Measurement of Gravitational Coupling between Millimeter-Sized Masses

Tobias Westphal,1 Hans Hepach,1, ∗ Jeremias Pfaff,2, ∗ and Markus Aspelmeyer1, 2, 3

1Institute for Quantum Optics and Quantum Information (IQOQI) Vienna,Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria2Vienna Center for Quantum Science and Technology (VCQ), Faculty ofPhysics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria

3Research Platform TURIS, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria

Gravity is the weakest of all known fundamental forces and continues to pose some of the mostoutstanding open problems to modern physics: it remains resistant to unification within the standardmodel of physics and its underlying concepts appear to be fundamentally disconnected from quantumtheory1–4. Testing gravity on all scales is therefore an important experimental endeavour5–7. Thusfar, these tests involve mainly macroscopic masses on the kg-scale and beyond24. Here we showgravitational coupling between two gold spheres of 1 mm radius, thereby entering the regime ofsub-100 mg sources of gravity. Periodic modulation of the source mass position allows us to performa spatial mapping of the gravitational force. Both linear and quadratic coupling are observed asa consequence of the nonlinearity of the gravitational potential. Our results extend the parameterspace of gravity measurements to small single source masses and small gravitational field strengths.Further improvements will enable the isolation of gravity as a coupling force for objects below thePlanck mass. This opens the way to a yet unexplored frontier of microscopic source masses, whichenables new searches of fundamental interactions25–27 and provides a natural path towards exploringthe quantum nature of gravity28–31.

The last decades have seen numerous experimentalconfirmations of Einstein’s theory of relativity, our bestworking theory of gravity, by observing massive astro-nomical objects and their dynamics5,8,9. This culmi-nated in the recent direct detection of gravitational wavesfrom the merger of two black holes10, and the directimaging of a supermassive black hole11. During thesame time, Earth-bound experiments have continuouslybeen increasing their sensitivity for gravity phenomenaon laboratory scales, including for general relativisticeffects12,13, tests of the equivalence principle6,14,15, preci-sion measurements of Newton’s constant16–18 or tests ofthe validity of Newton’s law at µm-scale distances19–21.While test masses in such experiments span the wholerange from macroscopic objects to individual quantumsystems12,13,22,23 the gravitational source is typically ei-ther Earth or masses on the kg-scale and beyond24.A yet unexplored frontier is the regime of microscopicsource masses, which enables new searches of fundamen-tal interactions25–27 and provides a natural path towardsexploring the quantum nature of gravity28–31. Here weshow gravitational coupling between two gold spheres of1 mm radius, thereby entering the regime of sub-100 mgsources of gravity.

Experiments with smaller source masses are exces-sively more difficult – the gravitational force generatedat a given distance by a spherical mass of radius Rshrinks with R−3 – and hence only few experiments todate have observed gravitational signatures of gram-scalemass configurations21,32,33. In one case, a hole-pattern

∗These authors contributed equally to this work

in a rotating, 5 cm diameter attractor disk made fromplatinum generated a periodic mass modulation of a fewhundred mg, which was resolved in a torsional balancemeasurement21. In another case, a single 700 mg tung-sten sphere was used to resonantly excite a torsionalpendulum33. Isolating gravitational interactions gener-ated by even smaller, single source-mass objects is achallenging task as it requires increasing efforts to shieldresidual contributions from other sources of acceleration,in particular of seismic and electromagnetic nature34.Also, resonant detection schemes, which are typically em-ployed to amplify the signal above readout noise, amplifydisplacement noise as well and hence don’t yield any gainin terms of separating the signal from other force noisesources. We are overcoming this limitation by combiningtime-dependent gravitational accelerations with an off-resonant detection scheme of a well-balanced differentialmechanical mode and independent noise estimation.

Experiment

In our experiment, the gravitational source is a nearlyspherical gold mass of radius r = (1.07 ± 0.04) mm andmass ms = (92.1±0.1) mg. A similarly sized gold sphereacts as test mass with mt = (90.7± 0.1) mg. The idea isthat a periodic modulation of the source mass positiongenerates a time-dependent gravitational potential atthe location of the test mass, whose acceleration ismeasured in a miniature torsion pendulum configuration(Figure 1). The experiment is conducted in highvacuum (6 × 10−7 mbar), which minimizes residualnoise from acoustic coupling and momentum transfer ofgas molecules34,35. To prevent non-linear coupling of

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Figure 1: (a) Schematic of the experiment. The torsion pen-dulum, which acts as transducer for gravitational accelera-tion, consists of two r ≈ 1 mm gold spheres held at 40 mmcenter distance by a glass capillary. One mass serves asmt = 90.7 mg test mass, the other as ma = 91.5 mg counter-balance that provides vibrational noise common mode rejec-tion. A 4µm diameter silica fiber provides a soft f0 ≈ 3.6 mHztorsional resonance separated well from other degrees of free-dom. The torsion angle is read out via an optical lever di-rected to a quadrant photodiode. The gravitational interac-tion is modulated by harmonically moving the ms = 92.1 mgsource mass by ≈ 3mm at a frequency fmod = 12.7 mHz wellabove the torsion resonance using a shear piezo. Direct elec-trostatic coupling is suppressed by discharging and a 150µmthick Faraday shield. (b) Source mass on a 1 Euro Cent coin.(c) Photo of the torsion pendulum and the mounted sourcemass.

high-frequency vibrations into the relevant low-frequencymeasurement band around the modulation frequencyfmod = 12.7 mHz, the pendulum support structure isresting on soft, vacuum-compatible rubber feet36.We optically monitor the angular deflection of thependulum, which provides a calibrated readout of thetest mass motion with a detector-noise limited sensi-tivity of ≈ 2 nm/

√Hz. Figure 2 shows an amplitude

spectrum of the test mass displacement. The torsionalmode resembles a damped harmonic oscillator withresonance frequency f0 = 3.59 mHz and mechanicalquality factor Q = 4.9. It is well decoupled from otheroscillation modes of the pendulum, which do not occurbelow 0.5 Hz. Diurnal variations in the low-frequencynoise-floor limit the times of best sensitivity to thosehours during the night where local public transportand pedestrian- and car-traffic are minimized (typicallybetween midnight and 5 a.m.). There, the test massoscillation was mainly governed by thermal noise35.

Figure 2: The rotation of the torsional oscillator measuredover a period of 13.5 hours is calibrated as test mass displace-ment (blue) and respective exerted force (light red). Themeasurement is limited by both thermal noise (dashed) and

diurnally varying white force noise below 100 mHz, whilewhite displacement readout noise (dotted) dominates above.The torsional motion is decoupled well from other oscillatormodes, starting at 0.5 Hz. Force residuals (yellow) representthe difference between measured and expected gravitational

force. The spectrum shows gravitational coupling atfmod = 12.7 mHz modulation frequency plus the nonlinearity

of the gravitational potential at 2fmod = 25.4 mHz wellabove the oscillator’s resonance frequency f0 = 3.59 mHz.

The flatness of the force residuals indicate that both signalsare in agreement with the expected gravitational signal.

Other sources of nonlinearity are found to be negligible35.

The corresponding force spectrum was obtained bydeconvolution of the test mass displacement time serieswith the inverse of the deduced mechanical susceptibility.It exhibits a flat noise floor that allows off-resonantdetection of test mass accelerations at frequencies upto 0.1 Hz at a sensitivity better than 2 × 10−11 m/s2

within half a day. We use this to obtain a live-estimateof the pendulum noise conditions, which is combinedwith inverse variance weighting of data to optimize theinformation obtained per experimental run.

