A micromechanical proof-of-principle experiment for measuring the gravitational force of milligram masses

Our system is composed of a spherical test mass loaded to a cantilever (test mass) and a spherical driving mass (source mass). In what follows we make the simplifying assumption that thermal noise contributions from internal friction of the cantilever [20] can be neglected compared to velocity-dependent (viscous) damping, which is legitimate for a single-mode, narrowband detection scheme close to resonance. The dynamics are then governed by the equations of a one-dimensional harmonic oscillator with equilibrium position $x^{\prime}$, internal damping rate $\gamma ^{\prime} ={\omega }_{0}^{\prime }/Q$, eigenfrequency ${\omega }_{0}^{\prime }$ and mass m that is driven in multiple ways: by motion (both deterministic and stochastic) of its support x_sup, by various deterministic forces F_i that may depend on the total distance d_tot and velocity ${\dot{d}}_{\mathrm{tot}}$ between the center of mass (COM) of the oscillator and the driving system, and by mean-zero stochastic noise terms N_i:

$$\begin{eqnarray}&&\ddot{x}^{\prime} +{\gamma }^{\prime }\;{\dot{x}}^{\prime }+{{\omega }_{0}^{\prime }}^{2}({x}^{\prime }-{x}_{\mathrm{sup}})={m}^{-1}\left(\displaystyle \sum _{i}{F}_{i}({d}_{\mathrm{tot}},{\dot{d}}_{\mathrm{tot}})+\displaystyle \sum _{i}{N}_{i}\right).\end{eqnarray} \tag{ 1 }$$

As shown in appendix A, splitting the time-dependent distance d_tot into a static part d₀ and a mean-zero time-dependent part allows us to linearize the force terms around the non-deflected position of the test mass. Both the test mass position and frequency are shifted due to the presence of the force terms. Specifically, in case that Newtonian gravity  ${F}_{{\rm{G}}}={{GmMd}}_{\mathrm{tot}}^{-2}$ is the dominant force, the new effective frequency and position are²

$$\begin{eqnarray*}&&{\omega }_{0}={({{\omega }_{0}^{\prime }}^{2}+2{{GMd}}_{0}^{-3})}^{1/2}\quad {\rm{and}}\quad x={x}^{\prime }-{{GMd}}_{0}^{-2}{\omega }_{0}^{-2}.\end{eqnarray*}$$

These are typically the observables of torsional-balance experiments. For example, usual Cavendish-type experiments use centimeter-to decimeter-size source masses and a torsional pendulum operating at a resonance frequency of some milli-Hertz. With distances d₀ on the order of the size of the masses, say ${d}_{0}=10\;\mathrm{cm}$, $M\propto {({d}_{0}/2)}^{3}$ and ${\omega }_{0}^{\prime }={10}^{-3}\;\mathrm{Hz}$, one expects frequency shifts ${\rm{\Delta }}\omega$ up to some hundreds of micro-Hertz and displacements up to millimeters, both of which can be reasonably resolved in precision measurements³ . One way of reducing the mass further would be to reduce the experimental dimension d₀, which is however accompanied by an increase in resonance frequency ${\omega }_{0}^{\prime }$ (if we assume an unaltered spring constant). This results in a highly unfavourable scaling of the observable effects, since ${\rm{\Delta }}\omega \propto 1/{\omega }_{0}^{\prime }$ and ${\rm{\Delta }}x\propto {d}_{0}^{-2}{\omega }_{0}^{-2}$. In particular, using ${d}_{0}=1\;\mathrm{mm}$ and ${\omega }_{0}^{\prime }=10\;\mathrm{Hz}$ yields effective frequency and position shifts of tens of nanohertz and picometers, respectively, which is significantly more challenging to measure. For this reason, simply scaling down a Cavendish experiment is not sufficient to measure the gravitational effects of small source masses.

