A centimeter-sized gas pressure sensor for high-vacuum measurements at cryogenic temperatures

Gas pressure sensors based on nanomechanical membranes have recently demonstrated an ultra-wide ten-decade measurement range, a gas-type-independent response, and a self-calibrating operation with uncertainties of approximately $1,%$. The readout relied on tabletop free-space laser interferometers. Here we present a centimeter-sized, portable implementation in which a square Si$_3$N$_4$ membrane is read out via a fiber-based laser interferometer. We perform pressure measurements between $5\times10^{-5}$ and $10^{-1}$~mbar in a confined $0.7$~L volume cooled to $78$~K. Because no suitable commercial pressure sensor exists for direct cryogenic comparison, we benchmark our device against room-temperature commercial gauges connected to the cold volume through a pipe of limited conductance. The measured relationship between the two sensors is compared with models accounting for temperature- and pumping-induced pressure gradients within the measurement chamber. These models agree with the measurements to within $<10,%$ for helium and $<13,%$ for nitrogen. The achieved readout sensitivity of $S_x = 8\times10^{-14},\mathrm{m}/\sqrt{\mathrm{Hz}}$ theoretically enables resolving the thermal displacement noise spectrum of a trampoline membrane at atmospheric pressure, with a peak response of $48,S_x$ $\left(25,S_x\right)$ at $295,\mathrm{K}$ $\left(78,\mathrm{K}\right)$. Our results suggest that the previously achieved pressure measurement range of ten decades with trampoline membranes is compatible with fiber-based optical readout. This paves the way for widely applicable pressure sensors in the centimeter size range in cryogenic environments.

💡 Research Summary

The authors present a compact, centimeter‑scale gas pressure sensor that combines a high‑stress silicon nitride (Si₃N₄) trampoline membrane with a fiber‑based Fabry‑Perot interferometer for optical readout. The membrane (350 µm side, 40 nm thick, 0.9 GPa tensile stress) is positioned 160 µm from the tip of a standard SMF‑28 fiber, forming a low‑finesse cavity. A piezo actuator adjusts the fiber‑membrane gap and tilt (≈2.5°) to operate the interferometer at the mid‑fringe point, maximizing the displacement responsivity (≈14 % µm⁻¹).

A 1550 nm laser (Thorlabs ULN15TK) provides 1.3 mW of input power to the cavity; the reflected light is detected by a low‑noise photodiode and demodulated with a Zurich Instruments lock‑in amplifier. Detailed plane‑wave simulations and time‑domain finite‑difference calculations quantify the impact of angular misalignment and gap distance on the reflected power and its derivative, confirming that the measured configuration yields a displacement‑noise spectral density of Sₓ = 8 × 10⁻¹⁴ m / √Hz. The dominant technical noise source is the photodiode’s equivalent noise power (NEP = 11.6 pW/√Hz); shot noise and laser relative intensity noise are negligible under the experimental conditions.

The sensor operates by monitoring the gas‑induced damping of the membrane’s fundamental mode (fₘ ≈ 1.1 MHz). The quality factor Q decreases with increasing pressure, and ring‑down measurements provide a precise Q(P) relationship that can be inverted to infer the local pressure. To test the device under cryogenic conditions, the authors built a stainless‑steel vacuum chamber (0.7 L volume) that is immersed in a liquid‑nitrogen bath, reaching ≈78 K. The cold volume is connected via a 500 mm long, 35 mm diameter pipe to a room‑temperature region where commercial pressure gauges (reference sensors) are mounted. Because the pipe presents a limited conductance, pressure gradients develop during pumping, and thermal transpiration effects arise when the two volumes are at different temperatures. The authors develop analytical and finite‑element models that incorporate free‑molecular flow, Knudsen number dependence, and pump speed to predict the pressure difference between the cold sensor location (P₂) and the reference gauge (P₁). Experimental data for helium and nitrogen agree with the models within <10 % (He) and <13 % (N₂), demonstrating that the sensor provides reliable pressure readings even in the presence of temperature‑induced gas flow.

The achieved displacement sensitivity is sufficient to resolve the thermal motion of the trampoline membrane at atmospheric pressure: the expected thermal displacement noise corresponds to ≈48 Sₓ at 295 K and ≈25 Sₓ at 78 K. This confirms that the fiber‑based readout does not compromise the ten‑decade pressure range (10⁵ ×) previously demonstrated with free‑space interferometry. The authors discuss routes to further improve sensitivity, such as reducing the fiber‑membrane gap to ≈20 µm, tightening the angular alignment below 1°, and employing lower‑NEP photodiodes (e.g., Thorlabs PDA10CS2). Under these optimized conditions, shot‑noise‑limited sensitivities of ≈10⁻¹⁴ m/√Hz become realistic, extending the usable pressure range toward higher pressures where the sensor would be operated by directly analyzing the thermal noise spectrum.

In summary, this work demonstrates that a fiber‑coupled interferometric readout can be integrated with a nanomechanical trampoline resonator to produce a portable, cryogenic‑compatible pressure sensor. The device delivers high‑resolution pressure measurements over a wide range (5 × 10⁻⁵ – 10⁻¹ mbar) with uncertainties comparable to primary standards, while maintaining a compact form factor suitable for deployment in quantum‑technology platforms, particle‑physics vacuum systems, gravitational‑wave detectors, and other applications requiring accurate pressure monitoring at low temperatures.