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Abstract
The temperature of atoms, coupled to several effects, plays an important role in high precision atom interferometry gravimeters. In this work, we present an ultra-cold 87Rb atom interferometry gravimeter, in which the atom source is produced by evaporative cooling in an all optical dipole trap to investigate the effects related to atom temperature. A condensate containing 4 × 104 atoms can be prepared within 3.2 s through an all-optical dipole trap composed of two reservoirs and a dimple. The fringe contrast of our atom interferometry gravimeter reaches up to 76(4)% due to the advantage of ultra-cold atom source even at a free evolution time of T=80 ms. A resolution of 6 μGal (1 μGal=1×10−8 m/s2) after 3000 s integration time with a sampling rate of 0.25 Hz is achieved in this atom gravimeter. The relationship between the fringe contrast and the atom temperature in the atom gravimeter is studied; in addition, the wavefront aberration effect in the atom gravimeter is also investigated by varying the temperature of atoms.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Atom interferometers have been widely used in precision measurements, such as the gravity measurement [1–5], the gravity gradient measurement [6–8], the rotation measurement [9,10], the Newtonian gravitational constant measurement [11,12], as well as testing of fundamental physical laws [13–16]. Interference with long interrogation time $T$ can effectively improve the sensitivity of interferometer [14,17,18], which is beneficial for high-precision measurements. However, the velocity distribution of atoms induced by the atom temperature cause inhomogeneous de-phasing in atom interferometry, which not only reduces the interference fringe contrast but also contributes systematic errors in precision gravity measurements [1,2,19,20], and one of the trickiest is the wavefront aberration effect [21,22].
Ultra-cold atoms, such as Bose-Einstein condensation(BEC), have been applied to interferometer due to their long coherent time and low thermal diffusion [23–27]. Using BEC for atom interferometry gravity measurement can effectively improve the contrast of interference fringe [4], and it also can dramatically increase the interference time and the transition efficiency, which can directly enhance the potential sensitivity of atom gravimeters [28,29]. In addition, it is also an effective way to reduce and evaluate various systematic errors [18], especially for the wavefront aberration effect [7]. However, the cycle time of most BEC-based gravimeters is about 15-20 s [4,23,24], which is not so conducive to gravity measurements compared with that using cold atoms [1,3] due to its low sampling rate.
In this work, ultra-cold $^{87}$Rb atoms are prepared in an all-optical dipole trap with double reservoirs and one dimple configuration, which can decrease the preparation time compared to the evaporative cooling in most magnetic trap or magneto-optic hybrid trap [30–32]. More than $4\times 10^{4}$ BEC atoms can be prepared within 3.2 s in this apparatus, and the gravity measurements based on ultra-cold atoms is presented with interference time 2$T$=160 ms. The total cycle time is less than 4 s, and the resolution of this atom gravimeter reaches 6 $\mu$Gal after 3000 s integration time. By modulating the temperature of atoms in a wide range, the effects related to the temperature of atoms in the atom gravimeter are studied. The interference fringe with a visibility of $76(4)\%$ at $T=80$ ms by using BEC atoms is obtained. The wavefront aberration effect in the atom gravimeter is studied by temperature modulation experiment, and the result is coincident with the evaluation based on a known wavefront distribution of Raman beams. In addition, the evaluation result is verified by the extrapolation of the $g$ value down to zero temperature [2].
The article is structured as follows, section 2 mainly introduces the principle of gravity measurement and the effects related to the temperature of atoms. The preparation process of ultra-cold atoms is described in detail in Section 3. In section 4, gravity measurement based on ultra-cold atoms has been realized and the effects related to the temperature of atoms have been investigated. Finally, conclusion of this work is given in section 5.
