Investigating the environmental dependence of ultralight scalar dark matter with atom interferometers
IInvestigating the environmental dependence of ultralight scalar dark matter withatom interferometers
Wei Zhao , Dongfeng Gao , ∗ Jin Wang , † and Mingsheng Zhan ‡ State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, APM,Chinese Academy of Sciences, Wuhan 430071, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China (Dated: February 5, 2021)We study the environmental dependence of ultralight scalar dark matter (DM) with linear inter-actions to the standard model particles. The solution to the DM field turns out to be a sum ofthe cosmic harmonic oscillation term and the local exponential fluctuation term. The amplitude ofthe first term depends on the local DM density and the mass of the DM field. The second term isinduced by the local distribution of matter, such as the Earth. Then, we compute the phase shiftinduced by the DM field in atom interferometers (AIs), through solving the trajectories of atoms.Especially, the AI signal for the violation of weak equivalence principle (WEP) caused by the DMfield is calculated. Depending on the values of the DM coupling parameters, contributions to theWEP violation from the first and second terms of the DM field can be either comparable or onelarger than the other. Finally, we give some constraints to DM coupling parameters using resultsfrom the terrestrial atomic WEP tests.
I. INTRODUCTION
A variety of astrophysical and cosmological observa-tions indicate the existance of DM and dark energy, al-though we have not directly discovered them [1, 2]. Itis commonly believed that about 5% of the cosmologi-cal energy density is contributed by the DM [3]. So farthe nature of DM is unknown except its gravitational ef-fects at the galactic scale and larger [4–6]. There hasbeen considerable efforts to search for a kind of particle-like DM candidate—weakly interacting massive particle(WIMP). Unfortunately, no evidences of WIMP darkmatter have been found [7–9]. In contrast, several ex-perimental strategies are proposed recently to search forlight, field-like DM using precision tools of atomic, molec-ular and optical physics, such as atomic clock [10, 11],atomic spectroscopy [12], accelerometers [13], and opti-cal cavities [14].In recent years, rapid technological progress in atom in-terferometry has been made. Atom interferometers (AIs)are realized by coherently manipulating atomic matterwaves [15]. The whole process mainly consists of prepar-ing an atomic wave packet in the initial state, coherentlysplitting the wave packet into two by applying the laserpulse, flipping the atomic states of the two wave packetsafter some drift time T , recombining these wave pack-ets after another drift time T , and finally measuring thephase shift of the detected fringes. AIs have already beenused in various precision measurements. For example,the value of the fine structure constant was determinedto be α − = 137.035999206(11) in the Rb-atom recoilexperiment [16], which is the most accurate measurement ∗ [email protected] † [email protected] ‡ [email protected] of α so far. AI has also been used to test WEP at quam-tum level. Recent results of quantum WEP test withAIs were reported by Zhou et al. [17] and Asenbaum etal. [18] with accuracies of 10 − -level and 10 − -level,respectively.Encouraged by the achievements AIs have made, peo-ple put forward several proposals of detecting ultralightDM with AIs [13, 19, 20]. The idea behind these pro-posals is the following. According to the popular scalarDM models [21, 22], the scalar DM may interact withstandard-model matters and change the fundamental pa-rameters, such as the mass of fermions, the electromag-netic fine structure constant and the QCD energy scale.This will lead to variations in atomic masses and atomicinternal energy levels, and finally end up with a change inthe mass of the Earth, resulting in a variation of the grav-itational acceleration. All these effects can be searchedby a net phase shift in AI experiments. But, in all theseproposals, only the cosmic harmonic oscillation part ofthe DM field has been considered.In this paper, we also work on the popular scalar DMmodels [21, 22]. After a thorough computation, the so-lution to the DM field is obtained. The new thing isthat the DM field is found to be a sum of the cosmicharmonic oscillation term and the local exponential fluc-tuation term. The second term comes from the localdistribution of mass, which has been ignored before. Wefurther calculate the signal for the WEP violation causedby the DM field in AIs. The calculation shows that con-tributions from the two terms of the DM field can beeither comparable or one larger than the other, depend-ing on the values of the DM coupling parameters.The paper is organized as follows. In Sec.II, The scalarDM model is briefly introduced. In Sec.III, we discuss theenvironmental dependence of the scalar DM, and the so-lution to the DM field near a local distribution of matter(such as the Earth) is obtained. In Sec.IV, we compute a r X i v : . [ phy s i c s . a t o m - ph ] F e b the phase shift in AI experiments under the influence ofthe scalar DM. In Sec.V, we discuss how to constrain theDM coupling parameters, using the newest atomic WEPtests. Finally, discussion and conclussion are made inSec.VI. II. THE SCALAR DARK MATTER MODEL
