Einstein's theory of general relativity predicts the gravitational redshift, where a clock at a higher gravitational potential ticks faster than one at a lower potential. While general relativity has been extensively tested, a unifying theory with quantum mechanics remains elusive. Precision tests of general relativity using quantum systems, like atomic clocks, offer a pathway to explore this interplay. Atomic clocks, with their increasing stability and accuracy, have enabled tests of the gravitational redshift across various length scales. Previous experiments involved comparing clocks separated by centimeters to hundreds of meters. This research aims to perform a more precise test at the sub-centimeter scale, enhancing the sensitivity to gravitational potential differences at a scale relevant for emerging applications.

Existing literature demonstrates successful tests of gravitational redshift using various methods and scales. These include comparisons between clocks separated by 30 cm, terrestrial clocks and orbiting clocks (yielding strong limits on deviations), and clocks separated by 450 m (providing the most precise terrestrial constraint). A recent study even observed the gravitational redshift gradient across a millimeter-scale atomic ensemble with high differential precision. These previous experiments, however, either lacked a blinded protocol, relied on techniques limited by local oscillator coherence times, or lacked a full systematic evaluation at the required precision level. This study addresses these limitations.

This study utilizes a spatially multiplexed optical lattice clock comprising five ⁸⁷Sr atomic ensembles arranged vertically, spaced 2.5 mm apart over a 1 cm height difference. Synchronous differential comparisons are conducted between all ensemble pairs, providing ten unique pairwise clock comparisons. Improvements over previous work include a deeper initial optical lattice for increased atom numbers and reduced temperature, leading to longer coherence times and reduced instability. A blinded measurement protocol is implemented to mitigate bias. The experiment employs synchronous Ramsey spectroscopy, which allows interrogation times beyond the local oscillator limit, enabling high-precision measurements. Systematic effects, such as density shifts, second-order Zeeman shifts, and black body radiation (BBR) shifts are meticulously evaluated and corrected for. A detailed systematic analysis is performed, including the characterization of the BBR gradient sensitivity, the density shift per atom number difference, and the second-order Zeeman gradient. These systematic effects were evaluated and corrected for individually for each ensemble pair to avoid bias. The measurement uncertainty is reported considering statistical and systematic uncertainties and including the covariance between correlated clock comparison pairs.

The experiment yielded a measured fractional frequency gradient of [-12.4 ± 0.7(stat) ± 2.5(sys)] × 10⁻¹⁹/cm, consistent with the expected redshift gradient of -10.9 × 10⁻¹⁹/cm. This result constrains deviations from the redshift predicted by general relativity to 0.13 ± 0.23 for millimeter to centimeter scale height differences. The statistical uncertainty was inflated by the square root of the reduced χ² statistic (χ² = 1.16). The measurement is inconsistent with zero gravitational redshift at a confidence level of 4.9σ. The study also demonstrated a gravitational redshift measurement resolution of 1.3 mm, primarily limited by the differential density shift. The results obtained are consistent across 14 independent measurement runs spanning 3 weeks. A secondary analysis method, averaging differential frequencies of ensemble pairs with the same height difference, yielded a consistent frequency gradient of (-11.9 ± 2.6) × 10⁻¹⁹/cm, further supporting the findings.

The results confirm the gravitational redshift at the sub-centimeter scale, consistent with general relativity predictions. The high precision achieved demonstrates the feasibility of relativistic gravitational potential measurements with millimeter-scale resolution. The study also showcases the advantages of synchronous Ramsey spectroscopy and spatially multiplexed ensemble techniques for achieving high accuracy and precision in optical lattice clocks without requiring state-of-the-art clock lasers. The results provide important insights for developing advanced applications utilizing spatially separated clocks. While this demonstrated the principle for short spatial separations, longer baselines present challenges.

This research provides a blinded, high-precision test of the gravitational redshift at the sub-centimeter scale, validating general relativity predictions. The use of a spatially multiplexed ensemble technique with synchronous Ramsey spectroscopy enables long coherence times and low differential instabilities, facilitating a comprehensive systematic evaluation at the 10⁻¹⁹ level. This opens opportunities for improved relativistic geodesy and exploration of the quantum-gravity interplay. Future research could expand the apparatus for larger spatial separations, further enhancing precision and extending the range of potential applications.

The primary limitation in this study is the 1.3 mm resolution, mainly due to the systematic uncertainty from the differential density shift. Extending the spatial separation to the meter scale could increase the magnitude of the redshift, potentially improving accuracy. While the experiment successfully demonstrates the principle of relativistic gravitational potential measurements with mm-scale resolution in a controlled lab environment, scaling it to geodetic applications over larger distances requires overcoming the challenges posed by non-common-mode systematic effects.