Sail membranes for optomechanical accelerometry

Technical Summary: Sail Membranes for Optomechanical Accelerometry

Problem Statement
Strained membrane resonators are established platforms for optomechanical force sensing and quantum experiments, typically achieving high quality factors ($Q$) and high $Q \times f$ products. However, a new generation of fundamental field searches (e.g., ultralight dark matter, gravitational waves) and distributed quantum sensing requires a different regime: large-area, mass-loaded membranes with reduced resonance frequencies ($f$) and increased $Q \times m$ products to minimize thermal acceleration noise ($S_{th}^a \propto f/Qm$).

While increasing the physical size ($L$) of a square membrane theoretically improves acceleration sensitivity via "hard-clamped" dissipation dilution scaling ($Q \propto L$), practical fabrication constraints have limited strained membranes to roughly centimeter scales. Furthermore, simply scaling up square membranes does not sufficiently optimize the $Q \times m / f$ ratio. There is a need for alternative geometries that can achieve lower frequencies while preserving the high $Q \times m$ product necessary for high-sensitivity accelerometry.

Methodology
The authors employed Bayesian optimization to explore the design space of strained silicon nitride ($\text{Si}_3\text{N}_4$) trampoline resonators. The optimization aimed to maximize the $Q \times m / f$ figure of merit.

• Geometry Exploration: Two "sail-like" trampoline geometries were investigated: one with square fillets and one with circular fillets. The designs feature a large central pad (mass loading) connected to the frame by narrow tethers (stress concentration).
• Simulation & Optimization: Finite-element simulations were used to calculate stress profiles ($\sigma$) and mode shapes ($u$) to estimate $Q$ and effective mass ($m$). The authors optimized parameters including pad width ($W$), tether width ($t$), and fillet radii ($R, r$).
• Fabrication: Devices were fabricated using a 100-nm-thick, double-side-coated $\text{Si}_3\text{N}_4$-on-Si wafer. Due to the fragility of centimeter-scale sails with high stress concentrations, a specialized two-stage wet-etching process was developed. This involved a selective back-etch to leave 20–40 $\mu$m of silicon, followed by a KOH wet-etch to release the membrane, ensuring the maximum von Mises stress remained below 3 GPa to prevent premature failure.
• Characterization: Mechanical properties were measured in a high-vacuum chamber ($\approx 10^{-8}$ mbar) using optical lever techniques for ringdown measurements and a scanning vibrometer for chip-mode analysis.
• Optomechanical Integration: A monolithic cavity optomechanical accelerometer was realized by vertically integrating a sail membrane atop a 200-$\mu$m-wide nanoribbon (serving as a rigid inertial reference). Vibration measurements were performed using a Fabry-Perot cavity formed by the double membrane, with displacement readout via a near-infrared laser.

Key Contributions and Results

1. Discovery of Sail Resonators: The optimization identified a class of "sail-like" resonators that reduce the fundamental frequency by an order of magnitude compared to square membranes while preserving the $Q \times m$ product.
2. Device Performance:
   • Frequency & Q: The authors fabricated centimeter-scale ($2.5 \times 2.5$ mm$^2$ pad) circular-fillet sails with frequencies around 7–8.5 kHz and quality factors up to $Q \sim 10^7$ (specifically $10 \times 10^6$).
   • Mass & $Q \times m$: The devices achieved an effective mass of $\sim 1.3$ $\mu$g, resulting in a $Q \times m$ product of $\sim 10$ g.
   • Thermal Noise: The measured thermal acceleration noise was approximately $43$ ng$_0$/ $\sqrt{\text{Hz}}$ for the best individual device, approaching the simulated limit of $\sim 10$ ng$_0$/ $\sqrt{\text{Hz}}$ for the optimized geometry.
3. Monolithic Accelerometer Demonstration: By integrating the sail with a nanoribbon, the team demonstrated a room-temperature optomechanical accelerometer.
   • Sensitivity: The device achieved a thermal noise floor of 40 ng$_0$/ $\sqrt{\text{Hz}}$.
   • Bandwidth: It resolved ambient vibrations of $\sim 1$ $\mu$g$_0$/ $\sqrt{\text{Hz}}$ over a 4 kHz bandwidth.
   • Displacement Imprecision: The system reached a displacement imprecision of $10^{-14}$ m/ $\sqrt{\text{Hz}}$ (specifically $\sim 9$ fm/ $\sqrt{\text{Hz}}$), limited by shot noise at higher optical powers.
4. Discrepancies: Measured $Q$ factors were lower than simulations by a factor of 5. The authors attribute this to potential gas damping (despite vacuum conditions) and mode hybridization with the fundamental flexural mode of the supporting 12-mm-wide chip, which was observed to resonate near the membrane frequency.

Significance and Claims
The paper claims that sail membranes represent a novel regime of sensitivity ($\sqrt{S_{th}^a} \le 100$ ng$_0$/ $\sqrt{\text{Hz}}$) previously unattained with $\text{Si}_3\text{N}_4$ membranes of any geometry. The work demonstrates that Bayesian optimization can successfully navigate complex design landscapes to discover geometries that decouple frequency reduction from $Q \times m$ degradation.

The authors position these devices as promising platforms for:

• Fundamental Physics: Searches for ultralight dark matter and high-frequency gravitational waves.
• Quantum Sensing: Distributed quantum sensing experiments and entanglement-enhanced force sensing protocols.
• Practical Applications: High-speed vibrometry and the study of radiation pressure effects in next-generation solar sails.

The paper concludes that cryogenic arrays of these sail membranes could be particularly attractive for new physics searches and distributed quantum sensing, leveraging their large area and low stiffness.