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Chinese Physics B

Resolving gravitational redshift with sub-millimeter height differences using spin-squeezed optical clocks

Deshui Yu, Jia Zhang, Shougang Zhang, Tiantian Shi, Jingbiao Chen

Published Apr 8, 2025• 0 citations
Publisher

Abstract

Abstract The phenomenon that a clock at a higher gravitational potential ticks faster than one at a lower potential, also known as gravitational redshift, is one of the classical tests of Einstein’s theory of general relativity. Owing to their ultra-high accuracy and stability, state-of-the-art optical lattice clocks have enabled resolving the gravitational redshift with a millimeter-scale height difference. Further reducing the vertical inter-clock separation down to the sub-millimeter level and especially shortening the required measurement time may be achieved by employing spin squeezing. Here, we theoretically investigate the spin-squeezing-enhanced differential frequency comparison between two optical clocks within a lattice-trapped cloud of 171Yb atoms. The numerical results illustrate that for a sample of 104 atoms, the atomic-collision-limited resolution of the vertical separation between two clocks can reach 0.48 mm, corresponding to a fractional gravitational redshift at the 10−20 level. In addition, the required averaging time may be reduced to less than one hundredth of that of conventional clocks with independent atoms. Our work opens a door to the future spin-squeezing-enhanced test of general relativity.

