# Current Status and Prospects on High-Precision Quantum Tests of the Weak Equivalence Principle with Cold Atom Interferometry

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## Abstract

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## 1. Introduction

## 2. Basic Theory

## 3. Developments and State of the Art

^{−11}g/shot reported by the Zhan group [105] and the potential acceleration sensitivity of 6.7 × 10

^{−12}g/shot given by the Kasevich group [104]. The above development of gravity measurement with atom interferometers has laid a good foundation for the WEP test.

#### 3.1. Dual Atomic Species

^{−7}, by increasing the free-falling time in 2020 [115]. Their accuracy was mainly limited by the second-order Zeeman effect and the wavefront curvature of the Raman beams. In 2015, the LP2N group proposed a dual-species fringe reconstruction by accelerometer correlation method to realize a common mode suppression ratio of 730 for the vibration noise and obtained an Eötvös parameter of $1.6\times {10}^{-6}$ per measurement at a free evolution time of $10\mathrm{m}\mathrm{s}$ [113]. The next year, they tested the WEP in a weightless environment produced during a parabolic flight [114]. The Eötvös parameter was measured with the uncertainty of $3.0\times {10}^{-4}$ in the microgravity, which is four times better than that in the standard gravity. In 2022, they improved the interrogation time to $T=20\mathrm{ms}$ and obtained the accuracy of $\eta =(0.9\pm 1.6)\times {10}^{-6}$ [116]. The statistical uncertainty of the Eötvös parameter is $7.8\times {10}^{-8}$ after $2.4\times {10}^{4}\mathrm{s}$ of integration. Tests with other atoms, such as ${}^{87}\mathrm{Rb}$ and ${}^{170}\mathrm{Yb}$ [127], and Cd and Sr [128], are still in progress.

#### 3.2. Dual Atomic Isotopes

^{−7}with a resolution of 2.5 × 10

^{−8}[117]. In 2014, the Tino group reported a new test of the WEP using two isotopes of strontium atoms, namely, the bosonic ${}^{88}\mathrm{Sr}$ and the fermionic ${}^{87}\mathrm{Sr}$ [118]. By measuring the Bloch frequencies of ${}^{88}\mathrm{Sr}$ and ${}^{87}\mathrm{Sr}$, they obtained $\eta =(0.2\pm 1.6)\times {10}^{-7}$. The main error sources come from the frequency shift of the Raman light and the Coriolis force. In 2015, the Zhan group proposed and implemented a four-wave double-diffraction Raman transition (FWDR) scheme to suppress the common-mode phase noises of the Raman lasers in the ${}^{85}\mathrm{Rb}$-${}^{87}\mathrm{Rb}$ dual-species atom interferometer [82]. The accuracy of the measured $\eta $ is (2.8 ± 3.0) × 10

^{−8}, and the statistical uncertainty is $0.8\times {10}^{-8}$ after $3200\mathrm{s}$ of integration. In 2021, the same group improved the accuracy of the WEP test to (0.8 ± 1.4) × 10

^{−10}[121]. In 2018, the Kasevich group suppressed gravity-gradient-induced phase differences by selecting the appropriate Raman pulse frequency shift with a relative precision of $\Delta g/g$ being about 6 × 10

^{−11}per shot [119]. In 2020, they demonstrated zero violation of the WEP between ${}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}$ with the accuracy at the level of 10

^{−12}[120], which is the highest accuracy so far by using microscopic particles. Further, proposals and ongoing experiments with dual-isotopes aim to achieve a precision of ${10}^{-15}$ or better [129,130].

#### 3.3. Dual Atomic Internal States

^{−7}[100]. Using the same atoms but with opposite-spin-oriented states, i.e., ${}^{85}\mathrm{Rb}$ atoms with ${m}_{F}=1$ and ${m}_{F}=-1$, a group from HUST carried out a test of the WEP with the measured Eötvös parameter being (0.2 ± 1.2) × 10

^{−7}[122]. In 2020, their result is improved to $\eta =(0.9\pm 2.7)\times {10}^{-10}$ by using the Bragg-diffraction atom interferometer with the hyperfine ground states $\left|F=1,{m}_{F}=0\right.\u232a$ and $|F=2,{m}_{F}=0\rangle $ of ${}^{87}\mathrm{Rb}$ atoms [124]. Recently, they further improved the upper bound of the WEP test to $2.9\times {10}^{-11}$ [125]. In 2017, the Tino group also realized quantum test of the WEP for the ${}^{87}\mathrm{Rb}$ atoms in coherent superposition of internal states $\left|F=1,{m}_{F}=0\right.\u232a$ and $\left|2,0\right.\u232a$ [123]. They used Bragg atom interferometers in a gravity gradiometer configuration and achieved a relative uncertainty of the Eötvös parameter at the low level of ${10}^{-9}$.

