Dependence of the sensitivity on the orientation for a free-fall atom gravimeter

Bin Wu; Dong Zhu; Bing Cheng; Liming Wu; Kainan Wang; Zhaoying Wang; Qing Shu; Rui Li; Helin Wang; Xiaolong Wang; Qiang Lin

doi:10.1364/OE.27.011252

1. Introduction

Mobile absolute gravimeters are of great interest for a wide range of field applications. For instance, mobile absolute gravimeters could increase the efficiency of mobile gravimetric surveys, which are helpful for geophysics and seismic investigation. Normally, an accuracy of 0.02 $mGal$ is required for the mobile gravimetric survey. For these applications, operating a compact absolute gravimeter on a truck allows the fast measurement of the local absolute gravity so that the flexibility and efficiency could be improved greatly. However, the mobile absolute gravimeter with small size, light weight, and capability of fast measurement is still unavailable.

The cold atom absolute gravimeters (CAAG) could be a potential candidate for the mentioned applications. In the past decades, the performances of CAAGs have been improved substantially. In terms of sensitivities, a few $μGal/ \sqrt{Hz}$ have been achieved [1–3], which are higher than the classical FG5s. As for the accuracy, about 3-5 $μGal$ have been demonstrated by operating these gravimeters in absolute gravity comparisons [2–8], which are slightly worse than that of FG5s (the best accuracy is about 2 $μGal$ ). More importantly, the technology maturity of CAAG has been improved remarkably in recent years such that the compact and robust CAAGs have been developed [9–12]. The commercial instruments have been produced [13] and the marine absolute gravimeter has become true [14].

Intrinsically, the CAAG is sensitive to its orientation, which would affect the accuracy and may be a problem for field applications. Basically, the orientation of CAAG determines the horizontal velocity of atom along the East–west direction, which makes the atom interferometer sensitive to Earth’s rotation through the Sagnac effect. This is also well known as the Coriolis effect, which would yield a phase shift at the output of atom interferometer and therefore induce an additional gravity bias. This gravity bias was first investigated with a laboratory-based CCAG by rotating the whole big atomic fountain apparatus [15]. To simplify the rotation setup, a tip-tilt mirror was used to remove the inﬂuence of the Coriolis effect [16–19]. The method of rotating the mirror could also be applied to phase-shear atom interferometer to measure the dispersion of Coriolis shift [20]. Indeed, the synchronous rotation compensation of the Earth’s rotation rate is effective for canceling the Coriolis shift for atomic fountain gravimeter. However, the experimental system and control process are usually complex and require careful adjustments. As for the free-fall atomic gravimeter, a simple method was developed to separate the Coriolis shift from other systematic shifts by carrying out measurements with opposite directions with respect to the Earth’s rotation vector direction. For this experiment, the apparatus should be inverted by 180 $°$ in a short time so that only the sign of Coriolis shift is changed. So far, the Coriolis shifts had been evaluated precisely during several absolute gravity comparisons by using this method [4,5,8,21] and a simple average between two opposite orientations could remove the Coriolis acceleration bias.

The influence of Coriolis effect on the absolute gravity measurement is complicated, which is the convolution of the atomic cloud with the detection area [2]. It is closely related with the distribution of atomic velocity and initial position, and the area of the detection beam and fluorescence collecting apparatus. In this paper, we consider it as an averaging effect and evaluate the influence of the orientation on the absolute gravity measurement. Especially, we focus on the simplification of free fall atom gravimeter and improvement of its mobility such that the fast adjustment and gravity measurement could be realized when the atom gravimeter is rotated to another position. Combined with a rotation platform, the variations of gravity as a function of orientation angles are measured. It is found that the orientation has a dominant influence on the gravity measurement. Besides, the average horizontal velocity of the detected atoms has been evaluated by this method. In particular, the gravity bias due to Coriolis effect has been estimated. The results may be helpful for the quick measurement of the absolute gravity in the practical applications.