DOI
10.1088/1674-1056/adca1d

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Abstract The phenomenon that a clock at a higher gravitational potential ticks faster than one at a lower potential, also known as gravitational redshift, is one of the classical tests of Einstein’s theory of general relativity. Owing to their ultra-high accuracy and stability, state-of-the-art optical lattice clocks have enabled resolving the gravitational redshift with a millimeter-scale height difference. Further reducing the vertical inter-clock separation down to the sub-millimeter level and especially shortening the required measurement time may be achieved by employing spin squeezing. Here, we theoretically investigate the spin-squeezing-enhanced differential frequency comparison between two optical clocks within a lattice-trapped cloud of 171Yb atoms. The numerical results illustrate that for a sample of 104 atoms, the atomic-collision-limited resolution of the vertical separation between two clocks can reach 0.48 mm, corresponding to a fractional gravitational redshift at the 10−20 level. In addition, the required averaging time may be reduced to less than one hundredth of that of conventional clocks with independent atoms. Our work opens a door to the future spin-squeezing-enhanced test of general relativity.
Rapid progress in the precision and accuracy of optical atomic clocks over the last decade has advanced the frontiers of timekeeping, metrology, and quantum science. However, the stabilities of most optical clocks remain limited by the local oscillator rather than the atoms themselves, leaving room for further progress. Here we implement a "multiplexed" one-dimensional optical lattice clock, in which spatially-resolved, movable ensembles of ultra-cold strontium atoms are trapped in the same optical lattice, interrogated simultaneously by a shared clock laser, and read-out in parallel. By performing synchronized Ramsey interrogations of ensemble pairs we observe atom-atom coherence times up to 26 seconds, a 270-fold improvement over the atom-laser coherence time, demonstrate a relative stability of 9.7(4)×10−18/τ9.7(4)\times10^{-18}/\sqrtτ9.7(4)×10−18/τ​ (where τττ is the averaging time in seconds), and reach a fractional uncertainty of 8.9(3)×10−208.9(3)\times 10^{-20}8.9(3)×10−20 after 3.3 hours of averaging. These results demonstrate that applications requiring ultra-high-precision comparisons between optical atomic clocks need not be limited by the stability of the local oscillator. With multiple ensemble pairs, we realize a miniaturized clock network consisting of 6 atom ensembles, resulting in 15 unique pairwise clock comparisons with relative stabilities below 3×10−17/τ3\times10^{-17}/\sqrtτ3×10−17/τ​. Finally, we demonstrate the capability to simultaneously load spatially-resolved, heterogeneous ensemble pairs of all four stable isotopes of strontium in a lattice. The unique capabilities offered by this platform pave the way for future studies of precision isotope shift measurements, spatially resolved characterization of limiting clock systematics, development of clock-based gravitational wave and dark matter detectors, and novel tests of relativity including measurements of the gravitational redshift at sub-centimeter scales.
Optical lattice clocks can now resolve the height difference below 1 mm within an atomic ensemble by means of gravitational redshift with integration over sufficient amount of time. Further improvement in the stability enables the clocks to resolve the height difference of subsystems within an atomic ensemble that is conventionally interrogated as a single coherent spin state in a single Ramsey sequence, resulting in the dephasing of the coherent spin state. This effect is observable with a clock of a stability of ≲10−21\lesssim 10^{-21}≲10−21 by introducing a single-layer-resolved imaging system for a three-dimensional optical lattice, and limits the 1 s stability of the clock around 10−1910^{-19}10−19 for an atomic ensemble distributing symmetrically for the three axes, which is also a signal of the decoherence. With atoms in an entangled state, this can be the first observation of the decoherence of a quantum state by a gravitational effect, and the suppression of other systematic shifts to observe this decoherence seems feasible.
Optical lattice clocks can now resolve the height difference below 1 cm by means of the gravitational redshift. Further improvement in the stability enables the clocks to resolve the height difference of subsystems within an atomic ensemble that is conventionally interrogated as a single coherent spin state, resulting in the dephasing of the coherent spin state. This effect is observable with a clock of the stability of ∼10−21\sim 10^{-21}∼10−21 by introducing single layer resolved imaging system for three dimensional optical lattice, and limits the 1 s stability of the clock at 10−1910^{-19}10−19 without proper compensations. This can be the first observation of the decoherence of a quantum state by a gravitational effect, and the suppression of other systematic shifts to observe this gravitational dephasing seems feasible.
Abstract The gravitational redshift (GR), as predicted by Einstein’s general theory of relativity, posits that two identical clocks situated at different gravitational potentials will tick at different rates. In this study, we explore the impact of the GR on a single-photon-based atom interferometer and propose a corresponding testing scheme. Our approach conceptualizes the atom interferometer as two coherent atomic clocks positioned at distinct elevations, which is referred to as an atomic clock interferometer, allowing us to derive the GR-induced phase shift. This effect becomes significant due to the notable energy difference between the two atomic internal states, comparable to other relativistic effects in single-photon-based atomic clock interferometers. Furthermore, our proposed scheme incorporates the velocity of the laser device to effectively mitigate other relativistic effects. The ensuing analysis indicates an anticipated GR test precision at the 10−5 level for our proposed approach.
Optical atomic clocks have spurred a range of precision frequency comparison experiments and proposals including tests of general relativity, gravitational wave detection, and relativistic geodesy. Current measurements using optical atomic clocks are often limited by the coherence of the clock laser rather than the atomic samples used as a frequency reference. In order to overcome the performance limits imposed by clock laser coherence, we realize a multiplexed optical clock system where multiple spatially resolved ensembles each consisting of thousands of strontium atoms trapped in a 1D optical lattice function as separate atomic frequency references that can be simultaneously interrogated with a shared clock laser. This scheme allows us to make differential frequency comparisons with a quantum projection noise limited fractional instability of 9.7(4)×10−18/ √τ and a differential fractional uncertainty of 8.9×10−20 after 3.3 hours of averaging. We then use this system to perform a lab-based test of the gravitational redshift at the centimeter scale, where we measure a fractional frequency gradient of [−12.4±0.7stat ±2.5sys]×10−19/cm, consistent with the predicted value of 10.9×10−19/cm. Having demonstrated that our differential comparisons are no longer limited by the local oscillator, we turn to fundamental limits on atom–atom coherence. We perform a direct decay measurement of the natural lifetime of the excited clock state 3P0 in 87Sr, in which we use measurements of simultaneously loaded high and low density atomic ensembles and compare decay rates of 87Sr and 88Sr to help isolate natural lifetime decay from other population dynamics. We measure a lifetime of 167+79 −40, which is consistent with previously reported experimental and theoretical values. Lastly, we demonstrate a hyperfine-resolved readout scheme in 87Sr that converts depolarization errors during interrogation into erasure errors, extending coherence times between ensembles in our optical lattice clock up to 150s using spin-echo measurements. These results taken together demonstrate the advantages of differential clock measurements that aren’t limited by clock laser coherence and how to characterize and improve on the performance limits that arise in that regime.
Many-body entangled states allow precision measurement beyond the standard quantum limit. Yet they have not been effectively exploited in optical lattice clocks except for proof-of-principle demonstrations. Under the current experimental conditions, taking into account the lattice perturbation, atom-atom interactions, and local oscillator noise, we theoretically evaluate the performance of spin-squeezing-based shallow-lattice Yb optical clocks. The numerical simulation shows that the stability of differential frequency comparison between two Yb ensembles in squeezed spin states potentially accesses the ${10}^{\ensuremath{-}19}$ level at the averaging time of 1 s. The resultant Wineland parameter may be as low as ${\ensuremath{\xi}}_{\text{W}}^{2}=0.027$, corresponding to a reduction of averaging time by a factor of 37, and is limited by the collision-induced degradation of spin squeezing. The metrological gain provided by spin squeezing opens up new opportunities for precision measurement and fundamental physics.