2. Theory
2.1 Principle of gravity measurements based on atom interferometry
The principle of atom interferometry gravimeter has been described in detail in Ref. [33]. As shown in Fig. 1, our atom gravimeter is based on stimulated Raman transition to split ($\pi /2$), reflect ($\pi$) and recombine ($\pi /2$) the atomic wave packet in a Mach-Zehnder configuration. The atoms are prepared in the $|F=1\rangle$ state firstly, after a sequence of interference pules, the number of the atoms in different state is measured by the fluorescence detection. The probability of atoms stay in $|F=2\rangle$ state can be described as
where $C$ is defined as fringe contrast, and $\Delta \varphi$ is the interference phase, which can be written as [5] where $\vec {k}_{eff}$ is the effective wave vector of the Raman beams, $T$ is the interval time between two pulses, and $\alpha$ is the chirp rate of the laser, which is used to compensate the Doppler shift of the free falling atoms, $\delta \varphi$ is the phase shift introduced by systematic errors, such as external electric field, magnetic field and the effects related to atom temperature. As shown in formula (3), the temperature of atoms corresponds to the velocity distribution of atoms [19] where $T_a$ is the temperature of atoms, $\sigma _v$ is the velocity distribution width of atoms. The atoms used for interferometry gravity measurement with non-zero temperature will not only limit the measurement sensitivity but also introduce systematic errors, such as the wavefront aberration effect.2.2 Decay of fringe contrast
The strength of interaction between atoms and laser can be characterized by the Rabi frequency, and we assume that the Rabi frequency of the atom at the center of the Raman beam is $\Omega _{max}$. The intensity distribution of Raman beams has the Gaussian distribution as shown in Fig. 2(b). Since the atoms have an initial space distribution and will diffuse during the free fall, it will cause that some atoms deviate from the center position, and the Rabi frequency for atom at position $r_{i}$ can be express as
where $r_{i}(t)^{2}=r_{0i}^{2}+\sigma _{vi}^{2}t^{2}$ is the atom position, and the $1/e^2$ radius of Raman beams $w$=13 mm. The initial radius of atomic cloud is 1 mm in the simulation. This causes that the atoms at different positions have different Rabi oscillation periods, which results in reducing the interference fringe contrast. The simulation result is shown in Fig. 3(a), and this effect is more obvious at larger interference time. So due to it’s low thermal diffusion, ultra-cold atoms can be used to obtain high contrast fringe and also can be used in experiment with long interrogation time.
Fig. 2. Effects coupled to the atom temperature. (a) Thermal diffusion of atoms during the free fall. (b) Intensity distribution of the Raman beam. (c) The wavefront distribution of Raman beams.
Fig. 3. (a) Numerical simulation of the relationship between the fringe contrast and the temperature of atoms: The red dotted line represents $T$=80ms and the black solid line represents $T$=200ms. (b) The wavefront aberration effect varies with the temperature of atoms at $T$=80ms, where we assume that the initial position of the atom cloud is located at the center of the optical element and the horizontal velocity is zero.
2.3 Wavefront aberration effect
For the Raman beams with non-planar wavefront distribution as shown in Fig. 2(c), the phase shift for an atom $i$ depends on its position $r_{i}$ within the beam profile and can be expressed as
The atoms will continue to diffuse during the free fall, which is determined by the temperature of atoms, and the initial position and velocity of atoms also should be considered [22]. In our device, only $\lambda /4$ wave plate and reflect mirror contribute to the phase shift, which can be expressed as
By measuring the figure of reflecting mirror and $\lambda /4$ wave plate, the parameters of atomic cloud, the wavefront aberration effect can be evaluated. Furthermore, it can be verified by modulating the temperature of atoms. When the wavefront of Raman beams is shown in Fig. 2(c), the numerical result of wavefront aberration effect varying with the temperature of atoms is shown in Fig. 3(b).
3. Process of preparing ultra-cold atoms
Due to high evaporative cooling efficiency and simple experimental configuration, evaporative cooling based on all-optical dipole traps is widely used to prepare ultra-cold atoms. In our apparatus, the optical dipole trap with double reservoirs and a dimple configuration is used to prepare ultra-cold atoms.
3.1 Apparatus structure
As shown in Fig. 4, the apparatus consists of three main regions: ultra-cold atoms preparation region, interference region and detection region. There are two chambers in ultra-cold atoms preparation region, two-dimensional magneto-optical trap(2D-MOT) chamber and three-dimensional magneto-optical trap (3D-MOT) chamber.
Fig. 4. The schematic diagram of the appratus structure. The appartus consists of three parts: Ultra-cold atoms preparation region, interference region and detection region.
The atoms are firstly pre-cooled in 2D-MOT, then pushed into 3D-MOT by push beam. Since two chambers are connected through a differential pumping, using 2D-MOT not only effectively increases the loading rate of 3D-MOT, but also maintains high vacuum [34]. In the apparatus, the pressure of 3D-MOT chamber is better than $1.0\times 10^{-8}$ Pa, which is more than $2.5$ orders of magnitude better than 2D-MOT. It is essential for the dipole trapping and evaporative cooling process, since they are carried out in 3D-MOT chamber.
