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Impact of the frequency error of direct digital synthesizers on the evaluation of two-photon light shift in atom gravimeters

Dong Zhu, Naike Liu, Bing Cheng, Pengfei Cao, Bin Wu, Kainan Wang, Li-Ming Wu, Kanxing Weng, Yin Zhou, J. G. Bian, Xiaolong Wang, Qiang Lin

Published Oct 8, 2024• 6 citations

Abstract

One major systematic error for free-fall atom gravimeters is the effect of two-photon light shift (LS2). In the process of evaluating LS2, the results can be affected by the residual frequency error of direct digital synthesizers due to necessary experimental parameter changes. In this paper, the impact of the coupling effect between the frequency error and LS2 has been investigated and analyzed, along with the related physical mechanisms. The parameters of the frequency error sequence, such as the time step of δt and the time delay of Δt, significantly affect the bias and uncertainty of LS2. Notably, when these parameters are significantly altered or misaligned, the impact can reach several tens of µGal. Conversely, when they remain within the optimal ranges, the impact can be minimal. Specifically, when δt is less than or equal to 10 µs, the impact on the LS2 bias is less than 0.3 µGal, with the contribution to the total uncertainty of the gravity value being approximately 0.1 µGal. Furthermore, after correcting the phase shift introduced by the frequency error, the evaluation results have a similar improvement effect on the whole. Compared to the LS2 theoretical value, the residual is generally on the order of µGal. Through the comprehensive analysis and optimization, as well as related experiments, this cross-effect is managed and decoupled, leading to accurate LS2 evaluation results. This improvement ensures a better accuracy in the obtained absolute gravity values. The analysis methods discussed can provide an effective strategy for enhancing the performance of atom gravimeters.

DOI: 10.1364/oe.535455

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One major systematic error for free-fall atom gravimeters is the effect of two-photon light shift (LS2). In the process of evaluating LS2, the results can be affected by the residual frequency error of direct digital synthesizers due to necessary experimental parameter changes. In this paper, the impact of the coupling effect between the frequency error and LS2 has been investigated and analyzed, along with the related physical mechanisms. The parameters of the frequency error sequence, such as the time step of δt and the time delay of Δt, significantly affect the bias and uncertainty of LS2. Notably, when these parameters are significantly altered or misaligned, the impact can reach several tens of µGal. Conversely, when they remain within the optimal ranges, the impact can be minimal. Specifically, when δt is less than or equal to 10 µs, the impact on the LS2 bias is less than 0.3 µGal, with the contribution to the total uncertainty of the gravity value being approximately 0.1 µGal. Furthermore, after correcting the phase shift introduced by the frequency error, the evaluation results have a similar improvement effect on the whole. Compared to the LS2 theoretical value, the residual is generally on the order of µGal. Through the comprehensive analysis and optimization, as well as related experiments, this cross-effect is managed and decoupled, leading to accurate LS2 evaluation results. This improvement ensures a better accuracy in the obtained absolute gravity values. The analysis methods discussed can provide an effective strategy for enhancing the performance of atom gravimeters.

We report on the study of the impact of the finite resolution of the chirp rate applied on the frequency difference between the Raman lasers beamsplitters onto the phase of a free fall atom gravimeter. This chirp induces a phase shift that compensates the one due to gravity acceleration, allowing for its precise determination in terms of frequencies. In practice, it is most often generated by a direct digital synthesizer (DDS). Besides the effect of eventual rounding errors, we evaluate here the bias on the

g

measurement due to the finite time and frequency resolution of the chirp generated by the DDS, and show that it can compromise the measurement accuracy. However, this effect can be mitigated by an adequate choice of the DDS chirp parameters resulting from a trade-off between interferometer phase resolution and induced bias.