2. Theoretical analysis of the gravity bias due to the orientation of CCAG

As for cold atom gravimeter, the atoms have certain velocity distribution after the process of molasses. Usually, the atoms will still have a non-vanishing average horizontal velocity due to the imperfect experimental conditions, such as the misalignment of laser beams of magneto-optical trapping (MOT), the power imbalance of MOT beams, the mismatching of centers of MOT magnetic field and laser beams. The atomic trajectory will be changed by this initial average horizontal velocity. Taking the Earth’s rotation into account, the atom interferometer is sensitive to the horizontal velocity component along the East–west direction due to Sagnac effect [15,22], which can be shown in Fig. 1. So that a certain spatial area would be yielded by the two different interferometer paths, and this will lead to an additional phase shift, namely Coriolis shift, which can be given by

Δ ϕ_{co} = 2 \vec{Ω} \cdot (\vec{ν_{e w}} \times \vec{k_{e f f}}) T^{2},

where

\vec{Ω}

denotes the Earth’s rotation angular velocity,

\vec{ν_{e w}}

is the initial average horizontal velocity along the East–west direction,

\vec{k_{e f f}}

is the effective wave vector of Raman beams,

T

is the time interval between two sequential Raman pulses. The resulted gravity bias due to the Coriolis shift can be written as

Δ g_{co} = 2 \vec{Ω} \cdot (\vec{ν_{e w}} \times \vec{v_{r e c}}),

where

\vec{v_{r e c}}

is the normal vector in the direction of the photon recoil. Assuming that the geographical latitude of the measured position is

