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

Abstract

-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...

Reducing detrimental electrostatic effects in Casimir-force measurements and Casimir-force-based microdevices

“…which is sufficiently precise for our purposes. As was shown recently [42][43][44], the relative correction to the proximity force approximation is approximately equal to 0.5a/R, i.e., varies from 0.19% to 0.33% when the separation increases from 235 to 400 nm. In Fig.…”

Examining the Casimir puzzle with an upgraded AFM-based technique and advanced surface cleaning

“…In doing so, special experimental efforts have been taken to avoid any impact of electrostatic patches on the measurement results for the Casimir force. On the theoretical side, due to the long-term efforts of several authors it was shown that the influence of deviations from the proximity force approximation [35,36], as well as the role of surface roughness [3,4,40,41], does not change the obtained results. Thus, the Casimir puzzle assumes a fundamental importance casting doubts on the basic assumption of quantum statistical physics that a material system responds similarly to electromagnetic fields with nonzero field strength and to fluctuating fields possessing zero field strength and nonzero dispersion.…”

Precision measurements of the gradient of the Casimir force between ultraclean metallic surfaces at larger separations

“…After the experiment [16] on measuring the difference Casimir force was performed, where the theoretical predictions of the Lifshitz theory using the Drude and plasma models differ by up to a factor of 1000, an exclusion of the former within this separation range was conclusively established. In so doing the role of different background effects, such as the surface roughness [49,50], variations in the optical data [51], patch potentials [52,53] etc., as well as the validity of calculation procedure (including the role of deviations from the proximity force approximation [54][55][56][57][58]), were investigated in detail.…”