In this section, we will briefly review the scalar DMmodel, introduced in Refs. [21, 22]. The microscopicaction of the model is the following, S = (cid:90) d x √− g κ (cid:34) R − g µν ∂ µ ϕ∂ ν ϕ − V ( ϕ ) (cid:35) + (cid:90) d x √− g (cid:34) L SM ( g µν , ψ i ) + L int ( g µν , ϕ, ψ i ) (cid:35) , (1)where d x ≡ dt d x , and κ = πGc . R is the Ricci scalar ofthe spacetime metric g µν , and ϕ denotes the dimension-less scalar DM field. The first line in Eq. (1) describesthe action for general relativity and the DM field, with V ( ϕ ) being the the potential term of ϕ . Here we onlyconsider the quadratic mass term in the potential, V ( ϕ ) = 2 c m ϕ (cid:125) ϕ , (2)where m ϕ is the mass of the DM field. L SM is the La-grangian density of the standard-model fields ψ i , and L int is the interactional Lagrangian density between theDM field and standard-model fields.To be specific, we focus on the linear coupling model, L int = ϕ (cid:34) d e e F µν F µν − d g β g F Aµν F Aµν − (cid:88) i = e,u,d ( d m i + γ m i d g ) m i ψ i ¯ ψ i (cid:35) , (3)where d e and d g are the couplings to the U (1) electromag-netic and SU (3) gluonic field terms, respectively. d m e , d m u and d m d are the couplings to the masses of electronand quarks. g is the QCD gauge coupling, and β isthe β -function for g . m i denotes the fermionic masses(eletron and quarks), γ m i is the anomalous dimensiondue to the renormalization-group running of the quarkmasses, and ψ i are the fermion spinors.It is easy to find that, in the linear coupling model, theLagrangian leads to the ϕ -dependence for the followingfive physical quantities, α ( ϕ ) = (1 + d e ϕ ) α Λ ( ϕ ) = (1 + d g ϕ )Λ m i ( ϕ ) = (1 + d m i ϕ ) m i , i = e, u, d (4)where α is the electromagnetic fine structure constant,and Λ is the QCD energy scale. Then, the physical meaning of the five coupling parameters ( d e , d g , d m e , d m u and d m d ) is very clear. They just introduce a linear ϕ -dependence to the corresponding physical quantities.For later discussion, it is convenient to rewrite themasses of up and down quarks into the form of symmetricand antisymmetric combinations,ˆ m = m u + m d , δm = m d − m u . (5)Their corresponding ϕ -dependence isˆ m ( ϕ ) = (1 + d ˆ m ϕ ) ˆ m, δm ( ϕ ) = (1 + d δm ϕ ) δm , (6)with d ˆ m = m u d m u + m d d m d m u + m d , d δm = m d d m d − m u d m u m d − m u . (7)From the action (1), it is straight to derive the fieldequations for the spacetime metric g µν and ϕ , which are R µν = κ [ T µν − g µν T ] + 2 ∂ µ ϕ∂ ν ϕ + 12 g µν V ( ϕ ) (8)and − c ¨ ϕ − (cid:52) ϕ = − κ ∂ L int ∂ϕ + V (cid:48) ( ϕ )4 . (9)The stress-energy tensor T µν is defined by T µν = − √− g δ √− g L mat δg µν , (10)where L mat denotes the Lagrangian for the mattersource.To solve the above field equations, one needs to writedown the phenomenological Lagrangian L mat for matter,in the spirit of the microscopic action (1). Since ordi-nary matter is made of atoms, which can be further de-composed into fundamental particles (photons, electrons,gluons and quarks), the problem is then reduced to writedown a phenomenological Lagrangian for atoms. In Ref.[23], such a phenomenological treatment of matter wasdeveloped, where the atom was modeled as a massivepoint particle. The phenomenological action for matterwas written as S mat [ g µν , ϕ ] = − c (cid:88) atom (cid:90) atom m A ( ϕ ) dτ , (11)where τ is the proper time along the atom’s worldline,and m A is the atomic mass. Since each atom has its owndecomposition, m A ( ϕ ) has different dependence on ϕ .In the paper [21], derived from the microscopic action(1), a dimensionless phenomenological factor α A is intro-duced to measure the coupling of DM field to the atom, α A ≡ ∂ ln m A ( ϕ ) ∂ϕ . (12)The expression for α A has been derived, α A = d g + [( d ˆ m − d g ) Q ˆ m + ( d δm − d g ) Q δm + ( d m e − d g ) Q m e + d e Q e ] , (13)where the dilaton charges are given by Q ˆ m = F A (cid:34) . − . A / − .
02 ( A − Z ) A − . × − Z ( Z − A / (cid:35) (14a) Q δm = F A (cid:34) . A − ZA (cid:35) (14b) Q m e = F A (cid:34) . × − ZA (cid:35) , (14c)and Q e = F A (cid:34) − . . ZA + 7 . Z ( Z − A / (cid:35) × − . (14d) Z is the atomic number, and A is the mass number ofatoms. The factor F A can be replaced by one in lowestapproximation. III. THE SOLUTION OF THE DM FIELDNEAR THE EARTH
To show the environmental dependence of scalar DMfield, we need to solve the field equation for ϕ neara distribution of ordinary matter, such as the Earth.For simplicity, we will regard the Earth as a sphericallysymmetric ball with radius R E , density ρ E , and mass M E = 4 πR E ρ E /
3. According to Eq. (11), the phe-nomenological action for the Earth is S E = (cid:90) L E √− g d x = − c (cid:90) ρ E ( ϕ ) √− g d x . (15)Let us dwell on the ϕ -dependence of M E , which comesfrom the ϕ -dependence of atoms. Since the five couplingparameters ( d e , d g , d m e , d ˆ m and d δm ) and ϕ are assumedto be very small, we could do Taylor expansion in ϕ forthe mass M E , M E ( ϕ ) = M E (cid:34) α E ϕ + ˜ α E ϕ + O ( ϕ ) (cid:35) , (16)with M E ≡ M E ( ϕ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ =0 , α E ≡ ∂ ln M E ( ϕ ) ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ =0 . (17)Note that, unlike other papers (such as [24]), we truncatethe Taylor expansion of M E at the second order in ϕ . Thephysical meaning of the ϕ -term will be clear soon. The calculation of α E is as follows. The Earth is madeof various elements, 49 .
83% Oxygen, 15 .
19% Iron, 15 . .
23% Silicon, 2 .
14% Sulfur, 1 .
