⇦Chinese Physics Letters, 2020, Vol. 37, No. 4, Article code 043701 Testing the Universality of Free Fall by Comparing the Atoms in Different Hyperfine States with Bragg Diffraction * Ke Zhang (张柯), Min-Kang Zhou (周敏康)**, Yuan Cheng (程源), Le-Le Chen (陈乐乐), Qin Luo (罗覃), Wen-Jie Xu (徐文杰), Lu-Shuai Cao (曹鲁帅), Xiao-Chun Duan (段小春), Zhong-Kun Hu (胡忠坤)** Affiliations MOE Key Laboratory of Fundamental Physical Quantities Measurement & Hubei Key Laboratory of Gravitation and Quantum Physics, PGMF and School of Physics, Huazhong University of Science and Technology, Wuhan 430074 Received 6 December 2019, online 24 March 2020 *Supported by the National Natural Science Foundation of China (Grant Nos. 11625417, 91636219, 11727809, 91736311, and 11922404).
**Corresponding author. Email: zmk@hust.edu.cn; zkhu@hust.edu.cn Citation Text: Zhang K, Zhou M K, Cheng Y, Chen L L and Luo T et al 2020 Chin. Phys. Lett. 37 043701    Abstract We perform a precision atom interferometry experiment to test the universality of free fall. Our experiment employs the Bragg atom interferometer with $^{87}$Rb atoms either in hyperfine state $\left| {F = 1,{m_F} = 0} \right\rangle $ or $\left| {F = 2,{m_F} = 0} \right\rangle $, and the wave packets in these two states are diffracted by one pair of Bragg beams alternatively, which is helpful for suppressing common-mode systematic errors. We obtain an Eötvös ratio ${\eta_{1 - 2}} = \left({ 0.9 \pm 2.7} \right) \times {10^{- 10}}$, and set a new record on the precision with improvement of nearly 5 times. This measurement also provides constraint on the difference of the diagonal terms of the mass-energy operator. DOI:10.1088/0256-307X/37/4/043701 PACS:37.25.+k, 03.75.Dg, 04.80.Cc © 2020 Chinese Physics Society Article Text The universality of free fall (UFF), which refers to any objects that are subjected to the same acceleration during free fall despite their classical or quantum nature, is a foundation of modern physics. The validity of UFF is one of the most fundamental postulations of general relativity (GR),[1] and the experimental tests of UFF hold the hope of (for example) unifying the fundamental interactions.[2,3] A huge amount of experiments have been carried out to test the validity of UFF[4–6] in various circumstances to search for the sign of the extended GR theory. The most accurate tests for UFF to date were provided by the MICROSCOPE satellite mission[7] at the relative precision of $10^{-14}$ level. Other space-born experiments have also been proposed.[8–11] The UFF test has also been extended to the domain of quantum technology based on matter-wave interferences.[12–25] Testing UFF with quantum method was performed between different atomic species, such as Rb and K,[16] or different isotopes of one species.[17–20] Experiments with much higher precision were proposed and are in progress.[26,27] The quantum test of UFF not only advances the potential improvements of precision but is also particularly interesting in the search for possible spin-gravity coupling and torsion of space time. Atoms possessing well defined spin properties, such as the fermionic and bosonic isotopes of Sr,[21] the $^{87}$Rb with opposite spin orientations,[22] the $^{85}$Rb in different hyperfine states,[18] were employed as test masses in the UFF experiments. An experimental implementation using entangled atoms of $^{85}$Rb and $^{87}$Rb has also been proposed.[23] In particular, a relative precision of low $10^{-9}$ has been achieved by $^{87}$Rb atoms prepared in two hyperfine states and in their superposition while considering the possible mass-energy operator.[24] In this Letter, we present an improved UFF test at precision of $2.7\times 10^{-10}$ through the comparison of the free fall of $^{87}$Rb atoms in different hyperfine states.
cpl-37-4-043701-fig1.png
Fig. 1. The experimental scheme, including (a) the level scheme of the internal state-labeled Bragg diffraction, (b) the space-time diagram of the internal state-labeled Bragg atom interferometer and (c) the experimental setup with optics and atom fountain.
