⚛️ quantum physics

Quantum-enabled measurement of gravitational forces at the milligram scale

This paper presents a breakthrough experiment conducted at millikelvin temperatures that successfully measures gravitational forces between milligram-scale masses using a microwave optomechanical system, achieving a two-order-of-magnitude improvement in sensitivity and establishing a viable platform for investigating gravitational interactions involving non-classical motional states.

Original author: Matthew Herbst, Ewa Rej, Joe Depellette, Richa Cutting, Jinhoon Jeong, Mika A. Sillanpää

Published 2026-11-09

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CC BY 4.0

Original author: Matthew Herbst, Ewa Rej, Joe Depellette, Richa Cutting, Jinhoon Jeong, Mika A. Sillanpää

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

Gravity is the force that keeps our feet on the ground and holds the planets in their orbits, yet it is the weakest of all the fundamental forces that govern the universe. While we understand how gravity shapes the cosmos on a grand scale, its behavior in the microscopic world remains a mystery. This is because the gravitational pull between tiny objects is so incredibly faint that it is usually drowned out by other, stronger forces like electricity or magnetism. For decades, scientists have struggled to measure gravity between small masses, and they have never been able to do so in an environment where the objects behave according to the strange rules of quantum mechanics. Bridging the gap between the physics of the very large and the very small is one of the greatest challenges in modern science, and doing so requires measuring the gravitational tug between objects that are light enough to be influenced by quantum effects.

A team of researchers at Aalto University in Finland has now taken a significant step toward this goal by successfully measuring the gravitational attraction between two small gold spheres in a highly controlled, ultra-cold environment. The experiment involved a source mass weighing 87 milligrams and a test mass weighing 1.3 milligrams. To make the measurement possible, the researchers placed the test mass on a thin, flexible membrane made of silicon nitride, which acted like a tiny drumhead. This entire setup was cooled to temperatures near absolute zero, a state where thermal jitters are minimized, allowing the delicate gravitational signal to emerge. The source mass was a gold sphere attached to a vibrating pole that moved up and down by a distance of only a few micrometers. As this sphere moved, its changing position exerted a tiny, rhythmic gravitational pull on the test mass, causing the membrane to vibrate in sync.

Detecting this motion required an extremely sensitive method, as the force involved was measured in the range of attonewtons, a unit so small it is difficult to comprehend. The researchers used a technique called microwave optomechanics, where they shined microwave signals into a superconducting cavity containing the membrane. The position of the vibrating membrane altered the properties of these microwaves, allowing the team to track the membrane's motion with incredible precision. By carefully shielding the experiment from electrical interference and vibrations, they were able to isolate the gravitational signal from the background noise. The data showed that the test mass was indeed responding to the movement of the source mass with a force that matched the predictions of Newton's law of gravity. The researchers confirmed this by moving the source mass to different locations and observing how the force changed, finding that the attraction grew stronger as the spheres got closer, exactly as expected for a gravitational interaction.

This achievement represents a major leap forward in the ability to measure gravity at small scales, improving the sensitivity of previous experiments by a factor of roughly one hundred. While the masses used were still large enough to be considered classical objects rather than quantum ones, the experiment was conducted in a setting designed to eventually support quantum states. The researchers demonstrated that their setup could detect the gravitational influence of a mass moving by a distance smaller than the width of a human hair. They also ruled out other potential causes for the observed motion, such as electrical forces or vibrations traveling through the support structure, by showing that the signal appeared only when the spheres were aligned in a way that maximized gravitational pull and disappeared when they were moved apart.

The success of this experiment opens the door to future studies where the masses involved could be placed into genuine quantum states, such as superposition, where an object exists in two places at once. If scientists can eventually measure gravity between objects in such states, it could provide the first experimental evidence of how gravity interacts with quantum mechanics, potentially leading to a unified theory of the universe. For now, the team has proven that it is possible to measure the gravitational pull between milligram-scale objects with high precision, turning a theoretical challenge into a practical reality. This work does not yet solve the mystery of quantum gravity, but it provides the necessary tools and confidence to keep searching for the answer in the quiet, cold depths of a laboratory.

Technical Summary: Quantum-enabled measurement of gravitational forces at the milligram scale

Problem Statement
Gravity is the weakest known fundamental interaction, making its experimental investigation at microscopic scales exceptionally difficult. While general relativity describes gravity and quantum mechanics describes the microscopic world, a unified theory remains elusive due to a lack of experimental data at the intersection of these regimes. Current efforts to probe the "gravity-quantum interface" require two simultaneous advancements: reducing the mass of gravitating objects to the milligram or microgram scale, and preparing these objects in non-classical motional states. Previous experiments have detected gravity from milligram masses using torsion pendulums, but these operate at millihertz frequencies and involve macroscopic thermal states. Conversely, quantum optomechanical systems have achieved ground-state cooling for much lighter masses (nanograms) but have not yet been used to measure gravitational coupling between distinct masses. The challenge lies in measuring the extremely weak dynamic gravitational force (in the low attonewton range) between two milligram-scale masses while suppressing overwhelming environmental noise and non-gravitational couplings.