## 4. Key Techniques and Systematic Effects

^{−9}g/$\sqrt{\mathrm{Hz}}$, which is the key obstacle that limits accuracy improvements.

#### 4.1. Preparation and Control of Laser Pulse

#### 4.2. Atom Trajectory and Signal Detection

#### 4.3. Major Systematic Effects

#### 4.3.1. Gravity Gradient and Coriolis Effect

#### 4.3.2. Wavefront Aberrations

^{−9}g, which strongly limits the accuracy of the WEP test. Wang et al. analyzed the influence of the wavefront curvature of Raman pulses by the method of a transmission matrix [168]. Schkolnik et al. presented a experimental analysis of wavefront curvature based on measured aberrations of optical windows. The uncertainty of the measured gravity is less than 3 × 10

^{−10}g [166]. Zhou et al. presented a detailed theoretical analysis of wavefront aberrations and measured the effect by modulating the waist of Raman beams [169]. Trimeche et al. used deformable mirrors to actively control the laser wavefront and achieve compensation for wavefront curvature [170]. Hu et al. proposed an expansion-rate-selection method to suppress the aberration phase noise in the WEP test using dual-species atom interferometers [167]. The simulations showed that the suppressed uncertainty to the Eötvös parameter is on the level of ${10}^{-14}$ for isotopic atoms and ${10}^{-13}$ for nonisotopic atoms. Better results can be obtained by using atoms with lower temperature. Karcher et al. established a thorough model to study the influence of wavefront curvature on atom interferometer and proposed a method to correct for this bias based on the extrapolation of the measurements down to zero temperature [171].

#### 4.3.3. Stark and Zeeman Effects

#### 4.3.4. Atoms Interaction and Self-Attraction Effect

^{−9}g in the measurement of gravity.

#### 4.4. Noise Suppression

#### 4.5. Integrated Packages

## 5. Prospect and Conclusions

^{−12}[120], and no violation is observed. To test the WEP with a higher accuracy in the future, we need to improve the sensitivity and accuracy of the atom interferometers. According to Equation (5), there are two major ways to improve the sensitivity: (i) to increase the evolution time T and (ii) to enlarge the momentum splitting k. A long-baseline setup, microgravity environment or a set of optical lattice can be used to increase the evolution time T. The main method of enlarging k is to use the Bragg diffraction. The research on the influence of the temperature and the entanglement on the sensitivity is reviewed and considered in this section.

^{−13}[127]. The atom interferometer build by the Kasevich group achieved an effective interference length of 8.2 m and an interrogation time $2T$ of 2.3 s [104,196]. They also proposed to establish a 100 m atom interferometer [34]. The Zhan group also realized a 10 m long-baseline atom interferometer towards the verification of the WEP [105,197]. In 2020, they proposed the ZAIGA plan to build a 300 m atom interferometer, which is expected to achieve a maximum integration time of 7.7 s and precision of 10

^{−15}for the WEP test [198].

^{−4}level under a 0 g environment [114]. Space missions such as the STE-QUEST plan [129,204] and the QTEST plan [130] are proposed, aiming at an accuracy of 10

^{−15}in the WEP test. The Cold Atom Lab (CAL) in the International Space Station was first powered operated in 2018, and the ${}^{87}\mathrm{Rb}$ ultracold BEC was prepared on board [207]. Last year, the microgravity scientific laboratory cabinet (MSLC) was launched to the China Space Station with the aim of testing the WEP in the level of ${10}^{-10}$ [208]. One can find more experimental details on the microgravity environment in space in [209].

^{−15}[26,27] and 10

^{−12}[120], respectively, we still have not observed any signs of the WEP breaking. However, as seen in this review, the potential advantages of using cold atoms to verify WEP have not been fully explored. Pushing the limits of the accuracy to higher levels with various microscopic atoms is the major research goal of the WEP test, though plenty of challenges and problems must be addressed [216]. Firstly, although WEP test experiments using non-isotopes may be more attractive [126,217], the experimental accuracy of WEP verification for non-isotopic atoms is generally low at present [112,113,114,115,116]. The main challenge is the difficulty in correcting system errors caused by different effective wave vectors for different atoms. Feasible methods are converting these noises into common-mode noise and reducing atom temperature to improve verification accuracy. Secondly, in a long-baseline atom interferometer, the error induced by the gravity gradient is a systematic error that is difficult to ignore due to the long distance of atoms falling. In addition to utilizing the method of Section 4 to reduce the error induced by gravity gradients, LMT technology and microgravity environments can also be developed to reduce the impact of gravity gradients and improve validation accuracy. Thirdly, the WEP test using large-scale molecules is still at its initial stage [96]. In its development, the corresponding cooling methods should be urgently put at the first priority. In the future, the controlling techniques of the multiple degrees of freedom, such as the chirality, the internal states and composition of different molecules, may also need to be developed in the large-scale WEP test using molecules. And what is more interesting and challenging in future WEP experiment tests is to use nonlocal correlations of atoms, such as atomic entanglement and squeezing [83,84,85,87]. Currently, there are relatively few experiments in this field, but it is potentially worthwhile to find possible evidence of the influence of entanglement in gravity. Also, further validation of LPI and LLI can be achieved using cold atoms, and some proposals and experiments have been proposed [218,219,220,221,222,223,224]. We believe future stringent tests of the WEP will open new doors to physics, such as modifying the GR theory, establishing a quantum gravity theory and searching for new forces or matter.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Acronym | Meaning | Section |