Here we perform Bragg interferometry measurements of the gravity acceleration difference between Rb atoms in states $\left| {5{S_{1/2}},F = 1,{m_F} = 0} \right\rangle $ and $\left| {5{S_{1/2}},F = 2,{m_F} = 0} \right\rangle $, termed as $\Delta g = {g_{F = 1}} - {g_{F = 2}}$. As shown in Fig. 1(a), $^{87}$Rb atoms are initially prepared in the magnetic-insensitive state $\left| {{m_F} = 0} \right\rangle $ either populated in state $\left| {5{S_{{1 / 2}}},F = 1} \right\rangle $ or $\left| {5{S_{{1 / 2}}},F = 2} \right\rangle $. Provided a proper laser frequency detuning $\Delta$, both states can be coupled to the same Bragg laser beam and manipulated between the momentum states $\left| {{p_0}} \right\rangle $ and $\left| {{p_0} + 2n\hbar k} \right\rangle $; i.e., the states with momenta ${p_0}$ and ${p_0} + 2n\hbar k$, respectively, via the Bragg diffraction.[28–35] We can then construct two Bragg atom interferometers with corresponding labeled hyperfine states in Fig. 1(b), and get the interference phase ${\phi _{F = 1}}$ and ${\phi _{F = 2}}$, respectively. The free fall acceleration ${g_{F = 1}}$ (or ${g_{F = 2}}$) with atoms in different energy states is proportional to the corresponding interference phase. The UFF signal can then be extracted from the phase difference (${\phi _{F = 1}} - {\phi _{F = 2}}$) of the two interferometers. The main advantages for our scheme are as follows: (i) the Bragg atom interferometer does not change the internal hyperfine state, which makes this method intrinsically insensitive to the noise arising from the external electromagnetic field; (ii) both of the interferometers labeled by $F=1$ and $F=2$ share the same Bragg beam, therefore some systematic effects coupled with the driven laser can be significantly commonly rejected. First, we offer a detailed description of the experimental setup. The key point for realizing the two Bragg atom interferometers with different hyperfine states is that we have to ensure the effective Rabi frequency ${{\it\Omega} _{\rm eff}}$ of $F=1$ equal to that of $F=2$ when both of them couple to the same Bragg laser beams. For the Gaussian shape Bragg pulses, the $n$th-order effective Rabi frequency[28] also depends on the normal two-photon Rabi frequency ${\it\Omega} $, where ${\it\Omega} $ is inversely proportional to the single photon frequency detuning $\Delta $.[36] We denote these two photon Rabi frequencies as ${{\it\Omega} _1}$ and ${{\it\Omega} _2}$ for the $F=1$ and $F=2$ states, respectively. For the requirements of ${{\it\Omega} _1} = {{\it\Omega} _2}$ and considering the couplings of hyperfine states in $\left| {5{P_{{3 / 2}}}} \right\rangle $, the detuning $\Delta $ should be set around 3.1817 GHz with respect to the ${5^2}{S_{1/2}}\left| {F = 2} \right\rangle \to {5^2}{P_{3/2}}\left| {F' = 3} \right\rangle $ transition. In this configuration, Bragg beams are red (blue) detuned for the $F=1$ ($F=2$) state. An atom interferometer with Bragg diffraction requires high power laser beams. The frequency doubling method[37] is employed to produce more than 8 W Bragg laser beam at 780 nm (Fig. 1(c)). A narrow-linewidth distributed feedback fiber laser at the telecom wavelength is amplified to 30 W by an erbium-doped fiber amplifier (EDFA, IPG photonics). Then the output beam from the EDFA passes through a periodically poled lithium niobate (PPLN) crystal, which can double the laser frequency from 1560 nm to 780 nm. The 780 nm laser is split into two beams (beam1 and beam2), which are frequency shifted by two pieces of acousto-optical modulators (AOM1 and AOM2) respectively. The Bragg beams consist of two perpendicular linear polarized beams[38] with adjusting the frequency of AOM2 to satisfy the resonance condition of the Bragg diffraction $\Delta \omega = {\omega _1} - {\omega _2} = {{\boldsymbol k}_{\rm eff}} \cdot {{\boldsymbol v}_{\rm a}} + 4n{\omega _{\rm r}}$. Here ${\omega _{\rm r}}$ is the single photon recoil frequency, ${{\boldsymbol k}_{\rm eff}}$ and ${{\boldsymbol v}_{\rm a}}$ represent the effective wave vector of Bragg beams and the atom free fall velocity, respectively. In order to maximize the diffraction efficiency, all the Bragg pulses are programmed to a Gaussian shape[29] by modulating the RF driver of AOM3. We add a fiber EOM into the optical path just before the EDFA to modulate the seed laser to produce the Raman beams. The EOM is driven by a 6.83 GHz microwave source and the Raman beams are employed on the velocity selection in the vertical direction. With turning off the AOM2, the Raman beams pass through the AOM1 and AOM3. Therefore the Bragg beams and Raman beams share a same optical path and are able to switch alternately by turning on or off the driven sources of EOM and AOM2. The Bragg beams and Raman beams are overlapped with the blow-away beams by a non-polarizing beam splitter. After coupling into a single mode polarization-maintaining (PM) fiber, all of the beams are aligned and injected into the vacuum chamber through the top window, passing through a quarter wave plate, and retro-reflected by a reference mirror on a vibration isolator. The ${e^{- 2}}$ diameter of both the Bragg and Raman beams is about 19 mm. The interference for the matter wave is implemented based on a cold $^{87}$Rb atom fountain that can be found elsewhere.[39] The total height of the fountain is 0.66 m. The initial state preparation that promises atoms either in $\left| {F = 1,{m_F} = 0} \right\rangle $ or in $\left| {F = 2,{m_F} = 0} \right\rangle $ is necessary before they fly into the interferometer chamber. This is realized with the microwave $\pi$-pulses between the hyperfine states, the repumping laser, the blow away beam of lower state ($\left| {5{S_{{1 / 2}}},F = 1} \right\rangle $ to $\left| {5{P_{{3 / 2}}},F = 0} \right\rangle $), and the blow-away beam of upper state ($\left| {5{S_{{1 / 2}}},F = 2} \right\rangle $ to $\left| {5{P_{{3 / 2}}},F = 3} \right\rangle $). By combining these microwave pulses and the blow away laser beams, we can prepare atoms into the two target states alternately shot by shot. A 80-µs-long Doppler-sensitive Raman $\pi$-pulse further prepares atoms in a narrow vertical momentum width less than $0.37 \hbar k$. The atoms then enter into the magnetic shielding region, and a series of Bragg pulse in the form of ${\pi / 2} - \pi - {\pi / 2}$ manipulate the atom wave packet regardless of their internal states. All the Bragg pulses are programmed to a Gaussian shape. A diffraction order $n=1$ is selected for both the states by carefully setting $\Delta \omega = {{\boldsymbol k}_{\rm eff}} \cdot {{\boldsymbol v}_{\rm a}} + 4{\omega _{\rm r}}$, and we ramp the frequency of AOM1 to compensate for the Doppler frequency shift due to the free fall. The laser frequency detuning $\Delta $ is well adjusted, thus interferences can happen between $\left| {F = 1,{p_0}} \right\rangle $ ($\left| {F = 2,{p_0}} \right\rangle $) and $\left| {F = 1,{p_0} + 2\hbar k} \right\rangle $ ($\left| {F = 2,{p_0} + 2\hbar k} \right\rangle $). The typical width for the $\pi $ pulse is about 42 µs, and the first order diffraction efficiency for both $\left| {F = 1} \right\rangle $ and $\left| {F = 2} \right\rangle $ can reach $88\% $ with a single Bragg $\pi $-pulse. The interrogation time $T$ between two Bragg pulses is 150 ms. When atoms fall back to the detection region, the two atomic clouds in the two interference paths are still partly overlapped in vertical direction, and can not be easily distinguished with the normal time of flight method. Therefore, we have developed a momentum-resolved detection method[40,41] to measure the population of the two momentum states $\left| {{p_0}} \right\rangle $ and $\left| {{p_0} + 2\hbar k} \right\rangle $ with the Doppler-sensitive