Instead, we periodically modulate the gravitational potential created by a small source mass in order to resonantly enhance the amplitude response of a cantilever test mass. The power spectral density S_xx of the (test mass) cantilever displacement is given by (appendix A)

$$\begin{eqnarray}{S}_{{xx}}(\omega ) & = & {| \chi ({\omega }_{{\rm{S}}})| }^{2}{| -2{GM}/{d}_{0}^{3}+{{\omega }_{0}^{\prime }}^{2}{T}_{{\rm{S}}}({\omega }_{{\rm{S}}})| }^{2}\displaystyle \frac{{d}_{{\rm{S}}}^{2}\pi }{2}(\delta (\omega -{\omega }_{{\rm{S}}})+\delta (\omega +{\omega }_{{\rm{S}}}))\\ & & +{| \chi (\omega )| }^{2}({{\omega }_{0}^{\prime }}^{4}{T}_{{\rm{E}}}^{2}(\omega ){S}_{{x}_{{\rm{E}}}{x}_{{\rm{E}}}}(\omega )+2\gamma {k}_{{\rm{B}}}T/m)\\ & & +{\rm{further\ deterministic\ forces}}+{\rm{further\ noise}},\end{eqnarray} \tag{ 2 }$$

where we define $\chi (\omega )={({\omega }_{0}^{2}-{\omega }^{2}+{\rm{i}}\gamma \omega )}^{-1}$ as the mechanical susceptibility. Here we take into account the Newtonian force F_G as well as thermal noise with power spectral density ${{S}_{{NN}}}_{\mathrm{th}}=2\;m\gamma {k}_{{\rm{B}}}T$. In addition, we assume a sinusoidal drive of the source mass with amplitude d_S.

The first contribution is the gravitational effect in which we are interested. The second term is the mechanical drive from the deflection of the source mass transmitted through the supporting structure between source mass and test mass with transfer function ${T}_{{\rm{S}}}(\omega )$. The third contribution is due to the environmental vibrational noise ${S}_{{x}_{{\rm{E}}}{x}_{{\rm{E}}}}(\omega )$ that is modified by a transfer function ${T}_{{\rm{E}}}(\omega )$. The last term describes Brownian motion of the test mass.

In any actual experiment the measurement time τ is finite. In the following we make the experimentally justifiable assumption that the measurement bandwidth ${\rm{\Gamma }}=2\pi /\tau$ is larger than the spectral width of the drive modulation and smaller than the mechanical width γ, which simplifies the following analytical treatment of the expected measurement signal. The measured displacement power ${P}_{{xx}}={\int }_{{\omega }_{{\rm{S}}}-{\rm{\Gamma }}/2}^{{\omega }_{{\rm{S}}}+{\rm{\Gamma }}/2}{S}_{{xx}}{\rm{d}}\omega$ in the frequency band around ${\omega }_{{\rm{S}}}$ can be written as

$$\begin{eqnarray*}&&{P}_{{xx}}={{P}_{{xx}}}_{{\rm{G}}}+{{P}_{{xx}}}_{{\rm{S}}}+{{P}_{{xx}}}_{{\rm{E}}}+{{P}_{{xx}}}_{\mathrm{th}}+{\rm{cross}}\;{\rm{terms}}+{\rm{further}}\;{\rm{contributions}}.\end{eqnarray*}$$

For resonant driving (${\omega }_{{\rm{S}}}={\omega }_{0}$) and weak coupling (${\omega }_{0}^{\prime }\approx {\omega }_{0}$) one finds

$$\begin{eqnarray}&&{{P}_{{xx}}}_{{\rm{G}}}=2\pi \displaystyle \frac{{Q}^{2}}{{\omega }_{0}^{4}}{({GM})}^{2}\displaystyle \frac{{d}_{{\rm{S}}}^{2}}{{d}_{0}^{6}}\qquad {\rm{gravity,}}\end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray}&&{{P}_{{xx}}}_{\mathrm{th}}=2\displaystyle \frac{Q}{{\omega }_{0}^{3}}\displaystyle \frac{{k}_{{\rm{B}}}T}{m}{\rm{\Gamma }}\qquad {\rm{thermal}}\;{\rm{noise,}}\end{eqnarray} \tag{ 3b }$$