ϕ_{l a t}

and there is an average horizontal velocity_{$v_{h o r}$ . If the angle of orientation is $θ_{o r}$ , the average horizontal velocity along the East–west direction $v_{e w} = v_{h o r} \cos θ_{o r}$ , then Eq. (2) can be simplified to
Fig. 1 The relative position of our laboratory with respect to the Earth’s rotation and the diagram of atomic velocity and trajectory during free fall. a) The atoms have the horizontal velocity component along the East–west direction; b) The atoms with different horizontal velocity have different atomic trajectories.
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$Δ g_{c o} = 2 Ω \cos ϕ_{l a t} v_{h o r} \cos θ_{o r}$ Usually, the Earth’s rotation rate could be considered as a constant ( $Ω$ = 72.9 $μrad/s$ ). The latitude of our laboratory is about 30.3 $°$ . The influence of orientation of CCAG on the high-precision gravity measurement can be simulated, and an average horizontal velocity of 10 mm/s would lead to a gravity change of about 250 $μGal$ , which can be shown in Fig. 2. Based on the calculated curves in Fig. 2, the direction of atomic velocity with respect to East–west direction could be obtained. In particular, the gravity bias results from the Coriolis effect (the amplitude value of theoretical curve) could be estimated easily. Theoretically, operating the CCAG on the bottoms or peaks of curve would be better than that of the slope since the gravimeter is not so sensitive to the small changes of orientation angle. More importantly, the measured gravity could be corrected by this curve if it has to work at different orientations, and this is valuable for the field application of atomic gravimeter.
Fig. 2 The variations of gravity with the orientation angles of CCAG at three different average horizontal velocity of atoms.
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3. Experimental apparatus
The cold atom absolute gravimeter is based on a free fall of cold ⁸⁷Rb atoms and the coherent manipulation of their wave packets with Raman laser pulses. The free-fall type of CCAG has been investigated in many literatures [7,9,13,23–25]. Compared with atomic fountain gravimeter, the experimental apparatus is simpler and more compact. Here, we give a brief description of the experimental setup and its main features. The CCAG is made of three parts: vacuum system, laser system, and control system. The vacuum system is built based on the material of glass to enlarge the optical access, and therefore the size of main vacuum chamber is reduced greatly. However, all the functions of atom gravimeter, including 2D-MOT, 3D-MOT, state selection, atom interferometer, and fluorescence detection, could be realized in this glass cell. In order to improve the compactness of vacuum chamber, the optical molasses is realized by splitting one laser beam to three and retro-reflecting them with 0 $°$ mirrors. As can been seen from Fig. 3, two pairs of molasses beams are located in the x-z plane and another one is along the y axis. The direction of detection beam is parallel to the y axis, and the fluorescence collection apparatus is in the x-y plane. There is only one collection system and its orientation angle with respect to x axis is about 45 $°$ . Besides, the vacuum chamber is directly mounted on an aluminum platform and surrounded by one layer of mu-metal for magnetic field shielding. The gravity sensor has a height of 0.6 m and a diameter of 0.6 m, and its weight is about 70 kg. In terms of laser system, the simple scheme based on only two lasers is used, which is similar to [26]. Two extended cavity laser diodes with a wavelength of 780 nm are implemented to produce all the required optical frequencies, and two tapered amplifiers are utilized to yield the powers for 2D-MOT, 3D-MOT, Push, Blow, Detection and Raman beams. As for the control system, all the electronic modules are situated in one custom 19” box with a size of 0.9 $\times$ 0.6 $\times$ 0.7 m³ (length $\times$ width $\times$ height). The vacuum system is connected to laser system with optical fibers, and communicates with the control system with several cables. All the three parts are designed to be compact and robust such that it can be transported easily.
Fig. 3 The schematic diagram of the experimental system.
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The schematic diagram of experimental measurement system is displayed in Fig. 3. The gravity sensor (mainly the vacuum system) lies on a custom rotation platform, which is used to change the orientation. The platform is fixed rigidly to the ground. The experimental procedure is as follows. At first, a two dimensional Magneto-Optical Trap (2D-MOT) is applied to prepared an intense cold atom beam, which is pushed to a 3D-MOT through a 3 mm differential tube by a transverse pushing beam. The atoms are cooled and trapped in the 3D-MOT, and approximate 10⁸ atoms are loaded within 300 ms. The magnetic field of MOT is turned off, then the power and detuning of the lasers are adiabatic switched off so that the temperature of atoms is cooled down to 6 $μK$ in a far detuning optical molasses. After that, the cold atoms are released from the molasses and fall freely in the gravitational field. We applied a vertical bias field of 50 $mG$ to define the quantization axis. Then a combination of microwave and optical Raman pluses are utilized to select the atoms in the magnetically