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

Abstract

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.

Figures (3)

• Figure 1: (a) Superfluid $^4$He gyrometer (left) consisting of a loop of superfluid $^4$He interrupted by a Josephson junction, connected in parallel with a controllable diaphragm to constitute a hydrodynamic Helmholtz resonator. Rotation perturbs the quantum phase difference across the junction, thereby modifying the Helmholtz resonant frequency, $\omega_H$. (b) Equivalent electrical circuit (right) showing the hydrodynamic inductance of the loop $L_l$, the junction $L_J(\phi_J)$, the hydrodynamic capacitance $C_d$, and the resistive losses $R_l$, $R_J$, and $R_d$.
• Figure 2: The angles $\theta$, $\chi$ and $\psi$ describe the relative orientations of the Earth angular velocity, $\vec{\Omega}_\oplus$, the gyrometer position relative to Earth center, $\vec{r}$, and the area vector normal to the gyrometer loop, $\vec{A}$.
• Figure 3: Noise spectral density on rotation rate (left vertical axis) and on proper time delay (right vertical axis) vs temperature for a single junction $^4$He SQUID, with diaphragm quality factors $Q_d=10^4-10^9$. The solid lines show the spectral density $\sqrt{S_\Omega}$ using dissipation from $R_l$, $R_J$, and $R_d$, while the dashed lines show the lower limit of spectral density achieved from $R_d$ alone. The lowest noise spectral density measured previously with superfluid helium is $1\times 10^{-7}~\mathrm{rad/s}/\sqrt{\mathrm{Hz}}$ (for a $^3$He gyrometer avenel2004superfluid), and the best achieved with a ring laser is $2\times 10^{-12}~\mathrm{rad/s}/\sqrt{\mathrm{Hz}}$di2024noise.