Raman spectroscopy. A Raman $\pi$-pulse is applied to make a coherent transition for atoms from hyperfine state $F=1$ to $F=2$. The frequency difference between the two momentum states due to the Doppler effect in the spectrum is about 30 kHz, while the resolution of our Raman spectroscopy can be better than 0.3 kHz, which is good enough to distinguish the $\left| {{p_0}} \right\rangle $ and $\left| {{p_0} + 2\hbar k} \right\rangle $ states and to measure their population. The maximal transition probability is obtained when both the Doppler shift and Zeeman shift of state $\left| {F = i,{p_0} + 2n\hbar k} \right\rangle $ ($i=1$,2 and $n=0$, 1) are compensated for by adjusting the Raman beam's frequency. Then the population ${N_{i\_n}}$ for state $\left| {F = i,{p_0} + 2n\hbar k} \right\rangle $ is proportional to the intensity of the corresponding peak in this momentum spectrum. Finally, we get the transition probability of the momentum state $\left| {F =i, {p_0} + 2\hbar k} \right\rangle $, which is given by ${P_i} = {{{N_{i\_1}}} / {\left({{N_{i\_0}} + {N_{i\_1}}} \right)}}$. Each measurement including the atom loading, state preparation, interference stage and detection takes 1 s.
cpl-37-4-043701-fig2.png
Fig. 2. (a) The alternate measured relative atom numbers of two momentum states for $F=1$ and $F=2$ states in the UFF test. The matter-wave interference is performed either in $F=1$ or $F=2$ state with a time separation of 2 s alternately. The black squares (red dots) represent for state $\left| {F=1, {p_0}} \right\rangle $($\left| {F=1, {p_0} + 2\hbar k} \right\rangle $), and the blue triangles (pink diamonds) represent for state $\left| {F=2, {p_0}} \right\rangle $($\left| {F=2, {p_0} + 2\hbar k} \right\rangle $). (b) Fringes of the two Bragg atom interferometers. The black triangles (red dots) are transition probabilities for atoms in $F=1$ ($F=2$) state, which are calculated from the above data, and the black line (red line) represents a sine curve fitting.
The probability of finding atoms in $\left| {{p_0} + 2\hbar k} \right\rangle $ state depends on the interferometry phase, and can be written as $P = {{\left({1 - \cos \left({\left({{k_{\rm eff}}g - \alpha } \right){T^2}} \right)} \right)} / 2}$, where $\alpha $ is the Bragg beam's frequency chirp rate for compensating the Doppler shift. The matter-wave interference is performed either in $F=1$ or $F=2$ state with a time duration of 2 s, respectively. Using the momentum-resolved detection method, we can directly detect the normalized population of atoms in $F=2$, denoted as ${P_{F = 2}}$ after the detecting Raman pulse as shown in Fig. 2(a). The population ${P_{F = 2}}$ represents the relative atom numbers of states $\left| {F = 1,{p_0}} \right\rangle $ and $\left| {F = 1,{p_0} + 2\hbar k} \right\rangle $, and $1-{P_{F = 2}}$ represents that of states $\left| {F = 2,{p_0}} \right\rangle $ and $\left| {F = 2,{p_0} + 2\hbar k} \right\rangle $ in the alternative measurements. The interference fringes are calculated and also presented in Fig. 2(b). Each fringe contains two periods corresponding to a phase interval of $4\pi $ and taking 160 s in total.
cpl-37-4-043701-fig3.png
Fig. 3. The long time measurements for the UFF test experiments. (a) The gravity accelerations measured with atoms in the $F=1$ (black triangles) and $F=2$ (red dots) states compared to the tide model (green line) with a common offset ${g_{\rm offset}}$. (b) The differential acceleration of the two states (blue squares). (c) The residual accelerations of atoms in the $F=1$ (black triangles) and $F=2$ (red dots) states after subtracting the tide, which is shown clearly and compared to ${g_{F = 1}} - {g_{F = 2}}$. (d) The Allan deviation of the differential acceleration ${g_{F = 1}} - {g_{F = 2}}$. The short-term sensitivity is $1.2 \times {10^{- 7}}g$/Hz${}^{1/2}$ from the $t^{-1/2}$ fitting.