insensitive sub-state ( $| F = 1, m_{F} = 0 〉$ ). Afterward, the Mach-Zehnder type atom interferometer is realized by three Raman pulses in a $π/2-π-π/2$ configuration separated by $T = 55$ ms, and the duration time of Raman $π$ pulse is 10 $μs$ . The Raman beams enter the vacuum chamber from the top though a polarization maintaining optical fiber with a lin-lin polarization configuration. A high-precision tilt meter is installed on the same plate as the Raman retro-reflected mirror to record the tilt angle. In the end, the populations are determined by normalized fluorescence detection. With the scanning of frequency chirp rate of Raman lasers, the fringe of atomic interference can be obtained, and therefore the gravity value could be derived from the phase of fringes. The total measurement time for a cycle is 0.5 s.
4. Experimental results
4.1 The performance of the free-fall atomic absolute gravimeter
We begin by introducing the performance of our CCAG, mainly including the sensitivity, accuracy and long-term stability. Since the CCAG here had participated in the pilot study of the International Comparison of Absolute Gravimeter (CCM.G-K2.2017), which was held by National Institute of Metrology (NIM) of China at Changping of Beijing in October 2017. The performance was evaluated and demonstrated by this comparison, and the related results of this comparison will be published recently by the organizer. The sensitivity and accuracy of our CCAG were estimated to be about 200 $μGal/ \sqrt{Hz}$ and 20 $μGal$ . In this experiment, the sensitivity has been evaluated to be 262 $μGal/ \sqrt{Hz}$ , which is slightly worse than that during the comparison due to that the vibration environment of our laboratory is not so good. The Allan deviation of the measured gravity can be seen from Fig. 4. Besides, the accuracy has been rechecked to be 20 $μGal$ by several other absolute gravity comparisons. As for the long-term stability, the experimental measurement of the gravity tide over 200 hours has been carried out and the measured results coincidence with the theoretical tidal model, which can be seen from Fig. 5.
Fig. 4 The allan deviation of the gravity data.
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Fig. 5 The demonstration of long-term stability by measurement of the gravity tide over 200 hours (MJD: Modified Julian Day). Each point denotes an average of 30 minutes. The scatters are the measured experimental data, and the red line is the theoretical curve calculated by tidal model.
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In order to carry out the experiment, we have improved the performance of mobility of our gravimeter such that the adjustment and movement are convenient. The experiment requires other systematic effects keep the same when we rotate the gravimeter to another position step by step. Therefore the measurements for each point should be finished as soon as possible. However, the high-precision atom gravimeters usually have complicated vibration isolation system to increase the sensitivity [3,9,23]. Also, the active tilt control systems are needed to keep the verticality. For instance, the passive vibration isolation platforms are normally used to suppress the ground vibration noise. However, it is difficult to keep the platform in a good condition and its tilt drift is also a problem for fast measurement. The authors claim that it takes about 2 hours to rotate the whole cold atom gravimeter by 180 $°$ in [7]. In our experiment, some measures are taken to enhance the mobility and reduce the measurement time. Firstly, we do not apply any vibration isolation system and active tilt control system for the sake of the simplification of experimental system. It is transportable and suitable for fast measurement. Secondly, the gravity sensor, rotation platform, and ground are connected rigidly, and the system structure is displayed in Fig. 3. The gravity sensor is fixed directly on the rotation platform. Its tilt could be adjusted conveniently by three supporting feet and recorded by a high-precision tilt meter in real-time. Besides, a rotation of 360 $°$ could be realized manually in 1 or 2 minutes. The verticality of gravity senor could be optimized in less than 5 minutes based on the value of tilt meter inside when the gravimeter is rotated to another position.
4.2 Measurement of the sensitivity to orientation of the gravimeter
Normally, the influence of Coriolis effect on absolute gravity measurement is complicated, which is related to the convolution of the atomic cloud with the detection area. Here, we do not focus on the experimental detailed parameters and just consider it as an averaging effect of all the detected atoms. Therefore, we take the gravimeter as a whole instrument and care about the sensitivity to its orientation.