AI | atom interferometry | 2 |

AOM | acousto-optic modulation | 4 |

BEC | Bose–Einstein condensate | 4, 5 |

CAL | Cold Atom Lab | 5 |

EEP | Einstein equivalence principle | 1 |

EOM | electro-optic modulation | 4 |

FWDR | four-wave double-diffraction Raman transition | 3, 4 |

GR | general relativity | 1, 5 |

HUST | Huazhong University of Science and Technology | 3, 4 |

I/Q | in-phase/quadrature | 4 |

LLI | local Lorentz invariance | 1, 5 |

LMT | large momentum transfer | 4, 5 |

LPI | local position invariance | 1, 5 |

LP2N | The Photonics, Numerical and Nanosciences Laboratory | 3 |

LUH | Leibniz Universität Hannover | 3 |

LENS | European Laboratory for Non Linear Spectroscopy | 3 |

MICROSCOPE | Micro-Satellite a traînée Compensée pour l’Observation du Principe d’Equivalence | 1 |

MPIQ | Max-Planck-Institut für Quantenoptik | 3 |

MSLC | microgravity scientific laboratory cabinet | 5 |

OPLL | optical phase lock-loop | 4 |

ONERA | The French Aerospace Lab | 3, 5 |

QTEST | Quantum Test of the Equivalence Principle in Space | 5 |

QUANTUS | QUANTen Gase Unter Schwerelosigkeit | 5 |

SM | Standard Model | 1 |

STE-QUEST | Space–Time Explorer and Quantum Equivalence principle Space Test | 5 |

UFF | University of Free Fall | 1 |

WEP | weak equivalence principle | 1, 2, 3, 4, 5 |

WIPM | Wuhan Institute of Physics and Mathematics | 3, 4 |

ZAIGA | The Zhaoshan Long-Baseline Atom Interferometer Gravitation Antenna | 5 |

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**Figure 1.**Schematic of the March–Zehnder atom interferometer using the $\frac{\pi}{2}$-$\pi $-$\frac{\pi}{2}$ Raman pulses. T represents the free evolution time between the Raman pulses. The three Raman pulses are used for splitting, reflection and recombination of the atomic wave packets. Path I and II mean the two arms of the interferometer. The gray line in the figure represents the classical interference path of the matter wave in the absence of gravity, and the black line represents the path in the presence of gravity. A and B label atoms of different natures for the test of the WEP. $\Delta {\Phi}_{\mathrm{A}}$ and $\Delta {\Phi}_{\mathrm{B}}$ are the phase change produced by gravity acceleration g, and the Eötvös parameter $\eta $ can be obtained from the comparison between them.

**Figure 2.**Measurement accuracy of the the Eötvös parameters $\eta $ in the WEP tests with cold atoms. The black, red and blue points represent results using dual-species (A1 [112], A2 [113], A3 [114], A4 [115], A5 [116]), dual-isotopes (B1 [100], B2 [117], B3 [118], B4 [82], B5 [119], B6 [120], B7 [121]), and dual-states (C1 [100], C2 [122], C3 [123], C4 [124], C5 [125]) atom interferometers respectively, as listed in Table 1.

**Figure 3.**Internal diagram of an I/Q modulator. ${E}_{\mathrm{in}}$ and ${E}_{\mathrm{out}}$: the input and output laser field; $\delta {\varphi}_{S}=\beta sin{\omega}_{m}t$ and $\delta {\varphi}_{C}=\beta cos{\omega}_{m}t$: the sine and cosine phase modulator; ${\Phi}_{1,2,3}$: optical phase shifter; MZM: Mach–Zehnder modulation.

**Figure 4.**Schematic of double-diffraction Raman transition (

**a**) and four-wave double-diffraction Raman transition (

**b**). ${k}_{1}$, ${k}_{2}$, ${k}_{3}$ and ${k}_{4}$ are wave vectors of the Raman beams, T is the free evolution time, and $2S$ is the enclosed area of the interference.

