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Abstract
Precise information of positions and sizes of atom clouds is required for atom-interferometry-based G measurements. In this work, characterizing atom clouds using a charge-coupled device (CCD) is presented. The parameters of atom clouds are extracted from fluorescence images captured by the CCD. For characterization, in-situ calibration of the magnification of the imaging system is implemented using the free-fall distance of atom clouds as the dimension reference. Moreover, influence of the probe beam on measuring the positions of atom clouds is investigated, and a differential measurement by reversing the direction of the probe beam is proposed to suppress the influence. Finally, precision at sub-mm level for characterizing atom clouds is achieved.
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1. Introduction
The measurement of $G$ with high precision is a challenge owing to the weakness and un-shielding property of gravitational interaction. The relative uncertainty of $G$ value as stated by the Committee on Data for Science and Technology (CODATA) in 2018 [1] is still 22 parts per million (ppm), although about $300$ experiments for $G$ measurements have been performed [2]. It is confusing that the difference in $G$ measurements reaches up to 550 ppm, while the combined uncertainty of individual $G$ measurement at the level of tens of ppm is achieved [1]. Performing $G$ measurements with different approaches is expected to explore possible unknown errors to explain the difference [3–6]. Currently, most experiments for $G$ measurements are based on torsion balance [2,7–10]. The $G$ measurement based on atom interferometry is a new approach that is intrinsically different from traditional ones [11–15].
In atom-interferometry-based $G$ measurements, cold atoms act as test masses, and the gravitational interaction between well-machined source masses and atom clouds is measured by atom interferometry. In analogy to traditional $G$ measurements using solid test masses [8–10], positions and sizes of atom clouds must be determined before extracting $G$ value from phase shift of the atom interferometer. Characterizing atom clouds for atom-interferometry-based $G$ measurements is not trivial as the clouds are not macroscopic solid bodies. Imaging atom clouds using charge-coupled devices (CCDs) is an effective way for characterization. CCDs have been widely used in cold atom experiments, ranging from imaging single atom [16–24], counting atom numbers [25–27] to acquiring the spatial information of atom clouds [28–31]. However, there are seldom applications where precision on measuring positions and sizes of atom clouds using a CCD is as stringent as that in atom-interferometry-based $G$ measurements. In Ref. [14], parameters of atom clouds have been measured at sub-mm level, which still contribute an uncertainty of 72 ppm to the $G$ measurement based on atom interferometry. Inspired by the work of Tino’s group [15], the characterization of atom clouds using a CCD is performed for our atom-interferometry-based $G$ measurements. $In$-$situ$ calibration of the magnification of the imaging system is realized, where the free-fall distance of atom clouds is used as dimension reference. A calibration precision of about $1\%$ is achieved. Furthermore, the influence of the probe beam on the measurement of horizontal positions of atom clouds is investigated. And a differential measurement by reversing the probe beam direction is explored to suppress the influence. The $in$-$situ$ calibration and the differential measurement improve the precision for characterizing atom clouds.
2. Principle and experimental setup
An atom gravity gradiometer based on dual magnetic–optical traps (MOTs) is developed to measure $G$ in our laboratory (Fig. 1). Two $^{87}$Rb atom clouds launched from the dual MOTs with a distance of about 0.3 m act as the test masses for the gradiometer. As discussed in our previous study [32], the gravity gradiometer contributes to first differential measurement, eliminating the major phase shift induced by gravity and suppressing common-mode noises. The close and far configurations of source masses contribute to second differential measurement, alleviating disturbances from nearby masses and errors caused by the gravity gradiometer. The two atom clouds, are coherently manipulated by the same Raman laser during their flights for simultaneous state preparation and interference. The spatial and velocity distributions of the atom clouds are assumed to follow Gaussian distributions along three dimensions [33]. For each dimension, the atom cloud can be characterized by four parameters at the initial moment just after launching, namely, the spatial center $r_{i0}$ ($i$ = $x$, $y$, and $z$, corresponding to the axes shown in Fig. 1), standard deviation $\sigma _{i0}$, and average velocity $v_{i}$, as well as corresponding standard deviation $\sigma _{vi}$. The spatial center of the cloud indicates its position, while the spatial standard deviation indicates the size. During flight, variations in the position and the size are expressed as [33]
Fig. 1. Scheme for our $G$ measurement based on a dual-MOTs atom gravity gradiometer. The red dots represent two atom clouds reaching the apex; 12 spheres are designed source masses with close configuration shown while the far configuration is shown in dotted line. The two atom clouds are captured when each cloud passes through the detection chamber, where a lenses group is used to collect the fluorescent photons. We note that in this work $z$ axis is along the vertical direction, $x$ axis is along the south-north direction, and $y$ axis is along the east-west direction. The detailed configuration of the imaging system is shown in the right tridimensional scheme.