The source mass is mounted on a 300µm diameter ti-tanium rod that is connected to a bending piezo, whichprovides more than 5 mm travel range in the vicinity ofthe test mass. The geometric separation of the masseswas determined with an accuracy of 20µm by linkingthe drive-piezo signal to position information from videotracking through a template matching algorithm35. Dur-ing the experiment, the center distance between sourceand test mass was varied between 2.5 mm and 5.8 mm,with a minimal surface distance of ≈ 0.4 mm.To isolate gravity as a coupling force we need to minimizeother influences on the test mass. In addition to seismicand acoustic effects these are predominantly electromag-netic interactions. We ground the source mass by directlyconnecting it to the vacuum chamber. The test mass isdischarged to less than 8 × 104 elementary charges us-ing ionized nitrogen37, a method that was developed forcharge mitigation in interferometric gravitational wave

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detectors10. Further shielding is provided by a 150µmthick conductive Faraday shield of gold-plated aluminumthat is mounted between source- and test mass. Inthat way, electrostatic forces were suppressed to well be-low 3 % of the expected gravitational coupling strength.Other shielding measures include housing the source-mass drive piezo inside a Faraday shield to suppress cou-pling via the applied electric fields, as well as gold-coatingand grounding most surfaces inside the vacuum chamberto exclude excitation from time-varying charge distribu-tions. We also rule out the presence of relevant mag-netic forces by independently measuring the magneticpermeabilities of the masses. As expected, permanentmagnetic dipoles are negligible in both (diamagnetic)gold and (paramagnetic) titanium. Coupling via inducedmagnetic moments, either from Earth’s magnetic field orfrom low-frequency magnetic noise originating in nearbyurban tram traffic, is also found to be orders of magni-tude below the expected gravitational coupling strength.In the current experimental geometry, Casimir forces arenegligible, although they will likely become a dominantfactor for significantly smaller masses38. Another rele-vant noise source is Newtonian noise, which is caused bynon-stationary environmental gravitational sources andcannot be shielded. For comparison, at a center separa-tion of 2.5 mm the static gravitational force between ourmasses is expected to be 9× 10−14 N. The same gravita-tional force is exerted on the test mass by a human exper-imenter standing at a distance of 2.5 m, or by a typicalVienna tram at 50 m distance from the laboratory build-ing. Consequently, our experiment experiences a com-plex low-frequency gravitational noise of urban origin.It is obvious that such gravitational noise sources willpose an increasing challenge for future experiments. Atpresent, our torsion pendulum is sufficiently small com-pared to the distance of typical environmental sources tobe insensitive due to common mode rejection.

Results

We observe gravitational coupling between the twomasses by harmonically modulating the source mass po-sition xs = d0 + dm cos(2πfmodt) at a frequency fmod =12.7 mHz, well above the fundamental torsional pendu-lum resonance (mean center-of-mass distance d0; mod-ulation amplitude dm). In this frequency regime thetest mass response becomes independent of the pendulumproperties and behaves essentially as a free mass. Accord-ing to Newton’s law the source mass generates a gravita-tional acceleration of the test mass of aG = G ms

xs(t)2(G:

Newton’s constant). The 1/r dependence of the grav-itational potential results in higher-order contributionsto the coupling at multiples of the modulation frequencyfmod. Figure 2 shows the measured force spectrum of thetest mass for a distance modulation of dm ≈ 1.6 mm at a

Figure 3: The probability density distribution of exertedforce versus source- test mass separation (a) shows the spa-tial non-linearity of the source-mass potential. This methodmakes full use of our measurement data by taking into ac-count weighting with the current noise conditions. The ap-parent width of the distribution is mainly given by noiseat frequencies other than the modulation. (b) The actualmeasurement precision becomes apparent by extracting themean of the force measured at given distances (red points)along with their standard deviation obtained by bootstrap-ping (red band). Note that the small statistical error col-lapses the error band to the narrow red line. A full weightedfit of the 13.5 h long data segment with Newtonian gravity(black line) yields Gfit = (5.89 ± 0.20) × 10−11 N m2/kg2

which is in agreement with our long-term combined value(see Figure 4). Comparison with the literature (CODATA)value GCODATA = 6.67 × 10−11 N m2/kg2 (dashed line) andour known systematic errors shows that influences other thangravity are below 10 %.

center separation of d0 ≈ 4.2mm, in which we resolve thelinear and quadratic acceleration modulations at fmod

and 2fmod, respectively. The flat noise residuals clearlyindicate that the observed peak heights agree well withthe expected force modulation due to Newtonian gravity.This confirms the gravitational origin of the interaction.To more accurately quantify the strength of the cou-pling we correlate the measured separation xs betweensource and test mass with the independently inferredforce on the test mass. This provides us with a position-dependent mapping of the gravitational force (Figure 3).All data evaluation is carried out in post-processing andusing non-causal zero-phase filtering35. Each data set,which consists of up to 13.7 hours long measurements,is fitted using the expression for Newtonian gravity toobtain a value for the measured coupling strength.

Figure 4 summarizes the results of a series of mea-surements that were taken during the seismically quietChristmas season. The weighted combination of mea-surement runs yields a coupling constant of Gcomb =(6.04 ± 0.06) × 10−11 m3kg−1s−2, with a statistical un-certainty of 1 %. The observed coupling deviates from

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Figure 4: Over a stretch of three weeks during the seismi-cally quiet Christmas season several measurement runs wereconducted, with electrostatic coupling being excluded dur-ing gray periods. The dataset presented in Figure 2 and3 is highlighted in black (run 1). The statistical error (1σerror bar) of the best fitting Newton-like coupling of eachdata segment (dark lines) varies due to highly non-stationarynoise in the measurement band. Combining individual mea-surements weighted by their respective data quality yieldsGcomb = (6.04±0.06)×10−11 N m2/kg2. The plotted system-atic uncertainty (red dashed) includes the identified system-atic influences from the experiment as summarized in Table II.We explicitly do not claim a significant systematic deviationof the combined coupling value from the CODATA recom-mended value (blue dashed). The experiment and its dataevaluation was mainly designed with the reduction of statis-tical errors in mind, and the deviation is fully covered by theknown experimental systematics.

the recommended CODATA value for Newton’s constant(GCODATA = 6.67430(15)×10−11 m3kg−1s−2) by around9 %. This offset is fully covered by the known systematicuncertainties in our experiment, which include unwantedelectrostatic, magnetic and gravitational influences fromthe masses and supports as well as geometric uncertain-ties in the center of mass distance due to the actual shapeof the masses (we provide a detailed list in the Methodssection35). Our results show that we are able to iso-late gravity of a single, small source mass, with otherinfluences being below the 10 % level. The small statis-tical error underlines the precision character of our mea-surement and our ability to measure even smaller sourcemasses with our approach in the future.

Discussion and Outlook

We have demonstrated gravitational coupling betweena test mass and a 90 mg spherical source mass. To ourknowledge this is the smallest single object whose grav-itational field has been measured. Our measurementsresolve an acceleration modulation of ≈ 3 × 10−10 m/s2

at the drive frequency fmod with high accuracy, quan-tified by a systematic uncertainty of ≈ 3 × 10−11 m/s2,and high precision, quantified by a statistical uncertaintyof ≈ 3 × 10−12 m/s2 over an integration time of 350 h.

Contributions of non-gravitational forces could be keptbelow 10 % of the observed signal. Our results extendgravity experiments to the regime of small gravitationalsource masses, with the potential of going significantlysmaller. As our current sensitivity is limited by the ther-mal noise floor of the torsion pendulum when operatedunder optimal ambient conditions, it can be readily im-proved by increasing the mechanical quality factor Q.Probing gravity of objects even smaller than the Planckmass mP ≈ 22µg = 2.4 × 10−4ms should become pos-sible for Q & 20, 00035. The thermal noise limited forcesensitivity of these miniature torsion pendula can be im-proved even further by dissipation dilution as recentlydemonstrated39. Another interesting route is to use lev-itated test masses, which provide almost perfect chargemitigation and similar acceleration sensitivities40–44. Tomake full use of improvements in thermal noise it is im-portant to understand and to mitigate other, predomi-nantly anthropogenic, low-frequency noise sources thatcurrently dominate our measurement performance mostof the time. Possible countermeasures include properchoice of a remote laboratory location or pushing the ex-periment to much higher pendulum frequencies34,45. Go-ing to smaller masses will involve additional challengesas other noise contributions will play an increasing role.For example, even for the ideal case of electrically neutralsource and test mass, Casimir forces will become relevantat distances below 100µm. Note however, that electro-magnetic shielding combined with a signal modulationcan overcome this particular challenge. We also note thatour method allowed us to directly measure the position-dependence of the gravitational force, which constitutesa 1D-mapping of the gravitational field strength. A morecomplex test mass geometry may enable a full mappingof the metric tensor of such small source masses46.Our experiment provides a viable path to enter and ex-plore a new regime of gravitational physics that involvesprecision tests of gravity with isolated microscopic sourcemasses at or below the Planck mass. This opens up newpossibilities. For example, such measurements offer a dif-ferent approach to determine Newton’s constant34, whichto date remains the least well determined of the funda-mental constants47. In general, miniaturized precisionexperiments may allow tests of the inverse square law ofgravity at significantly smaller scales than possible to-day (the best reported value is on the 50µm-scale21),thereby probing, for example, fundamental features ofstring theory such as extra dimensions and masslessscalar particles48. Small source masses also allow to testconsequences of new speculative scalar fields that havebeen discussed in the context of dark energy25–27. Morestringent parameter constraints for so-called Chameleonforces may already be possible for source masses at thelevel investigated here25. Another example is experimen-tal access to gravitational accelerations at or even belowthe level of galaxy rotations. It has been suggested to