$$\begin{eqnarray}&&{{P}_{{xx}}}_{{\rm{S}}}=\displaystyle \frac{\pi }{2}{Q}^{2}{T}_{{\rm{S}}}^{2}({\omega }_{0}){d}_{{\rm{S}}}^{2}\qquad {\rm{source}}\;{\rm{mass}}\;{\rm{drive,}}\end{eqnarray} \tag{ 3c }$$

$$\begin{eqnarray}&&{{P}_{{xx}}}_{{\rm{E}}}={Q}^{2}{T}_{{\rm{E}}}^{2}({\omega }_{0}){S}_{{x}_{{\rm{E}}}{x}_{{\rm{E}}}}{\rm{\Gamma }}\qquad {\rm{environmental}}\;{\rm{vibrations,}}\end{eqnarray} \tag{ 3d }$$

which are the relevant contributions to our expected signal.

When neglecting all other noise sources, thermal noise becomes a fundamental hurdle for seeing the effect of gravitation on a small scale. A few interesting insights can be obtained by comparing the scaling of the gravitational contribution (3a) with the one of thermal noise (3b). First, while the thermal noise decreases for larger test masses m, the gravitational contribution does not depend on the size of the test mass but only of the source mass M (as is expected from the weak equivalence principle). Therefore, to ensure that gravity dominates over the thermal noise, either source mass or test mass (or both) can be increased. In our case we want to keep the source mass small and hence can increase the test mass. One trade-off that needs to be considered in this case is the strong scaling of the gravitational contribution with the COM distance between test and source mass (${d}_{0}^{-6}$), which is likely to increase when increasing the size of the test mass. Second, the thermal noise scales linearly with the bandwidth, whereas the gravitational contribution does not—in other words, a longer measurement time will decrease the stochastic thermal noise when compared to the steady-state oscillatory signal. Third, because of the explicit Q-dependence of the Brownian force noise (due to the fluctuation–dissipation theorem), both contributions scale differently with mechanical quality Q: quadratic for the gravitation and linear for the thermal noise part. Hence, increasing Q will not only lift the absolute thermal noise level but will also improve the signal to noise ratio between gravitational signal and thermal noise. For the same reason, scaling is also different in mechanical frequency, specifically ${\omega }_{0}^{-4}$ for the gravitational and ${\omega }_{0}^{-3}$ for the thermal noise part. Operating at lower mechanical resonance frequencies is therefore favorable. In summary, our gravitational sensing approach should allow for achieving a good signal to noise ratio. Although the relative noise contribution increases with smaller (test) masses and larger COM distances d₀, this can be compensated for by larger mechanical quality factors Q and longer measurement times.

With the expressions derived in section 2.1, we now assess the feasibility of the experiment for a realistic parameter regime. Figure 2 shows the signal contribution of thermal noise and gravity as a function of the source mass radius. Here we assume a test mass of the same size as the source mass, a test mass cantilever of frequency ${\omega }_{0}=50\;\mathrm{Hz}$ and mechanical quality factor $Q=2\times {10}^{4}$, and an integration time of one hour, i.e.  $\tau =2\pi /{\rm{\Gamma }}=3600\;{\rm{s}}$. The material of choice is gold due to its high density (${\rho }_{\mathrm{Au}}=1.93\times {10}^{4}\;\mathrm{kg}\;{{\rm{m}}}^{-3}$), purity and homogeneity [21]. In addition, we assume that the minimal distance between the surfaces of source and test mass is $\epsilon =0.5\;\mathrm{mm}$ and we choose an optimal drive amplitude for the source mass modulation (see section 3.4). For these (conservative) settings, a signal to noise ratio of 1 is reached for a source mass radius of $500\;\mu {\rm{m}}$, which in case of gold corresponds to a source mass weight of about $10\;\mathrm{mg}$. For our further considerations we want to leave some overhead for unaccounted experimental noise sources and hence choose a source mass radius of $1\;\mathrm{mm}$, where the gravitational contribution is about six times higher than the thermal noise. A gold sphere of this size has a volume of $4.2\;{\mathrm{mm}}^{3}$ and a mass of $80.9\;\mathrm{mg}$, which is still three orders of magnitude smaller than the smallest reported attractor masses in a laboratory based experiment [6, 7]. Figure 2 also shows the contribution of other residual forces. For unwanted Coulomb forces we assume 200 surface charges per mass with opposing charges located at the closest position on each sphere⁴ . The London-Van der Waals force contribution that is shown is estimated for the worst possible material properties and the effects of residual gas scattering is shown for a pressure of ${10}^{-8}\;\mathrm{mbar}$. Details of the calculations are shown in appendix B. Another possible effect is non-contact friction due to time-dependent electric fields (patch potentials), which is a known effect for conducting surfaces [22, 23]. Due to the relatively large distance $\epsilon$ between the test and source mass surfaces, such fields can be shielded by a membrane between the two masses [24].