The two atom clouds are imaged when each cloud passes through the detection chamber upwards and downwards to get the information of atomic parameters. For small optical depth clouds, fluorescence imaging is suitable [34], where atom cloud is illuminated by probe beam and the emitted fluorescent photons are collected by detectors. Our imaging system is shown in Fig. 2, where a CCD (ProEM-HS: 1024BX3) is used as the photon detector. For the imaging system, there are two sets of lenses to collect fluorescent photons, since there is insufficient space to place the CCD if only one set of lenses is used in our experiment. Each set includes two lenses. The focal lengths of the two lenses are 150 and 100 mm respectively, and the objective set has a numerical aperture (NA) of 0.1. With this configuration, the magnification of the imaging system is estimated to be about 0.4. While the response of CCDs is not instantaneous, the probe beam is pulsed to determine the effective exposure time for imaging. With regard to short pulse time for the probe beam, photons accumulated by the CCD form an image of atom clouds. The typical image of the cloud passing through the detection chamber upwards is shown in Fig. 3(a). A Gaussian fit can be explored to extract the center and standard deviation of the image along each dimension, for example, $z$ and $y$ axes in Fig. 3(b). The cloud is identically characterized along $x$ axis with the probe beam and the imaging system reinstalled.
Fig. 2. Top view of the imaging system. Atom cloud is illuminated by the probe beam and imaged onto the CCD through a lens system. The information of the cloud along $y$ axis can be obtained from the captured image with the scheme as shown. For characterizing the cloud along $x$ axis, the probe beam will be reinstalled along $x$ axis, and accordingly, the imaging system will be reinstalled along $y$ axis. With the probe beams either along $x$ axis or $y$ axis, the cloud along $z$ axis can be characterized.
Fig. 3. (a) A typical image of atom cloud passing through the detection chamber upwards. (b) Two-dimensional Gaussian fitting of the image to determine the spatial center and standard deviation of the image.
3. Magnification calibration of imaging system
Since $G$ measurements require precise measurements of the position and size of atomic clouds, the actual magnification must be calibrated independently. The typical size of an atom cloud in atom interferometers is about 5 mm. In Ref. [14], a precision of 0.1 mm for measuring the cloud size along the vertical dimension contributes a relative uncertainty of 56 ppm for the $G$ measurement. Thus, a relative precision of $2\%$ for magnification is required if a precision of 0.1 mm to measure the size is necessary. The magnification is usually calibrated using objectives with known dimensions, e.g., a USAF 1951 test card [35–38]. In Ref. [39], two parts split from one atom cloud are imaged using a CCD, and the distance between the two parts (about 4 mm) defines a dimension reference to calibrate magnification. Inspired by the aforementioned study, free-fall distance of the atom cloud is selected as a dimension reference here to realize an $in$-$situ$ calibration of magnification, where the free-fall distance within the detection chamber is about 20 mm.
Taking the imaging of the upper atom cloud as an example, the distance between the upper MOT and the detection chamber is $0.2\; \textrm{m}$, and it takes about $47\; \textrm{ms}$ for the upper cloud to arrive at the detection chamber during its upward flight. To calibrate the magnification, several images of the cloud passing through the detection chamber are obtained, with a pulse duration of $50\; \mu\textrm{s}$ and a time step of $0.5\; \textrm{ms}$ for the probe beam (Fig. 4(a)). The vertical center of every image can be obtained from Gaussian fitting, while the actual vertical position is calculated based on Eq. (1). The vertical velocity ${v_{\textrm{z}}}$ is $4.6632(2)\; \textrm{m}/\textrm{s}$, which is measured by Raman spectroscopy [32]. And the local gravity acceleration is $9.7934(1)\; \textrm{m}/\textrm{s}^2$, which has been obtained from the previous measurement. As expected, the center of the image and the calculated position form a linear relationship (Fig. 4(b)). The slope determined from a linear fit denotes the magnification, which is $0.439(3)$ for the upper cloud. The magnification of the lower cloud is calibrated as well, which is $0.420(1)$. For the same imaging system, there is a difference of the magnification at a level of $4\%$ for the two clouds, which is possibly originated from different objective distance of the two clouds at the imaging moments. This difference also explains the importance of $in$-$situ$ calibration of magnification. Once the magnification is determined, the actual size of the cloud can be obtained from the image.