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modify Newtonian dynamics in this regime as an alterna-tive to dark matter scenarios49, and some of these modelscan therefore be directly tested50. Finally, the ongoingdiscussion on the quantum nature of gravity has revived agedankenexperiment by Feynman to probe gravitationalcoupling between quantum systems28–30. The underlyingquestion is whether it is possible to probe gravitationalphenomena that cannot be explained by a purely classi-cal source mass configuration1. Such experimental testshave thus far been elusive, since they require the abil-ity to prepare quantum states of motion of the sourcemass. Since decoherence phenomena scale dramaticallywith system size, isolating gravity of microscopic massesis a necessary prerequisite for such future experiments.Our result provides a first step in this direction – al-though we stress that, with current quantum experimentsusing sub-micron sized objects of 10−17 kg51,52, addingquantum coherence to experiments in the relevant massregime is yet a completely different experimental chal-lenge.

Acknowledgements We thank Eric Adelberger,Aaron Buikema, Peter Graham, Nikolai Kiesel, NorbertKlein, Davide Racco and Jonas Schmole for stimulat-ing discussions. We are grateful for the suspension fiberprovided by Arno Rauschenbeutel and Thomas Hoinkesand for the exceptional mechanical design assistance byMathias Dragosits. This project was supported by theEuropean Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation program(Grant Agreement No. 649008, ERC CoG QLev4G),by the Austrian Academy of Sciences through the Inno-vationsfonds Forschung, Wissenschaft und Gesellschaft,by the Alexander von Humboldt foundation through aFeodory Lynen fellowship (T.W.) and by the AustrianFederal Ministry of Education, Science and Research(project VCQ HRSM).

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[65] Shimoda, T. and Ando, M., Nonlinear vibration trans-fer in torsion pendulums. Classical and Quantum Gravity36(12), 125001 (2019).

[66] Lide, D. R., CRC handbook of chemistry and physics,volume 85, (CRC Press 2004).

[67] Shih, J. W., Magnetic properties of gold-iron alloys.Physical Review 38(11), 2051 (1931).

[68] Henry, W. and Rogers, J., XXI. The magnetic suscepti-bilities of copper, silver and gold and errors in the Gouymethod. Philosophical Magazine 1(3), 223–236 (1956).

[69] Sushkov, A., Kim, W., Dalvit, D., and Lamoreaux, S.,Observation of the thermal Casimir force. Nature Physics7(3), 230–233 (2011).

[70] Emig, T., Graham, N., Jaffe, R. L., and Kardar, M.,Casimir forces between arbitrary compact objects. Phys-ical Review Letters 99(17), 2–5 (2007).

[71] Beer, W. et al., The METAS 1 kg vacuum mass com-parator - adsorption layer measurements on gold-coatedcopper buoyancy artefacts. Metrologia 39(3), 263–268(2002).

[72] Glaser, M. and Borys, M., Precision mass measurements.Reports on Progress in Physics 72(12), 126101 (2009).

8

SUPPLEMENTARY MATERIAL

TORSION PENDULUM

Geometry

The torsion pendulum consists of the test mass (mt =90.7 mg) and a counterbalance mass (ma = 91.5 mg).The masses are rigidly connected by a 37.9 mm long,square-shaped glass capillary with an outer dimension of0.4 mm and 0.1 mm wall thickness (9.6 mg). It is sup-ported by a 35 mm long quartz fiber pulled down to3.6µm diameter over a 20 mm section at the center formaximal torsional compliance. The 5 − 10 mm long, upto 50µm wide taper region below this torsion section in-creases pitch- and roll-stiffness for better mode separa-tion and provides a bigger cross section for attachment tothe capillary. The attachment point was tuned to mini-mize static roll to < 10 mrad. A 5.85 mm wide, 0.3 mmthin, hand ground and gold coated mirror (27.0 mg) wasattached underneath the capillary for deflection readoutof the torsion (yaw) angle. This turned out to pointdownwards by 50 mrad in pitch, potentially coupling non-torsion motion, in particular roll, into the yaw readoutas well. All connections were established by UV-curingadhesive. The fundamental mode frequency f0 is deter-mined by the stiffness of the suspension fiber and theeffective mass meff = 183.8 mg ≈ mt + ma = 182.2 mg.The effective mass is almost minimal, i.e. hugely dom-inated by the two gold masses for the balanced, differ-ential mode used here. It can only be reduced furtherby using a single mass in a non-differential setup, whichwould then, however, be subject to the full environmentaltranslational acceleration.

Pendulum characterization

Since we aim to characterize the gravitational force ex-erted by our source mass, we specify the test mass motionby its spatial displacement rather than by the pendulumdeflection angle. The amplitude spectrum of the testmass displacement (shown in Figure 2) follows the slopeof the mechanical susceptibility of a damped harmonicoscillator1 which is solely defined by its eigenfrequencyf0, mechanical quality factor Q, and mass meff . Thefirst two were obtained by a fit to the spectrum to bef0 = 3.59 mHz and Q = 4.9. Both parameters were ver-ified during measurement runs and independently by a

1 Due to the low frequency of observation and the extremely goodfrequency separation of mechanical modes of a torsion pendulumin general, all transfer functions except the mechanical suscepti-bility of the yaw mode can be neglected.

ringdown measurement with relatively small amplitude.For larger amplitudes, we observe a significant decreasein mechanical losses. Values in excess of Q = 2×104 havebeen observed for larger amplitudes (≈ π rad) and differ-ent rest positions. The effective mass has been deter-mined by weighing and modelling all components exceptfor the little amounts of glue that have been used.

The drive frequency fmod was chosen to lie wellabove the torsion resonance (≈ 3.6 mHz), yet far awayfrom other, highly excited modes (e.g. around 500and 700 mHz) and sufficiently below the readout noise(≈100 mHz) such that the 2fmod (and eventually the3fmod) signal could be resolved. The intention of theexact choice of 12.7 mHz to be such an odd number is toprevent frequency mixing occurring in nonlinear systemssuch as discretization, clipping etc. of narrow band sig-nals to accidentally match with the gravitational signal.

By choosing the signal frequency in such a way, thetest mass effectively behaves like a free mass (meff )2.Hence, in this regime the response is mostly independentof the resonant enhancement factor Q and the resonancefrequency f0.

Prospects towards smaller masses

The experiment presented in this publication verifiesour approach to isolate gravitational coupling of 100 mgsized objects. Going forward, the achieved signal to noiseratio (SNR) of 6.04/0.06 ≈ 100 already allows measuringgravity exerted by 1 mg sized objects with SNR ≈ 1 at ameasurement duration of ≈ 150 h. Note that our currentmeasurement runs undergo diurnal variations in the noisefloor, to which the thermal noise of our Q = 4.9 oscilla-tor assembly poses a hard limit. In order to make betteruse of our measurement time (SNR ∝ Tmeas) we have toimprove our understanding and the reduction of couplingof daytime environmental noise. Furthermore, mechan-ical quality values in excess of Q = 2 × 104 have beenobserved even for this particular pendulum. As the off-resonant thermal noise amplitude decreases with 1/

√Q,

this provides a factor√

20000/5 ≈ 65 improvement overthe current sensitivity.

Combined, this shows the potential of our approachfor sensing the gravitational field of a Planck mass sizedobject (mP = 22µg).

2 More precisely, if the signal frequency is well above the mechan-ical linewidth γ0/2π = f0/Q, i.e. fmod � γ0/2π ≈ 0.7 mHz

9

CALIBRATION

Optical lever readout

The angular position of the torsion balance is read outby means of an optical lever using 1.8 mW of InnolightMephisto laser light at an angle of incidence of≈ 50 mrad.The signal is obtained as differential photo voltage of anoff the shelf Thorlabs QPD80A quadrant photo detec-tor (QPD), which provides high dynamic range (0.2 −0.5 Vx/Vsum) and low noise (≈ 2×10−9 Vx/Vsum/

√Hz)

in its approximately linear region of 0.5 mrad. The datais sampled by a Picoscope 4824 at 250 and 286 Samples/s(anti-aliasing filter at 132 Hz) and digitally downsam-pled by a factor 10. The ADC provides 12 bit resolution(±10 V full scale → 4.9 mV resolution). Although weare oversampling the signal already, the high frequencyquantization noise was shown to be improved when us-ing even higher data rates. Reducing the full range wouldalso help, yet we require it to cover large low frequencyamplitudes, in particular during noisy times of data ac-quisition.