After prepared, the dipole trap beams are turned off, and the atoms freely fall to the interference region. The earth’s magnetic field is shielded by magnetic shielding material and the bias field is provided by a solenoid winding on interference region. When the atoms pass through the interference region, the Raman beams interact with the atoms to finish the interference process. Finally, the atoms in different state is detected by fluorescence detection in detection region. The reflecting mirror and the $\lambda /4$ wave plate at the bottom are used to configure Raman beams.
3.2 BEC preparation
As shown by the insert in Fig. 5(a), the dipole trap consists of double reservoir beams and a dimple beam, with $1/e^{2}$ beam waist of 70 $\mu$m and 30 $\mu$m, respectively. The dipole trap beam is emitted from a high-power fiber laser with a wavelength of 1064nm (IPG:YLR-100-1064-LP). A splitter is used to divide the beam into two paths, used as reservoir beams and dimple beam respectively. Then, the beams enter into high power acousto-optic modulators (AOM) (ISOMET-M1135-T80L-3) separately, the first order laser diffracted from the beam is expanded by a beam expander, and a plano-convex lens is used to focus it at the center of 3D-MOT to form a cross dipole trap. The plano-convex lens are placed on a two-dimensional translation platform to accurately adjust the position of the beam waist. The double reservoirs come from the same AOM diffraction beam and separated by a beam splitter. By adjusting the reflection mirror, we can make them parallel before injecting into the plano-convex lens, and they will be recombined by a focusing lens at the center of 3D-MOT. By using double reservoir beams, the loading rate of the dipole trap is obviously larger than single reservoir beam at the same optical power as shown in Fig. 5(b).
Fig. 5. (a) Schematic diagram of the dipole trapped beam. The dipole trapped beams consists of three beams, of which two reservoir beams are used to improve the loading rate of the trap, and the dimple beam is used to accelerate the evaporative cooling. (b) The relationship between the load rate of the dipole trap and the total optical power under different configurations.
$^{87}$Rb atoms are pre-cooled in 2D-MOT, then they are pushed into 3D-MOT, and about $3\times 10^{9}$ atoms are trapped in 3D-MOT within 1 s. The dipole trap beam is turned on in the middle of the MOT trapping process, and the power of reservoir beams is ramped up to 20 W in 500 ms. Then we compress the MOT by increasing the magnetic field gradient from 8 Gs/cm to 16 Gs/cm, and reduce the repump beam from 2 mW to 100 $\mu$W simultaneously, and then turn off the magnetic field. After that, by increasing the detuning and reducing the power of the MOT beams, 30 ms static molasses process is applied to further increase the phase-space density of the atoms. Finally, the repump beam is turned off 5 ms before the MOT beams to prepare all atoms in $F=1$ state. These steps are necessary for dipole trap loading, and more than $1\times 10^{7}$ atoms are loaded into the dipole trap. The whole preparation process of ultra-cold atoms is shown in Fig. 6.
Evaporative cooling of atoms is realized by fast ramping down the optical power of the dipole trap. With the presence of dimple beam, the atoms can be evaporated effectively to the pure BEC state within 2.2 s. In the experiment, absorption imaging method is employed at different moment of flight to obtain relevant information of atoms, such as the size, number, temperature, phase space density.
Absorption images after 20 ms free fall with different cooling parameters are shown in Fig. 7. With the increase of the evaporative cooling time, the temperature of the atoms decreases and optical density ($OD$) becomes larger. Partial condensed atoms can be observed at 1.9 s, and all atoms are condensed within 2.2 s. There are about $4\times 10^{4}$ atoms, the temperature of atoms is lower than 30 nK, the phase space density is more than 15. In addition, the temperature of atoms can be varied from 30 nK to 6 $\mu$K by controlling the cooling parameters, so the effects related to atom temperature in the atom gravimeter can be studied.
Fig. 7. The result of evaporative cooling. This is the absorption image of atoms after 20 ms free fall under different evaporative cooling steps, the atoms appeared partially condensed when evaporated for 1.9 s and further condensed for 2.2 s.