Fig. 4. (a) Images of the cloud passing through the detection chamber upwards, the pulse duration of the probe beam is $50\; \mu\textrm{s}$. The timing of the probe beam pulse changes with a step of 0.5 ms when capturing the images. The number in each image denotes the corresponding order. (b)The variation of the center of the image versus the calculated position with the timing of the probe beam pulse changed. The slope of data represents the magnification of imaging system.
4. Influence of probe beam
For imaging the atom clouds using the CCD based on the fluorescence imaging method, the probe beam propagates horizontally, for example, along $y$ direction in Fig. 2, when the position of the atom cloud along $y$ direction is measured. As a result, the position measurement may be affected owing to the interaction between the atom cloud and the probe beam. In order to investigate the influence, the pulse duration of the probe beam is modulated. As shown by the red circle dots in Fig. 5, the measured horizontal position of the cloud indeed changes with the pulse duration. According to the red circle dots in Fig. 5, for pulse duration longer than $200\; \mu\textrm{s}$, the atom clouds are pushed away along the direction of the probe beam due to the power attenuation of the retroreflected beam induced by absorption and imperfect retroreflection. The variation of the horizontal position versus the pulse duration is $3.6(1)\times 10^{-4}\; \textrm{mm}/\mu\textrm{s}$, which is educed by a linear fit of red circle dots with pulse durations longer than $200\; \mu\textrm{s}$. In order to suppress the influence of the probe beam, the direction of the probe beam is reversed, and the corresponding variation of the position is manifested by the black square dots in Fig. 5. As expected, the black square dots change oppositely with regard to the red circle dots due to opposite directions of the probe beam. And the corresponding slope is $-2.6(3)\times 10^{-4}\; \textrm{mm}/\mu\textrm{s}$.
Fig. 5. Horizontal position of the atom cloud for different pulse durations with the direction of the probe beam reversed. The solid lines are corresponding linear fit lines using data from $200\; \mu\textrm{s}$ to $700\; \mu\textrm{s}$, and the dotted lines are corresponding extensions of the linear fit lines. The influence of the probe beam on the position measurement is suppressed by about one order by reversing the direction of the probe beam to make a differential measurement.
It is surprising that the two groups of data intersect at $200\; \mu\textrm{s}$ rather than at $0\; \mu\textrm{s}$. It is assumed that for very short pulse durations, the illumination of the cloud by the probe beam is incomplete, and the atoms meeting the probe beam earlier scatter more photons than those meeting later, causing a visual shift of position for the image. Therefore, the push effect of the probe beam dominates for pulse durations longer than $200\; \mu\textrm{s}$ while the visual shift dominates for quite short pulse durations. As a result of the balance, the two groups of data intersect at about $200\; \mu\textrm{s}$. The differential measurement to reverse the direction of the probe beam indicates the actual horizontal position (blue triangle data in Fig. 5), where the influence of the probe beam is considerably suppressed. The result of the differential measurement shows a peak-to-peak residual variation at a level of 0.03 mm, which is attributed as the uncertainty.