For future experiments, the actual (not quantizationnoise limited) readout sensitivity can be improved by us-ing interferometric methods, although at the cost of lim-ited absolute range. Systematic calibration errors can besignificantly reduced by exchanging the current read-outwith a perfectly linear, self-calibrating auto collimator atthe expense of increased complexity and limited oversam-pling capability53.

Position calibration

Position calibration of the test mass motion was doneby measuring the pendulum yaw deflection both viathe optical lever and, independently, via video tracking.For the latter measurement, the torsion pendulum wasrecorded from the top with a resolution of 24µm/px withboth gold spheres being pattern tracked to obtain thedeflection (compare Section ). Each method contributestheir own source of error: for small amplitudes, the videotracked deflection angle is relatively noise, while at largedeflections the optical transfer function of the QPD read-out becomes nonlinear. For our calibration, the pendu-lum was excited to an amplitude of ≈ 125µm, largelyexceeding the linear detection range.

In the ideal case the rotation angle and the normal-ized QPD signal are related via the Gauss error func-tion. In our case, clipping and other effects deterio-rate this relation at large amplitudes. Since our exper-iment is confined to small deflections we only require agood calibration in the central QPD range. Therefore, a3rd order polynomial was fitted to the data surroundingVx/Vsum ≈ 0.09 to serve as yaw angle calibration for allour acquired QPD data.

Figure 5: The video tracked yaw angle of the torsion pendu-lum is compared to the QPD readout (horizontal differencesignal normalized by the total sum signal to reduce laser inten-sity noise coupling). A third order polynomial fit to this dataprovides our QPD signal calibration. The occurrence proba-bility accumulated over all our data runs shows the region inwhich a calibration is required, i.e. Vx/Vsum ∈ [−0.4, 0.2].

Force calibration and force-noise estimate

To infer the force exerted on the test mass we decon-volve the test mass position time series with the inverse ofthe previously deduced mechanical susceptibility. A com-pensation low-pass has been applied at 1 Hz to preventinstabilities from infinite response around the Nyquistfrequency or above.

A major feature of off-resonant detection is in our casethe flatness of the force noise floor around the signal.This allows to obtain a live-estimate of the current noiseconditions without additional sensors. Known narrowband signals such as fmod and harmonics are removedfrom the inferred force signal and the instantaneous am-plitude in the 5−75 mHz band is obtained by the absolutevalue of the signal plus its Hilbert transform. Specifi-cally, evaluating amplitude (real part of complex signal)and change of amplitude (imaginary part) gives a con-tinuous estimate for the noise coupling into the system.The goal of the algorithm is to identify excess excitationsegments of the data that occur over a long duration,in particular to minimize the data degradation from thesometimes up to tenfold increased noise condition duringdaytime. We are therefore low-pass filtering (moving av-erage, 1/fmod ≈ 79 s) the instantaneous amplitude anduse the result for inverse variance weighting of the mea-surement data. Compared to completely vetoing dataduring bad conditions this allows to make use of the fullmeasurement time under these extremely non-stationaryconditions. Employing this method we could optimallycombine a total of 350 hours of non-stationary measure-ment time which are worth ≈ 150 hours of higher qualitydata as presented in Figure 2 and 3 into the single eval-uation shown in Figure 4.

10

Non-stationary environmental conditions

Each data chunk contains up to 13:53 h of data (5 ×104 s). In order to display the time evolution of our noisewe perform a spectral density estimate on 1 h segmentsusing Welch’s method with 20 min long segments and50 % overlap (Figure 6)3. The low frequency noise (be-

10 3 10 2 10 1 100Frequency / Hz10 3

10 1

101

ASD

/ (m

/Hz

)

a

17-18 h18-19 h19-20 h20-21 h21-22 h

22-23 h23-00 h00-01 h01-02 h02-03 h

03-04 h04-05 h05-06 hThermalReadout

03:15 h +300 s +600 s +900 s +1200 s +1500 s

0.5

0.0

Volta

ge /

V

b

Photodiode (PD)PD(5-75 mHz) × 20Drive Voltage

Figure 6: (a) The displacement spectral density estimate ofrepresentative data taken between 26th and 27th of Decembershows stationary readout noise at high frequencies and non-stationary white force noise at low frequencies. The level ofthe latter varies up to tenfold, consistently being the lowestat 0 − 5 a.m., in particular in the night after public holidaysor Sundays / before normal workday. During such a quietperiod 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 frequencyrange from 5−75 mHz with a 6th order butterworth filter andmagnified by a factor of 20.

low ≈ 100 mHz) is dominated by actual motion of ourtorsion pendulum. While we reach the theoretical ther-mal noise limit of the damped harmonic oscillator duringquiet times, we observe up to 10 times higher noise levelsduring other times of the day. The origin of this noisehas not been fully identified yet (see also Section andfor more details). Still, it always exhibits the same pat-tern: within 1 h segments the noise appears to be whitein terms of force exerted onto the test mass. Yet its levelslowly varies. The noise power is well correlated with thediurnal variability of anthropogenic noise, be it in higherfrequency seismic activity (evidenced by the excitation of

3 Note that the spectral resolution is inverse proportional to theintegration time and therefore worse for the segmented spectrathan in 2. Also the gravitational signal is smaller for these shortsegments as less signal is integrated. If we chose the amplitudespectrum representation instead, our signal would remain at thesame height but the noise floor would rise as less noise is averagedout.

the 0.7 Hz mode), or in 10 mHz magnetic fields. Nightsare better than daytimes, even with nobody being presentin the lab. Weekdays present worse conditions comparedto weekends and public holidays. Therefore, the Christ-mas 2019 time was exceptionally well suited for the lownoise measurements presented in this publication.

At higher frequencies above ≈ 100 mHz the noise flat-tens out (except for higher order modes) at a readoutnoise level of 2 × 10−9 m/

√Hz. Since this noise further

improves with increased sampling frequencies it is sus-pected to be quantization noise.

DATA EVALUATION

Data handling

All data is sampled by a Picoscope 4824 at 250 or286 Samples/s using the internal clock. The ADC pro-vides 12 bit resolution (±10 V full scale → 4.9 mV reso-lution). A typical data trace spans just 5× 104 s so justshort of 14 hours. An analog anti-aliasing filter at 132 Hzis applied to the horizontal difference of the photodiodequandrants to get rid of high 50 Hz harmonics (roomlight) and other narrow band large amplitude signals.The QPD signals are digitally de-rotated by ≈ 20 radwhich maximizes the sensitivity to the torsion motionwhile minimizing the impact of pitch motion (verticalspot motion). The calibration procedure described ear-lier provides a signal in terms of test mass displacement.From this, the force signal is deduced as described be-low. At this point the momentary noise estimation signalis deduced processed alongside with the other channels.Only thereafter, the signals are digitally down-sampledby a factor 10 (a 10th order Chebyshev Type 1 filter at0.8fs,new with the response at fmod normalized to unityis applied before the decimation) to reduce the amountof data. The initial oversampling reduces digitizationnoise (optimally by a factor

√10). Finally, the signals

are cropped, i.e. usually the first 5 % are being removed.Besides eliminating general edge-effects of any filters, thismainly reduces the ring-in effect of the force estimation.

Force inference

In general, a damped harmonic oscillator can be rep-resented by a 2 nd order low pass filter. The mechani-cal susceptibility χ(ω) is defined as the transfer functionfrom externally applied force Fext(ω) to induced displace-ment x(ω).

χ(ω) =x(ω)

Fext(ω)=

1

meff

(−ω2 − iγω + ω2

0

)−1(1)

At a given frequency ω, χ is defined by the (viscous)damping rate γ = ω0/Q, the (un-damped) resonance fre-

11

quency ω0 and the effective mass meff . We map this toa zero pole gain (zpk) representation

χ!= k (s− p0)(s− p∗0) (2)

in Laplace domain (s = σ+iω, choosing σ = 0) by meansof a pair of complex conjugate poles (p0, p

∗0) and a DC-

response k (which contains the unit conversion). We find

k =1

meffp0 = γ/2±

√w2

0 − (γ/2) . (3)

As long as the zpk representation of a system is known,then the time domain response x(t) can be inferred fromthe driving force Fext(t) without solving the equationof motion by the following recipe e.g. using MATLAB ’sbuilt-in functions:

1. convert the zpk model to its transfer functionform by [den,num] = zp2tf(z,p,k) (appropriatek must be chosen to take into account the re-normalization by placing poles/zeros!)