Zoom In Zoom Out Reset image size

The last two decades have seen dramatic improvements in the fabrication and performance of nano- and micro-mechanical devices [25, 26]. For our experiment, we are considering a micromechanical system that is mass-loaded with a $1\;\mathrm{mm}$-radius gold sphere, thereby forming a test mass cantilever at the target frequency of ω₀ ≈ 50–100 Hz (see appendix C). An outstanding question is the achievable mechanical quality for such a structure. In the context of atomic force microscopy (AFM) with colloidal probes, polystyrene microbeads of glass, polystyrene, polyethylene and other materials have been successfully attached to cantilevers while maintaining typical AFM cantilever quality factors in the ten-thousands to millions [27]. Our experiment deals with significantly more massive objects, which will require a relatively large attachment area. As we could not find consistent values for the bulk quality factors of the high-density metals gold and lead, a rough estimate was gathered in a piezomechanical S21 gain/loss measurement (appendix C). With a measured value of $Q\approx 450$ for gold we assumed 100 as a worst case estimate. With such low mechanial quality it is important to avoid deformation of the test mass as a mode shape contribution of the relevant COM mode. This requires a careful design of the cantilever geometry. Finite element modeling (FEM) methods provide the means to analyze mode shapes and from that estimate effective Q-values of compound cantilever systems, which can be used to optimize geometries with regard to test mass deformation. It turns out that in a simple cantilever-geometry the mechanical quality of the material directly at the bonding surface between cantilever and test mass can have a huge negative impact on the overall quality of the compound system. This can be circumvented by changing the cantilever geometry such that deformation of the bonding surface is avoided, or by attaching the test mass to the cantilever using an adhesion layer with low internal losses [28]. A specific example is presented in appendix C. Assuming Qs of 30 000 for an AlGaAs cantilever [29], 300 for the adhesive [28] and 100 for the test mass one obtains the overall dissipation by adding up the loss angles and at the same time scaling their contribution with the mode stress derived from FEM simulations, which yields quality factors of the mass-loaded structure of at least $Q\approx 24\;000$. A brief estimation of additional damping through Brownian noise from gas impacts is presented in appendix D.

The test mass cantilever displacement is subject to additional external noise sources, in particular seismic noise of the environment, ${S}_{{x}_{{\rm{E}}}{x}_{{\rm{E}}}}$, and mechanical backaction of the source mass displacement, which is coupled through the mechanical support structure via the transfer functions T_E and T_S, respectively (see (2)). Their contributions can therefore be damped by additional vibration isolation of the test mass cantilever. Achieving a suppression well below the thermal noise limit in the measured signal power requires ${P}_{{xx},T}\gt {P}_{{xx},E},{P}_{{xx},D}$ (see (3b)–(3d)), yielding

$$\begin{eqnarray*}&&{T}_{{\rm{E}}}(\omega )\lt {\left(\displaystyle \frac{2{k}_{{\rm{B}}}T}{{Qm}{\omega }_{0}^{3}}\right)}^{1/2}{S}_{{x}_{{\rm{E}}}{x}_{{\rm{E}}}}^{-1/2}\approx {10}^{-7},\end{eqnarray*}$$

where we assume typical laboratory noise of ${S}_{{x}_{{\rm{E}}}{x}_{{\rm{E}}}}^{1/2}\approx {10}^{-8}\;\;{\rm{m}}\;{\mathrm{Hz}}^{-1/2}$ at $50\;\mathrm{Hz}$, and

$$\begin{eqnarray*}&&{T}_{{\rm{S}}}({\omega }_{{\rm{S}}})\lt 2{\left(\displaystyle \frac{{k}_{{\rm{B}}}T{\rm{\Gamma }}}{\pi {Qm}{\omega }_{0}^{3}}\right)}^{1/2}{d}_{{\rm{S}}}^{-1}\approx {10}^{-18}.\end{eqnarray*}$$