A test of the UFF with atoms in different hyperfine states is then performed by continuously measuring the gravity acceleration with this state-labeled atom interferometer. As shown in Fig. 3, data for about 63 hours are recorded by the apparatus. Each point in this data is the mean result of 160 s. Both the interferometers with atoms in $F=1$ and $F=2$ states can precisely map the gravity tides, which are displayed in Fig. 3(a). What we care about in the UFF test is the differential acceleration $\Delta g$ shown by blue squares in Fig. 3(b). By averaging all the data, we get $\Delta g = \left({ - 1.2 \pm 2.6} \right) \times {10^{- 10}}g$, where the uncertainty is the standard deviation of the weighted mean. The short-term sensitivity of ${g_{F = 1}} - {g_{F = 2}}$ is $1.2 \times {10^{- 7}}g/{\rm{H}}{{\rm{z}}^{{1 / 2}}}$ as shown in Fig. 3(c).
cpl-37-4-043701-fig4.png
Fig. 4. The UFF test by modulating the magnetic bias field of the interference region. (a) In different magnetic fields, the $\Delta g$ directly measured by the atom interferometers (red dots) agree with the evaluated quadratic Zeeman effect based on the magnetic field distribution from Raman spectroscopy experiments (black squares). The blue line is the quadratic polynomial fit result. (b) The orange diamonds represent the $\Delta g$ after correcting the quadratic Zeeman effect, and the magenta line is the average value of $\left({ 1.6 \pm 2.4} \right) \times {10^{- 10}}g$.
In our experiment, atoms in the $F=1$ and $F=2$ states are prepared in the same way, which possess the same temperature, initial velocity and position, and only one pair of Bragg beams is employed to diffract atoms. Therefore some systematic effects can be commonly rejected by differential measurement, such as the gravity gradient effect, the Coriolis effect and the wavefront aberration. The fluctuations of these effects between two states due to the alternate measurements only contribute to the noise of the differential measurement, which are included in the statistical uncertainty. The main systematic effects for $\Delta g$ are evaluated as follows. The magnetic field inhomogeneity contribute with a bias to $\Delta g$, because of the spatial separation of the two arms of the Bragg atom interferometer and the opposite sign of the Landé $g$-factor for the $F=1$ and $F=2$ states. As atoms are prepared in the ${m_F} = 0$ state, we only must consider the quadratic Zeeman effect. The magnetic field in the interferometry region is measured precisely by the Raman spectroscopy method.[42] With the magnetic bias field of 29 mG in the interferometry region, the Zeeman effect on the UFF test due to the spatial separation of the wave packets is evaluated to be $\left({ -2.1 \pm 0.5} \right) \times {10^{- 10}}g$ giving a dominant systematic impact. This effect is also confirmed with modulation experiments by measuring the differential acceleration $\Delta g$ in different magnetic bias fields. As shown in Fig. 4(a), when increasing the magnetic field, the value of $\Delta g$ performs as a quadratic increase, which is consistent with the evaluated Zeeman effects based on the magnetic field distribution. The data in Fig. 4(a) also describe five measurements of the UFF test in the case of different magnetic fields. After correcting the quadratic Zeeman effect, the average of these measurements in Fig. 4(b) is obtained to be $\Delta g = \left({ 1.6 \pm 2.4} \right) \times {10^{- 10}}g$, and the uncertainty is the standard deviation