2. convert continuous time model to filter coefficientsof discrete time model sampled at frequency fs by[bd,ad] = bilinear(den,num,fs)

3. apply the filter by x = filt(bd,ad,F)

While steps 1&2 are more or less technical re-definitions,step 3 convolves the force time series with the impulseresponse in a very resource friendly manner.

We use this method in reverse by inverting the me-chanical susceptibility, which is equivalent to exchangingpoles and zeros p0 → z′0, z0 → p′0 and k → 1/k′ to inferthe force Fext required to produce the measured displace-ment x(t). In order to avoid an un-physical (infinite)response at or above the Nyquist frequency fN = fs/2from the two zeros, we have to limit the response bya pair of compensation poles. By allowing them to becomplex as well, their phase loss around the detectionband can be limited. In step 2, we apply pre-warpingat 100 mHz to reduce malicious effects onto the transferfunction. The compensation poles were chosen to lie atwcomp = 2π× 1 Hz with Qcomp = 3. This choice was ver-ified not to influence the measured gravitational signalsignificantly (compensation pole in Table II).

Data visualization

The spectrum estimate shown in Figure 2 was pro-duced by Welch’s method (pwelch). Each time series isdivided into five blocks with 50 % overlap and taperedby a Hanning window. The spectrum estimates of theblocks are then averaged. Hence, the noise floor in thefigure is elevated but smoothed out compared to perform-ing a single spectral estimate on the whole trace at once.

The residuals shown in the plot are a spectral estimateof the difference of the measured force signal (red trace)and the expected/nominal gravity signal. By perform-ing the subtraction before the spectral (power) estimate,these residuals become sensitive to the relative phase ofthe two signals.

As our goal is to quantify a (nonlinear, but stationary)relation between source-test mass separation and exertedforce, plotting one against the other comes with the im-plicit assumption of equal significance of each data point.As our momentary noise estimation channel, however,provides us with means of judging the significance (intherms of signal to noise ratio) of each point, we chose toweight the data points in Figure 3 with this significanceto obtain the most general visual representation of theforce versus separation relation. The hidden assumptionhere is that the momentary noise level estimation pro-cedure does not introduce systematics into the quantita-tive data evaluation (see next step). The authors triedto verify this by checking that there is no quantitativedependence of the results from noise estimation filter pa-rameters.

Only after this most general data representation, therelation is subjected to assumptions by using Newton’slaw to quantify the relation. Hence, the width of thesignal band is determined by all noise within the band-width of evaluation (5 − 75 mHz bandpass, see above)and not only at the signal frequency or its harmonics.While subjecting data observed in dedicated high preci-sion experiments to the rigid corset of assumptions (fit-ting parameters) is a well established method, we believethe least assumptions data representation approach ofFigure 3 to be extremely valuable in a yet unexploredregime of gravitational sensing.

Data evaluation

In order to quantify the measured gravitational forcewe assume the validity of Newton’s law and treat themasses as point masses concentrated at their respectivecenters of mass. Strictly speaking, this assumption isvalid only for perfectly rotationally symmetric masses.Possible deviations due to bumpyness are considered assystematic errors. We allow the gravitational constantas fitting parameter resembling the strength of 1/x2s cou-pling. We have to further allow for a distance indepen-dent force resembling the measured force averaged overone signal period as the experiment was not designed tomeasure the DC-attraction, i.e. the force with relation tothe source mass being infinitely far away.The statistical uncertainty of data can be inferred by re-sampling it. While often bootstrapping is used, it turnsout that without further pre-conditioning high frequencytemporal correlations such as high frequency vibrationalmodes spoil the results as the data points cannot be

12

regarded as sufficiently independent. Subsampling, incontrast, holds under weaker assumptions and appearsto work for our data as short term correlations remainwithin the same segment.Each time trace is cut into N equally long segments.For each segment, the coupling strength Gi is fittedwith inverse variance weight factors described in the pre-vious section non-stationary environmental conditions.Furthermore, a common force offset F0 is fitted to allsegments (allowing each segment to have its own DC-force F0,i leads to similar results, but is less physi-cal as this force shouldn’t drift). The weighted meanG =

∑iGi/σ

2i , where 1/σ2

i are the normalized inversevariance weights for the segment, gives the same resultas a weighted fit to the whole trace. From the (inversevariance weighted) standard deviation stdw(Gi) we in-fer the statistical uncertainty of each Gi. The uncer-tainty for the whole trace is then inferred by meansof std(G) = stdw(Gi)/

√Neff with an effective number

Neff of segments with independent noise contributing tothe weighted standard deviation. The number N of seg-ments per trace is varied between 11 and 30. For small N ,drifts within the segments and non-stationarity of noisewithin the segments may become significant, while forlarge N the exact resonance frequency and quality factorof the torsion resonance gain importance as the segmentlength approaches a torsion period. To exclude such ef-fects, for each trace the results were verified not to de-pend on N in order not to introduce systematic errorsby its exact choice and the fit parameters were averagedover the full range of segmentations.

This evaluation gives the independent estimate of thecoupling constant and estimate for statistical uncertaintyfor each trace presented in Figure 4. All deduced cou-pling constants agree well within their statistical errorbudgets. A coupling constant with even lower statisticaluncertainty is obtained by averaging the coupling con-stants of all traces while using the respective deducedstatistical uncertainties as inverse variance weights. Thepower in the time resolved representation of Figure 4is that it shows that there are no major (monotonous)time-dependent systematics such as adsorption, continu-ous electrical charging etc.

VACUUM

The experiment has been conducted in high vacuum(10−6− 10−7 hPa) to reduce excitation from residual gasmolecules and direct momentum transfer. The vacuumchamber is pumped by a turbo molecular pump (PfeifferTMU 071 P) running at 25 Hz during measurement runs.These narrow band vibrations have not been observed toproduce sensitivity degradation of the gravitational sig-nal. The pressure could not be monitored permanentlyduring measurement runs as the Pirani pressure gauge

(Pfeiffer PKR 251 ) had to be dismantled from the cham-ber. It had been observed to constantly charge the tor-sion balance when in use and its relatively strong mag-netic field was suspected to cause magnetic interactionbetween the gold masses.

Impacts of residual gas molecules cause a Brownianforce noise that is derived as an additional dampingrate54:

γwall =P

mvairr2

π3/2r2√2d2ln(r2/d2 + 1)

(4)

(P: pressure; vair = kBT/mair: average velocity of ’airmolecules’ of mass mair; d: separation between the sur-face of the cylinder and the EM-shield; r: cylinder ra-dius; m: cylinder mass; kB : Boltzmann constant; T:Temperature). The additional force noise damping dueto the proximity to the EM-shield is taken into ac-count. Moreover, it assumes the worst case scenarioof a cylindrical test mass, which would trap consider-ably more gas molecules compared to a sphere of sim-ilar size close to the wall. Using our experimental pa-rameters (P=6 × 10−7 mBar, mair = 4.8 × 10−26 kg,r = 1 mm, m = 91mg, d = 150µm, T = 300 K, ) weobtain an additional damping of γwall ≈ 5.1 × 10−8,which is much smaller than the natural damping rateγmt

= ω0/Q ≈ 5.0 × 10−3 of our test mass oscillator.Looking ahead, even for a projected experiment in thePlanck-mass regime, which would require a mechanicalquality Q > 20, 000, the effect of force noise from gas col-lisions would be at the 5% level in this pressure regime- and can be further reduced by pushing vacuum levelsinto the UHV.

VIBRATION ISOLATION

The vacuum chamber is placed on an optical table withdeflated air springs. While inflating the springs did at-tenuate high frequency noise as expected55, the pendu-lum yaw mode in contrast was observed to get heavilyexcited. We suspect a relation to table tilt, which wehave independently observed to couple strongly into theexperiment (3 radyaw/radtilt). In the inflated state airdensity fluctuations act differently onto the air springsand thereby can cause excessive table tilt at low frequen-cies, which could explain the excessive yaw excitation.Operating our experiment with inflated air springs willrequire an active tilt stabilization system as described inRef.56

High frequency vibrations are known to couple non-linearly into lower frequency bands, where the measure-ment takes place36. To decouple the pendulum from suchvibrations, for example as generated by the turbo pump,the 420 g pendulum support structure is resting on softvacuum compatible rubber feet (Viton®).