38% Alu-minum and 1% Calcium [25]. We first calculate the α A for each element, using Eqs. (13, 14a-14d). Then, the α E is given by taking the atomic average over the Earth’sisotopic composition, α E = d g + [0 . d ˆ m − d g ) + 2 . × − ( d δm − d g )+ 2 . × − ( d m e − d g ) + 1 . × − d e ]=0 . d g + 0 . d ˆ m + 2 . × − d δm . × − d m e + 1 . × − d e . (18)Note that the α E here is slightly different from theone used in Refs. [21, 22, 24], where α E (cid:119) d ∗ g = d g + 0 . d ˆ m − d g ) + 0 . d e + 0 . d m e − d g ).The reason is that the authors made some simplificationand omitted the d δm -dependence in the previous papers.In this paper, we keep all the d i ’s in α E , which will beimportant later.According to the microscopic action (1), there areonly linear interactions between the DM field and thestandard-model fields. Thus, the ϕ -term in M E shouldcome from the one-loop Feynman-diagram calculationwith action (1). The exact calculation is very tricky andlengthy, which is beyond the scope of this paper. Butwe can make some simplification. Since the Earth’s massis mainly contributed by nucleons (proton or neutron),and the nucleon’s mass mainly comes from the gluonicinteractions, then it is natural to suppose that ˜ α E is ap-proximated by ˜ α E (cid:119) d g . (19)Based on the above discussion, a phenomenological ac-tion, describing ϕ near the Earth, is given by S = (cid:90) d x √− g κ (cid:34) R − g µν ∂ µ ϕ∂ ν ϕ − V ( ϕ ) (cid:35) + S E . (20)It is straightforward to write down the field equation for ϕ , − c ¨ ϕ + (cid:79) ϕ = κ ρ E c α E + V (cid:48) eff , (21)with V eff ( ϕ ) = V ( ϕ ) + κρ E c ˜ α E ϕ = 2 c (cid:125) (cid:32) m ϕ + 12 κρ E (cid:125) ˜ α E (cid:33) ϕ . (22)Note that a Minkowski spacetime metric g µν =diag( − , , ,
1) is assumed. It is clear that the poten-tial for ϕ is changed from V ( ϕ ) to the effective potential V eff ( ϕ ), due to the appearance of the Earth.To solve the field equation (21), let us first consider thecase without the Earth. Obviously, Eq. (21) becomes − c ¨ ϕ + (cid:79) ϕ = V (cid:48) ( ϕ )4 . (23)It is easy to find out the solution, ϕ bg ( t, x ) = ϕ cos( k · x − ωt + δ ) , (24)where ϕ is the amplitude, ω = | k | c + m ϕ c / (cid:125) , and δ is the initial phase. The solution is a plane wave, andwe call it the background of ϕ .As in Ref. [10], the harmonic oscillation background ϕ bg will be identified as the DM. The wave vector isthen given by k = m ϕ v vir / (cid:125) , where v vir (cid:119) − c is theEarth’s velocity with respect to the DM. ϕ bg will con-tribute an energy density, π m ϕ ϕ M P l , where M P l =1 . × GeV is the Planck mass. Using the DM en-ergy density ρ DM = 0 . / cm at the solar system,the amplitude is calculated to be ϕ = 7 . × − eV m ϕ . (25)It is easy to see that ϕ (cid:28) − eV (cid:46) m ϕ (cid:46) ϕ bg . Let us decompose ϕ into ϕ = ϕ bg + δϕ , where δϕ is the fluctuation around ϕ bg . Insert it into Eq.(21), we have (cid:79) δϕ − c (cid:125) m eff δϕ = κ ρ E c α E , (26)with m = m ϕ + 4 πG (cid:126) c ρ E ˜ α E . (27)Now, it is clear that keeping the ϕ -term in M E resultsin a change in the mass of δϕ to the effective mass m eff ,which depends on the density ρ E and ˜ α E . Inserting thenumber ρ E = 5 . × kg / m , one gets m = m ϕ + (1 . × − eV) ˜ α E . (28)Inside the Earth ( r (cid:54) R E ), the solution of Eq. (26) is δϕ = − α E I ( rλ eff ) GM E c e − rλ eff r − α E GM E λ c R E r (cid:34) ( r + λ eff ) e − rλ eff − ( R E + λ eff ) e − REλ eff (cid:35) sinh( rλ eff ) , (29)where I ( x ) = 3 x cosh( x ) − sinh( x ) x , sinh( x ) = e x − e − x andcosh( x ) = e x + e − x . The effective wavelength of the DMfield is defined to be λ eff = (cid:126) m eff c . Outside the Earth ( r (cid:62) R E ), the solution of Eq. (26)is δϕ = − α E I ( R E λ eff ) GM E c e − rλ eff r . (30)So, in the neighborhood of the Earth where the terres-trial AI experiment is performed, the full solution to ϕ is ϕ = ϕ cos( k r − ωt + δ ) − α E I ( R E λ eff ) GM E c e − rλ eff r . (31)Similar result was also obtained in the paper [24]. Thedifference is that we have λ eff in the exponential term,instead of λ ϕ = (cid:126) m ϕ c . IV. THE DM SIGNAL IN ATOMINTERFEROMETERS
The theory of AIs can be found in many papers, suchas Ref. [26, 27]. A typical π - π - π Raman atom inter-ferometer is shown in Fig. 1. The stimulated Ramantransitions are realized by two counter-propagating laserbeams. One is called the control laser beam, with fre-quency ω and wave vector k . The other one is calledthe passive laser beam, with frequency ω and wave vec-tor k . The cold atom beam, prepared in the | g (cid:105) state,is loaded into the AI with the launch velocity v L . Attime t=0, the first Raman π /2-pulse is applied, and co-herently splits the atomic wave packet into a superposi-tion of states | g (cid:105) and | e (cid:105) , with a momentum difference of k eff = k - k . After a drift time T, Raman π pulses are ap-plied, which transit the state | g (cid:105) to | e (cid:105) , and the state | e (cid:105) to | g (cid:105) , respectively. After another drift time T, the twowave packets overlap, and the final Raman π /2-pulses areapplied to make the two wave packets interfere. Then,the phase shift can be measured by detecting the numberof atoms in either | g (cid:105) or | e (cid:105) states.According to the discussion in previous sections, be-cause of its interaction with standard-model fields, thescalar DM will change the fundamental parameters, suchas