of the weighted mean of these five measurements, showing that the experimental system is quite stable and the statistic error can go down to the $10^{-10}$ level. Because atoms are in a same internal state for Bragg type interferometer, the ac Stark shifts caused by the spatial intensity gradients of the Bragg lasers[24] and the intensity fluctuation between the first and third Bragg pulses[30] are both less than $1 \times {10^{- 11}}g$ in our experiment. Limited by the accuracy of absolute frequency of the Bragg lasers, the maximum frequency deviation of 1 MHz from the detuning $\mathit{\Delta}=3.1817$ GHz will contribute less than $1 \times {10^{- 11}}g$ due to the two-photon light shift.[43,44] The local gravity variation due to the tides will induce a systematic error as the measurement in the $F=1$ state is always 2 s after the $F=2$ state. This effect is evaluated at the level of $3 \times {10^{- 12}}g$ and can be neglected in the present test. As we select the first order Bragg diffraction to manipulate atoms without other unwanted momentum states, there should be no parasitic interference.[31,45] The $20\% $ difference of atom numbers for two states caused by the efficiency of microwave $\pi$-pulse only affects the detection noise of measuring the gravity without contributing to $\Delta g$. After considering and correcting of these systematic effects described above which are summarized in Table 1, the Eötvös ratio given by $$ {\eta _{1 - 2}} = 2{{\left({{g_{F = 1}} - {g_{F = 2}}} \right)} / {\left({{g_{F = 1}} + {g_{F = 2}}} \right)}}~~ \tag {1} $$ is finally determined to be ${\eta _{1 - 2}} = \left({0.9 \pm 2.7} \right) \times {10^{- 10}}$, which means that the UFF between atoms in different hyperfine states is still valid at precision of $10{^{-10}}$ level. The diagonal terms of the possible breaking operator of a UFF violation is also estimated to be ${r_1} - {r_2}{\rm{ = }}(0.9 \pm 2.7) \times {10^{{\rm{ - 10}}}}$ according to the mass-energy equivalence, which corresponds to an improvement over previous results with nearly a factor of 5.[24,46] In conclusion, we have implemented Bragg atom interferometers with different hyperfine states, and demonstrated its application in high precision measurements of gravitational acceleration. Due to the property of coupling to the same Bragg beams, various systematic effects can be commonly rejected in the present precision. With this state-labeled $^{87}$Rb atom interferometer, a precise quantum test on the UFF between different hyperfine states is performed at ${10^{- 10}}$ level. We can gain about 5 times improvement in the accuracy and still see no violation of UFF. We also set a new constraint on the difference of the diagonal elements of the mass-energy operator.[24] The experimental scheme demonstrated here can be further developed to construct two atom interferometers simultaneously, which can be applied in the UFF test with the isotope of rubidium or other species,[21,19] paving a way for high precision quantum test of UFF better than ${10^{-10}}$ level.
Table 1. Main contributions to the differential acceleration measurements with the magnetic bias field of 29 mG.
$\Delta g$ (${10^{- 10}}$g) Uncertainty (${10^{- 10}}$g)
Statistical uncertainty $-1.2$ 2.6
Quadratic Zeeman shift $-2.1$ 0.5
AC Stark shift 0 0.2
Tide effect 0 0.03
Corrected 0.9 2.7
