Open Access

Michael Hartmann

• We study the Casimir interaction between a sphere of radius R separated by a distance L from a plane. We focus on the proximity force approximation (PFA) which has been exclusively used to analyze experiments in this geometry. Our results can be divided in three parts: (i) the derivation of the proximity force approximation as the first term in an asymptotic expansion, (ii) the extension of the numerics to aspect ratios R/L~5000 used in typical experiments, and (iii) the comparison of the PFA with numerical results and the computation of corrections to the PFA. We derive the proximity force approximation expression for the Casimir free energy as the leading asymptotic result in the limit of large aspect ratios R/L. To leading order, only the direct reflection term in the Debye expansion of the WKB Mie scattering amplitudes contributes. The trace over a number of round-trip matrices is evaluated within the saddle-point approximation. The saddle point corresponds to the conservationWe study the Casimir interaction between a sphere of radius R separated by a distance L from a plane. We focus on the proximity force approximation (PFA) which has been exclusively used to analyze experiments in this geometry. Our results can be divided in three parts: (i) the derivation of the proximity force approximation as the first term in an asymptotic expansion, (ii) the extension of the numerics to aspect ratios R/L~5000 used in typical experiments, and (iii) the comparison of the PFA with numerical results and the computation of corrections to the PFA. We derive the proximity force approximation expression for the Casimir free energy as the leading asymptotic result in the limit of large aspect ratios R/L. To leading order, only the direct reflection term in the Debye expansion of the WKB Mie scattering amplitudes contributes. The trace over a number of round-trip matrices is evaluated within the saddle-point approximation. The saddle point corresponds to the conservation of the wave-vector component parallel to the plane. Therefore, the leading-order contribution results from specular reflection in the vicinity of the points of closest distance between the sphere and the plane. Our derivation holds for arbitrary materials and temperatures. As an important consequence, we find that no polarization mixing contributes to leading order. From a more theoretical perspective, our results help understanding why local approaches such as the derivative expansion are capable of providing both the leading and next-to-leading-order terms in several situations of interest. The standard approach to computing the Casimir free energy within the scattering approach has been plagued with ill-conditioned round-trip matrices resulting in numerical difficulties. These difficulties can be eliminated by a symmetrization of the round-trip operator. Moreover, the determinant of the symmetrized round-trip operator can be evaluated using a state-of-the-art algorithm suited for hierarchical matrices. This significantly reduces the computational time and thus allows us to perform calculations in the experimentally relevant regime with aspect ratios up to R/L~5000, an improvement of almost two orders of magnitude compared to the largest aspect ratios treated so far. With large aspect ratios being numerically accessible, we assess the quality of the PFA by determining its deviations from the exact result. For zero temperature, we confirm the leading-order correction to the PFA which is linear in L/R. Commonly, it is believed that the next-to-leading-order correction is either quadratic in the inverse aspect ratio or contains logarithmic terms. However, we find that for perfect reflectors and metals described by the plasma and the Drude model, the next-to-leading-order correction is proportional to (L/R)^(3/2). This suggests that for the next-to-leading-order correction to the PFA non-local effects and/or the far side of the sphere become important. As a consequence, the next-to-leading-order correction to the PFA cannot be obtained within the framework of the derivative expansion. Furthermore, we compare numerically exact results obtained using the scattering formula with the PFA for parameters corresponding to typical experiments. We show that the dissipationless plasma prescription leads to a violation of the upper bound for the PFA correction obtained experimentally by Krause et al. This could have been expected, since dissipation is present in the gold coatings used in the experiment. However, all experiments performed with coated microspheres with aspect ratios R/L~10²-10³ agree with the plasma prescription but not with the Drude prescription. When taking the Drude prescription, the correction is smaller but still in violation of the experimental bound for small distances between sphere and plane. Moreover, we directly compare our numerical results with the experimental data of Krause et al. While the plasma prescription within the proximity force approximation agrees well with the experimental data, the exact results obtained within the scattering approach deviate from the experimental data when the aspect ratio is small. Our results suggest a discrepancy between Casimir experiments on the one hand, and the theoretical description on the other hand. The theoretical results presented here, taking the sphere curvature fully into account, indicate that experiments probing the Casimir interaction beyond the PFA regime could provide new insight into the role of dissipation in Casimir physics.…