13

SOURCE MASS POSITION DRIVE

The position of the source mass gold sphere is mod-ulated by a long range bending piezo (PI PL140.10,PICMA® Bender). Applying a voltage of 0−60 V bendsthe piezo and provides on the order of 1 mm travel rangeat the piezo tip, which is connected to a 0.31 mm diame-ter non-magnetic titanium rod carrying the source mass(Figure 7).

Figure 7: The source mass drive consisting of a bending piezoand a titanium rod is able to modulate the source mass po-sition by more than 5 mm peak-peak at frequencies as highas 1 Hz. The geometry of the titanium rod amplifies andtranslates the piezo deformation into an approximately linearmotion.

The rod was inserted only halfway into the source massso that its contact potential facing towards the test masswas effectively shielded by the surrounding gold.

Upon actuation, the piezo bending is translated to amostly linear motion of the source mass of up to 5 mmand a rotation around its center, which can be neglected.The position of the source mass is modulated harmoni-cally in order to avoid DC-noise of the torsion pendulumsuch as slow thermal drifts.

DISTANCE MEASUREMENT

A major source of uncertainty for the measurement ofthe gravitational coupling G is the absolute separationbetween source mass and test mass. We measure the dis-tance with a DSLR camera (Canon EOS 60D) that ismounted on top of the vacuum chamber and is lookingdown vertically onto the experiment. This means thatonly a 2D projection of the assembly can be monitored.Videos were taken with a 60 mm macro lens and exten-sion tubes. The positions of the masses were trackedusing a template matching algorithm based on the open-source image and video processing library OpenCV.

At the beginning of each distance measurement, tem-plates of the masses are defined in the first frame of avideo. Their positions are tracked in subsequent framesby integrating the squared pixel value distance over the

dimensions of the template

R(x, y) =∑

i∈{r,g,b}

∑x′,y′

(Ti(x′, y′)− Ii(x+ x′, y + y′))

2

.

(5)Here, Ti(x, y) designates the 8-bit pixel value of thecolour channel i of the template image, and Ii(x, y) thesame for the video frame to be evaluated. The minimummin(R) = R(x, y) then corresponds to the position of thebest match (x, y) for the template. This method is lim-ited to pixel accuracy with 1 px ≈ 24µm. Therefore, foreach dimension a Gaussian fit over the five neighboringpixels centered at the best match position is performedto improve on the tracking resolution.

In order to limit the effect of changes in lab lighting aswell as the influence of reflexes on the source mass sur-face, only the outer perimeter of the masses is tracked.We verify the tracking quality by visual inspection ofthe residuals, i.e. by looking at a video with each framerealigned to the tracked position similar as in a digi-tally stabilized video. The resolution and accuracy ofthe tracking is limited by the (manual) determination ofthe edges of the masses in the template, the low contrastinside the vacuum chamber due to its gold plated interiorand the resolution of the DSLR and is taken to be 20µm.

We measured the separation of source and test mass viavideo at the beginning and the end of each measurementrun.

Furthermore, the source to test mass separation is notonly defined by the momentary source mass position, butalso by residual test mass motion. In particular, the cen-ter of mass motion of the pendulum varies the distanceby more than 100µm under bad conditions. Therefore,the gravitational force is averaged over these distances.The separation modulation changing from this test massmotion cannot be extracted from the optical lever signalto be calibrated out. In future continuous video trackingwill provide this information.

Piezo calibration

The piezo is operated without feedback. As the po-sition could not be monitored continuously, the piezo’sposition response to the applied voltage needs to be char-acterized and modeled. The response of the piezo canbe modeled as an ideal mechanical transducer combinedwith an additional low-pass filter57. Fundamentally, thisreflects electron mobility / diffusion into the piezo mate-rial. As a consequence, the step response is delayed andthe drive shows a slow creep towards its final positionas well as hysteresis when driven cyclically. The creep(knowledge of the exact position) is of particular interestfor the charge measurements while the hysteresis (knowl-edge of the exact amplitude and phase) is important for

14

the gravity measurements, as it results in the source massposition lagging behind the applied voltage signal andreduces the amplitude. The stationary creep (the stepresponse of the piezo) depends on the size of the appliedvoltage step and the starting position (the voltage lev-els applied to the piezo before and after) and can reachup to 30µm over a period of 15 minutes. As the creepdepends on parameters we have little control over, thecreep has been fitted for every measurement separatelyrather than using global piezo parameters. The observedtime lag depends on the drive frequency, but does notshow any dependence on the modulation amplitude. Fora modulation frequency of 12.7 mHz we find a time lagof (509± 3) ms by correlating the drive voltage with thesource mass position as measured by video tracking.

We also observe a drift over time in both the mean po-sition and the modulation amplitude of the source massdrive. A possible reason is continuous charging of thepiezo material, similar to the observed creep for a stepresponse in quasi-static operation even in dynamic op-eration. In contrast to the stationary case where creepinduced piezo deflection reached a final state after aboutan hour, under cyclic operation the mean deflection andamplitude continued to drift over a measurement periodof 100 hours. For a peak-to-peak modulation of about2 mm, the amplitude reduced by 13µm, while the meanposition drifted by 55µm. Although this long term driftresembles the stationary creep as described in57 with amuch longer time constant, we cannot fit the correspond-ing function with only the start and end videos. The driftof the mean position as well as the modulation depth,therefore, is taken to be linear over the period of a mea-surement run.

Additionally, the source mass position modulation ex-hibits higher harmonics contributions. We measure therelative amplitudes and phases of these higher harmonicswith respect to the fundamental mode in the start andend videos. We infer the actual position modulation forthe gravitational measurement runs from the drive signalby taking the time lag, the amplitude change and meanposition drift and the higher harmonics into account. Forexample, the amplitude of the 2nd harmonic of the driveposition has been fitted to and found to be below 0.3 %of the fundamental amplitude. Remaining higher ordercontributions are estimated to be below < 0.1 % and aretherefore irrelevant for our experiment.

CHARGE MEASUREMENTS

We are discharging the test mass to less than 8 × 104

elementary charges using a nitrogen gas discharge andion diffusion as developed in Ref.58. The nitrogen gasis channeled over a set of high-voltage AC tips. Afterpassing through an aperture, the nitrogen ions neutral-ize the surfaces of the vacuum chamber by diffusion. Af-

ter discharging, the unshielded electrostatic interaction isstill approximately three times stronger than gravity (seeFigure 10). Note that the interaction is enhanced over apure Coulomb interaction by induced surface charges ofthe source and test mass.

Figure 8: Without electrostatic shielding, the probability den-sity distribution of inferred force versus source- test massseparation requires three constituents to describe: Newto-nian gravity (see Figure 3), electrostatic interaction betweencharged (≈ 8× 104 e+) test- and grounded source mass59 andan unexplained force proportional to the separation.

Because of that, even if the source mass was perfectlygrounded, a charged test mass induces charge separationresulting in parasitic attraction. While most torsion ex-periments striving to measure small forces conductivelycoat their suspension filament and thereby sacrifice me-chanical quality (statistical error) for charge control (sys-tematic error), we explicitly chose not to do so. One rea-son is that when down-scaling the torsion balance furtherthe proportion of lossy coating (surface) grows with re-spect to fiber material (volume). The other is the longterm prospect to contact-less levitate the test mass. Thiswill not allow for any sort of electrical contacting at all.

It could be shown by finite element simulations (Figure9), however, that electrostatic coupling between sourcemass position and test mass force can be suppressed wellbelow 3 % of the expected gravitational signal by meansof a 150µm thick conductive Faraday shield inserted inbetween source and test mass (see Figure 10). The shieldcomes at the added benefit of reducing other electrody-namic interactions such as short range Casimir forces aswell.

