We propose a laboratory-scale experiment to locally measure the general relativistic frame-dragging effect on Earth using the macroscopic quantum properties of a novel superfluid ${}^4\mathrm{He}$ single Josephson junction gyrometer. We derive the frame-dragging and related geodetic and Thomas effects in the superfluid gyrometer and present a procedure for their experimental measurement. We compute the expected thermal noise floor and find that very high sensitivity can be expected at millikelvin temperatures, where near-future Josephson junctions using nanoporous 2D materials are expected to operate. Assuming utilization of the lowest-mechanical-loss materials, we find a noise spectral density of $5\times {10}^{-17}\mathrm{rads}/\mathrm{s}/\sqrt{\mathrm{Hz}}$ at 10 mK, which is sufficient to resolve the frame-dragging rate to 0.2% within one second of measurement, giving a rotational sensitivity of 1 revolution in 4 Byrs. This extreme sensitivity to rotation corresponds to a measurement of proper time differences as small as ${10}^{-35}\mathrm{s}$.

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• General relativity
• Hydrodynamic noise
• Laboratory studies of gravity
• Quantum sensing
• Helium-4 superfluids