Quantum Sensing of Gravitational Frame-Dragging with a Superfluid $^4$He Gyrometer

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 $^4$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 $5\times 10^{-17}$ rads/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 $10^{-35}$ s.