The drive piezo was housed in a Faraday shield in orderto suppress coupling of the applied electric fields onto thependulum e.g. via residual charges or polarization effects.During an 8.5 h measurement without a source mass at-tached but piezo modulation active, we did not observe atest mass actuation. While this verified the effectivenessof the shield, a full analysis of this drive piezo contri-

15

test masssource mass

visualcutaway

electrostatic shield

Figure 9: The charge distribution induced by a 3 × 104 e+

charged source mass onto the grounded test mass was deter-mined by a finite element simulation (COMSOL, exemplarilyshown for 1 mm surface separation). With the grounded, con-ductive electrostatic shield in place a lot of mirror charges areinduced in the unmodulated shield, resulting in a DC-force.The induced surface charge density (color coded) on the po-sition modulated source mass is suppressed approximately bya factor 100. These charges are screened by the shield result-ing in further suppression of the exerted force. The actualsuppression could not be quantified due to numerical inaccu-racies.

bution to the systematic error of our experiment wouldrequire the same amount of effective integration time asin the gravity run, i.e. a measurement run would have tolast for weeks under quiet conditions as well.

As long term drifts from charge diffusion on non-conductive materials were observed to degrade our lowfrequency sensitivity, all surfaces inside the vacuumchamber (except for optical viewports) were made fromconductive materials and gold plated wherever possi-ble to equalize surface potentials and mitigate thermo-electric potentials.

Figure 10: Our finite element simulation of electrostatic inter-action of a charged test mass (105 e+) and a grounded sourcemass was validated using the analytic method described inRef.59. When inserting a 150µm thick, conductive electro-static shield in between, the electrostatic force exerted ontothe test mass is dominated by test mass mass to shield interac-tion, i.e. becomes source mass position independent. Residualfluctuations of the force are on the 3 − 10 % level of gravitywithout a clear source position dependence and are expectedto stem from numerical errors.

Charge characterization

Static charges on our test mass can be characterized bymeans of induced mirror charges. They can be quantifiedby a force versus separation measurement without elec-trostatic shield and with the source mass being groundedas shown in Figure 8. This is effectively a macroscopicrealization of a Kelvin probe force microscope, which istypically used to contact-free measure (surface) poten-tials. The theory for the electrostatic interaction includ-ing polarization effects of extended conductive spheres ispresented in Ref.59. In addition to the expected signals(electrostatic, gravity) we observe an unidentified cou-pling which is proportional to the sphere separation.

To overcome this and other experimental challengessuch as unknown contact potentials we chose to isolatethe effect induced by residual test mass charges. Wetherefore quantify them by biasing the source mass po-tential and observing the test mass charge vs. sourcemass potential interaction at a set of discrete separations.Table I summarizes the results of the intermittent chargemeasurements over the whole duration of the measure-ments. A more detailed description of the charge deter-mination will be presented in an upcoming publication.

In conjunction with the electrostatic shield between themasses and our finite element simulation we have therebyverified that electrostatic coupling is suppressed to below3 % of gravity over the whole measurement period.

In conjunction with the electrostatic shield between themasses and our finite element simulation we have therebyverified that electrostatic coupling is suppressed to below3 % of gravity over the whole measurement period.

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12/28/2019 12/23/2019 12/27/2019 01/03/2020 01/07/2020 01/15/2020

surplus charges [ 103 e+] 34.7 25.0 44.4 78.8 38.8 52.8accuracy [ 103 e+] 0.7 2.6 9.4 0.4 3.9 1.1

Table I: The surplus charges on the test mass have been measured in between our measurement runs during the time periodsmarked as grey bands in Figure 4. All measured charges stay well below 8 × 104 e+, which we take as an upper limit in ourestimation of electrostatic coupling between source and test mass.

SEISMIC NOISE

Even during the environmentally quiet Christmas sea-son our fundamentally thermal noise limited sensitivityis dominated by a yet unidentified contribution with astrong diurnal variation (see Figure 6). Reduction of thisnoise is important to advance towards smaller detectablesource masses.

Long period seismic ground motion in an urban envi-ronment is a major concern for extremely sensitive lowfrequency experiments60–62. For ideal harmonic motion,the rotational mode of an oscillator will not be directlyexcited by translational ground motion. Unfortunately,unavoidable imperfections in the construction of the pen-dulum cause asymmetries that couple horizontal groundmotion into rotational motion63,64. Furthermore, downconversion of noise at higher frequencies such as crosscoupling from other modes cannot be neglected65.

To monitor horizontal and vertical ground displace-ment down to 100 mHz, a Raspberry Shake 3D seismome-ter has been installed in close proximity to the exper-iment. Low frequency seismic displacements at the ex-periment site were monitored with a RefTek Observer 60sbroadband seismometer. In addition, we have access tothe broadband data of the ZAMG seismic station (Streck-eisen STS2 ) at Hohe Warte, Vienna, roughly 3 km awayfrom the lab.

Correlating strong seismic events like the 6.7 magearthquake in Turkey on January, 24th 2020 in the dataof the local seismometer as well as the STS2 with theresponse of our oscillator provides a rough estimate ofseismic coupling, which suggests that the diurnal low fre-quency noise observable in our data is not due to linearcoupling of horizontal or vertical ground motion.

This claim is supported by directly comparing the non-stationary low frequency noise in the STS2 data in Vi-enna (Figure 11) and the torsion balance (Figure 6).There is a strong variability of the seismic noise floorin the 30 mHz band but a relatively steady primary mi-croseismic peak around 70 mHz. The daytime dependentchange in the 30 mHz noise floor is consistent in ampli-tude and time with the varying white force noise thatwe see in our test mass below 100 mHz. Yet, the fre-quency distribution is very different. In particular, themicroseismic peak at 70 mHz is a very strong, stationaryseismic feature within the frequency band of the exper-iment. Since we do not see it in the torsion pendulum

Figure 11: The time resolved amplitude spectral densityof the horizontal ground motion in Vienna recorded by anSTS2 broadband seismometer shows a strong frequency de-pendence of the variability. A rise of the noise floor is ob-served throughout the entire frequency range, starting frommidnight (dark blue) to noon (light blue), but is most promi-nent at 30 − 40 mHz. By contrast, the noise floor at the mi-croseismic peak, at around 70 mHz, varies only slightly. Themodulation frequency of the drive mass is marked by the ver-tical dashed line.

data, we conclude that our rotational mode is not domi-nantly driven by linear coupling of the horizontal seismicnoise.

MECHANICAL COUPLING

Recoil of the source mass drive is of major importancefor higher modulation frequencies such as 50 Hz34. Herewe argue that due to its strong frequency dependencethis effect can be neglected for our experiment. A 100 kgfreely suspended optical table for example would move by1 nm per mm of 100 mg source mass motion from recoilwhich is on the order of our observed signal. The couplinginto pendulum yaw is attenuated further by the pendu-lum’s common mode rejection, i.e. how well balanced theassembly is. Yet, already a table suspension resonance of1 Hz which can roughly be assumed for the air springs,attenuates this by another factor (fm/f

table0 )2 ≈ 1/6000.

A deflated table, in contrast, typically exhibits internalresonances well beyond the 100 Hz region yielding evenmuch higher resistance to recoil.

Another possible coupling at low frequencies is thecyclic deformation of the environment under the changingload distribution of the drive setup. Again, imagine the1 m sized 100 kg optical table suspended by air springs

17

(≈ 4×1 kN/m stiffness). A 1 mm horizontally shifted100 mg mass causes 1µN differential load and therebyinduces approximately 1 nrad tilt. In conjunction withthe above mentioned tilt to yaw coupling this can resulton the order of 1 nm test mass motion per mm of 100 mgsource mass motion, which is on the order of the observedsignal and dominating the recoil effect. In practice, theair springs employ active tilt feedback at low frequen-cies, stiffening the table tilt motion. When deflated, thetable leg stiffness may increase to 1 MN/m, providing afactor 1000 suppression of source mass induced table tilt.Internal table flexing modes even in the low 100 Hz re-gion would not allow for increased tilt coupling at themodulation frequency. For the future a more in depthinvestigation of tilt to yaw coupling and cancellation ofboth effects by means of a counter-acting mass will beinvestigated.

Therefore, both effects are indeed of importance formassive source masses and high accuracy measurementsaimed to determine G very accurately. But to our currentunderstanding they can be neglected in the presented ex-periment and similar future low frequency designs aimingfor further reduced source masses.

MAGNETIC INTERACTION

To minimize influence from magnetic coupling the ex-periment was designed without ferromagnetic materials.All masses are made of diamagnetic gold, and the sourcemass is mounted on a paramagnetic titanium rod. As aconsequence, Earth’s magnetic field (≈ 50µT at our loca-tion) as well as anthropogenic magnetic noise in an urbanenvironment induce dipoles in the gold spheres leading toa repulsive force between them.