The authors gratefully acknowledge Časlav Brukner for the inspiring discussions on this work. References Testing the equivalence principle: why and how?Extended Theories of GravityTorsion-balance tests of the weak equivalence principleRelativistic tests with lunar laser rangingTest of the Equivalence Principle with Chiral Masses Using a Rotating Torsion PendulumMICROSCOPE Mission: First Results of a Space Test of the Equivalence PrincipleFeasibility for Testing the Equivalence Principle with Optical Readout in SpacePrecision Gravity Tests with Atom Interferometry in SpaceSTE-QUEST—test of the universality of free fall using cold atom interferometryDual matter-wave inertial sensors in weightlessnessAtomic interferometry with internal state labellingMeasurement of gravitational acceleration by dropping atomsMicro-Gal level gravity measurements with cold atom interferometryComparison between two mobile absolute gravimeters: optical versus atomic interferometersQuantum Test of the Universality of Free FallSimultaneous dual-species matter-wave accelerometerAtomic Interferometer with Amplitude Gratings of Light and Its Applications to Atom Based Tests of the Equivalence PrincipleTest of Equivalence Principle at 1 0 − 8 Level by a Dual-Species Double-Diffraction Raman Atom InterferometerUnited test of the equivalence principle at $10^{-10}$ level using mass and internal energy specified atomsTest of Einstein Equivalence Principle for 0-Spin and Half-Integer-Spin Atoms: Search for Spin-Gravity Coupling EffectsTest of the Universality of Free Fall with Atoms in Different Spin OrientationsProposal for a Quantum Test of the Weak Equivalence Principle with Entangled Atomic SpeciesQuantum test of the equivalence principle for atoms in coherent superposition of internal energy statesTesting the universality of free fall with rubidium and ytterbium in a very large baseline atom interferometerEffective Inertial Frame in an Atom Interferometric Test of the Equivalence PrincipleAtom Interferometry with up to 24-Photon-Momentum-Transfer Beam SplittersAtom-wave diffraction between the Raman-Nath and the Bragg regime: Effective Rabi frequency, losses, and phase shiftsPrecision atomic gravimeter based on Bragg diffractionHigh-Resolution Atom Interferometers with Suppressed Diffraction PhasesLarge-momentum-transfer Bragg interferometer with strontium atomsBragg interferometer for gravity gradient measurementsSimultaneous Precision Gravimetry and Magnetic Gradiometry with a Bose-Einstein Condensate: A High Precision, Quantum SensorMeasurement of the fine-structure constant as a test of the Standard ModelMeasurement of the gravitational acceleration of an atom with a light-pulse atom interferometer11 W narrow linewidth laser source at 780nm for laser cooling and manipulation of RubidiumLow-phase noise and high-power laser for Bragg atom interferometerDemonstration of an ultrahigh-sensitivity atom-interferometry absolute gravimeterAtomic velocity selection using stimulated Raman transitionsMomentum-resolved detection for high-precision Bragg atom interferometryPrecisely mapping the magnetic field gradient in vacuum with an atom interferometerOff-resonant Raman transition impact in an atom interferometerLight shifts in atomic Bragg diffractionControlling the multiport nature of Bragg diffraction in atom interferometryQuantum formulation of the Einstein equivalence principle
[1]Misner C, Thorne K and Wheeler J 1973 Gravitation (San Francisco: Freeman)
[2] Damour T 1996 Class. Quantum Grav. 13 A33
[3] Capozziello S and De M 2011 Phys. Rep. 509 167
[4] Wagner T A, Schlamminger S, Gundlach J H and Adelberger E G 2012 Class. Quantum Grav. 29 184002