the mass of fermions, the electromagnetic fine struc-ture constant and the QCD energy scale. Subsequently,this will induce variations in the atomic masses, atomicinternal energy levels, and the mass of the Earth, re-sulting in a variation of the gravitational acceleration.All these effects will cause a net phase shift in AI ex-periments, which signals the existance of the scalar DM.But, in previous proposals [19, 20], only the harmonicoscillation term of the DM field (31) has been consid-ered. The paper [19] considered the DM effects on theatomic masses and the mass of the Earth, while the au-thors in Ref. [20] focused on the change in the atomicmasses and atomic internal energy levels. In the follow-ing, we will give a complete computation for the phaseshift due to varations in atomic masses, atomic internal FIG. 1. Schematic diagram for a typical π - π - π Raman atominterferometer. The left part shows the stimulated Ramantransition between two atomic hyperfine ground states | g (cid:105) and | e (cid:105) . The atomic population is resonantly transferred between | g (cid:105) and | e (cid:105) if the frequency difference ω - ω is close to ω hfs .The right part shows the sequence of laser pulses and thepaths of atoms. energy levels, and the mass of the Earth, based on thefull DM solution of ϕ .To calculate the DM-induced phase shift, we need todetermine the atom’s trajectories of the upper and lowerarms, compute the phases along each arm, and take thephase difference between the two arms. The trajectory isdetermined by the atomic equation of motion, which canbe derived from the non-relativistic approximation of theLagrangian (11), L = − m A c + 12 m A ˙ z − m A gz , (32)where z ≡ r - R E and g is the Earth’s gravitational accel-eration. The DM-dependence of the Lagrangian (32) isencoded in m A and g . To be explicit, let us write down m A and g , m A ( ϕ ) = m (1 + α A ϕ )= m (cid:34) α A ϕ cos( kr − ωt + δ ) − α A α E I ( R E λ eff ) GM E c e − rλ eff r (cid:35) (33)and g ( ϕ ) = GM E ( ϕ ) /R E = g (cid:34) α E ϕ cos( kr − ωt + δ ) − α E I ( R E λ eff ) GM E c e − rλ eff r + O ( ϕ ) + O ( d i ) (cid:35) , (34)where m and g denote the atomic mass and the gravi-tational acceleration in the absence of DM, respectively.Higher order terms in ϕ and d i are neglected. As pointed out in Ref. [20], we also need to considerthe DM effect on atomic internal energy levels (i.e. | c (cid:105) , | g (cid:105) and | e (cid:105) in Fig. 1), which comes from the change inthe electronic mass and the electromagnetic fine struc-ture constant (4). The change in atomic internal energylevels accordingly affects the stimulated atomic Ramantransitions. In the end, the effective photon momentumtransfer k eff in stimulated atomic Raman transitions ischanged to k eff ( ϕ ) = k eff (cid:34) d m e + ξd e ) ϕ (cid:35) = k eff (cid:34) d m e + ξd e ) ϕ cos( kr − ωt + δ ) − ( d m e + ξd e ) α E I ( R E λ eff ) GM E c e − rλ eff r (cid:35) , (35)where k eff denotes the unperturbed value, and ξ (=2.34for the Rb atom) is the relativistic correction factor 2 + K rel given in Ref. [28]. Then, through the laser pulse’sinteraction with atoms, this effect is finally transferredto the atomic recoil velocity v R ( ϕ ) = (cid:126) k eff ( ϕ ) m A ( ϕ ) (cid:119) v R (cid:34) − ( α A − ˜ d ) ϕ cos( kr − ωt + δ )+ ( α A − ˜ d ) α E I ( R E λ eff ) GM E c e − rλ eff r (cid:35) , (36)where v R = (cid:126) k eff /m denotes the unperturbed value, and˜ d ≡ d m e + ξd e .Now, it is straightforward to write down the atomicequation of motion from the Lagrangian (32) m A ¨ z = − ∂m A ∂z c + 12 ∂m A ∂z ˙ z − ˙ m A ˙ z − ∂m A ∂z gz − m A ∂g∂z z − m A g . (37)Solving Eq. (37) is very lengthy, and the full result willbe given in Eqs.(A4, A5) in Appendix A.The total phase shift can be written as a sum of threecomponents [27], the propagation phase shift, the laserphase shift, and the separation phase shift,∆ φ = ∆ φ prop + ∆ φ laser + ∆ φ sep . (38)The calculation of the total phase shift is quite long, andwill be given in Eqs. (B6-B8) in Appendix B.To show that our result is a complete result, let usdiscuss two cases. First, consider effects induced by thelocal exponential fluctuation term δϕ , which turns out tobe the ∆ φ δϕ term (B7) of the total phase shift ∆ φ . Inthe λ eff → ∞ limit, the ∆ φ δϕ is reduced to˜∆ φ δϕ = − g T k eff (cid:20) (cid:18) v L ( v R + v L )2 c (cid:19) α A α E (cid:21) − g T k eff (cid:20) − g T c + g (2 v L + v R ) T − g R E c (cid:21) α E . (39)Then, it is easy to find that the E¨otv¨os parameter couldbe written as˜ η δϕ (cid:39) ( α a − α b ) α E − v L ( v R + v L )2 c ( α a − α b ) α E , (40)for atomic species a and b . The first line in Eq. (40)exactly reproduces the formula used in Ref. [29]. Thesecond line in Eq. (40) gives small corrections. For coldatoms, the velocity v L is about several m / s. Thus, thecorrections are about 10 − times smaller than the firstline. For hot atoms, the corrections can be much bigger.The second case is to focus on effects caused by thethe cosmic harmonic oscillation term ϕ bg , which finallycontributes the ∆ φ bg term (B8) in the total phase shift.If we ignore terms originated from the mc term inthe Lagrangian (32) and omit terms involving v L,R /c or v L,R
T /R E , the ∆ φ bg is reduced to˜∆ φ bg = − α A g k eff Tω ϕ (sin ωT − sin 2 ωT )+ ( α E + 2 α A ) g k eff ω ϕ (1 − ωT + cos 2 ωT )+ α A ( k eff ( v L + v R / ω ) ϕ (2 sin ωT − sin 2 ωT )(41)One can see that we reproduce the result used in Ref.[19]. V. CONSTRAINTS ON THE COUPLINGPARAMETERS