4. Gravity measurements and atom temperature related effects
After the preparation, the ultra-cold atoms are free falling, then Raman beams and blowaway beam for state preparation are used before the interference, as shown in Fig. 6. All atoms are prepared in $|F=1, m_{F}=0\rangle$ state, which is insensitive to external magnetic field. When the atoms reach interference region, a sequence of Raman pulses is applied to the atoms with an interval time of $T$=80 ms, and the power of the Raman beams is about 80 mW. Firstly, Rabi oscillation is performed in order to determine the interaction time of $\pi$ pulse, and the result is shown in Fig. 8(a). The Rabi efficiency can reach more than $80\%$ by using ultra-cold atoms, the duration time of $\pi$ pulse is 16 $\mu$s. After the interference process, atoms in different states are detected in detection region. By changing the chirp rate of the laser frequency, the interference fringes can be obtained, as shown in Fig. 8(b).
Fig. 8. (a) Rabi oscillation with BEC. (b) Interference fringes of the gravity measurement: The blue dots are the interference fringes of BEC with temperature of 30 nK, and the red rectangular dots are the interference fringes of normal cold atoms with temperature of 2 $\mu$K.
The resolution and stability of the apparatus are optimized, and the results is shown in Fig. 9(a). The sensitivity of the gravimeter can reach 210(10) $\mu$Gal/$\sqrt {\mathrm {Hz}}$, it can reach a resolution of 6 $\mu \mathrm {Gal}$ after 3000 s integration time. As shown in Fig. 9(b), the repeatability of the instrument is better than 5 $\mu$Gal.
Fig. 9. Data of gravity measurements. (a) The Allan deveaion of gravity measurements based on BEC atoms. (b) Results of repeated measurements carried out under the same experimental parameters in six days.
4.1 Influence on the fringe contrast
The interference fringes of different temperatures at the same interval time $T$=80 ms are performed in experiment, and the result is shown in Fig. 10(a). The contrast of the interference fringe increases significantly as the temperature decreases. By using ultra-cold atoms with temperature of 30 nK, the fringe contrast can reach $76(4)\%$, while the fringe contrast is only $51(3)\%$ using cold atoms with temperature of 2 $\mu$K. The reason for that is the higher the temperature of atoms, the more inhomogeneous intensity of Raman beams irradiating atoms, this result in lower efficiency of beam splitting. Therefore, higher contrast interference fringes can be obtained by using ultra-cold atoms. However, the fringe contrast is still smaller than $100\%$, although ultra-cold atoms have been employed in the atom interferometer. The direct reason for that is the efficiency of Raman transition is only $80\%$ when using ultra-cold atoms, which is shown in Fig. 8(a). In addition, many other experimental factors affect the fringe contrast as well, such as dephasing caused by the inhomogeneity of magnetic field, background atoms, stray light. In conclusion, based on the result in Fig.10(a), the temperature of atoms can be considered as a dominant factor that affects the fringe contrast.
Fig. 10. (a) Interference fringe contrast at different atom temperatures. (b) The interference fringe contrast varied with interference time $T$ using atoms at different temperatures: the point is the experimental data and the straight line is the result of linear fitting.
In addition, the relationship between the fringe contrast and the interference time $T$ at different atom temperatures is also studied. It can be seen from Fig. 10(b) that the fringe contrast does not change significantly when $T$ varies from 10 ms to 80 ms with ultra-cold atoms. However, when the atom temperature increase, the fringe contrast decreases obviously with the increase of interference time. The higher the temperature of atoms, the faster the fringe contrast decreases with the increase of interference time. It shows that the ultra-cold atoms can effectively improve the contrast of interference fringe, which is especially beneficial for the measurements with long interference time.
4.2 Evaluation of the wavefront aberration effect
The use of ultra-cold atoms can not only improve the fringe contrast, but also reduce some systematic errors, especially for wavefront aberration effect, which is related to the temperature of atoms and the wavefront distribution of Raman beams. In our apparatus, the wavefront distribution of Raman beam is mainly determined by the surface distribution of the $\lambda /4$ wave plate and the reflecting mirror. The surface distribution of the mirror and the wave plate is measured by a figure measurement device based on wavefront sensor (Imagine Optics HASO3-128GE2), and the result is shown in Fig. 11(a) and (b). Both of them have high surface quality, the peak to valley (PV) value of the surface is less than $\lambda$/20 within a diameter of 15 mm, and the root mean square (RMS) value of the surface distribution is less than $\lambda$/80. The initial position of the atoms are determined by the position of the dipole trap which is near the center of 3D-MOT, and the uncertainty is less than 1 mm. The horizontal velocity of the atoms is measured by absorption imaging, which is 0.0(7) mm/s. So the wavefront aberration effect of our apparatus can be evaluated, which is (0.2$\pm$0.4) $\mu$Gal when the atom temperature $T_a$=30 nK.
