A Mass-Energy United Test of the Equivalence Principle

The equivalence principle (EP) is one of the basic assumptions of general relativity. Almost all new theories 1 that attempt to unify gravity with the standard model 2 require the EP be broken. Experimental tests of EP provide opportunities for 10 verification of different theoretical models and emergence of new physics.

respectively. The violation parameters of mass and internal energy are constrained to 1  0 = (-0.8  1.4)  10 -10 and  = (-0.6 6.9)  10 5 . This work opens a door for united 2 tests of EP and MEE in large energy range with quantum systems.

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Mass tests of EP have been performed in different ways including Lunar laser 5 ranging 15 , torsion balance 16,17 , satellite 3 and AIs 4-9 . However, it is unknown whether the 6 energy behaves the same as mass in a gravitational field although MEE holds at least at 7 10 -7 level 18 . To parameterize possible contributions of mass and energy to EP violation, 8 under the condition that MEE is valid, we express the gravitational mass m g of a test body 9 as a sum of different types of mass-energy and their EP violation terms 1 10   violation coefficient or the energy violation coefficient. Up to now, no one has combined 23 mass and energy in one platform to precisely extract potential tiny difference between 1 mass and energy violation coefficients in noisy environments. The main obstacles lie in 2 the technical complexity when putting dual-species atoms and specific quantum states 3 together in AIs. It is rather difficult to satisfy the following requirements in one 4 experiment: (1) keeping the same specified quantum state in an AI; (2) suppressing 5 common mode noise for different species of atoms; (3) adjusting internal energy of atoms. 6 Here, we improve the 4WDR dual-species 87 Rb-85 Rb AI 7 and perform a united mass and 7 internal energy test of EP. Mass and internal energy specified AIs are realized by 8 deterministically manipulating the internal states of 87 Rb and 85 Rb atoms. Eötvös 9 parameters are measured for the four paired combinations, constraints to mass and energy 10 violation are then given respectively (see Table 1).

Table 1 | Measured Eötvös parameters of EP tests with atoms.
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The experimental setup is shown in Fig. 1(c), which it is upgraded from that in our 11 previous work 7 . It is briefly described as follows. The three-dimensional magneto-optical  10 -10 , and the slope value  = (-0.6  6.9) × 10 5 . The experimental data for four combination pairs are shown in Fig. 2. Fig. 2(a1) 9 shows 640 measurements of  1 value using 87 Rb|F=1-85 Rb|F=2 atom pair, where each The Allan deviations are shown in Fig. 2(b). The red squares are data of  1 , the 5 statistical uncertainty is 0.6  10 -10 at an average time of 35840 s. The black dots, green 6 boxes, and blue triangles are data of  2 ,  3 and  4 , respectively. The corresponding 7 statistical uncertainties at an average time of 17920 s are 1.8  10 -10 , 2.5  10 -10 and 1.8  8 10 -10 , respectively. 9 The systematic errors are suppressed by correcting the wave vector, optimizing MIR, 10 compensating the rotation of the Raman beams mirror, calibrating the gravity gradient 11 error, the quadratic Zeeman shift and suppressing the wave-front error.

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The uncertainty of wave vectors is suppressed to 0.5  10 -10 by precise frequency 13 control. The MIR is optimized from 1.0: 1.0: 3.1: 14 in our previous experiment 7 to 1.00:  The wave-front-aberration error is analyzed and estimated by modulating the size of 7 detection beams. The error and uncertainty of the wave-front aberration analyzed from the 8 actual experimental parameter is (0.5  0.5)  10 -10 (Extra Data Fig.7). 9 The contribution of other systematic error terms is evaluated less than 1.0  10 -10 .

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where  1 and  2 ( 1 = − 2 ) are two-photon detunings of  1 and  2 , respectively. The phase 1 shift of dual-species AI is where ∆k eff is the difference of effective wave vectors of the dual-species atoms,  is the 4 duration of the /2 Raman pulses. As shown in Fig.3 (a)  can be decreased to a large extent. In addition, the interference loop for one species atom 11 keeps at the same internal state, thus the influence of different internal states is also 12 reduced.