In this section, we discuss how to constrain the fivecoupling parameters ( d e , d g , d m e , d ˆ m and d δm ) by re-cent results of quantum WEP test with Rb- Rb duel-species AIs. η =( − . ± . × − was reported in Ref.[17] and η =(1 . ± . × − was obtained in Ref. [18].For comparison, η =( − ± . × − , measured by theMICROSCOPE space mission [30] using macroscopic Ti-Pt objects, is also discussed.Since v L,R (cid:28) c and v L,R T (cid:28) R E , we can omit allthe terms involving v L,R /c or v L,R
T /R E in Eq. (B6-B8).Then, the full result for ∆ φ can be simplified into∆ φ (cid:119) − g T k eff − k eff c kα A ϕ ω (cid:34) sin( kR E − ωT + δ ) − kR E − ωT + δ ) + sin( kR E + δ ) (cid:35) + ( α E + 2 α A ) g k eff ω ϕ (cid:34) cos( kR E + δ ) − kR E − ωT + δ ) + cos( kR E − ωT + δ ) (cid:35) − T k eff (cid:34) g T − (2 v L + v R ) Tλ eff + (1 + R E λ eff ) (cid:35) I ( R E λ eff ) GM E R E α A α E e − REλ eff . (42)For Rb and Rb atoms, the DM-induced acceler-ation of gravity is given by − ∆ φT k eff . Accordingly, theE¨otv¨os parameter is defined to be η ≡ g − g g + g . (43)We find that η can be written as a sum of a static com-ponent η δϕ and an oscillatory component η bg . η = η δϕ + η bg . (44) The δϕ -contribution to η is given by η δϕ = (1 + R E λ eff ) I ( R E λ eff )( α − α ) α E e − REλ eff , (45)where α = 9 . × − d g + 8 . × − d ˆ m + 2 . × − d δm + 2 . × − d m e + 2 . × − d e . (46) α = 9 . × − d g + 8 . × − d ˆ m + 2 . × − d δm + 2 . × − d m e + 2 . × − d e . (47)The ϕ bg -contribution to η is given by η bg = (cid:34) c kg ω T (cid:32) sin( kR E − ωT + δ ) − kR E − ωT + δ ) + sin( kR E + δ ) (cid:33) − ω T (cid:32) cos( kR E + δ ) − kR E − ωT + δ ) + cos( kR E − ωT + δ ) (cid:33)(cid:35) ( α − α ) ϕ . (48)Note that η bg oscillates in time because δ (the phaseof the DM field ϕ bg at the time of applying the first π -pulse) is oscillatory. The other notable thing is that η bg is linearly proportional to the d i parameters, while η δϕ isquadratic in them. According to Eq. (18) and (25), onecan see that η δϕ and η bg can be either comparable or onelarger than the other depending on the values of d i ’s.We first constrain only one parameter each time withthe other four parameters set to zero. This method iswidely used in many papers [13, 24, 29]. The result isshown in Fig. 2. It is clear that the MICROSCOPE’sresult gives better constraints on all five parameters thanAI experiments since it keeps the best precision on WEPtest. From Figs. 2(a)-2(e), we can see that constraints on d g and d ˆ m are better than constraints on the other threeparameters. As explained in Refs. [21, 22], the reasonis because the gluonic interaction (i.e. the strong inter-action) and quark masses make the most contribution tothe atomic masses, which can be seen from the coeffi-cients of d i ’s in Eqs. (13) and (18). At current precisionlevel on WEP test, the oscillatory component η bg makesneglectable contributions to constraints on the five DMparameters.According to Eqs. (13) and (18), d g dominates thecontribution to α A and α E , if one assumes that d e , d g , d m e , d ˆ m and d δm are of the same order. To investigate thecorrelation between d g and the other four parameters, weassume d g always nonzero and set one of the other fourparameters nonzero each time. Then, we can draw theconstraints on the four pairs ( d g - d ˆ m , d g - d m e , d g - d e and d g - d δm ) in Fig. 3, where the result from Ref. [18] is used.From Figs. (3(a))-(3(d)), due to the loose constraints weget, we could not see whether there exist correlationsbetween d g and the other four parameters, or whether d g is of the same order as the other four parameters. VI. CONCLUSSION AND DISCUSSION
In this paper, we first generalize the linear couplingscalar DM model to the appearance of a central massivebody, such as the Earth. We find that the DM field obtains a local exponential fluctuation term besides thecosmic harmonic oscillation term. Our method can beapplied to more general scalar DM models. The case ofthe quadratic coupling scalar DM model has been studiedin Ref. [24].According to Eq. (27), m eff is proportional to the den-sity of the central massive body. For more dense bodiesthan the Earth, the difference between m eff and m ϕ be-comes more remarkable. On the other hand, accordingto Eqs. (29) and (30), the δϕ is proportional to the totalmass of the central body. Thus, for more massive bodiesthan the Earth, the local exponential fluctuation termbecomes more important.We then use our solution for the DM field to calculatethe DM-induced phase shift in atom interferometers. Theresulting phase shift is a sum of a static term and an os-cillatory term. Accordingly, for the WEP test with AIs,the E¨otv¨os parameter η is also a sum of a static com-ponent and a time-varying component. The two compo-nents can be either comparable or one larger than theother depending on the values of d i ’s. For current WEPtest experiments, the oscillatory component η bg makesneglectable contributions in constraining the five DM pa-rameters. For future improved precision of WEP tests,the oscillatory component will become more and moreimportant. ACKNOWLEDGMENTS
This work was supported by the National Key Re-search and Development Program of China under GrantNo. 2016YFA0302002, and the Strategic Priority Re-search Program of the Chinese Academy of Sciences un-der Grant No. XDB21010100.
