Gravity is the weakest of all known fundamental forces and poses some of the most important open questions to modern physics: it remains resistant to unification within the standard model of physics and its underlying concepts appear to be fundamentally disconnected from quantum theory1,2,3,4. Testing gravity at all scales is therefore an important experimental endeavour5,6,7. So far, these tests have mainly involved macroscopic masses at the kilogram scale and beyond8. Here we show gravitational coupling between two gold spheres of 1 millimetre radius, thereby entering the regime of sub-100-milligram sources of gravity. Periodic modulation of the position of the source mass allows us to perform a spatial mapping of the gravitational force. Both linear and quadratic coupling are observed as a consequence of the nonlinearity of the gravitational potential. Our results extend the parameter space of gravity measurements to small, single source masses and low gravitational field strengths. Further improvements to our methodology will enable the isolation of gravity as a coupling force for objects below the Planck mass. This work opens the way to the unexplored frontier of microscopic source masses, which will enable studies of fundamental interactions9,10,11 and provide a path towards exploring the quantum nature of gravity12,13,14,15.
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The data that support the findings of this study are available from the corresponding author upon reasonable request.
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We thank E. Adelberger, A. Buikema, P. Graham, N. Kiesel, N. Klein, D. Racco and J. Schmöle for discussions. We are grateful for the suspension fibre provided by A. Rauschenbeutel and T. Hoinkes and for the mechanical design assistance by M. Dragosits. This project was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 649008, ERC CoG QLev4G), by the Austrian Academy of Sciences through Innovationsfonds Forschung, Wissenschaft und Gesellschaft, by the Alexander von Humboldt Foundation through a Feodor Lynen Fellowship (T.W.) and by the Austrian Federal Ministry of Education, Science and Research (project VCQ HRSM).
The authors declare no competing interests.
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Extended data figures and tables
The video-tracked yaw angle of the torsion pendulum is compared to the QPD readout (horizontal difference signal normalized by the total sum signal to reduce laser intensity noise coupling). A third-order polynomial fit to these data provides our QPD signal calibration. The occurrence probability accumulated over all our data runs shows the region in which a calibration is required, that is, Vx/Vsum ∈ [−0.4, 0.2].
a, The displacement amplitude spectral density (ASD) estimate of 1-h data chunks of representative data taken between 26 and 27 December shows stationary readout noise at high frequencies and non-stationary white force noise at low frequencies. The level of the latter varies up to tenfold, consistently being the lowest at 0–5 a.m., particularly in the night after public holidays or Sundays, that is, before normal workdays. During such quiet periods, the thermal noise of the pendulum is reached. b, Time-series data spanning half an hour during the 3–4 a.m. segment. The data trace is bandpass-filtered in the frequency range 5–75 mHz with a 6th-order Butterworth filter and magnified by a factor of 20.
The source-mass drive, consisting of a bending piezo and a titanium rod, can modulate the source-mass position by more than 5 mm peak to peak at frequencies as high as 1 Hz. The geometry of the titanium rod amplifies and translates the piezo deformation into an approximately linear motion.
Without electrostatic shielding, the probability density distribution of inferred force versus source–test mass separation requires three constituents to describe: Newtonian gravity (see Fig. 3), electrostatic interaction between charged (~8 × 104e+; e, elementary charge) test mass and grounded source mass56, and an unexplained force proportional to the separation.
The charge distribution induced by a 3 × 104e+ charged source mass on the grounded test mass was determined by a finite-element simulation (COMSOL; shown for 1 mm surface separation as an example). With the grounded, conductive electrostatic shield in place, many mirror charges are induced in the unmodulated shield, resulting in a d.c. force. The induced surface charge density (colour-coded) on the position-modulated source mass is suppressed by a factor of approximately 100. These charges are screened by the shield, resulting in further suppression of the exerted force. The actual suppression could not be quantified owing to numerical inaccuracies.
Our finite-element simulation of the electrostatic interaction of a charged test mass (105e+) and a grounded source mass was validated using the analytical method described in ref. 56. When inserting a 150-μm-thick, conductive electrostatic shield between them, the electrostatic force exerted onto the test mass is dominated by the test mass–shield interaction, that is, it becomes independent of the source-mass position. Residual fluctuations of the force are on the 3–10% level of gravity, without a clear source position dependence, and are expected to stem from numerical errors.
a, The time-resolved amplitude spectral density of the horizontal ground motion in Vienna, recorded by an STS2 broadband seismometer, shows a strong frequency dependence of the variability. A rise of the noise floor is observed throughout the frequency range, starting from midnight (dark blue) to noon (light blue), but is most prominent at 30–40 mHz. By contrast, the noise floor at the microseismic peak, at around 70 mHz, varies only slightly. The modulation frequency of the drive mass is marked by a vertical dashed line. Each spectrum corresponds to a single 2-h measurement. b, The amplitude spectral density of magnetic field variations at the site of the experiment were measured with a three-axis magnetometer (Stefan Mayer FLC3-70; z = vertical, y = along source–test mass axis). The signal at 10 mHz is explained by the period of the traffic light at a nearby crossing, regulating car traffic as well as trams. The inferred force acting on the pendulum (red line), shows no sign of this signal. Each spectrum corresponds to a single 8,000-s measurement.
a, 2D projection of the test mass. The centre of mass determined by the geometric centre of mass (mean over all image dimensions), as well as the centre of mass calculated by our shape evaluation, are indicated (‘Volume COM’). The centre of mass used in the determination of G is indicated by a blue cross. b, The radius measured from the geometric COM over all angles is shown.
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Westphal, T., Hepach, H., Pfaff, J. et al. Measurement of gravitational coupling between millimetre-sized masses. Nature 591, 225–228 (2021). https://doi.org/10.1038/s41586-021-03250-7