Nanofabricated torsion pendulums for tabletop gravity experiments

TL;DR

This work addresses the challenge of measuring gravitational interactions at tabletop scales by leveraging dissipation dilution in nanofabricated, high‑stress Si$_3$N$_4$ suspensions to achieve ultra‑high coherence in torsion pendulums. The authors demonstrate a 87 g pendulum supported by a 1.8 μm Si$_3$N$_4$ ribbon, marking the largest thin‑film Si$_3$N$_4$ oscillator to date and establishing a platform for both classical gravity tests and quantum‑gravity aspirations, including gravity‑induced entanglement. They develop analytical models for gravitational mode splitting, define a figure of merit η = Q/ω0^3, and present realistic designs for two‑pendulum coupling with predictions for detectable mode splitting (e.g., Δω ≈ 0.7 μHz) and entanglement requirements (Δω > Γ_th) under cryogenic conditions. The paper also discusses practical obstacles—identical resonance matching, electrostatic shielding, and loss channels—that must be overcome to realize tabletop gravity experiments, while outlining scalable pathways toward mHz‑frequency, kg‑scale resonators capable of probing classical and quantum gravitational phenomena. Overall, this work provides a concrete nanofabrication route to ultra‑coherent torsion systems and a roadmap for near‑term and long‑term gravity experiments at tabletop scales.

Abstract

Measurement of mutual gravitation on laboratory scales is an outstanding challenge and a prerequisite to probing theories of quantum gravity. A leading technology in tabletop gravity experiments is the torsion balance, with limitations due to thermal decoherence. Recent demonstrations of lithographically defined suspensions in thin-film silicon nitride with macroscale test masses suggest a path forward, as torsion pendulums dominated by gravitational stiffness may achieve higher mechanical quality factors through dilution of material losses. Here we demonstrate a 250 micron by 5 mm by 1.8 micron torsion fiber supporting 87 grams and forming a Cavendish-style torsion pendulum with tungsten test masses that -- to our knowledge -- is the largest thin-film silicon-nitride-based oscillator to date. Torsion pendulums with thin-film, nanofabricated suspensions provide a test bed for near-term tabletop experiments probing classical and quantum gravitational interaction between oscillators.

Figures (4)

• Figure 1: Compilations of torsion balances and thin-film silicon nitride resonators. The plots are underlaid by the mechanical design figure of merit $\eta=Q/\omega_0^3$ and include contours indicating the cryogenic cooling necessary for gravitational entanglement to overcome thermal decoherence, assuming spherical tungsten test masses. The torsion pendulum presented in this work is marked 'x', with measured frequency (theoretical $Q$) marked by a brown, solid vertical (dashed horizontal) line. a) Compilation of torsion balances with quartz/silica downsbrough1937dampingzhao2023experimentalli2018measurementsli2014gyang2009directross2025probingliu2025amplitudeshaw2022torsionwestphal2021measurementhagedorn2006qualitycavalleri2009newagafonova20241fu2025studycatano2020highsu2024influence and metallic hu2000amplitudechen2009nanonewtonli2014gyang2009directquinn1995stressrichman1999preliminarynewman2014measurementnewman1999determiningbantel2000highfleischer2022cryogenicubhi2025demonstrationtu2010performancetu2009electrostaticyang2012torsionlin2025newxu2025measuringSchlamminger2008gundlach1997shorttan2016newKapner2007yang2012testrajalakshmi2008torsionabercrombie2016developmentcarbone2005characterizationhueller2002torsionciani2017newcavalleri2009newbai2015improvingxu2022measuringdong2023coupling suspensions. b) Compilation of Si$_3$N$_4$ thin-film resonators liu2021gravitationalsadeghi2020frequencywilson2009cavitythompson2008strongvillanueva2014evidencereinhardt2016ultralowchowdhury2025optomechanicaltsaturyan2017ultracoherentghadimi2018elasticchen2020entanglementxia2023entanglementthomas2021entanglementzwickl2008highzhang2012synchronizationanetsberger2009nearhyatt2025ultrahighwilson2015measurementsudhir2017quantumbereyhi2022hierarchicalgisler2022softfischer2016opticalyuan2015siliconnorte2016mechanicalchowdhury2023membranecupertino2024centimeterhodges2023characterization including torsion ribbons shin2025activepratt2023nanoscalehyatt2025ultrahigh. Brown points indicate mass-loaded torsion pendulums bsaibes2025lithographicallymanley2024microscalecondos2025ultralow
• Figure 2: Macroscopic torsion pendulum with a nanofabricated suspension. a) A 5.6 mHz (180 s period) torsion pendulum formed by suspending an 87 gram mass from a 1.8 $\upmu$m thick Si$_3$N$_4$ ribbon suspension. Cylindrical test masses are used for convenience; spherical masses are preferable for the gravitational coupling experiment described in the main text. b) Equivalent schematic, defining variables and design parameters. c) Image of the nanofabricated chip containing the suspension. The Si$_3$N$_4$ ribbon spans a window etched from a Si chip, with alignment and screw holes for clamping and temporary supports to protect the suspension during fabrication and mounting. The test mass is released by severing the breakout tabs and removing the supports.
• Figure 3: Design of a gravitational coupling experiment between two torsion pendulums. a) General configuration. Inset: Illustration of bifilar suspension. b) Visualization of design considerations, where brown curves indicate scaling of the pendulum quality factor and resonance frequency with suspension parameters, thickness $h$, length $l$, width $w$, and separation $s$, relative to the device in Fig. \ref{['fig:SiN_TB']}. The suspension is assumed monofilar except for along the curve of increasing $s$. Blue (green) lines indicate contours of constant $T_\text{ent}$ ($\eta$), where in both cases the optimal designs favor high $Q$ and low $\omega_0$. c) Hypothetical improvement for the design described in Section \ref{['sec:gravEnt']}.
• Figure 4: Numerical simulation of coupled oscillators with thermal noise. Simulation results are compared to the analytical models presented in Appendix \ref{['app:classicalModels']} for a) the modal PSDs, b) the oscillator position CSD, and c) the individual mode frequency resolution. d) Simulations of a single harmonic oscillator with different starting amplitude show divergence from the frequency resolution model---as occurs in c)---for oscillation amplitudes at or below the thermal motion $\left<\theta_\text{th}^2\right>=k_\text{B}T/I\omega_0^2$.