Based on the improved atom gravimeter and a custom rotation platform, the measurements of sensitivity to orientation have been taken. In advance, the tilt modulation experiments are carried out to find the true vertical direction for the gravimeter, and the experimental procedures and more details can be found in our previous paper [25]. The verticality of gravimeter can be finally marked by specific values of the tilt meter, such as ( $t x_{0}$ , $t y_{0}$ ). Then the orientation of gravimeter is changed step by step via the rotation platform. In order to guarantee the verticality for each measurement, the tilt of gravimeter is adjusted to ( $t x_{0}$ , $t y_{0}$ ) by three supporting legs within 5 minutes when the gravimeter is moved. Besides, the overlaps of Raman beams inside the vacuum chamber have been re-checked by coupling the retro-reflected light back into the same fiber and monitoring the power after the fiber with an optical power meter. Finally, the gravity is measured for about 15 minutes with the method of reversing the direction of effective Raman wave-vector [7], which allows us to reject most of the systematic effects that do not depend on the sign of $k_{e f f}$ . Meanwhile, the drifts of tilt are recorded in real time during the gravity measurement, and the resulted gravity biases have been calculated and corrected to each measured gravity value. One measurement takes about 5 minutes for verticality adjustment and 15 minutes for gravity measurement.
Figure 6 displays the gravity values measured with respect to the orientation angles. The scatters are the experimental data and the different colors denote the measured data at three different days. The red line is the theoretical curve calculated by Eq. (3). It is found that the experimental data agree well with the theoretical calculation, and stability of the measurement is good as well, which can be seen from the measurements in different days. Some interesting results can be obtained from the measured data. On the one hand, it is found that the orientation angles of the gravimeter yield a dominant influence on the absolute gravity measurement (the peak to peak value is about 415.4(3.8) $μGal$ . The magnitude of average horizontal velocity of the detected atoms is estimated to 16.5(0.3) $mm/s$ according to the Eq. (3). The direction of average horizontal velocity of the detected atoms with respect to East–west direction is also derived, where peaks or bottoms denote the parallel between the direction of atomic average horizontal velocity and East–west orientation. On the other hand, the gravity bias due to Coriolis effect could be estimated conveniently from the fitted curve, which is about 207.7(3.8) $μGal$ for Fig. 6.
Fig. 6 Measurements of the sensitivity to orientation for the mobile atomic gravimeter. The scatters are the experimental data and the different colors denote the measured data at three different days; the red line is the fitted curve based on a cosine function.
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In order to check the influence of atomic velocity on this experiment, we have optimized the overlap of MOT beams such that the amplitude of atomic average horizontal velocity is reduced. Then, the gravity values are recorded as an adjustment of the orientation angles, and the angle is changed with a step of 30 $°$ . Figure 7 shows the experimental results, the peak to peak value becomes 203.6(3.7) $μGal$ and therefore the magnitude of average horizontal velocity of the detected atoms is changed to 8.1(0.3) mm/s. Besides, the positions of peaks move as well, which means that the direction of average horizontal velocity of the detected atoms varies. The angle change for atomic horizontal velocity direction is estimated to be 96.8(1.3) $°$ , which can be obtained from the fitted curves. In principle, the atomic average horizontal velocity could be optimized to be zero through this method, and this is useful for atom gravimeter to reduce its sensitivity to orientation.
Fig. 7 The measured gravity variations as a function of the orientation angles after adjustment of the overlap of horizontal cooling beams. The scatters are the experimental data, and the black line is fitted curve.
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4.3 Evaluation of the gravity bias due to the Coriolis effect
Measurement of the East–west direction is helpful for the evaluation of gravity bias due to the Coriolis effect. Usually, we can use a high-precision compass to determine the true North of the measured site. Then the careful alignments of the gravimeter to this direction are required such that the detection system is orientated to the East–west direction. However, the compass is easy to be disturbed by other stray magnetic field. Instead, the gyro-theodolite has been utilized in gravity comparison to measure the direction of true North with a high accuracy [6]. The procedures for precise alignments are usually complicated and take a long time.
Here, we measure the East–west direction by using the atom gravimeter itself rather than the external auxiliary measurements. When the gravimeter is orientated to a direction where the atomic horizontal velocity direction is parallel to the East–west direction, there will be peaks or bottoms in Fig. 7. Therefore, the East–west direction could be deduced based on the fitted curve of Fig. 7. The orientation angles of the gravimeter are 28 $°$ and 208 $°$ for the bottom and peak, which denote the desired direction for the experiment. The measurements here provide a method for finding the true North especially in the limited space, such as the small room in the basement, truck, cave, and so on.