Appendix A: Calculation of velocity and position ofatoms
Solving Eq. (37) is as follows. We first integrate onboth sides to get the velocity,˙ z ( t (cid:48) ) = ˙ z ( t i ) − g ( t (cid:48) − t i ) − α E I ( R E λ eff ) GM E c (cid:90) t (cid:48) t i (cid:34) α A (cid:32) c + ( ˙ z (0) ( t )) g z (0) ( t ) (cid:33)(cid:32) R E + z (0) ( t )) + 1( R E + z (0) ( t )) λ eff (cid:33) (a) Constraint on d g (b) Constraint on d ˆ m (c) Constraint on d e (d) Constraint on d m e (e) Constraint on d δm FIG. 2. Constraints on the five DM parameters, d e , d g , d m e , d ˆ m and d δm . The blue solid line (and the corresponding yellowshaded area) is the constraint set by the MICROSCOPE’s result [30]. The red dot-dashed line is the constaraint set byAsenbaum’s result[18] and the black dotted line is the constaint set by Zhou’s result[17]. In addition, the red solid line is theconstraint set by Asenbaum’s result, considering only the oscillatory component η bg . + g α E (cid:32) z (0) ( t )( R E + z (0) ( t )) + 1 R E + z (0) ( t ) (1 + z (0) ( t ) λ eff ) (cid:33) e − RE + z (0)( t ) λ eff (cid:35) dt + ϕ (cid:90) t (cid:48) t i (cid:34) k (cid:32) ( c − ( ˙ z (0) ( t )) α A + g z (0) ( t )( α A + α E ) (cid:33) sin (cid:32) k ( R E + z (0) ( t )) − ωt + δ (cid:33) − g ( α A + α E ) cos (cid:32) k ( R E + z (0) ( t )) − ωt + δ (cid:33)(cid:35) dt − α A ϕ (cid:32) cos (cid:16) k ( R E + z (0) ( t (cid:48) )) − ωt (cid:48) + δ (cid:17) ˙ z (0) ( t (cid:48) ) − cos( k ( R E + z (0) i ) − ωt i + δ ) ˙ z (0) ( t i ) (cid:33) . (A1) (a) Constraint on d g - d ˆ m pair (b) Constraint on d g - d e pair(c) Constraint on d g - d δm pair (d) Constraint on d g - d m e pair FIG. 3. Constraints on the four pairs ( d g - d ˆ m , d g - d m e , d g - d e and d g - d δm ), where the shaded area is the allowed region. We havetaken a generic value 10 − eV for m ϕ . Here, z (0) ( t ) denotes the unperturbed atomic trajectory,which is nothing but the freefall trajectory z (0) ( t ) = z (0) i + v (0) i ( t − t i ) − g ( t − t i ) (A2)where t i is the initial time, z (0) i and v (0) i are respectivelythe initial position and velocity for each segment of thefreefall trajectory.To finish the integration in Eq. (A1), we do the fol-lowing approximation, e − RE + z (0)( t ) λ eff (cid:39) (1 − z (0) ( t ) R E ) e − REλ eff R E + z (0) ( t ) (cid:39) R E (1 − z (0) ( t ) R E )1( R E + z (0) ( t )) (cid:39) R E (1 − z (0) ( t ) R E ) (A3)Then we can get the result˙ z ( t ) = 110 ( t − t i ) (cid:34)
514 ( t − t i ) α E (cid:18) λ eff + R E (cid:19) g + ( t − t i ) (cid:18) α E v (0) i t i − α E tv (0) i + λ eff α E + α E R E − (cid:16) α E + α A (cid:17) z (0) i (cid:19) (cid:18) λ eff + R E (cid:19) g − t − t i ) (cid:32) − (cid:18)(cid:16) α E − α A (cid:17) ( v (0) i ) − c α A (cid:19) (cid:18) λ eff + R E (cid:19) t i − (cid:18)(cid:18)(cid:18) − α E α A (cid:19) ( v (0) i ) + 25 c α A (cid:19) t + v (0) i (cid:16) α E R E + λ eff α E − α A z (0) i − α E z (0) i (cid:17)(cid:19) (cid:18) λ eff + R E (cid:19) t i − (cid:18) (cid:16) α E − α A (cid:17) ( v (0) i ) − c α A (cid:19) (cid:18) λ eff + R E (cid:19) t + v (0) i (cid:16) α E R E + λ eff α E − α A z (0) i − α E z (0) i (cid:17) (cid:18) λ eff + R E (cid:19) t (cid:18) − α E R E + 43 α E z (0) i + 23 α A z (0) i (cid:19) λ eff2 + (cid:18) − α E R E +2 (cid:16) α E + α A (cid:17) z (0) i R E − α E ( z (0) i ) − α A ( z (0) i ) (cid:19) λ eff +23 z (0) i R E (cid:18)(cid:16) α E + α A (cid:17) R E − α E z (0) i − α A z (0) i (cid:19) (cid:33) g + (cid:32) v (0) i (cid:16)(cid:16) α E − α A (cid:17) ( v (0) i ) − c α A (cid:17) (cid:18) λ eff + R E (cid:19) t i + (cid:18) − v (0) i (cid:16)(cid:16) α E − α A (cid:17) ( v (0) i ) − c α A (cid:17) (cid:18) λ eff + R E (cid:19) t + (cid:18)(cid:18) − α A + 203 α E (cid:19) ( v (0) i ) − c α A (cid:19) λ eff2 + (cid:18)(cid:18)(cid:18) α E − α A (cid:19) ( v (0) i ) − c α A (cid:19) R E − z (0) i (cid:18)(cid:16) α E + α A (cid:17) ( v (0) i ) − c α A (cid:19)(cid:19) λ eff +103 (cid:18)(cid:18)(cid:16) α E − α A (cid:17) ( v (0) i ) − c α A (cid:19) R E − z (0) i (cid:18)(cid:16) α E + α A (cid:17) ( v (0) i ) − c α A (cid:19)(cid:19) R E (cid:19) t i + (cid:18) v (0) i (cid:16)(cid:16) α E − α A (cid:17) ( v (0) i ) − c α A (cid:17) (cid:18) λ eff + R E (cid:19) t + (cid:18) (cid:18)(cid:18) − α E + 103 α A (cid:19) ( v (0) i ) + 203 c α A (cid:19) λ eff2 + (cid:18)(cid:18)(cid:18) − α E + 103 α A (cid:19) ( v (0) i ) + 203 c α A (cid:19) R E + 40 z (0) i (cid:18)(cid:16) α E + α A (cid:17) ( v (0) i ) − c α A (cid:19)(cid:19) λ eff − (cid:18) (cid:18)(cid:16) α E − α A (cid:17) ( v (0) i ) − c α A (cid:19) R E − z (0) i (cid:18)(cid:16) α E + α A (cid:17) ( v (0) i ) − c α A (cid:19) (cid:19) R E (cid:19) t +10 v (0) i (cid:18) (cid:16) α E R E − z (0) i (cid:16) α E x + α A (cid:17)(cid:17) λ eff2 + (cid:18) α E R E − (cid:16) α E + α A (cid:17) z (0) i R E + 3 ( z (0) i ) (cid:18) α E + 23 α A (cid:19)(cid:19) λ eff − z (0) i R E (cid:18)(cid:16) α E + α A (cid:17) R E − z (0) i (cid:18) α E + 23 α A (cid:19)(cid:19) (cid:19)(cid:19) t i − v (0) i (cid:16)(cid:16) α E − α A (cid:17) ( v (0) i ) − c α A (cid:17) (cid:18) λ eff + R E (cid:19) t + (cid:18) (cid:18)(cid:18) − α A + 20 α E (cid:19) ( v (0) i ) − c α A (cid:19) λ eff2 + (cid:18) (cid:18)(cid:18) α E − α A (cid:19) ( v (0) i ) − c α A (cid:19) R E − z (0) i (cid:18)(cid:16) α E + α A (cid:17) ( v (0) i ) − c α A (cid:19) (cid:19) λ eff + 103 (cid:18) (cid:18)(cid:16) α E − α A (cid:17) ( v (0) i ) − c α A (cid:19) R E − z (0) i (cid:18)(cid:16) α E + α A (cid:17) ( v (0) i ) − c α A (cid:19) (cid:19) R E (cid:19) t − v (0) i (cid:18) (cid:16) α E R E − z (0) i (cid:16) α E + α A (cid:17)(cid:17) λ eff2 + (cid:18) α E R E − (cid:16) α E + α A (cid:17) z (0) i R E +3( z (0) i ) (cid:18) α E + 23 α A (cid:19)(cid:19) λ eff − z (0) i R E (cid:18)(cid:16) α E + α A (cid:17) R E − z (0) i (cid:18) α E +23 α A (cid:19)(cid:19) (cid:19) t +10 (cid:16) − z (0) i + λ eff (cid:17) (cid:16)(cid:16) α E R E − z (0) i (cid:16) α E + α A (cid:17) R E +2( z (0) i ) ( α A + α E ) (cid:17) λ eff − z (0) i R E (cid:16) R E − z (0) i (cid:17) ( α A + α E ) (cid:17) (cid:33) g +10 α A (cid:18) c + 12 ( v (0) i ) (cid:19) (cid:18) −
23 ( v (0) i ) (cid:18) λ eff + R E (cid:19) t i − v (0) i (cid:18) − v (0) i (cid:18) λ eff + R E (cid:19) t + λ eff2 + (cid:16) R E − z (0) i (cid:17) λ eff +12 R E (cid:16) R E − z (0) i (cid:17) (cid:19) t i −
23 ( v (0) i ) (cid:18) λ eff + R E (cid:19) t + v (0) i (cid:18) λ eff2 + (cid:16) R E − z (0) i (cid:17) λ eff + 12 R E (cid:16) R E − z (0) i (cid:17)(cid:19) t − (cid:16)(cid:16) R E − z (0) i (cid:17) λ eff + R E (cid:16) R E − z (0) i (cid:17)(cid:17) (cid:16) − z (0) i + λ eff (cid:17) (cid:19)(cid:35) g c R E λ α E I ( R E λ eff )e − REλ eff + 14 ω (cid:34) (cid:32) (cid:16) g ( t − t i ) − v (0) i (cid:17) (cid:16) g ( t − t i ) +( − t +2 t i ) v (0) i − z (0) i (cid:17) kα A ω + (cid:18) (cid:16) g ( t − t i ) +( − t + 2 t i ) v (0) i − z (0) i (cid:17) · (cid:18) − ( t − t i ) (cid:16) α A + α E (cid:17) g + (cid:16)(cid:16) (2 t − t i ) v (0) i + z (0) i (cid:17) α A + (cid:16) v (0) i ( t − t i )+ z (0) i (cid:17) α E (cid:17) g + α A (cid:18) c −
12 ( v (0) i ) (cid:19)(cid:19) k +2 g ( α A + α E ) (cid:19) ω − (cid:16) g ( t − t i ) − v (0) i (cid:17) g (cid:18) α A + 23 α E (cid:19) kω − (cid:18) − t − t i ) (cid:16) α A + α E (cid:17) g + (cid:32)(cid:16) (12 t − t i ) v (0) i +3 z (0) i (cid:17) α A + 6 (cid:32) v (0) i ( t − t i ) + z (0) i (cid:33) α E (cid:33) g + (cid:18) c −
92 ( v (0) i ) (cid:19) α A − α E ( v (0) i ) (cid:19) g k ω − g (cid:16) α A + α E (cid:17) k (cid:33) ϕ sin ( R E k − ωt + δ )+4 ωϕ (cid:32) (cid:16) g ( t − t i ) − v (0) i (cid:17) α A ω + (cid:18) −
32 ( t − t i ) (cid:18) α A + 23 α E (cid:19) g + (cid:16)(cid:16) (3 t − t i ) v (0) i +2 z (0) i (cid:17) α A + 2 (cid:16) v (0) i ( t − t i )+ z (0) i (cid:17) α E (cid:17) g + α A (cid:18) c −
12 ( v (0) i ) (cid:19) (cid:19) kω − (cid:16) g ( t − t i ) − v (0) i (cid:17) k · (cid:18) − t − t i ) (cid:16) α A + α E (cid:17) g + (cid:16)(cid:16) (4 t − t i ) v (0) i +3 z (0) i (cid:17) α A +2 (cid:16) v (0) i ( t − t i )+ z (0) i (cid:17) α E (cid:17) g + α A (cid:18) c −
12 ( v (0) i ) (cid:19)(cid:19) ω +3 g (cid:18) α A + 23 α E (cid:19) kω − (cid:16) g ( t − t i ) − v (0) i (cid:17) g (cid:16) α A + α E (cid:17) k (cid:33) cos ( R E k − ω t + δ ) + 4 (cid:32) − α A ω kv (0) i z (0) i + (cid:18) z (0) i (cid:18) z (0) i ( α A + α E ) g + α A (cid:18) c −
12 ( v (0) i ) (cid:19)(cid:19) k − g ( α A + α E ) (cid:19) ω − v (0) i g (cid:18) α A + 23 α E (cid:19) kω + (cid:18) z (0) i (cid:18) α A + 23 α E (cid:19) g + (cid:18) c −
92 ( v (0) i ) (cid:19) α A − α E ( v (0) i ) (cid:19) g k ω +12 g (cid:16) α A + α E (cid:17) k (cid:33) ϕ sin ( R E k − ωt i + δ ) − ωϕ (cid:32) − α A ω v (0) i + k (cid:18)(cid:18) c +2 z (0) i g −
12 ( v (0) i ) (cid:19) α A +2 z (0) i g α E (cid:19) ω + k v (0) i (cid:18) (cid:18) c +3 z (0) i g −
