Gradient of the Casimir force between Au surfaces of a sphere and a plate measured using an atomic force microscope in a frequency-shift technique

We present measurement results for the gradient of the Casimir force between an Au-coated sphere and an Au-coated plate obtained by means of an atomic force microscope operated in a frequency-shift technique. This experiment was performed at a pressure of $3\times {10}^{-8}$ Torr with a hollow glass sphere of 41.3 $\mu$m radius. Special attention is paid to electrostatic calibrations, including the problem of electrostatic patches. All calibration parameters are shown to be separation independent after the corrections for mechanical drift are included. The gradient of the Casimir force was measured in two ways with applied compensating voltage to the plate and with different applied voltages and subsequent subtraction of electric forces. The obtained mean gradients are shown to be in mutual agreement and in agreement with previous experiments performed using a micromachined oscillator. The obtained data are compared with theoretical predictions of the Lifshitz theory including corrections beyond the proximity force approximation. An independent comparison with no fitting parameters demonstrated that the Drude model approach is excluded by the data at a 67$\%$ confidence level over the separation region from 235 to 420 nm. The theoretical approach using the generalized plasmalike model is shown to be consistent with the data over the entire measurement range. Corrections due to the nonlinearity of oscillator are calculated and the application region of the linear regime is determined. A conclusion is made that the results of several performed experiments call for a thorough analysis of the basics of the theory of dispersion forces.

Figure 1

(a) Layout of the vacuum FM-AFM setup used in precision dynamic measurements of the gradient of the Casimir force. (b) Schematic of the force measurements microscope. (1) is a Au plate placed on the AFM piezo (2). (3) is the plate movement interferometer detection fiber. For monitoring the cantilever (4) oscillations the second interferometer was used, the detection fiber end is shown as (5). The end was fixed in the fiber holder (6), which was placed in the XYZ stage and can move in the XYZ direction for adjusting the signal from cantilever. The cantilever chip (7) was connected to two piezoelectric actuators [(8) and (9)] and clutched in the home-made cantilever holder (10) as shown in the picture.

Figure 2

The sphere-plate configuration with patches that were put on the top of a plate.

Figure 3

The residual potential difference between the sphere and the plate as a function of separation for (a) patches of fixed sizes $6\times 6$ $\mu {\text{m}}^2$ with different distances between patches and (b) fixed distances between patches equal to 600 nm with different sizes of patches.

Figure 4

(a) The drift in experimental curves for the frequency-shift signal as a function of the change in sphere-plate separation for eight repetitions of the same applied voltage to the plate. (b) The change in the plate position at one frequency-shift signal value as a function of time.

Figure 5

(a) The residual potential difference between Au-coated sphere and plate as a function of separation. (b) The systematic error of each individual $V_0$, as determined from the fit, versus separation.

Figure 6

The dependences of (a) the closest sphere-plate separation and (b) the coefficient $C$ in Eq. (5) on the end point of the fit.

Figure 7

Mean measured gradients of the Casimir force as a function of separation are shown by solid lines. Gray dots indicate all 40 individual force gradients plotted with a step of 5 nm (1 nm in the inset).

Figure 8

The histograms for the measured gradients of the Casimir force at separations (a) $a=235$ nm and (b) $a=275$ nm. $f$ is the fraction of 40 data points having the force values in the bin indicated by the respective vertical lines of the histogram. The corresponding Gaussian distributions are shown by the dashed black lines. The solid and dashed vertical lines show the theoretical predictions from the plasma and Drude model approaches, respectively.

Figure 9

The random, systematic, and total experimental errors in the measured gradients of the Casimir force determined at a 67$\%$ confidence level are shown by the short-dashed, long-dashed, and solid lines, respectively. The measurement scheme with applied compensating voltage is used.

Figure 10

Magnitudes of the mean Casimir pressure previously measured[23, 24] are shown by (a) black and (b) white lines as functions of separation. Magnitudes of the mean Casimir pressure measured here are indicated as crosses. The arms of the crosses are determined by errors in the measurement of separations and force gradients.

Figure 11

(a) Mean measured gradients of the Casimir force as a function of separation are shown by solid lines. Gray dots indicate all 40 individual force gradients plotted with the step of 5 nm (1 nm in the inset). (b) The random, systematic, and total experimental errors in the measured gradient of the Casimir force determined at a 67$\%$ confidence level are shown by the short-dashed, long-dashed, and solid lines, respectively. The measurement scheme with different applied voltages is used.

Figure 12

Comparison between the experimental data for the gradient of the Casimir force (crosses plotted at a 67$\%$ confidence level) and theory (black and gray bands computed using the Drude and plasma model approaches, respectively) within different separation regions. The experimental data are obtained with the compensating voltage applied to the plate.

Figure 13

Comparison between the experimental data for the gradient of the Casimir force (crosses plotted at a 67$\%$ confidence level) and theory (black and gray bands computed using the Drude and plasma model approaches, respectively) within different separation regions. The experimental data are obtained with different voltages applied to the plate.

Figure 14

Maximum allowed amplitudes of oscillations of the AFM cantilever in the linear regime as a function of separation are shown by the solid line (the plasma model approach) and by the dashed line (the Drude model approach). The allowed regions of the $(a,A_z)$ plane lie beneath the lines.