Light-pulse atom gravimetry has emerged as a powerful technique for precisely measuring absolute gravitational acceleration. Raman lasers are commonly used in this technique to coherently manipulate the atomic wave packet, of which the effective frequency is chirped continuously to compensate for the Doppler shift. In this study we investigate an additional phase shift in the Raman lasers that arises due to the frequency chirp. We directly measure this phase shift by recording the beat signal of the Raman lasers with a high-speed oscilloscope, providing an independent evaluation of the resultant error in atom gravimeters caused by this phase shift. Concurrently, we detect the influence of the additional phase shift on gravity measurements using our atom gravimeter, and the results are in good agreement with the independent evaluation. Notably, there may still be a residual error at the microgal level when performing differential measurements by reversing the direction of the effective wave vector of the Raman lasers. We propose that this residual error can in principle be eliminated by selecting the same chirp frequency range for both directions in the differential measurement.

Differential light shift (DLS) is an important error term that limits the atom interferometer's measurement precision, especially for the case of the electro-optic modulator (EOM)-based scheme, where multiple laser sidebands exist, and their ratios are hard to control synchronously. This article carried out an experimental and theoretical study on this subject. By conducting long-term gravity measurement, we find that the gravity exhibits drifts of about 13.13 μGal, and is strongly correlated to the Raman laser's sidebands. A model of the DLS-induced gravity error is established and a DLS compensation method is proposed to suppress the gravity drift to 2.54 μGal. Besides the compensation method, we propose a Dual-Sideband Ratio Locking scheme to more robustly eliminate the gravity measurement drift. By feeding back to both the EOM microwave power and the tapered amplifier's temperature, this method locks both the ±1 order sideband to a stability level of 10-5, which corresponds to a gravity error of less than 0.1 μGal. Long-term gravity measurement is carried out after the locking method, showing a long-term stability of 1.6 μGal. The proposed methods will benefit the suppression of the DLS effect for high-precision atom interference measurement.

The propagation time delay due to the finite speed of light (FSL) in atom gravimeters introduces a bias in the gravity measurement, as well as that in classical free-falling corner-cube gravimeters, which is usually termed the FSL effect. For a typical atom gravimeter, the FSL time delay is about several nanoseconds, resulting in the FSL effect, a non-negligible bias in the gravity-acceleration measurement. However, a time delay of about several microseconds, achieved by controlling the Raman-pulse timing directly, contributes a negligible effect. This interesting phenomenon motivates us to make clear two questions: first, what are the origins of the FSL effect in atom gravimeters, and second, what is the difference between the two time delays? Our analysis shows that the FSL effect in atom gravimeters is not just a matter of FSL time delay to a great extent but also the change in the effective wave vector; moreover, the FSL time delay can be quantitatively regarded as the same as the pulse time delay since both actually affect the gravity measurement by changing the two interferometer pulse separations.

The mechanism of light-shift-related uncertainty in an atom-interferometry gravimeter is investigated systematically, which arises from the inhomogeneous intensity of Raman beams and the Gaussian distribution of an atomic cloud. Experiments using atom interferometry are performed to verify this systematic error by directly modulating the integration time

T

and the intensity ratio of Raman laser beams. Using our method, the uncertainty of gravity measurements contributed by the single-photon light shift in an atom gravimeter can be corrected to a $\mathrm{sub}\ensuremath{-}\ensuremath{\mu}\mathrm{Gal}\phantom{\rule{0.28em}{0ex}}(1\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{Gal}=1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}\phantom{\rule{0.28em}{0ex}}\mathrm{m}/{\mathrm{s}}^{2})$ level.

The alignment of Raman beams is important for high-precision atom interferometry gravimeters, and tilt modulation experiments are usually carried out to calibrate the direction of Raman beams. However, the Coriolis effect could be changed during the tilt modulation experiment, thus introducing a deviation in the alignment of Raman beams. In this work, a method of rotating the sensor head by 90° to eliminate the impact of the Coriolis effect on Raman beams' alignment in a fountain atom gravimeter was proposed and demonstrated experimentally. Using this method, the alignment uncertainty of the Raman beams in the atom gravimeter was reduced to 11 μrad, which contributed an uncertainty of less than 0.1 μGal (1 μGal = 1 × 10-8 m/s2) to the gravity measurement. Further simulations explained the experimental results and pointed out a way to reduce the Coriolis effect in atom gravimeters.