Measurement of the Casimir Force between Two Spheres

Complex interaction geometries offer a unique opportunity to modify the strength and sign of the Casimir force. However, measurements have traditionally been limited to sphere-plate or plate-plate configurations. Prior attempts to extend measurements to different geometries relied on either nanofabrication techniques that are limited to only a few materials or slight modifications of the sphere-plate geometry due to alignment difficulties of more intricate configurations. Here, we overcome this obstacle to present measurements of the Casimir force between two gold spheres using an atomic force microscope. Force measurements are alternated with topographical scans in the $x\text{−}y$ plane to maintain alignment of the two spheres to within approximately 400 nm ($\sim 1\%$ of the sphere radii). Our experimental results are consistent with Lifshitz’s theory using the proximity force approximation (PFA), and corrections to the PFA are bounded using nine sphere-sphere and three sphere-plate measurements with spheres of varying radii.

Figure 1

(a) Schematic of the experimental configuration with one gold-coated sphere held directly above another. (b) Topographical scans are used to position the top sphere directly above the bottom sphere (scan speed: $10\mu \mathrm{m}/\mathrm{s}$, $64\times 64\mathrm{pixels}$). (c) Spatial derivative of the force measured between two spheres as a function of separation. During the measurement, the hydrodynamic force (normalized by the shake amplitude) is separated from the spatial derivative of the Casimir force through the phase of the force signal. All the individual measurements (light dots) are shown ($\approx 20000$ points). The force gradients and separations of individual measurements are binned into groups of $\approx 200$ points and averaged (dark squares). The inset shows the cantilever’s response to the Casimir (red) and hydrodynamic (blue) forces. These data are collected with the spheres shown in (a) and (d) of Fig. 3.

Figure 2

Representative measurements of the spatial derivative of Casimir force for both sphere-plate (blue) and sphere-sphere (red) measurement geometries. Results are in agreement with calculated values of the Casimir force derivative for two gold spheres with a 4.9 nm rms perturbative roughness correction (black line). Gray shaded region shows the uncertainty in the roughness correction due to the uncertainty in the orientation of the spheres [32]. The sphere-plate force is measured with the sphere in Fig. 3, and the sphere-sphere data are collected between it and the one in Fig. 3.

Figure 3

(a)–(c) Measurements of the spatial derivative of the Casimir force as a function of separation for nine sphere-sphere combinations. Colored data points correspond to measurements between a top sphere (insets) and the three different bottom spheres, color matched to the topography maps shown in (d)–(f). The error bars in (a)–(c) are dominated by the uncertainties in the ambient water layer thickness ($x$ axis) and from the stray light effect ($y$ axis). Black lines correspond to Lifshitz theory with the PFA, measured optical data, and roughness corrections.

Figure 4

(a) Measured $(1/R^{\prime })F_C^{\prime }$ versus $1/R^{\prime }$ from combined data sets of all measurements corresponding to separations of approximately 72 nm (purple), 118 nm (red), and 265 nm (green). For each separation, a line is fit to the data, and Eq. (2) is used to obtain the slope, $m=2\pi F_{\mathrm{pp}}{\beta }^{\prime }d$, and the intercept, $b=2\pi F_{\mathrm{pp}}$. The graphic insets depict sphere-plate measurements on the left and sphere-sphere measurements on the right (shaded). (b) ${\beta }^{\prime }$ is calculated from each line fit and plotted as a function of separation, $d$. Limits are placed on the overall value of ${\beta }^{\prime }$ (shaded region), so that any individual ${\beta }^{\prime }$ within falls within the $2\sigma$ confidence interval of the ${\beta }^{\prime }$ estimate at every separation.