Methodology
The authors present an experiment conducted at millikelvin temperatures utilizing a cavity-optomechanical setup to measure the gravitational interaction between an 87 mg source mass and a 1.3 mg test mass.

Experimental Apparatus: The core device is a superconducting microwave optomechanical system housed in a cryogen-free dilution refrigerator. The test mass ( $m_t \approx 1.26$ mg) is a gold sphere attached to a high-stress silicon nitride ( $Si_3N_4$ ) membrane, forming a mechanical oscillator with a resonance frequency of $\approx 2.14$ kHz. This oscillator is coupled to a 3D superconducting microwave cavity, allowing for the detection and manipulation of its vibrational state via microwave tones.
Source Mass Actuation: The source mass ( $m_s \approx 86.6$ mg), also a gold sphere, is mounted on a carbon fiber pole driven by a piezo buzzer. It oscillates vertically with an amplitude of $\approx 2.5$ $\mu$ m at a frequency matching the test mass resonance.
Noise Suppression and Shielding: To isolate the weak gravitational signal (expected $\approx 10$ $\approx 10$ aN), the setup employs multiple layers of protection:
- Vibration Isolation: Both masses are suspended from the mixing chamber via acoustic filters (copper cubes and titanium ring springs) providing $\sim 40$ dB attenuation at 2 kHz.
- Electromagnetic Shielding: The test mass is enclosed in a superconducting aluminum 3D cavity. A movable Faraday shield (two aluminum layers) separates the driving stage from the sensing stage. Magnetic fields are suppressed by $\sim 70$ dB using non-magnetic materials and static magnetic shielding.
- Distance Control: Nanopositioners allow precise adjustment of the source mass position, achieving a minimal surface-to-surface distance of $> 270$ $\mu$ m.
Measurement Scheme: The system operates in the unresolved sideband regime ( $\kappa \gg \omega_m$ ). A microwave pump tone near the cavity resonance ( $\approx 6.82$ GHz) is used to read out the mechanical motion via homodyne detection. The gravitational force from the oscillating source mass drives the test mass, creating sidebands in the reflected microwave signal.

Key Contributions and Results

Sensitivity Improvement: The experiment achieves a gravitational sensitivity of $\sqrt{S_g} \approx 61$ nGal/ $\sqrt{\text{Hz}}$ ( $6.1 \times 10^{-10}$ m/s $^2$ / $\sqrt{\text{Hz}}$ ) at kHz frequencies. This represents a 35 dB improvement over previous state-of-the-art measurements at similar frequencies and a two-order-of-magnitude improvement in the performance metric ( $m_t m_s^2$ ) compared to recent milligram-scale gravity experiments.
Detection of Dynamic Gravity: The authors successfully detected the dynamic gravitational force between the two masses. By driving the source mass at the resonance frequency of the test mass and coherently averaging 1-hour time traces, they observed a force consistent with Newtonian gravity.
Spatial Mapping:
- Horizontal Scan: Scanning the source mass in the horizontal plane revealed a "hotspot" of maximum force at a specific coordinate, matching the expected alignment of the mass centers. The force dropped off with a characteristic length scale of $\sim 1$ mm.
- Vertical Scan: Measuring the force at varying vertical separations yielded a fitted gravitational constant of $G_{fit} = (7.26 \pm 0.43 \text{ (stat)} +0.73_{-0.70} \text{ (syst)}) \times 10^{-11}$ m $^3$ kg $^{-1}$ s $^{-2}$ . This value is consistent with the CODATA literature value ( $6.67 \times 10^{-11}$ ) within the 1 $\sigma$ systematic uncertainties.
Validation of Gravitational Nature: The data followed the expected $d^{-3}_{com}$ distance dependence across the measurement range. The authors ruled out alternative explanations such as electromagnetic coupling (which would show a different distance dependence) or inertial actuation via suspension vibrations (which would be distance-independent).

Significance and Claims
The paper claims to demonstrate a "plausible approach" for studying gravitational interactions between masses prepared in non-classical motional states. While the current experiment operates with the test mass in a thermal state (phonon occupancy $n_T \approx 3.6 \times 10^6$ ), the authors argue that the setup is "quantum-ready." They posit that with lower bath temperatures and feedback cooling, the system could approach the motional ground state.

The authors characterize this work not as a demonstration of "true quantum gravity," but as a step toward "Noisy Intermediate Scale Quantum Gravity" (NISQ-Gravity). They suggest that the ability to couple milligram-scale masses with high-frequency, high-sensitivity optomechanical readout removes a major technological barrier to future experiments where both masses are in non-classical states. This would allow for the indirect probing of the quantum nature of gravity, a goal that has previously been hindered by the inability to access the necessary mass and quantum control scales simultaneously. The work establishes that gravitational coupling between milligram masses can be measured with sufficient precision to be compatible with future quantum control protocols.

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