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Gravitomagnetic Sagnac Effect for Matter Waves: Atom Interferometry in the Lense–Thirring Field of a Rotating Mass

Norbert Levente Kis

Published Aug 3, 2026• 0 citations
Publisher

Abstract

We derive the phase shift of a matter-wave interferometer in the gravitomagnetic field of a rotating body. By combining the Lense–Thirring metric in theweak-field, slow-rotation limit with the path-integral formulation of atom interferometry, we obtain an exact expression for the gravitomagnetic Sagnac phase:∆ϕ = (2m/ℏ) Ag · dl, where Ag is the gravitomagnetic vector potential. Foran Earth-orbiting atom interferometer, the predicted phase shift is ∆ϕ ∼ 10−1102 rad depending on the interferometer geometry and atomic species, placing itwithin reach of current cold-atom technology. A space-based implementation wouldimprove the sensitivity to the Lense–Thirring effect by two to three orders of magnitude compared to Gravity Probe B, providing a direct test of general relativitywith quantum systems

DOI
10.5281/zenodo.21771425

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We derive the phase shift of a matter-wave interferometer in the gravitomagnetic field of a rotating body. By combining the Lense–Thirring metric in theweak-field, slow-rotation limit with the path-integral formulation of atom interferometry, we obtain an exact expression for the gravitomagnetic Sagnac phase:∆ϕ = (2m/ℏ) Ag · dl, where Ag is the gravitomagnetic vector potential. Foran Earth-orbiting atom interferometer, the predicted phase shift is ∆ϕ ∼ 10−1102 rad depending on the interferometer geometry and atomic species, placing itwithin reach of current cold-atom technology. A space-based implementation wouldimprove the sensitivity to the Lense–Thirring effect by two to three orders of magnitude compared to Gravity Probe B, providing a direct test of general relativitywith quantum systems
The Sagnac effect, manifest in the phase shift of a rotating interferometer proportional to its physical area and to the rotation rate of the frame, plays a crucial role in modern physics being generalized to any kind of interferometer. However, as compared to light, the accurate experimental validation of the Sagnac effect for matter-waves remains challenging. We report the accurate measurement of the Sagnac phase shift, induced by the Earth’s rotation, using a large-area cold atom interferometer. We probe the atomic Sagnac phase shift along the two orthogonal horizontal axes in various orientation of the sensor with respect to geographic north and obtain an agreement with the theoretical prediction at the 20 ppm level, improving by twenty five-fold over the previous achievements for matter waves. Our results underline the universality of the Sagnac effect and open new horizons for testing other fundamental theories, as well as for a number of technological applications in seismology, geodesy and inertial navigation systems.
A rotation-rate sensor was developed using atom interferometry and the Sagnac effect for matter waves. This device has a short-term sensitivity of 6× 10−10 rad/sec for a 1 second integration time, which is the best publicly reported value to date. It has high intrinsic accuracy because it relies on interactions between atoms and laser light that is stabilized to an atomic transition. Long-term stability is promising and characterization is currently underway. Potential applications include inertial navigation, geophysical studies, and general relativity tests such as verification of the Lense-Thirring and geodetic effects. The principle of operation is as follows: A thermal cesium atomic beam crosses three laser interaction regions where two-photon stimulated Raman transitions between cesium ground states transfer momentum to atoms and divide, deflect, and recombine the atomic wavepackets. Rotation induces a phase shift between the two possible trajectories and causes a change in the detected number of atoms with a particular internal state. Counterpropagating atomic beams form two interferometers using shared lasers for common-mode rejection, and the rotation phase shifts have opposite sign since the phase shift is proportional to the vector velocity of the atoms. Therefore, subtracting or adding the interferometer signals discriminates between rotation and transverse acceleration summed with laser arbitrary phase. Furthermore, in the inertial reference frame of the atoms, rotations Doppler-shift the Raman lasers. Adding Raman frequency shifts to cancel these Doppler-shifts allows operation at an effective zero rotation rate, improving sensitivity and bandwidth. Modulating these frequency shifts facilitates sensitive lock-in detection readout techniques.
The Sagnac effect enables interferometric measurements of rotation with high precision. Using matter waves instead of light promises resolution enhancement by orders of magnitude that scales with particle mass. So far, the paradigm for matter wave Sagnac interferometry relies on de Broglie waves and thus on free propagation of atoms either in free fall or within waveguides. However, the Sagnac effect can be expressed as a proper time difference experienced by two observers moving in opposite directions along closed paths and has indeed been measured with atomic clocks flown around Earth. Inspired by this, we investigate an interferometer comprised of a single atomic clock. The Sagnac effect manifests as a phase shift between trapped atoms in different internal states after transportation along closed paths in opposite directions, without any free propagation. With analytic models, we quantify limitations of the scheme arising from atomic dynamics and finite temperature. Furthermore, we suggest an implementation with previously demonstrated technology.
A rotating interferometer with paths that enclose a physical area exhibits a phase shift proportional to this area and to the rotation rate of the frame. Understanding the origin of this so-called Sagnac effect has played a key role in the establishment of the theory of relativity and has pushed for the development of precision optical interferometers. The fundamental importance of the Sagnac effect motivated the realization of experiments to test its validity for waves beyond optical, but precision measurements remained a challenge. Here, we report the accurate test of the Sagnac effect for matter waves, by using a Cesium atom interferometer featuring a geometrical area of 11 cm 2 and two sensitive axes of measurements. We measure the phase shift induced by Earth’s rotation and find agreement with the theoretical prediction at an accuracy level of 25 parts per million. Beyond the importance for fundamental physics, our work opens practical applications in seismology and geodesy.
Since the first atom interferometry experiments in 1991, measurements of rotation through the Sagnac effect in open-area atom interferometers have been investigated. These studies have demonstrated very high sensitivity that can compete with state-of-the-art optical Sagnac interferometers. Since the early 2000s, these developments have been motivated by possible applications in inertial guidance and geophysics. Most matter-wave interferometers that have been investigated since then are based on two-photon Raman transitions for the manipulation of atomic wave packets. Results from the two most studied configurations, a space-domain interferometer with atomic beams and a time-domain interferometer with cold atoms, are presented and compared. Finally, the latest generation of cold atom interferometers and their preliminary results are presented.
We reports improvements to our Sagnac effect matter-wave interferometer gyroscope. This device now has a short-term rotation-rate sensitivity of 6 × 10 -10 rad s -1 over 1 s of integration, which is the best publicly reported value to date. Stimulated Raman transitions are used to coherently manipulate atoms from counterpropagating thermal beams, forming two interferometers with opposite rotation phase shifts, allowing rotation to be distinguished from acceleration and laser arbitrary phase. Furthermore, electronically compensating the rotation-induced Doppler shifts of the Raman lasers allows operation at an effective zero rotation rate, improving sensitivity and facilitating sensitive lock-in detection readout techniques. Long-term stability is promising but not yet fully characterized. Potential applications include inertial navigation, geophysical studies and tests of general relativity.