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

TL;DR

Problem: extend portable cryogenic sensing to centimeter-scale devices with a wide dynamic range for gas pressure measurements. Approach: integrate a silicon nitride membrane with a fiber-based interferometric readout inside a cryogenic vacuum chamber, enabling local pressure sensing in constrained volumes. Key results: achieved measurements from $5\times10^{-5}$ to $10^{-1}$ mbar in a $0.7$ L volume at $78$ K, with relative deviations $<10\%$ for He and $<13\%$ for N$_2$, and displacement sensitivity about $8\times10^{-14}$ m/√Hz, compatible with prior trampoline-based ten-decade performance. Significance: demonstrates portable, centimeter-scale, gas-type–independent pressure sensing in cryogenic environments, opening pathways for integration with quantum technologies and precision metrology.

Abstract

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.

Figures (6)

• Figure 1: Nanomechanical gas pressure sensor with fiber-optical readout. (a) Microscopic image of the square Si$_3$N$_4$ membrane (M) with 350 µm side length, which is suspended from a periodically-patterned silicon chip for acoustic isolation. A cleaved optical fiber of circular cross section (F) is placed at a distance of 160 µm behind the membrane (M), as schematically shown in inset (i) thereby forming a Fabry-Perot interferometer. (b) Photograph of the pressure sensor assembly, with dimensions of approximately 4 cm in each direction. The assembly includes two piezoelectric actuators: one for setting the fiber–membrane separation, contacted by red and black cables, and one for membrane actuation (located on the back side). The optical fiber with its white buffer coating is also visible. (c) Schematic of the fiber-coupled interferometric readout. Optical (electrical) signals are indicated by blue (black) arrows. Laser light at 1550 nm is directed to the sensor inside a cryogenically cooled vacuum chamber (78–295 K), and the back-reflected signal is detected with a photodiode and lock-in amplifier to read out the membrane motion. The lock-in amplifier output is applied to a piezoelectric actuator to excite membrane oscillations.
• Figure 2: Readout sensitivity of the fiber interferometer. (a) Angular misalignment $\theta$ between the fiber tip and the membrane (Mem) leads to an increasing lateral offset and mode mismatch with each reflection between the fiber and the reflected light. (b, upper) Reflected light power fraction as a function of fiber–membrane distance $L$ for different misalignment angles. Colored surfaces indicate the extent of the interference fringe; the dashed white (gray) line marks the fiber tip reflectivity (plane-wave Fabry–Pérot limit, Eq. \ref{['eq:plane_wave_limit']}). (b, lower) Gradient of the reflected power fraction with respect to displacement $L$, which defines the interferometric responsivity. (c) Measured displacement noise of the membrane at its intrinsic $Q$ factor [see Fig. \ref{['fig:cryostat']}(b)] at 295 K (gray). Modeled thermal displacement noise of the membrane at 295 K and 78 K is shown in orange and blue, respectively. Also shown are contributions from the photodetector noise-equivalent power (NEP, red), shot noise (green), and laser relative intensity noise (RIN, purple). The black curve (RSS) denotes the root-sum-square of all individual noise contributions for the demonstrated sensor ($L=160$ µm, $\theta=2.5^\circ$), while the brown curve (RSS mod.) indicates the predicted readout noise level for an optimized sensor (see text).
• Figure 3: Cryogenic vacuum setup for sensor characterization. (a) The vacuum chamber is mounted inside a liquid-nitrogen-filled (LN) dewar (78 K) and includes a thermal radiation shield, a gas inlet for He and N$_2$, and the membrane pressure sensor at the cold bottom. A thin pipe of diameter $D=35$ mm connects the cold volume to the room-temperature top, where a commercial pressure gauge provides reference readings. The system is evacuated through a bellow and angle valve. (b) Measurement (blue) of the membrane's intrinsic quality factor $Q_\text{in}$ versus temperature together with a fit function (orange; see text). The relative deviation between data and fit is shown above.
• Figure 4: Pressure measurements without active gas pumping or injection. (a) Simulated helium pressure distribution inside the vacuum chamber [see Fig. \ref{['fig:cryostat']}(a)] at $T_1 = 295$ K and $T_2 = 78$ K in the absence of gas injection or pumping. Symmetry boundary conditions are applied to model one half of the chamber; a similar distribution is obtained for nitrogen (not shown). The resulting pressure gradient originates from thermal transpiration and is consistent with the free-molecular-flow expectation $P_2/P_1 = \sqrt{T_1/T_2} = 1.94$. (b, lower) Measured relations between $P_1$ and $P_2$ (green dots) for helium (left) and nitrogen (right) under the same conditions. The corresponding predictions of the Takaishi–Sensui (TS) model are shown as solid green lines, smoothly transitioning between the free-molecular limit ($P_2/P_1 = \sqrt{T_1/T_2}$, dashed line) and the viscous limit ($P_2 = P_1$, solid black line). (b, upper) Quantities plotted relative to the measured $P_2$: the deviation between measured values and the TS model ($+$), the random measurement error ($\times$), and the root-sum-square of these two contributions ($\circ$); see text for details.
• Figure 5: Pressure measurements with active gas pumping and injection. (a) Simulated helium pressure distribution inside the vacuum chamber [see Fig. \ref{['fig:cryostat']}(a)] at $T_1 = 295$ K and $T_2 = 78$ K (half of the chamber simulated using symmetry boundary conditions). A local pressure maximum appears in the connecting pipe due to the combination of upward-directed gas flow from injection and opposing pressure increase in the upper volume from thermal transpiration. Similar distributions are obtained for nitrogen (not shown). (b, lower) Measured and modeled relations between $P_1$ and $P_2$ for helium (left, $T_2 = 78$ K) and nitrogen (right, $T_2 = 85$ K) at $T_1 = 295$ K. Finite-element (FEM) results for the free-molecular flow regime and a two-parameter fit at higher pressures are shown for comparison. Data points are color-coded for molecular-flow (blue) and higher-pressure (red) regimes. (b, upper) Deviation between measurements and models ($+$), random measurement error ($\times$), and root-sum-square of these contributions ($\circ$), all expressed relative to measured $P_2$ values; see text for details.