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Physical Review A

Time delay and the effect of the finite speed of light in atom gravimeters

Yu-Jie Tan, Cheng-Gang Shao, Zhong-Kun Hu

Published Aug 2, 2017• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.96.023604

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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.
We present here an analysis of the influence of the frequency dependence of the Raman laser light shifts on the phase of a Raman-type atom gravimeter. Frequency chirps are applied to the Raman lasers in order to compensate gravity and ensure the resonance of the Raman pulses during the interferometer. We show that the change in the Raman light shift when this chirp is applied only to one of the two Raman lasers is enough to bias the gravity measurement by a fraction of $\ensuremath{\mu}\mathrm{Gal}$ ($1\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{Gal}\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}{10}^{\ensuremath{-}8}\phantom{\rule{0.28em}{0ex}}\mathrm{m}/{\mathrm{s}}^{2}$). We also show that this effect is not compensated when averaging over the two directions of the Raman wave vector kkk. This thus constitutes a limit to the rejection efficiency of the kkk-reversal technique. Our analysis allows us to separate this effect from the effect of the finite speed of light, which we find in perfect agreement with expected values. This study highlights the benefit of chirping symmetrically the two Raman lasers.
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 ggg 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.
In the presence of Earth gravity and gravity-gradient forces, centrifugal and Coriolis forces caused by the Earth rotation, the phase of the time-domain atom interferometers is calculated with accuracy up to the terms proportional to the fourth degree of the time separation between pulses. We considered double-loop atom interferometers and found appropriate condition to eliminate their sensitivity to acceleration to get atomic gyroscope, or to eliminate the sensitivity to rotation to increase accuracy of the atomic gravimeter. Consequent use of these interferometers allows one to measure all components of the acceleration and rotation frequency projection on the plane perpendicular to gravity acceleration. Atom interference on the Raman transition driving by noncounterpropagating optical fields is proposed to exclude stimulated echo processes which can affect the accuracy of the atomic gyroscopes. Using noncounterpropagating optical fields allows one to get a new type of the Ramsey fringes arising in the unidirectional Raman pulses and therefore centered at the two-quantum line center. Density matrix in the Wigner representation is used to perform calculations. It is shown that in the time between pulses, in the noninertial frame, for atoms with fully quantized spatial degrees of freedom, this density matrix obeys classical Liouville equations.
The technology of electro-optic modulation is one of the several methods of generating the Raman beams. The experimental system based on this method is simple and much easier to implement, and the environmental adaptability is strong as well. However, this kind of modulation technology will produce additional laser lines, which may affect the measurement accuracy of cold atom gravimeter. Based on a homemade transportable cold atom gravimeter, the influence of Raman sideband effect on the accuracy of cold atom gravimeter is investigated in this paper. We analyze in detail the relationship between Raman sideband effect and some experimental parameters, such as the height of Raman retro-reflection mirror, the time of free fall of the atoms, the detuning of Raman laser, etc. It is found that those parameters have a dominant influence on the measured gravity resulting from Raman sideband effect. Besides, it is also found that the gravity measurements will be sensitive again to some experimental parameters in the case of Raman sideband effect while these parameters are usually insensitive in case of laser system without sideband effect. Finally, we investigate the relationship between Raman sideband effect and Raman detuning, and presente a method of evaluating the gravity induced by Raman sideband effect. The experimental results in this paper can provide a reference for reducing the influence of Raman sideband effect on the accuracy evaluation of cold atomic gravimeter.
We investigate the dependence of the sensitivity of atom interferometer gravimeter on the Raman laser pulse sequence. The sensitivity of atom gravimeter can be adjusted by changing the pulse duration. Under the standard pulse configuration condition, the atom interferometer with four pulses is insensitive to the gravity if only gravity and its first order gradient are taken into account. The three-pulse Raman atom interferometer gravimeter displays best sensitivity. Atom interferometers with more than four Raman pulses show poorer sensitivity.