Figure 1

Schematic drawing of the experimental setup (not to scale). The membrane frames, the source mass, and its driving stage were mounted on a 6-degree-of-freedom stage (not shown here), which was used to adjust the relative positions and orientation. The protrudent left surface of the alignment glass was used to touch the fiber by moving the source mass in $X$ direction to acquire the relative positions between the pendulum and the source mass. The pendulum twist was measured by the autocollimator I, and controlled by applying differential voltages to the two capacitive actuators. The separation between the tungsten test and source masses was modulated by driving a motor translation stage with frequency $f_s$, and the sensitivity of the pendulum was calibrated by rotating the copper cylinder at frequency $f_c$. The autocollimator II was used to monitor the changes of the source mass’s attitude synchronously due to the yaw and pitch motions of the driving stage around its shaft.Reuse & Permissions

Figure 2

The typical PSD of the torque experienced by the open-loop (top panel) and closed-loop (middle and bottom panel) pendulum. The gravitational calibrating signal occurred at $f_c=2.5\mathrm{mHz}$ in each graph with an amplitude of $(5.6\pm 0.4)\times {10}^{-16}\mathrm{N}\mathrm{m}$. Top panel: The typical PSD curve of the free pendulum with the pendulum-membrane gap being 3 mm. The $f_0$ ($=1.47\mathrm{mHz}$) is the natural frequency of the pendulum. The dash line shows the thermal noise limit inferred from the finite quality factor, $Q\approx 2000$. Middle panel: The typical PSD curve of the null experiment with the gap ranging from 0.4 to 1.0 mm, and the expected non-Newtonian signal at $f_s$ did not appear with the sensitivity of approximately $0.5\times {10}^{-16}\mathrm{N}\mathrm{m}$. Bottom panel: The typical PSD curve of the non-null experiment with the gap ranging from 1.1 to 1.7 mm, and the measured change of the Newtonian torque at $f_s=2.0\mathrm{mHz}$ is $(5.7\pm 1.0)\times {10}^{-16}\mathrm{N}\mathrm{m}$, in accord with the theoretical calculation of $(5.5\pm 0.8)\times {10}^{-16}\mathrm{N}\mathrm{m}$.Reuse & Permissions

Figure 3

Comparison of the change of the Newtonian torque ($\Delta \tau$) between the theoretical calculations (dash) and the experimental results (dot) in each 0.6 mm-range experiment. The gap between the two dashes shows an uncertainty of less than $2\times {10}^{-16}\mathrm{N}\mathrm{m}$, yielded by the dimension errors of the apparatus. The $\tau (\sim 1\times {10}^{-12}\mathrm{N}\mathrm{m})$, accounting for the net Newtonian torque between the pendulum and the source mass platform, is a constant on the lever of 3% in the total experimental region (0.4–4.3 mm), and a perfect cancellation of it was designed at the gap of 0.7 mm. The inset is a magnified view of the result obtained in 3.1–3.7 mm.Reuse & Permissions

Figure 4

Upper limits on Yukawa violations of the Newtonian $1/r^2$ law. The shaded region is excluded with 95% confidence level. The heavy lines labeled Stanford 2008 [12], Colorado 2003 [13], Eöt-wash 2004 and 2007 [14], HUST 2007 [15], This work, and Irvine [16] show the experimental constraints, respectively. Lighter lines showed various theoretical predictions summarized in 7.Reuse & Permissions