5. Atom clouds position and size
With regard to horizontal dimensions (namely $x$ and $y$ axes), two images of each cloud are obtained when it passes through the detection chamber upwards and downwards, from which the four parameters, including the initial position, initial size, average velocity and standard deviation of velocity distribution, can be determined. The standard deviation of the velocity distribution along $x$ or $y$ axis is about 14 mm/s for both atom clouds, corresponding to a temperature of about $2\; \mu\textrm{K}$. The measured horizontal positions and sizes of each cloud are listed in Table 1. We note that the parameters of the clouds along $x$ and $y$ axes in the table are obtained using two images without state preparation. In our atom gravity gradiometer prepared for $G$ measurements [32], the Raman laser is explored to simultaneously implement the state preparations for both clouds based on two-photon stimulated Raman transitions [40]. The state preparation helps to select out atoms in magnetic insensitive sublevel within a narrow vertical velocity range from the clouds. With state preparation, only one image of each cloud can be used when it passes through the detection chamber downwards as the state preparation has not been implemented when the clouds pass through the detection chamber upwards. In an ideal situation, the state preparation should not affect the horizontal distribution of the clouds for either spatial or momentum space. However, it is reported that horizontal parameters are affected by the state preparation [15]. The velocity distributions along $x$ and $y$ axes will be measured by Raman spectroscopy for characterizing the atom clouds with state preparation for our formal $G$ measurements in the near future. In this way, the influence of the state preparation on the horizontal spatial and velocity distributions can be studied.
Table 1. Measured positions and sizes of atom clouds. The positions in the table correspond to the apex moment, which are calculated from the initial positions and average velocities.
For vertical dimensions (namely $z$ axis), only one image for each cloud with state preparation is obtained when it passes through the detection chamber downwards. Raman spectroscopy with the Raman laser propagating along the vertical dimension is applied after the state preparation to measure the corresponding velocity distribution, thereby obtaining the average velocity and standard deviation of the velocity distribution along $z$ axis. Combined with the obtained image, initial position and initial size for the selected atoms can be determined, as shown in Table 1. With a falling velocity of about $4\; \textrm{m}/\textrm{s}$ for the cloud passing through the detection chamber, atoms will fall about 0.2 mm during the exposure time of $50\; \mu\textrm{s}$ , which contribute the major uncertainty to determine the vertical positions of atom clouds. The influence of the probe beam on the characterization of the clouds along $z$ axis is negligible as the probe beam propagates along horizontal direction.
The measured positions listed in Table 1 are referenced to the center of the CCD. What really matters is the relative positions between the clouds and source masses. During the imaging of the clouds, an image of the window of the detection chamber is obtained. The window is farther from the lens system than the clouds, thus corresponding to a smaller magnification. The actual magnification is about 0.26, which allows the imaging of the whole window into the active area of the CCD (the diameter of the window is about 50 mm, and the active area of the CCD is 13.3 mm $\times$ 13.3 mm). Therefore, the connection of the CCD center and the window center can be developed. Once our source masses are prepared, the position of the source masses relative to the window can be measured using a laser tracker with a precision of $10\; \mu\textrm{m}$. In this way, the measured positions here are relatable to the positions of the source masses. Based on the measured parameters in Table 1, uncertainties related to the clouds to measure $G$ can be predicted according to the designed scheme reported in [32]. The contributed uncertainty related to the parameters along $z$ axis is predicted to be 38 ppm. For the parameters along $x$ and $y$ axes, if the same measurement accuracy can be achieved for atom clouds with state preparation, the corresponding contributed uncertainty is expected to be 24 ppm for our $G$ measurement. The combined uncertainty related to both vertical and horizontal dimensions of the clouds is 45 ppm.
6. Discussion and conclusion
In our first stage of $G$ measurement, parameters of the dual MOTs will be modulated in the near future, trying to decrease the sizes and the horizontal velocities of the clouds. The images obtained using the CCD will be a convenient reference for the modulation. For future high-precision $G$ measurements, the optical lattice is expected to provide better control of clouds during transport compared with the simple launch using optical molasses.
In conclusion, the characterization of clouds for atom-interferometry-based $G$ measurements using a CCD is presented in this study. Different from other applications that use CCDs to image atoms, $G$ measurement requires precise evaluation of positions and sizes of clouds. In this study, $in$-$situ$ calibration of the magnification of the imaging system is implemented, and the influence of the probe beam is investigated. A total uncertainty related to the clouds at a level of 45 ppm is expected.
Funding
National Natural Science Foundation of China (11574099, 11625417, 11727809).
Acknowledgment
We thank Dr. Zhongkai Ai for his help in setting up the CCD. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11727809, 11625417, and 11574099).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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