In order to extract the gravity bias due to the Coriolis effect, the method of reversing the direction of effective Raman wave-vector is used to remove most of the systematic phase shifts that do not depend on the sign of k_eff. Then the gravity measurements for two opposite orientations are carried out to derive the gravity bias due to Coriolis effect. Since the gravity sensor could be rotated by 180 $°$ in 5 minutes and the stability of our gravimeter is good enough. It is reasonable to consider that all other systematic phase shifts remain the same, except for the Coriolis phase shift that changes the sign. Therefore we can measure the gravity signal for two opposite orientations (28 $°$ and 208 $°$ ), and the experimental results are shown in Fig. 8. We can see that the gravity values are stable when the alternate measurements are carried out for 20 times. Each measured point takes about 20 minutes. The gravity biases due to Coriolis effect could be derived by half the difference of gravity signals of two opposite orientations, which can be shown in the right of Fig. 8. Take all the data into account, the gravity bias results from Coriolis effect is 106.6(3.5) $μGal$ .
Fig. 8 Measurement of the gravity bias due to Coriolis effect. Left: measurement of the gravity values for two opposite orientation (28 $°$ and 208 $°$ ). Each point takes about 20 minutes; Right: gravity biases due to Coriolis effect are obtained by half the difference of gravity signals.
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In our experiments, the amplitude of Coriolis bias is much larger than that of other state of the art gravimeters. The sources of Coriolis effect and resulted amplitude are discussed in the following. Basically, the systematic bias resulted from Coriolis effect is mostly related to the average horizontal velocity and position of the detected atoms, the area of detection beam, the homogeneity of detection system, and so on. Firstly, the average horizontal velocity and position of the detected atoms may be not the main source since the molasses beam is retro-reflected with 0 $°$ mirrors. The drift velocity of about 1 cm/s is large and cannot be caused only by molasses beam imbalance, which can also be proved by the experimental of [2,7]. Secondly, the large amplitude of Coriolis effect may be due to the detection method, especially for our single-side detection system. In order to verify this effect, the experiments of Coriolis effect have been carried out by changing the transverse widths of the detection beam, and the results are shown in Fig. 9. It is found that the width of detection beam has a dominant influence on gravity bias resulted from Coriolis effect. The narrow velocity distribution of detected atoms will be selected with a small slit and thus produce a reduced Coriolis bias. Furthermore, the gravity values as a function of the orientation angles have been measured when the width of detection beam is 4 mm. The amplitude of resulted gravity bias due to Coriolis effect is estimated to be 19.1(3.6) μGal, which is much smaller than that of the previous experiments. In summary, the amplitude of Coriolis effect in our experiment is mainly due to the inhomogeneity of detection system, which could be reduced with a narrow transverse width of detection beam.
Fig. 9 Measurements of the gravity variations due to Coriolis effect when the transverse width of the detection beam is changed. Red and black scatters represent two different measurements.
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5. Conclusions
We have analyzed the influence of orientation on high-precision absolute gravity measurement for atom gravimeter, and shown the theoretical simulations of the gravity changes for three different atomic horizontal velocities. In order to verify this simulation, we have improved the performance of our free-fall cold atom gravimeter, especially the mobility. Even without any vibration isolation system, the sensitivity of 262 $μGal/ \sqrt{Hz}$ is achieved for a small $T$ (55 ms). Besides, the experimental setup is simplified so that the fast adjustment and measurement could be carried out within 20 minutes. Based on a rotation platform, we have measured the gravity changes as a function of orientation angles, which agree well with the theoretical prediction. Moreover, it is found that changes of the orientation angle of our gravimeter yields a dominant influence on the absolute gravity measurement. Based on the experimental data, the amplitude and direction of average horizontal velocity of the detected atoms are deduced from the fitted curve. In particular, the direction of East–west could be measured by the atom gravimeter itself such that gravity bias arising from Coriolis effect is estimated to be 106.6 (3.5) $μGal$ .
Funding
National Key Research and Development Program of China (2016YFF0200206, 2017YFC0601602); National Natural Science Foundation of China (NSFC) (61727821, 61475139, 11604296); Research and application experiment of carrying measurement and support technology for ocean route (DD20189831).
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Dependence of the sensitivity on the orientation for a free-fall atom gravimeter

Abstract

1. Introduction

2. Theoretical analysis of the gravity bias due to the orientation of CCAG

3. Experimental apparatus

4. Experimental results

4.1 The performance of the free-fall atomic absolute gravimeter

4.2 Measurement of the sensitivity to orientation of the gravimeter

4.3 Evaluation of the gravity bias due to the Coriolis effect

5. Conclusions

Funding

References

Cited By

Figures (9)

Equations (3)

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