Using literature values66 for the permeability of com-mercial grade gold and titanium we estimate dipole forces

Fmag =6µ0m1m2

4πd4c(6)

between two magnetic dipoles mi at center distance dcsuch as between our two diamagnetic spheres as well asthe force between the titanium rod and one gold sphere.The strength of the dipoles is determined by the strengthof the externally applied field, the susceptibility and thevolume and geometry of the material.

Iron impurities in the gold are suggested to dominatethe magnetic susceptibility67. Yet gold-iron alloys willact diamagnetically up to 1000 ppm iron, paramagnet-ically up to 10 % iron and only then start acting as aferromagnetic material. The gold used in the experimentis specified to have impurities below 1000 ppm, but ac-cording to Ref.68 commercial grade gold has iron impu-rities between 10 and 100 ppm. As a rough experimentalverification for the permeability, a 1 cm cube NeFeB mag-net with a remanent magnetization of around 1.4 T was

brought within 7 mm distance of test and feedback mass.The literature value of XAu is confirmed within a factorof ≈ 4.

Projecting the dipoles induced by earth’s magneticfield to exerted forces by means of Equation 6, they areabout 4 orders of magnitudes weaker than gravity in thissetup.

Figure 12: The amplitude spectral density of magnetic fieldvariations at the site of the experiment were measured witha 3-axis magnetometer Stefan Mayer FLC3-70 (z = vertical,y = along source-test mass axis). The signal at 10 mHz is ex-plained by the period of the traffic light at a nearby crossing,regulating car traffic as well as trams. The inferred force act-ing on the pendulum (red line), shows no sign of this signal.

To estimate magnetic signals induced by anthro-pogenic fields we measured the magnetic field close to theexperiment with a 3-axis magnetic sensor (Stefan MayerFLC3-70 ), and two different Samsung phones contain-ing fluxgate magnetometers (see Figure 12). The 10 mHzfrequency comb is correlated with a close-by traffic lightperiod. We verified that it is a magnetic signal by mea-suring with different magnetometers using different sam-pling ADCs. Moreover, the signal strength does not de-pend on the exact location in the lab. Yet, it is notvisible in the force signal of the experiment. We there-fore conclude that the masses do not couple magneticallyvia dipoles induced from anthropogenic magnetism.

CASIMIR FORCE

The steady state Casimir force for a plane-sphere ge-ometry can be calculated69 using

FCasimir =rtkBTζ(3)

8d2s. (7)

where ζ is the Riemann zeta function. For our setup, theparameters are T = 300 K, rt = 1 mm and ds ≈ 250µm.Thus, the Casimir force between the test mass and theFaraday shield is more than four orders of magnitudesmaller than Newtonian gravity70. For extending the ex-periment towards smaller source masses, however, theCasimir force will dominate the gravitational attraction

18

of a Planck mass sized source object for shield-sphere sep-arations below ≈ 100µm. As in our approach the gravi-tational potential is modulated, even in that source massregime we will be able to distinguish Casimir forces ex-erted onto the test mass (which are static in the presenceof an electrostatic shield) from gravity. Furthermore, forreduced shield separations the Casimir force might be-come relevant as a static non-linearity or instability ofthe torsion balance potential. For a 183 mg oscillatorwith a natural frequency of 1 mHz the Casimir force willrender the oscillator unstable by surpassing the restoringforce only for surface distances below ≈ 100 nm.

For any set up without a Faraday shield Casimir forcesset a lower bound for test mass to source mass separationwhich in turn sets an ultimate lower limit on detectableNewtonian force. For our scheme these limits are approx-imately 2µm for the symmetric setup with 90 mg massesand 20µm for a source object with m = mPlanck.

SYSTEMATIC DEVIATIONS

In this section we list identified sources of systematicover- or underestimation of the measured gravitationalforce. Neither of these has been corrected for in the eval-uation as most can only be guessed. We present Table IIto assess the most relevant contributions considered. Ingeneral, we find that most effects scale with the sourcemass. Hence, they limit the accuracy of our G-estimationbut will not pose a limit to measure smaller source massesin future.

All involved masses except for the tiny amounts of glueused have been measured at a 0.1 mg accuracy (KernABS120-4 ). While in the evaluation we treated oursource and test mass as point sources of gravity, theyalso act gravitationally onto the other components suchas capillary, glue etc. as well. Gravity between the sourcemass support rod (titanium gravity) and the test mass isnumerically simulated by integrating over (10µm)3 ele-ments. Since its mass distribution deviates from spheri-cal symmetry it cannot be summarized as effective point-mass co-moving with the source mass. Instead, the mod-ulation caused over a range of 2.4− 6 mm center of massseparation is put into relation to the gravity modulationcaused by source/test mass modulation over the samerange. Similarly, the influence of the spatial extent ofthe hole for mounting the titanium rod inside the sourcemass is estimated. The difference between a sphericallysymmetric source mass with a 300µm hole and a spher-ically symmetric source mass without a hole but withthe same mass is evaluated and put into relation withnominal source/test mass gravity. The modulation ofgravity between the counterbalance mass of the torsionpendulum and the source mass is quantified over the same2.4−6 mm range. The calibration of the QPD signal wasdone once on best effort basis, results are shown in Fig-

ure 5. The quality was checked by varying assumptions,but cannot be tested independently or continuously. Thesource drive position is calibrated to a 1 pixel accuracy(≈ 20µm) at the beginning and end of each measurementrun. The accuracy of the interpolation in between cannotbe quantified as the camera could not be operated con-tinuously. Instead of only a 2D projection the 3D massseparation will be monitored continuously in future, im-proving the data for close approaches such as in Figure8 by giving height offset and high frequency pendulumand roll mode excitation information. The 3D mass dis-tribution of our gold ’spheres’ is not known as we canonly observe 2D projections. From microscopic imagingunder different angles the used masses are known to berather bumpy (source mass radii vary between 1.03 and1.11 mm) while in high accuracy experiments aiming forthe determination of G even tiniest shape and densitydeviations are known to be relevant.

effect influence / FG

hole gravity +8.4E-3

titanium gravity +6.1E-3

capillary gravity +4.5E-3

glue gravity +3E-3

counterbalance grav. −1.1E-4

electrostatic ±3E-2

magnetic ±1E-4

calibration ±6E-3

drive calibration ±1.6E-2

mass separation ±1E-3

ms accuracy ±1.1E-3

mt accuracy ±1.1E-3

mglue accuracy +3E-3

height offset ±1.5E-2

ms roundness ±3.2E-2

mt roundness ±1.0E-2

Q accuracy ±5E-3

bandpass 1fm +1.6E-2

bandpass 2fm -7.6E-2

downsampling ±1E-3

compensation pole ±1E-5

anti aliasing filter ±2E-6

upper limit +15.9E-2

lower limit −11.8E-2

Table II: identified systematic deviations in units of expectedNewtonian gravity between source and test mass.

The masses used in the experiment deviate from per-fect spheres. In order to quantify the influence of theshape deviations, we took photographs of the test sphere

19

0 1 2X / mm

0

1

2Y

/ mm

a

Volume COMGeometric COMEvaluation COM

0 200Angle / °

1.00

1.05

1.10

Radi

us /

mm

b

Figure 13: (a) 2D projection of the test mass: The center ofmass determined by the geometric center of mass (mean overall image dimensions) as well as the center of mass calculatedby our shape evaluation are indicated (Volume COM). Thecenter of mass used in the determination of G is indicated ablue cross. (b) The radius measured from the geometric COMover all angles is shown.

under a microscope and different angles as shown in Fig-ure 13. We extract the angle resolved radius from the geo-metric center of mass of each projection. We assume, thateach combination of angle and radius defines a sphericalwedge and calculate the 2D deviation of the combinedcenter of masses of the wedges from the geometric centerof mass of the projection. We repeat the evaluation forevery projection and take the mean deviation from thegeometric center of mass as a correction for the centerof mass distance used in the evaluation. Comparing fitsto the uncorrected and corrected center of mass distanceresults in an estimated systematic uncertainty of 4.2 %for both, test and source mass shape.

An additional damping rate caused by residual gasmolecule impacts was estimated according to54. Usingour experimental parameters and taking into account theclose proximity of the EM-shield to the test mass, we ob-tain a damping rate 5 orders of magnitude smaller thanthe natural damping rate of our test mass oscillator.

A worst case estimate for sorption effects on uncleanedgold surfaces71,72, results in a relative mass change in theorder of 10−7 and can therefore be neglected. We alsonote that we do not observe time-dependent systematiceffects in agreement with the absence of significant sorp-tion effects.