Plasma versus Drude Modeling of the Casimir Force: Beyond the Proximity Force Approximation

We calculate the Casimir force and its gradient between a spherical and a planar gold surface. Significant numerical improvements allow us to extend the range of accessible parameters into the experimental regime. We compare our numerically exact results with those obtained within the proximity force approximation (PFA) employed in the analysis of all Casimir force experiments reported in the literature so far. Special attention is paid to the difference between the Drude model and the dissipationless plasma model at zero frequency. It is found that the correction to PFA is too small to explain the discrepancy between the experimental data and the PFA result based on the Drude model. However, it turns out that for the plasma model, the corrections to PFA lie well outside the experimental bound obtained by probing the variation of the force gradient with the sphere radius [D. E. Krause et al., Phys. Rev. Lett. 98, 050403 (2007)]. The corresponding corrections based on the Drude model are significantly smaller but still in violation of the experimental bound for small distances between plane and sphere.

Figure 1

(a) Sphere of radius $R$ separated from the plane by a distance $L$. (b) Frequency dependence of the permittivity of gold used in the numerical calculations. The vertical solid line indicates the first Matsubara frequency ${\xi }_1$ while the dotted line indicates the plasma frequency ${\omega }_P$.

Figure 2

Beyond-PFA corrections to (a) the force and (b) the force gradient are shown as function of the distance between sphere and plane. We multiply the correction by the aspect ratio $R/L$. The upper three and lower three lines refer to the Drude ($D$) and plasma ($P$) prescription, respectively. The dashed horizontal lines indicate the results for perfect reflectors at zero temperature, corresponding to the constant coefficients ${\beta }_{T=0}^{\mathrm{PR}}$ and ${{\beta }^{\prime }}_{T=0}^{\mathrm{PR}}$ for the force and force gradient, respectively, as defined in the text. In the lower panel, the grey area marks the parameter range for the force gradient excluded experimentally at the 95% confidence level [20].

Figure 3

Beyond-PFA corrections to the (a),(b) force and (c),(d) force gradient are shown as function of the inverse sphere radius. While in (a) and (c) the relative corrections are displayed, the data in (b) and (d) have been scaled by $R/L$. The sphere-plane distances are $L=200\mathrm{nm}$ (∘), 400 nm (□), and 600 nm (△). Solid lines with open symbols refer to the Drude prescription while dashed lines with filled symbols refer to the plasma prescription.