New Test of the Gravitational Inverse-Square Law at the Submillimeter Range with Dual Modulation and Compensation

By using a torsion pendulum and a rotating eightfold symmetric attractor with dual modulation of both the interested signal and the gravitational calibration signal, a new test of the gravitational inverse-square law at separations down to $295\mu \mathrm{m}$ is presented. A dual-compensation design by adding masses on both the pendulum and the attractor was adopted to realize a null experiment. The experimental result shows that, at a 95% confidence level, the gravitational inverse-square law holds ($|\alpha |\le 1$) down to a length scale $\lambda =59\mu \mathrm{m}$. This work establishes the strongest bound on the magnitude $\alpha$ of Yukawa-type deviations from Newtonian gravity in the range of $70–300\mu \mathrm{m}$, and improves the previous bounds by up to a factor of 2 at the length scale $\lambda \approx 160\mu \mathrm{m}$.

Figure 1

Schematic drawing of the experimental setup (not to scale). The attractor and the alignment glass are mounted on a 6-degree-of-freedom stage (not shown here). A rotary stage outside the vacuum chamber is used to rotate the attractor through a feedthrough. The pendulum twist is measured by an autocollimator, and controlled by two differential capacitive actuators. The sensitivity of the pendulum is calibrated synchronously by rotating a copper cylinder outside the vacuum chamber.

Figure 2

(a) The front view of the bilateral symmetric pendulum and the eightfold attractor. (b) The residual Newtonian torque ${\tau }_N$ (dotted line) and the expected Yukawa torque ${\tau }_Y$ (solid line), which are calculated from the geometric parameters of the masses with the separation of $295\mu \mathrm{m}$. The Yukawa torque changes as a sinusoidal curve with a frequency of $8{\omega }_d$ and an amplitude of $3.4\times {10}^{-17}\mathrm{N}·\mathrm{m}$ ($\alpha =0.12$, $\lambda =0.1\mathrm{mm}$ are used according to the previous bound [10]). The residual Newtonian torque varies mainly at the fundamental frequency due to the eccentricity of the attractor. From the Fourier transform, the $8{\omega }_d$ Newtonian torque is $(0.7\pm 0.5)\times {10}^{-17}\mathrm{N}·\mathrm{m}$ (the in-phase) and $(-0.1\pm 0.5)\times {10}^{-17}\mathrm{N}·\mathrm{m}$ (the quadrature), respectively.

Figure 3

(a) The typical torque PSD of the pendulum with the attractor at rest. It is closed to the thermal noise limit at $\sim \mathrm{mHz}$, but increased at higher frequency due to the readout noise. (b) Comparison of the torque PSDs of the pendulum with the attractor rotating at separations of 295 and $1095\mu \mathrm{m}$. Both of them have a similar structure in the full frequency range. No significant signals appear at $8{\omega }_d$. The obvious signals at $1{\omega }_d-4{\omega }_d$ are caused by both the Newtonian torque and the disturbance of the shield membrane, and the $15{\omega }_d$ and $17{\omega }_d$ signals are Newtonian torques due to the imperfect assembly of the attractor. (c)–(e) The measured torques at $8{\omega }_d$ at 295, 695, and $1095\mu \mathrm{m}$ separations, respectively. Each solid dot represents a value obtained from a 5-rotation-period data segment, and the squares represent the mean values (shown in brackets) with $2\sigma$ error.

Figure 4

Non-null experimental results of the $8{\omega }_d$ (upper) and $16{\omega }_d$ (lower) Newtonian torque. The shaded belt represents the theoretical calculations, and the little squares are the measured values, both with $2\sigma$ error. The $8{\omega }_d$ theoretical Newtonian torque shows a smaller error bar compared with the $16{\omega }_d$ one due to the dual compensation design. Inset: schematic drawing of the displacement of the attractor with respect to the pendulum.

Figure 5

Constraints on Yukawa violation of the Newtonian $1/r^2$ law. The shaded region is excluded at a 95% confidence level. The heavy lines labeled Stanford 2008 [13], Colorado 2003 [14], Eöt-Wash 2007 [10], HUST 2007 and 2012 [23, 24], This work, and Irvine [11] show the experimental constraints, respectively. Light lines show various theoretical predictions summarized in Ref. [8].