gratings

Reflection of electromagnetic waves from a periodic grating can be described in terms of a discrete coupled multichannel scattering problem. By modeling the grating as a space- and frequency-dependent dielectric, it is possible to use a variable phase method, applied to a generalized Helmholtz equation incorporating both transverse and longitudinal modes, to efficiently compute the scattering $S$ matrix. The projection onto transverse modes of this result, evaluated for imaginary wave vector, provides the information necessary for a Casimir energy calculation. This approach is of particular interest for gratings with deep corrugations, which can limit the applicability of techniques based on the Rayleigh expansion. We demonstrate the method by calculating the Casimir interaction energy between sinusoidal grating profiles as a function of separation and lateral displacement.

Finite temperature Casimir interaction between spheres in(D+1)-dimensional spacetime: Exact computations and asymptotic expansions

We consider the finite temperature Casimir interaction between two Dirichlet spheres in $(D+1)$-dimensional Minkowski spacetime. The Casimir interaction free energy is derived from the zero temperature Casimir interaction energy via the Matsubara formalism. In the high temperature region, the Casimir interaction is dominated by the term with zero Matsubara frequency, and it is known as the classical term since this term is independent of the Planck constant $\ensuremath{\hbar}$. Explicit expression of the classical term is derived and it is computed exactly using appropriate similarity transforms of matrices. We then compute the small separation asymptotic expansion of this classical term up to the next-to-leading order term. For the remaining part of the finite temperature Casimir interaction with nonzero Matsubara frequencies, we obtain its small separation asymptotic behavior by applying certain prescriptions to the corresponding asymptotic expansion at zero temperature. This gives us a leading term that is shown to agree precisely with the proximity force approximation at any temperature. The next-to-leading order term at any temperature is also derived and it is expressed as an infinite sum over integrals. To obtain the asymptotic expansion at the low and medium temperature regions, we apply the inverse Mellin transform techniques. In the low temperature region, we obtain results that agree with our previous work on the zero temperature Casimir interaction.

Exact classical sphere-plate Casimir interaction in(D+1)-dimensional spacetime

We consider the high-temperature limit of the Casimir interaction between a Dirichlet sphere and a Dirichlet plate due to the vacuum fluctuations of a scalar field in $(D+1)$-dimensional Minkowski spacetime. The high-temperature leading term of the Casimir free interaction energy is known as the classical term, since it does not depend on the Planck constant $\ensuremath{\hbar}$. From the functional representation of the zero-temperature Casimir interaction energy, we use Matsubara formalism to derive the finite-temperature Casimir free energy and obtain the classical term. It can be expressed as a weighted sum over logarithms of determinants. Using similarity transforms of matrices, we reexpress this classical term as an infinite series. This series is then computed exactly using a generalized Abel-Plana summation formula. From this, we deduce the short-distance asymptotic expansions of the classical Casimir interaction force. As expected, the leading term agrees with the proximity force approximation. The next two terms in the asymptotic expansion are also computed. It is observed that the ratio of the next-to-leading-order term to the leading-order term is proportional to the dimension of spacetime. Hence, a larger correction to the proximity force approximation is expected in spacetime with higher dimensions. This is similar to a previous result deduced for the zero-temperature case.

Casimir interaction between spherical and planar plasma sheets

We consider the interaction between a spherical plasma sheet and a planar plasma sheet due to the vacuum fluctuations of electromagnetic fields. We derive the TGTG formula for the Casimir interaction energy and study its asymptotic behaviors. In the small separation regime, we confirm the proximity force approximation and calculate the first correction beyond the proximity force approximation. This study has potential application to model Casimir interaction between objects made of materials that can be modeled by plasma sheets such as graphene sheets.

Casimir interaction between spheres in (D + 1)-dimensional Minkowski spacetime

We consider the Casimir interaction between two spheres in $(D+1)$-dimensional Minkowski spacetime due to the vacuum fluctuations of scalar fields. We consider combinations of Dirichlet and Neumann boundary conditions. The TGTG formula of the Casimir interaction energy is derived. The computations of the T matrices of the two spheres are straightforward. To compute the two G matrices, known as translation matrices, which relate the hyper-spherical waves in two spherical coordinate frames differ by a translation, we generalize the operator approach employed in [IEEE Trans. Antennas Propag. \textbf{36}, 1078 (1988)]. The result is expressed in terms of an integral over Gegenbauer polynomials. Using our expression for the Casimir interaction energy, we derive the large separation and small separation asymptotic expansions of the Casimir interaction energy. In the large separation regime, we find that the Casimir interaction energy is of order $L^{-2D+3}$, $L^{-2D+1}$ and $L^{-2D-1}$ respectively for Dirichlet-Dirichlet, Dirichlet-Neumann and Neumann-Neumann boundary conditions, where $L$ is the center-to-center distance of the two spheres. In the small separation regime, we confirm that the leading term of the Casimir interaction agrees with the proximity force approximation, which is of order $d^{-\frac{D+1}{2}}$, where $d$ is the distance between the two spheres. Another main result of this work is the analytic computations of the next-to-leading order term in the small separation asymptotic expansion. This term is computed using careful order analysis as well as perturbation method. We find that when $D$ is large, the ratio of the next-to-leading order term to the leading order term is linear in $D$, indicating a larger correction at higher dimensions.