The first term requires an isolation of the test mass cantilever platform from seismic noise by at least $70\;\mathrm{dB}$ at around 50–100 Hz, which is clearly within current state of the art. For example, a combination of multiple passive and actively controlled suspension stages in gravitational wave detectors routinely achieve seismic isolations of $100\;\mathrm{dB}$ and better at even lower frequencies. For our case, already a dual-stage passive spring-pendulum system should be sufficient to achieve the required levels of isolation at $50\;\mathrm{Hz}$ [30].

The second term seems to impose a significant challenge, but one should bear in mind that we consider here the contribution of the source mass displacement that is due to mechanical backaction on the support structure of the experiment. There are several strategies to minimize this. First, mechanical unbalance can be compensated for by having a second mass counter-moving against the first (section 3.4). Second, the source mass drive platform can be physically separated from the test mass cantilever platform. This requires a separate vibration isolation (spring pendulum) system, which couples to the test mass cantilever platform only via the large mass of the vacuum tank that hosts the experiment. Finally, one can even envision a complete mechanical isolation of the source mass by levitating and driving it in external fields [31].

For our envisioned parameter regime, gravitational driving will result in signal noise powers of the test mass displacement of ${S}_{{xx}}^{1/2}({\omega }_{0})\approx {10}^{-10}\;{\rm{m}}\;{\mathrm{Hz}}^{-1/2}$ on resonance and ${S}_{{xx}}^{1/2}(\omega =0)\approx {10}^{-14}\;{\rm{m}}\;{\mathrm{Hz}}^{-1/2}$ off resonance. Optical interferometry is a convenient and well-established way to read out such small signal levels. In essence, the displacement $\delta x$ is converted into an optical phase modulation $\delta \phi =2\pi \delta x/\lambda$, which can be measured either directly as amplitude modulation in a balanced interferometer or via optical homodyne detection [32]. The challenge for our experiment is to obtain this sensitivity at small frequencies in the audio band. One has to consider the following noise sources: classical amplitude- and phase noise of the laser source, quantum noise (shot noise and backaction), electronic noise from the detection circuit, and residual amplitude modulation in the readout architecture.

The optical source noise is composed of amplitude noise, which we can circumvent by optical homodyning and measuring the phase qudrature, and phase noise, which can essentially be converted to amplitude noise using polarization optics. Quantum noise in an optical position measurement is due to intrinsic photon number fluctuations in a laser beam (shot noise), which contributes statistical noise both in photon counting and in actual displacement due to radiation-pressure induced momentum transfer (backaction). This results in the well-known standard quantum limit for continuous position sensing, which resembles the working point of maximal achievable sensitivity in the presence of quantum noise [33–35]. For reasons of practicality, we will discuss a readout scheme that does not invoke a cavity configuration but consists only of a two-path interferometer operated at an optical input power P. In this case, the added shot-noise contribution to the displacement measurement is [35] ${{S}_{{xx}}}_{\mathrm{shot}}={\hslash }\lambda c{(32\pi P)}^{-1}$, which is an effect of photon counting and hence does not depend on the mechanical susceptibility of the test mass. In contrast, the backaction contribution to the displacement noise is ${{S}_{{xx}}}_{\mathrm{back}}={| \chi ({\omega }_{{\rm{S}}})/2| }^{2}{({\hslash }/m)}^{2}S{{}_{{xx}}}_{\mathrm{shot}}^{-1}$, which is amplified by the mechanical response. Figure 3(a) compares the noise contributions of photon shot noise and backaction (at $\omega ={\omega }_{0}$) to the thermal noise floor on and off the mechanical resonance. Because of the large mass of the test mass system backaction noise can be neglected for all reasonable parameter regimes.