[5] Hofmann F and Müller J 2018 Class. Quantum Grav. 35 035015
[6] Zhu L et al 2018 Phys. Rev. Lett. 121 261101
[7] Touboul P et al 2017 Phys. Rev. Lett. 119 231101
[8] Gao F, Zhou Z B and Luo J 2011 Chin. Phys. Lett. 28 080401
[9] Tino G M et al 2013 Nucl. Phys. B 243 203
[10] Aguilera D et al 2014 Class. Quantum Grav. 31 115010
[11] Barrett B, Laura A M, Chichet L, Battelier B, Lévèque T, Landragin A and Bouyer P 2016 Nat. Commun. 7 13786
[12] Bordé C J 1989 Phys. Lett. A 140 10
[13] Peters A, Chung K Y and Chu S 1999 Nature 400 849
[14] Zhou M K, Duan X C, Chen L L, Luo Q, Xu Y Y and Hu Z K 2015 Chin. Phys. B 24 050401
[15] Merlet S, Bodart Q, Malossi N, Landragin A, Santos F P D, Gitlein O and Timmen L 2010 Metrologia 47 L9
[16] Schlippert D, Hartwig J, Albers H, Richardson L L, Schubert C, Roura A, Schleich W P, Ertmer W and Rasel E M 2014 Phys. Rev. Lett. 112 203002
[17] Bonnin A, Zahzam N, Bidel Y and Bresson A 2013 Phys. Rev. A 88 043615
[18] Fray S, Diez C A, Hänsch T W and Weitz M 2004 Phys. Rev. Lett. 93 240404
[19] Zhou L et al 2015 Phys. Rev. Lett. 115 013004
[20] Zhou L et al 2019 arXiv:1904.07096 [quant-ph]
[21] Tarallo M G, Mazzoni T, Poli N, Sutyrin D V, Zhang X and Tino G M 2014 Phys. Rev. Lett. 113 023005
[22] Duan X C et al 2016 Phys. Rev. Lett. 117 023001
[23] Geiger R and Trupke M 2018 Phys. Rev. Lett. 120 043602
[24] Rosi G, D'Amico G, Cacciapuoti L, Sorrentino F, Prevedelli M, Zych M, Brukner C and Tino G M 2017 Nat. Commun. 8 15529
[25]Wang J and Zhan M S 2018 Acta Phys. Sin. 67 160402 (in Chinese)
[26] Hartwig J, Abend S, Schubert C, Schlippert D, Ahlers H, Posso-Trujillo K, Gaaloul N, Ertmer W and Rasel E M 2015 New J. Phys. 17 035011
[27] Overstreet C, Asenbaum P, Kovachy T, Notermans R, Hogan J M and Kasevich M A 2018 Phys. Rev. Lett. 120 183604
[28] Müller H, Chiow S, Long Q, Herrmann S and Chu S 2008 Phys. Rev. Lett. 100 180405
[29] Müller H, Chiow S and Chu S 2008 Phys. Rev. A 77 023609
[30] Altin P A et al 2013 New J. Phys. 15 023009
[31] Estey B, Yu C, Müller H, Kuan P C and Lan S Y 2015 Phys. Rev. Lett. 115 083002
[32] Mazzoni T, Zhang X, Del Aguila R, Salvi L, Poli N and Tino G M 2015 Phys. Rev. A 92 053619
[33] D'Amico G, Borselli F, Cacciapuoti L, Prevedelli M, Rosi G, Sorrentino F and Tino G M 2016 Phys. Rev. A 93 063628
[34] Hardman K S et al 2016 Phys. Rev. Lett. 117 138501
[35] Parker R H, Yu C, Zhong W, Estey B and Müller H 2018 Science 360 191
[36] Kasevich M and Chu S 1992 Appl. Phys. B 54 321
[37] Sané S S, Bennetts S, Debs J E, Kuhn C C N, McDonald G D, Altin P A, Close J D and Robins N P 2012 Opt. Express 20 8915
[38] Cheng Y, Zhang K, Chen L L, Xu W J, Luo Q, Zhou M K and Hu Z K 2017 AIP Adv. 7 095211
[39] Hu Z K, Sun B L, Duan X C, Zhou M K, Chen L L, Zhan S, Zhang Q Z and Luo J 2013 Phys. Rev. A 88 043610
[40] Kasevich M, Weiss D S, Riis E, Moler K, Kasapi S and Chu S 1991 Phys. Rev. Lett. 66 2297
[41] Cheng Y, Zhang K, Chen L L, Zhang T, Xu W J, Duan X C, Zhou M K and Hu Z K 2018 Phys. Rev. A 98 043611
[42] Zhou M K, Hu Z K, Duan X C, Sun B L, Zhao J B and Luo J 2010 Phys. Rev. A 82 061602(R)
[43] Gauguet A, Mehlstäubler T E, Lévèque T, Le Gouët J, Chaibi W, Canuel B, Clairon A, Santos F P D and Landragin A 2008 Phys. Rev. A 78 043615
[44] Giese E, Friedrich A, Abend S, Rasel E M and Schleich W P 2016 Phys. Rev. A 94 063619
[45] Parker R H, Yu C, Estey B, Zhong W, Huang E and Müller H 2016 Phys. Rev. A 94 053618
[46] Zych M and Brukner C 2018 Nat. Phys. 14 1027