Probing modified gravity with magnetically levitated resonators

We present an experimental procedure, based on Meissner effect levitation of neodymium ferromagnets, as a method of measuring the gravitational interactions between milligram masses. The scheme consists of two superconducting lead traps, with a magnet levitating in each trap. The levitating magnets behave as harmonic oscillators and, by carefully driving the motion of one magnet on resonance with the other, we find that it should easily be possible to measure the gravitational field produced by a 4 mg sphere, with the gravitational attraction from masses as small as $30\mu \mathrm{g}$ predicted to be measurable within a realistic measurement time frame. We apply this acceleration sensitivity to one concrete example and show the abilities of testing models of modified Newtonian dynamics.

Figure 1

A schematic of the experimental setup. Neodymium magnet spheres are levitated in two independent lead wells, which are separated by a thin divide. Each trap also has a lead lid. The entire setup is cooled to around 300 mK in a ${}^3\mathrm{He}$ sorption refrigerator. Both levitating magnets have their position recorded using SQUIDs, and one of the magnets (source mass) has its motion driven, using electromagnetic forces, to an amplitude of $\mathrm{\Delta }x/2$. This motion modulates the gravitational field experienced by the other magnet (test mass), which influences the motion and creates a detectable signal. The test mass trap is suspended by passive vibration isolation systems, which decouple the motion of the two traps from each other and isolate against background seismic noise. Earth’s gravity is shown as $g$.

Figure 2

Expected detectable acceleration as a function of measurement time for two realistic experimental scenarios. The blue dashed line represents the minimum detectable acceleration for a $Q$ factor of $Q=5500$ (experimentally measured with a 0.5 mm radius neodymium magnet [33]). The blue region represents accelerations that can be measured with these experimental parameters. The orange dashed line shows the minimum detectable acceleration for $Q=5\times {10}^5$, which is the predicted $Q$ factor with magnetic shielding. The orange region represents accelerations which can be measured with these experimental parameters. The blue and orange dots show the minimum measurement time required to resolve the motion of the oscillator for each $Q$ factor. The acceleration modulation that will be generated by driving the source mass magnet at an amplitude of $\mathrm{\Delta }x/2=500\mu \mathrm{m}$ and $50\mu \mathrm{m}$ is shown with the dotted and dash-dotted horizontal lines.

Figure 3

Predicted acceleration (in units of ${10}^{-10}\mathrm{m}/{\mathrm{s}}^2$) imprinted on a test particle by a particle of mass $M=4\mathrm{mg}$, as a function of particle separation $r$ in millimeters. The blue dotted line is the expected acceleration in gravitational Newtonian dynamics, while the green dashed and orange solid lines are the predicted gravitational acceleration results for MOND based on Eq. (5) [the green shaded region brackets the simple MOND model when one includes the EFE via Eq. (6) with $e=\pm 0.1$; see the text for details]. The red dot-dashed line is the acceleration from Eq. (5) in the deep MOND limit, i.e., when $\mu [x]=x$.