Fig. 11. The wavefront aberration effect of the instrument. (a) Surface distribution of the reflecting mirror. (b) Surface distribution of the $\lambda /4$ wave plate. (c) Numerical calculation results and experimental data of the wavefront aberration effect varying with atom temperature.
The wavefront aberration effect at different temperatures is calculated based on the measured wavefront, shown as red line in Fig. 11(c), the gray area is the uncertainty introduced by the initial position and initial velocity uncertainty of the atoms. By changing the parameters of evaporative cooling, the temperature of atoms is adjusted in a wide range from 30 nK to 6 $\mu$K, the experimental results are shown as blue dots in Fig. 11(c), which are consistent with the calculation result within the error bar.
There are also some other effects that varied with the temperature of atoms, such as Coriolis effect, second-order Zeeman effect, light shift and interaction between atoms. In our experiment, the method of rotating the mirror to compensate the rotation of earth is employed to reduce the influence of Coriolis effect. Moreover, the method of alternating the wave vector of Raman beams is also employed to reduce the impact of $k$ independent systematic errors, such as second-order Zeeman effect and light shift. We also calculated the influence of the interaction between atoms, and its contribution is less than 0.1 $\mu$Gal, which can be neglected at the current situation.
Since the wavefront aberration effect is related to the ballistic expansion of the atomic source through its motion during the interaction with Raman beams, the contribution of this effect will be disappeared when the temperature of atoms is zero. Extrapolating $g$ value to zero temperature according to the experimental results of atom temperature modulation is an effective method to evaluate the effect of wavefront aberrations [2]. Therefore, this method is also employed to verify our evaluating results. The wavefront aberrations $\delta z$ can be decomposed onto the basis of Zernike polynomials, and the diameter of Raman beams used in calculation is 15 mm. The contributions of different orders of Zernike polynomials with the same peak to valley value of 20 nm varying with atom temperature are calculated, shown in Fig. 12(a). The value of $g_m$ combined contribution of different orders Zernike polynomials can be expressed,
The fitting result is shown in Fig. 12(b) by using weighted least square adjustment, and the extrapolated value $g_{bias}$ is (−2.1$\pm$0.9) $\mu$Gal, the correlation coefficient $R^2$=0.82. The measured value at $T_a$=30 nK is (−2.5$\pm$4.7) $\mu$Gal, so the wavefront aberration effect in the atom gravimeter is (−0.4$\pm$4.8) $\mu$Gal when it works at $T_a$=30 nK, and the uncertainty is mainly limited by the statistical error. This result is consistent with our evaluation result (0.2$\pm$0.4) $\mu$Gal by measuring the surface distribution of the Raman beams within the uncertainty.
Fig. 12. Evaluating the wavefront aberration effect by extrapolating $g$ value to zero temperature. (a) Wavefront aberration effects contributing by different orders of Zernike polynomials vary with atom temperature. (b) Experimental data and fitting result.
5. Conclusion
In this work, an all-optical dipole trap based on double reservoir beams and a dimple beam has been used in an atom gravimeter, which can produced pure BEC within 3.2 s. It can effectively solve the disadvantage of low sampling rate of atom gravimeter based on ultra-cold atoms, making it compatible with normal cold atoms. The atom gravimeter based on ultra-cold atoms has reached the resolution of 6 $\mu$Gal after the integration time of 3000 s. By changing the parameters of evaporative cooling, the temperature of atoms can be easily adjusted to study the effect related to atom temperature in the atom gravimeter.
The interference fringe with a contrast of $76(4)\%$ at $T$=80 ms has been obtained by using ultra-cold atoms, and the fringe contrast varying with atom temperature has also been studied in this work. Moreover, the wavefront aberration effect in this atom gravimeter has been studied by modulating the temperature of atoms in the range from 30 nK to 6 $\mu$K. And the wavefront aberration effect has been evaluated by two methods, and the evaluation results are consistent with each other within the error range.
Funding
National Key Research and Development Program of China (2020YFC2200200); National Natural Science Foundation of China (11625417, 11727809, 11922404, 12004128).
Acknowledgments
Authors would like to thank Professor Zehuang Lu, Jie Zhang, Chenggang Shao, Xiaochun Duan and Dr. Xiaobing Deng, Lele Chen for enlightening discussions.
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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