Zoom In Zoom Out Reset image size

For resonant detection it is sufficient to suppress the frequency-independent shot noise well below the thermal noise contribution at the mechanical frequency, i.e. ${{S}_{{xx}}}_{\mathrm{thermal}}({\omega }_{0})\gg {{S}_{{xx}}}_{\mathrm{shot}}$, yielding

$$\begin{eqnarray*}&&P\gg \displaystyle \frac{{\hslash }\lambda c}{64\pi }\displaystyle \frac{m{\omega }_{0}^{3}}{{k}_{{\rm{B}}}T}\displaystyle \frac{1}{Q}\approx 5\;\mathrm{fW}.\end{eqnarray*}$$

It may also be useful to fully resolve the thermal noise of the oscillator, allowing for example for off-resonant detection schemes. This requires suppression of the shot noise contribution well below the off-resonance thermal noise, resulting in

$$\begin{eqnarray*}&&P\gg \displaystyle \frac{{\hslash }\lambda c}{64\pi }\displaystyle \frac{m{\omega }_{0}^{3}}{{k}_{{\rm{B}}}T}Q\approx 2\;\mu {\rm{W}},\end{eqnarray*}$$

which was derived using the thermal noise contribution from (2) at $\omega =0$.

Although detection in the audio-band is challenging because of unavoidable low-frequency fluctuations in the setup [36–38] compact interferometric readout schemes have been demonstrated that operate with the desired sensitivity [39–41]. For example, by combining a robust homodyning architecture with a large-bandwidth probe laser a recent experiment has reported a displacement sensitivity of $4\times {10}^{-14}\;{\rm{m}}\;{\mathrm{Hz}}^{-1/2}$ above $20\;\mathrm{Hz}$ at laser powers of $10\;\mu {\rm{W}}$ [40].

We set the minimal distance $\epsilon$ between the test mass and source mass surfaces to a fixed value. One can then derive an optimal distance d₀ and driving amplitude d_s. By using ${d}_{0}={d}_{{\rm{S}}}+{r}_{{\rm{T}}}+{r}_{{\rm{S}}}+\epsilon$ (${r}_{{\rm{T}}/{\rm{S}}}$: test/source mass radius) and the fact that the Newtonian contribution to the measurement signal scales with ${d}_{{\rm{S}}}^{2}\;{d}_{0}^{-6}$, we find the optimum to be

$$\begin{eqnarray*}&&{{d}_{0}}_{\mathrm{opt}}=\displaystyle \frac{3}{2}({r}_{{\rm{T}}}+{r}_{{\rm{S}}}+\epsilon ),\quad {{d}_{{\rm{S}}}}_{\mathrm{opt}}=\displaystyle \frac{1}{2}({r}_{{\rm{T}}}+{r}_{{\rm{S}}}+\epsilon ).\end{eqnarray*}$$

For the parameters discussed above ($\epsilon =0.5\;\mathrm{mm},{r}_{{\rm{T}}}={r}_{{\rm{S}}}=1\;\mathrm{mm})$ the optimal values are ${{d}_{0}}_{\mathrm{opt}}=3.75\;\mathrm{mm}$ for the COM distance and ${{d}_{{\rm{S}}}}_{\mathrm{opt}}=1.25\;\mathrm{mm}$ for the actuation amplitude. Achieving smooth driving of the source mass at around $50\;\mathrm{Hz}$ at such amplitude, with sufficiently large lifetime (10⁸–10⁹ cycles) and without adding significant stray fields represents a substantial engineering challenge. State-of-the art piezoelectric actuators fall short of the required drive amplitude by at least one order of magnitude and available actuated positioning platforms do not achieve the desired accelerations between $123\;{\mathrm{ms}}^{-2}$ ($50\;\mathrm{Hz}$) and $493\;{\mathrm{ms}}^{-2}$ ($100\;\mathrm{Hz}$). One possible drive mechanism could be a spring-mounted electromagnet, which in principle allows actuation at small input power and therefore small stray fields and heating [42]. Alternatively, one can operate at a smaller drive amplitude. Figure 3(b) shows the resulting decrease in signal strength. With the current parameter settings, an overall signal to noise ratio of 1 can be achieved with a drive amplitude of around $70\;\mu {\rm{m}}$.

