Cavity Optomechanical Probe of Gravity Between Massive Mechanical Oscillators

TL;DR

This work introduces a microwave cavity optomechanical scheme to sense gravity between milligram-scale oscillators by modulating a driven source mass and reading out the induced changes in optomechanically induced transparency. The key idea is that dynamic gravitational driving adds a coherent force on the test mass, leading to a measurable modification of the OMIT transmission spectrum, quantified by the peak-height factor $|1 + r e^{i\phi}|^2 - 1$. Under plausible parameters, the approach yields a peak-height variation up to about $2.3\%$ and a force responsivity on the order of $10^{18}$ to $10^{19}\ \text{N}^{-1}$, with the potential to reach even higher sensitivity at increased probe power. This demonstrates the feasibility of probing gravity at the milligram scale within a cavity optomechanical platform, offering a pathway to test Newtonian gravity at small masses and to explore gravity in macroscopic quantum regimes as well as possible deviations from Newtonian gravity due to extra dimensions.

Abstract

Exploring gravitational interactions between objects with small masses has become increasingly timely. Concurrently, oscillators with masses ranging between milligrams and grams in cavity optomechanical systems sparked interest for probing gravity, and even investigating gravity within macroscopic quantum systems. Here we present a measurement scheme for probing gravity in a microwave optomechanical setup that incorporates periodic gravitational modulation between the test mass and the driven source mass at the milligram scale. Optomechanically induced transparency (OMIT) can be utilized to sense the gravitational interactions between test masses and source masses. Specifically, the relative variation in the height of the OMIT peak, expressed as $|1 + re^{iφ}|^2 - 1$, where $r$ represents the ratio of the amplitude of the gravitational driving force to the radiation pressure force of the probe tone, and $φ$ denotes their phase difference, can reach up to 2.3\% under plausible experimental conditions. This work may facilitate cavity-optomechanical probing of gravitational coupling between milligram-scale mechanical oscillators, a mass regime where quantum and gravitational effects converge.

Figures (9)

• Figure 1: The values of source masses (blue marked area) used in gravitational constant measurement experiments and the masses of mechanical resonators (yellow marked area) used in cavity optomechanical experiments from 1790 to 2025 according to Table \ref{['tab:tableG']} and Table \ref{['tab:tableOPTO']} in Appendix \ref{['sec:tablesGandOPTO']}. The horizontal axis represents the year, and the vertical axis represents the mass. It can be seen from the figure that the mass scales involved in both types of experiments overlapped in 2021. The mass scale in our scheme is presented in the figure by a blue "$\times$" marker.
• Figure 2: This figure illustrates the sphere-sphere gravitational interaction component in the setup under investigation. Fig. \ref{['fig:fig2']}(a) shows the two membranes without the loaded masses. In Fig. \ref{['fig:fig2']}(b), after loading the test mass $M_1$ and source mass $M_2$, the two membranes bend slightly towards each other due to gravitational attraction, with the equilibrium center of mass distance between the two masses being $d$. Fig. \ref{['fig:fig2']}(c) depicts the small dynamic displacements $x_1$ and $x_2$ of the respective masses $M_1$ and $M_2$ away from equilibrium.
• Figure 3: This figure shows the studied setup, which consists of an optomechanical cavity with length $L=48\ mm$, width $W=20\ mm$, and height $H=20\ mm$ with two coaxially aligned prestressed membranes (membrane 1 and membrane 2), loaded with a test mass $M_1$ and a source mass $M_2$, forming oscillators 1 and 2. These components are aligned along a parallel axis from top to bottom, with metalized membrane 1 and the cavity forming a 3D rectangular microwave cavity. Oscillator 2, consisting of mass $M_2$ loaded on membrane 2, is positioned on a piezoelectric disk. When an alternating voltage is applied, the disk drives oscillator 2 into vibration, causing $M_2$ to undergo harmonic motion with amplitude $x_s$ and frequency $\omega_s$. The input and output microwave ports are also indicated. Both the pump and probe fields are coupled through the same in/out SMA connector. A gray partition between the two spheres represents a conducting Faraday shield with thickness of $50\ \mu m$, which suppresses Coulomb interactions due to stray charges while leaving the modulated gravitational interaction unaffected.
• Figure 4: The effect of the cavity field's radiation pressure and the gravitational interaction of mass $M_2$ on oscillator 1. The radiation pressure from the cavity field acting on oscillator 1 is equivalent to a spring with spring constant $k_p$ (which equals zero when $\omega = -\overline{\Delta}$) and a periodic driving force $F_p \cos(\omega t + \phi_p + \arg\overline{a})$, where the former corresponds to the optical spring effect and the latter to the driving force induced by the probe tone $\alpha_p$. Similarly, the gravitational force from $M_2$ acting on oscillator 1 can be equivalently represented as a spring with spring constant $k_G$ and a periodic driving force $F_G \cos(\omega t + \phi_s + \pi)$, where the former corresponds to the spring effect due to the gravitational gradient of $M_2$ and the latter originates from the vibration of $M_2$.
• Figure 5: The figures demonstrate the transmission spectra $|t_p|^2$ in the presence (red line, labelled as 'Driven') and absence of gravitational driving (blue dashed line, labelled as 'Unriven') under plausible experimental parameters from Table \ref{['tab:table1']}. Fig. \ref{['fig:fig5']}(a) demonstrates the overall transmission spectra for both cases. In Fig. \ref{['fig:fig5']}(b), the spectra near $\omega = \omega'_1$ are magnified by a factor $\delta = 5\times 10^{-7}$ in the $x$-direction.