Improved constraints on non-Newtonian forces at 10 microns

Several recent theories suggest that light moduli or particles in “large” extra dimensions could mediate macroscopic forces exceeding gravitational strength at length scales below a millimeter. Such new forces can be parameterized as a Yukawa-type correction to the Newtonian potential of strength $\alpha$ relative to gravity and range $\lambda$. To extend the search for such new physics we have improved our apparatus utilizing cryogenic micro-cantilevers capable of measuring attonewton forces, which now includes a switchable magnetic force for calibration. Our most recent experimental constraints on Yukawa-type deviations from Newtonian gravity are more than 3 times as stringent as our previously published results and represent the best bound in the range of $5–15\mu \mathrm{m}$, with a 95% confidence exclusion of forces with $|\alpha |>14,000$ at $\lambda =10\mu \mathrm{m}$.

Figure 1

A schematic (not to scale) showing the cantilever, test and drive masses, shield, and piezo-electric bimorph actuator. Figure is adapted from Ref. 16. Not shown are the metallization of the shield and drive mass ground plane. A Fabry-Perot cavity is formed between the bottom of the optical fiber and top of the test mass and is used to interferometrically measure the cantilever displacement. The coordinate axes shown in the lower inset are used throughout the paper to describe the orientation of the drive mass and test mass. Upper inset: SEM micrograph of FIB-fabricated test mass cut from $27\mu \mathrm{m}$ thick gold foil attached to cantilever.

Figure 2

Calculated vertical component of the Yukawa ($\alpha =1000$, $\lambda =18\mu \mathrm{m}$) force amplitude and phase at the third harmonic of the drive frequency ($3\omega$), for $119\mu \mathrm{m}$ of bimorph amplitude and $31.5\mu \mathrm{m}$ edge-to-edge $z$ separation between masses with $6\mu \mathrm{m}$ tilt across the drive mass in the scanning ($y$) direction. The horizontal axis shows the $y$-equilibrium position of the drive mass oscillation. For reference, the $100\mu \mathrm{m}$-wide gold bars are centered in the calculation at $y=-250\mu \mathrm{m}$, $y=-50\mu \mathrm{m}$, $y=150\mu \mathrm{m}$, and $y=350\mu \mathrm{m}$, with silicon bars in between.

Figure 3

Calculated magnetic force at the third harmonic of the drive frequency, per unit magnetic moment per ampere of current through the meander for varying $y$-equilibrium position of the drive mass oscillation. Results are shown for $119\mu \mathrm{m}$ of bimorph amplitude and $30\mu \mathrm{m}$ face-to-face $z$ separation, for net magnetic moment in the $z$, $y$, and $x$ directions, respectively.

Figure 4

Magnetic moment measured by MPMS-SQUID for deposition of $3\AA \mathrm{Co}/15\AA \mathrm{Pt}/3\AA \mathrm{Co}/30\AA \mathrm{Pt}$ on gold foil substrate, including seed layer of 100 Å Pt deposited at 350 C, for in-plane and out-of-plane magnetization.

Figure 5

Main data set showing statistical and systematic uncertainties in the measured force. Diamonds show signal with current off. Circles show signal with 10 mA current in the drive mass. Lines are guides to the eye. Error determination is described in Sec. 6. The $y$ coordinates shown for the data set on the horizontal axis are the displacements relative to one particular magnetic force minimum chosen as $y=0$. We note that these coordinates vary by an unknown offset with respect to those shown in the calculations in Figs. 2, 3.

Figure 6

Histogram of best-fit $\alpha$ results for $\lambda =10\mu \mathrm{m}$.

Figure 7

Real and imaginary parts of measured force. A typical Monte Carlo fit result (best-fit Yukawa signal plus offset) for $\lambda =10\mu \mathrm{m}$, $\alpha =4450$ is shown as a solid line. Error bars ($1\sigma$) on the measured forces are also shown.

Figure 8

Amplitude and phase of measured force. A typical Monte Carlo fit result (best-fit Yukawa signal plus offset) for $\lambda =10\mu \mathrm{m}$, $\alpha =4450$ is shown as a solid line. Error bars ($1\sigma$) on the measured forces are also shown. Dotted line shows fit result with best-fit offset force subtracted. The amplitude of the best-fit offset force is significantly larger than the amplitude of the Yukawa component.

Figure 9

New limits on Yukawa forces. The shaded area referred to as “Stanford 3” represents the 95% exclusion region yielded with the present data set, listed in Table . Experimental limits derived in this work are on $|\alpha |$. The previous results labeled “Stanford 1” and “Stanford 2” are presented in Refs. 6, 7, respectively. Also shown are theoretical predictions and other experimental results as indicated in Refs. 22, 23, 24, 25, 26, 27.