**Table 1.**Summary of the main experimental results of the WEP test with cold atoms performed in the past two decades.

Properties of the Test Bodies | Year | Accuracy ($\mathit{\eta}$) | Group & Reference | |
---|---|---|---|---|

Dual-species | ${}^{87}\mathrm{Rb}{-}^{39}\mathrm{K}$ | 2014 | $\left(0.3\pm 5.4\right)\times {10}^{-7}$ | LUH [112] |

${}^{87}\mathrm{Rb}{-}^{39}\mathrm{K}$ | 2015 | $\left(-\phantom{\rule{4.pt}{0ex}}-\pm 1.6\right)\times {10}^{-6}$ per shot | LP2N [113] | |

${}^{87}\mathrm{Rb}{-}^{39}\mathrm{K}$ | 2016 | $\left(0.9\pm 3.0\right)\times {10}^{-4}$ @ $0g$ | LP2N [114] | |

${}^{87}\mathrm{Rb}{-}^{39}\mathrm{K}$ | 2020 | $\left(1.9\pm 3.2\right)\times {10}^{-7}$ | LUH [115] | |

${}^{87}\mathrm{Rb}{-}^{39}\mathrm{K}$ | 2022 | $\left(0.9\pm 1.6\right)\times {10}^{-6}$ | LP2N [116] | |

Dual-isotopes | ${}^{85}\mathrm{Rb}{-}^{87}\mathrm{Rb}$ | 2004 | $\left(1.2\pm 1.7\right)\times {10}^{-7}$ | MPIQ [100] |

${}^{85}\mathrm{Rb}{-}^{87}\mathrm{Rb}$ | 2013 | $\left(1.2\pm 3.2\right)\times {10}^{-7}$ | ONERA [117] | |

${}^{88}\mathrm{Sr}{-}^{87}\mathrm{Sr}$ | 2014 | $\left(0.2\pm 1.6\right)\times {10}^{-7}$ | LENS [118] | |

${}^{85}\mathrm{Rb}{-}^{87}\mathrm{Rb}$ | 2015 | $\left(2.8\pm 3.0\right)\times {10}^{-8}$ | WIPM [82] | |

${}^{85}\mathrm{Rb}{-}^{87}\mathrm{Rb}$ | 2018 | $\left(6\pm -\phantom{\rule{4.pt}{0ex}}-\right)\times {10}^{-11}$ per shot | Stanford [119] | |

${}^{85}\mathrm{Rb}{-}^{87}\mathrm{Rb}$ | 2020 | $\left(1.6\pm 3.8\right)\times {10}^{-12}$ | Stanford [120] | |

${}^{85}\mathrm{Rb}{-}^{87}\mathrm{Rb}$ | 2021 | $\left(0.8\pm 1.4\right)\times {10}^{-10}$ | WIPM [121] | |

Dual-states | ${}^{85}\mathrm{Rb},\left|2\right.\u232a-\left|3\right.\u232a$ | 2004 | $\left(0.4\pm 1.2\right)\times {10}^{-7}$ | MPIQ [100] |

${}^{87}\mathrm{Rb},{m}_{F}=\pm 1$ | 2016 | $\left(0.2\pm 1.2\right)\times {10}^{-7}$ | HUST [122] | |

${}^{87}\mathrm{Rb},\left|1\right.\u232a-\left|2\right.\u232a$ | 2017 | $\left(1.0\pm 1.4\right)\times {10}^{-9}$ | LENS [123] | |

${}^{87}\mathrm{Rb},\left|1\right.\u232a-\left|2\right.\u232a$ | 2020 | $\left(0.9\pm 2.7\right)\times {10}^{-10}$ | HUST [124] | |

${}^{87}\mathrm{Rb},\left|1\right.\u232a-\left|2\right.\u232a$ | 2022 | $\left(0.9\pm 2.9\right)\times {10}^{-11}$ | HUST [125] |

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Yuan, L.; Wu, J.; Yang, S.-J.
Current Status and Prospects on High-Precision Quantum Tests of the Weak Equivalence Principle with Cold Atom Interferometry. *Symmetry* **2023**, *15*, 1769.
https://doi.org/10.3390/sym15091769

**AMA Style**

Yuan L, Wu J, Yang S-J.
Current Status and Prospects on High-Precision Quantum Tests of the Weak Equivalence Principle with Cold Atom Interferometry. *Symmetry*. 2023; 15(9):1769.
https://doi.org/10.3390/sym15091769

**Chicago/Turabian Style**

Yuan, Liang, Jizhou Wu, and Sheng-Jun Yang.
2023. "Current Status and Prospects on High-Precision Quantum Tests of the Weak Equivalence Principle with Cold Atom Interferometry" *Symmetry* 15, no. 9: 1769.
https://doi.org/10.3390/sym15091769