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The 4WDR-e scheme is explained by taking 85 Rb atoms as an example.  Therefore, a blow away pulse is added to clear atoms in |F=3 state.

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For UGS AI, the -blow away pulses are added for states preparation and 20 velocity-selection. After the atoms interact with the first /2 pulse, they are transited from 21 |F=2 to |F=3. A repumping pulse is used to pump the atoms from |F=2 to |F=3.

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Although this process cannot completely clean up the atoms in |F=2 state, it can make 23 them only as background without participating in the interference process. Due to the 24 presence of a repumping pulse, the atom number in the background increases sharply. In 1 order to decrease the background, we use a blow away-repumping pulse sequence after the 2 last Raman /2 pulse, and only detect the atoms in |F=2 sate which participate in the 3 interference loop. Note that, when the detected state is inconsistent with the interference 4 state, a  phase correction is required.

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For dual-species AI with different atom pairs, atoms are initially prepared to states

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(b) 87 Rb|F=2-85 Rb|F=2 dual-species AI (Extended Data Fig. 1(b) are the same as those described in Extended Data Fig. 1(a) Fig. 1(c)). A -blow 9 away--Repumping pulses sequence is applied to 87 Rb atoms and a -blow 10 away- c -repumping--blow away pulses sequence is applied to 85 Rb atoms for states 11 preparation and velocity-selection. The purpose of laser pulses are the same as those 12 described in Extended Data Fig. 1(a) Fig. 1(d)). A -blow away 21 pulses sequence is applied to 87 Rb atoms and 85 Rb atoms for states preparation and  Table 1, where T=203.164 ms, =31 s, and the uncertainty of 17 wave vector correction is 5  10 -11 . The difference of f rec value is caused by the frequency 18 change of ꞷ 3 after applying co-propagating state selection pulses.

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Extended Data (2) 87 Rb|F＝2-85 Rb|F＝2 1 (3) 87 Rb|F＝1-85 Rb|F＝3 2 (4) 87 Rb|F＝2-85 Rb|F＝3 3 4 (b) ac Stark shift. The ac Stark shift is caused by Raman lasers, blow away and 5 repumping lasers. In our previous experiment 7 , the MIR of the Raman lasers was set to I 1 :  Fig. 4). According to Extended Data Fig. 1(b), (c), and (d), the blow away The Coriolis effect is expressed as where  E is the Earth rotation,  0 is the velocity difference between two species atom 8 clouds. Since the detectors are fixed, the phase difference mainly depends on the overlap 9 degree of the atom clouds. In our experiments, the Coriolis effect is reduced mainly by 10 two ways, one is to overlap the atom clouds and to reduce atom temperature; the other is 11 to compensate the rotation of the Raman lasers mirror 33,34 . We design and implement 26 a 12 two-dimensional rotation compensation system for Raman lasers mirror. We perform The phase shift caused by the velocity difference between the two atoms clouds is    Rb|F=2-85 Rb|F=3 dual-species AI.  Raman beams with frequencies of  1 ,  2 ,  3 , and  4 . The red dots are data before 6 improvement when changing the Raman laser intensity from 10% to 100%, the 7 maximum residual frequency shift is 6 kHz. The blue squares are data after 8 improvement, and the frequency shift is less than 500 Hz. The error bars are obtained 9 using Gaussian fitting.      Schematic diagram of 4WDR-e 87Rb-85Rb dual-species AI. (a) Relevant sub-levels covering energy interval from micro-eV to giga-eV. Raman lasers with frequencies of ω1, ω2, and ω3 are used for 85Rb atoms, while that with frequencies ω1, ω2, and ω4 are for 87Rb atoms; ω1 is the detuning of ω 1, δ2 is the detuning of ω2. ω1 and ω2 are detuned to the blue side of transitions 85Rb|F = 3> to |F′= 4>with a detuning of Δ1 and 87Rb|F =2> to |F′= 3> with a detuning of Δ2. (b) The 4WDR-e con guration for 87Rb-85Rb dual-species AI. The blue dash lines represent LGS atoms, and red solid lines represent UGS atoms.  Dependence of η values on energy, the intercept value of the tted straight line η0 = (-0.8 ± 1.4) × 10-10, and the slope value β = (-0.6 ± 6.9) × 105.