Corrections beyond the proximity force approximation

Teo, Lee-Peng; Bordag, M.; Nikolaev, V. M.

doi:10.1103/physrevd.84.125037

Phys. Rev. D

2011

DOI: 10.1103/physrevd.84.125037

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Corrections beyond the proximity force approximation

Lee-Peng Teo

M. Bordag

V. M. Nikolaev

Abstract: We recalculate the first analytic correction beyond proximity force approximation for a sphere in front of a plane for a scalar field and for the electromagnetic field. We use the method of Bordag and Nikolaev [J. Phys. A 41, 164002 (2008)]. We confirm their result for Dirichlet boundary conditions whereas we find a different one for Robin, Neumann and conductor boundary conditions. The difference can be traced back to a sign error. As a result, the corrections depend on the Robin parameter. Agreement is found… Show more

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Cited by 51 publications

(99 citation statements)

References 10 publications

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“…The weighted KramersKronig dispersion relations [50] was used to compute precisely (iξ n ) starting from the real-frequency optical data given by Palik. The coefficients θ andθ can be calculated also using the method developed by Bordag and Nikolaev [10][11][12], and we have verified that this alternative method gives for θ andθ the same results as the DE.…”

supporting

confidence: 55%

“…Below we prove that a very accurate formula for the Casimir force and its gradient can be obtained by a combined use of certain limiting exact solutions [16] and asymptotic formulae [10][11][12]15] that have been recently reported in the literature.…”

mentioning

confidence: 67%

“…The PFA has been put on a firmer basis recently, by showing rigorously that it coincides with the leading term of the asymptotic expansion of the exact scattering formula for large R/a (at fixed a) [10]. By the same approach, it has been also possible to estimate the O(a/R) corrections to the PFA [11,12]. An alternative and equivalent method to compute curvature corrections to the PFA is based on a derivative expansion (DE) of the Casimir energy in powers of derivatives of the curved surface height profile [13][14][15].…”

mentioning

confidence: 99%

“…Recently, important progress in the understanding of the sphere-sphere and sphere-plate force has been made in the non-retarded or van der Waals regime by using transformation optics [9]. By a combination of asymptotic techniques [10][11][12][13][14][15] with a partial exact solution valid in the classical limit [16], here we derive a new semi-analytic formula for the complete retarded sphere-plate Casimir force. Comparison with high precision numerical simulations reveals that the formula is remarkably accurate at all separations.…”

mentioning

confidence: 99%

“…The expressions for the Casimir force F = −dF/da and its gradient F = dF/da are obtained by taking derivatives of Eq. (2) [10][11][12]. It turns out that to estimate the Casimir force and its gradient with an accurary of 10 −4 , as we seek, one needs l max 6R/a, m max 6 R/a, n max 10 λ T /a.…”

mentioning

confidence: 99%

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Going beyond PFA: A precise formula for the sphere-plate Casimir force

Bimonte¹

2017

EPL

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-Quantum fluctuations of the electromagnetic field in the medium surrounding two discharged macroscopic polarizable bodies induce a force between the two bodies, the so called Casimir force. In the last two decades many experiments have accurately measured this force, and significant efforts are made to harness it in the actuation of micro and nano machines. The inherent many body character of the Casimir force makes its computation very difficult in nonplanar geometries, like the standard experimental sphere-plate configuration. Here we derive an approximate semi-analytic formula for the sphere-plate Casimir force, which is both easy to compute numerically and very accurate at all distances. By a comparison with the fully converged exact scattering formula, we show that the error made by the approximate formula is indeed much smaller than the uncertainty of present and foreseeable Casimir experiments.The Casimir effect [1] is the tiny force acting between two (or more) discharged polarizable objects, that originates from quantum and thermal fluctuations of the electromagnetic field in the medium surrounding the bodies. It represents one of the rare manifestations of the quantum at the macroscopic scale, similar to black-body radiation, superfluidity and superconductivity. Reviews can be found in Refs. [2][3][4][5]. Recent years witnessed an impetuous resurgence of interest in the Casimir effect, triggered by a series of precision experiments [6,7] and by the exciting perspective of harnessing this force in the nanoworld [8]. The Casimir force is notoriously difficult to compute in non-planar geometries, like the standard sphere-plate geometry adopted in almost all experiments (see Fig. 1). Recently, important progress in the understanding of the sphere-sphere and sphere-plate force has been made in the non-retarded or van der Waals regime by using transformation optics [9]. By a combination of asymptotic techniques [10-15] with a partial exact solution valid in the classical limit [16], here we derive a new semi-analytic formula for the complete retarded sphere-plate Casimir force. Comparison with high precision numerical simulations reveals that the formula is remarkably accurate at all separations. The new formula thus provides a simple and yet fully reliable tool to interpret present and future experimental data.In his famous 1948 paper [1], Hendrik Casimir discovered that the ground state energy of the quantized electromagnetic (em) field is modified by the presence of material bodies that interact with the em field. By carefully adding up the zero-point energies of the em modes of a planar cavity consisting of two perfectly conducting plates of (large) area A at distance a, he obtained his celebrated formula for the force acting on the plates at zero temperature:This simple formula reveals the main features of the Casimir force: the presence of Planck's constant indicates its quantum character, while the presence of the speed of light c shows that it is a relativistic effect. The a −4 dependence s...

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supporting

confidence: 55%

mentioning

confidence: 67%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Going beyond PFA: A precise formula for the sphere-plate Casimir force

Bimonte¹

2017

EPL

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Classical and fluctuation‐induced electromagnetic interactions in micron‐scale systems: designer bonding, antibonding, and Casimir forces

et al. 2014

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Whether intentionally introduced to exert control over particles and macroscopic objects, such as for trapping or cooling, or whether arising from the quantum and thermal fluctuations of charges in otherwise neutral bodies, leading to unwanted stiction between nearby mechanical parts, electromagnetic interactions play a fundamental role in many naturally occurring processes and technologies. In this review, we survey recent progress in the understanding and experimental observation of optomechanical and quantumfluctuation forces. Although both of these effects arise from exchange of electromagnetic momentum, their dramatically different origins, involving either real or virtual photons, lead to different physical manifestations and design principles. Specifically, we describe recent predictions and measurements of attractive and repulsive optomechanical forces, based on the bonding and antibonding interactions of evanescent waves, as well as predictions of modified and even repulsive Casimir forces between nanostructured bodies. Finally, we discuss the potential impact and interplay of these forces in emerging experimental regimes of micromechanical devices.

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

Teo

2014

J. High Energ. Phys.

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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. 36, 1078(1988]. The result is expressed in terms of an integral over Gegenbauer polynomials. In contrast to the D = 3 case, we do not re-express the integral in terms of 3j-symbols and hyper-spherical waves, which in principle, can be done but does not simplify the formula. 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 − 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. In the case the radius of one of the sphere goes to infinity, we find that the results agree with the one we derive for sphere-plate configuration. When D = 3, we also recover previously known results. 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. The methodologies employed in this work and the results obtained can be used to study the one-loop effective action of the system of two spherical objects in the universe.

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Corrections beyond the proximity force approximation

Cited by 51 publications

References 10 publications

Going beyond PFA: A precise formula for the sphere-plate Casimir force

Going beyond PFA: A precise formula for the sphere-plate Casimir force

Classical and fluctuation‐induced electromagnetic interactions in micron‐scale systems: designer bonding, antibonding, and Casimir forces

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

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