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.