In order to position the source mass, an optical scheme can be employed to read out the relative distance between the test mass and the source mass in all spatial directions. This can be accomplished, for example, with quadrant diodes at the source mass stage paired with focused light beams from the test mass stage. If higher precision is required, one might think of interferometric schemes or even techniques involving optical cavities, which would in principle allow to push relative positioning errors into the sub-nanometer regime. The displacement of the source mass could be read out in a similar fashion, however the technical implementation will depend on the mechanism used for the source mass drive.

The accurate determination of Newton's constant G has become a highly debated subject [45, 46]. Although some experiments are now reaching precision levels up to ${\rm{\Delta }}G/G\approx 1\times {10}^{-5}$ [19], different implementations continue to disagree in the absolute value of G by multiple standard deviations [18, 47, 48]. A significant, if not dominant, contribution to the error budget of most of these measurements is due to uncertainties associated with the manufacture of the macroscopic source masses and their incorporation and use in the experimental arrangements. This includes suspension noise [20] as well as inaccuracies in the COM distance between test and source mass due to, for example, inhomogeneities or temperature fluctuations. In addition, long integration times require a detailed understanding of all long-term systematics in these experiments. Alternative approaches for measuring G may therefore provide helpful insights. One is cold-atom interferometry [49, 50], where a precision of ${\rm{\Delta }}G/G\approx 1\times {10}^{-4}$ has recently been demonstrated that was mainly limited by the position measurement of the atoms with respect to a macroscopic tungsten source mass [50].

Our approach involves a centimeter-scale experimental architecture, a microscopic source mass and short integration times of only hours. This combination reduces conventional sources of errors in precision measurements of G, since the small volumes enable a better control of positioning and density inhomogeneities of the masses as well as of temperature fluctuations. In addition, short integration times may allow for a systematic study of the influence of fluctuations of other spurious external forces that give rise to systematic errors in long-term experiments. A remaining challenge is to achieve a measurement precision that is competitive with experiments involving macroscopic source masses. One straight-forward way to achieve this is to increase the size of the masses, which will both boost the gravitational signal and decrease the thermal cantilever noise. For example, following our analysis above and using 10 mm radius spheres instead of 1 mm with otherwise unaltered parameters would result in a signal to noise ratio beyond 10⁶, i.e. a precision of ${\rm{\Delta }}G/G\lt 1\times {10}^{-6}$. Ultimately, operating the experiment in a low-temperature environment would in principle allow for even higher precision, provided that all technological challenges of low-frequency, cryogenic vibration isolation can be addressed in future experiments. Some third-generation gravitational wave detector designs have already been studying cryogenic scenarios [51, 52].

Although the predictions of both quantum theory and general relativity are extremely well confirmed by experiment, interfacing these two theories belongs to one of the outstanding big challenges of modern science. Notwithstanding the conceptual and mathematical hurdles in writing down a full quantum theory of gravity, the number of available experiments that probe the interface between quantum physics and gravity is also extremely sparse. One type of experiments focus on observations over astronomical distances, which may reveal imprints of quantum gravity effects [53, 54]. The other type of experiments exploit the availability of continuously improving high-precision lab-scale experiments [55, 56]. The latter ones fall essentially into two categories: they are either genuine quantum tests in the limit, where Newtonian gravity acts as a constant classical background field [57–61], or they are tests of genuine gravity effects measured through high-precision quantum experiments [62–64]. In other words, thus far all of these laboratory scale experiments have been using quantum systems as test masses in external gravitational fields. Using quantum systems as gravitational source masses would establish a qualitatively new type of experiment. Obviously, this will require quantum control over the motion of sufficiently massive objects and at the same time the experimental sensitivity to their gravitational forces. In this context, our micromechanics platform presented in this paper can be seen as a top-down approach for designing such future experiments. The lowest masses and shortest timescales above which gravitational coupling can be observed will be an important benchmark for both mass and coherence time of future quantum experiments. In the most optimistic scenario, the combination of force sensitivity and coherence time will eventually enable the quantum regime of gravitational source masses, for example by demonstrating gravitationally induced entanglement as suggested by Feynman [65, p 250].