12 ( v (0) i ) (cid:19) α A +2 z (0) i g α E (cid:19) ω +3 g (cid:18) α A + 23 α E (cid:19) kω + 12 v (0) i g (cid:16) α A + α E (cid:17) k (cid:33) cos ( R E k − ω t i + δ ) (cid:35) + ˙ z ( t i ) − g ( t − t i ) . (A4)For later use, we give the following velocities. At thetime of applying the π -pulse, the velocity for atoms in thelower arm is ˙ z l ≡ ˙ z ( t ) | t = T with t i = 0 and ˙ z ( t i ) = v L ,while the velocity for atoms in the upper arm is ˙ z u ≡ ˙ z ( t ) | t = T with t i = 0 and ˙ z ( t i ) = v L + v R ( ϕ ( t i )). At thetime of applying the second π -pulse, ˙ z l ≡ ˙ z ( t ) | t =2 T with t i = T and ˙ z ( t i ) = ˙ z l + v R ( ϕ ( t i )), while ˙ z u ≡ ˙ z ( t ) | t =2 T with t i = T and ˙ z ( t i ) = ˙ z u − v R ( ϕ ( t i )).Next, we do the time integration on Eq. (A4) to getthe solution for the trajectory z ( t ) = z ( t i ) + (cid:90) tt i ˙ z ( t (cid:48) ) dt (cid:48) . (A5)For later use, we give the following positions. At thetime of applying the π -pulse, the position for atoms inthe lower arm is z l ≡ z ( t ) | t = T with t i = 0, z ( t i ) =0 and ˙ z ( t i ) = v L , while the position for atoms in theupper arm is z u ≡ z ( t ) | t = T with t i = 0, z ( t i ) = 0 and˙ z ( t i ) = v L + v R ( ϕ ( t i )). At the time of applying the second π -pulse, z l ≡ z ( t ) | t =2 T with t i = T , z ( t i ) = z l and˙ z ( t i ) = ˙ z l + v R ( ϕ ( t i )), while z u ≡ z ( t ) | t =2 T with t i = T , z ( t i ) = z u and ˙ z ( t i ) = ˙ z u − v R ( ϕ ( t i )). Appendix B: Calculation of the DM-induced phaseshift in AI experiments
The total phase shift can be written as a sum of threecomponents [27], the propagation phase shift, the laserphase shift, and the separation phase shift,∆ φ = ∆ φ prop + ∆ φ laser + ∆ φ sep . (B1) For each segment of the atomic trajectory, the atomaccumulates a propagation phase φ prop = (cid:90) t f t i L dt , (B2)where t f is the final time for each segment, and L is theLagrangian (32). The propagation phase shift ∆ φ prop isthe difference in the propagation phase between the twoarms, ∆ φ prop = (cid:88) upper φ prop − (cid:88) lower φ prop . (B3)The laser phase shift comes from the interaction oflaser pulses with atoms. At each interaction point, thelaser field transfers its phase to the atom. Then, ∆ φ laser is the difference in the accumulated laser phase betweenthe upper and lower arms∆ φ laser = (cid:88) upper φ laser − (cid:88) lower φ laser = c (cid:90) zic k eff ( t ) dt − c (cid:90) T + z uc T k eff ( t ) dt − c (cid:90) T + z lc T k eff ( t ) dt + c (cid:90) T + z lc T k eff ( t ) dt , (B4)where z i is the initial position of atoms at the time ofapplying the first π -pulse.Since the two arms do not exactly intersect at the fi-nal laser pulse, then the separation phase shift ∆ φ sep φ sep = m A (cid:126) ( ˙ z u − v R + ˙ z l )( z l − z u ) . (B5)With Eqs. (A4) and (A5), we can calcualte φ prop along each segment of the lower and upper arms, and thus com-pute ∆ φ prop . Similarly, ∆ φ laser and ∆ φ sep can also becomputed. Summing them together, one can get the fi-nal result for the DM-induced phase shift. We find that∆ φ consists of a static component ∆ φ δϕ , an oscillatorycomponent ∆ φ bg , and the well-known term − g T k eff ,∆ φ = − g T k eff + ∆ φ δϕ + ∆ φ bg . (B6)The δϕ -contribution to ∆ φ is given by∆ φ δϕ = − g T k eff (cid:34)(cid:32) g T − (2 v L + v R ) Tλ eff + (1 + R E λ eff ) + v L ( v R + v L )2 c (cid:33) α A + (cid:32) g (2 v L + v R ) T − g R E c − g T c (cid:33) α E + 1 λ eff c (cid:32)(cid:16)(cid:0) v L ( v L + v R ) − v R (cid:1) g T − v L ( v L + v R ) (cid:0) ( v L + 12 v R ) T − R E (cid:1)(cid:17) α A + (cid:16) − T g + 92 (cid:0) v L + v R (cid:1) g T − (cid:0) g R E + 72 v L ( v L + v R ) + v R (cid:1) T + R E (cid:0) v L + v R (cid:1)(cid:17) · g T α E (cid:33)(cid:35) I ( R E λ eff ) α E e − REλ eff + g k eff T (cid:34) c (cid:32) g T + 12 ( − v L T − T v R + 2 R E ) g + (4 v L + v R ) ( v R + v L ) (cid:33) + 1 λ eff c (cid:32) g T − (cid:18) v L + 3762 v R (cid:19) g T + 12 g (cid:0) g R E + 138 v L + 168 v R v L + 49 v R (cid:1) T − (cid:0) g R E v L + 15 g R E v R + 32 v L + 60 v R v L + 34 v R v L + 6 v R (cid:1) T + (4 v L + v R ) R E ( v R + v L ) (cid:33)(cid:35) I ( R E λ eff ) ˜ dα E e − REλ eff (B7)The ϕ bg -contribution to ∆ φ is given by∆ φ bg = − k eff c kα A ϕ ω (cid:32) sin( kR E − ωT + δ ) − kR E − ωT + δ ) + sin( kR E + δ ) (cid:33) + α A g k eff Tω ϕ (cid:32) sin( kR E − ωT + δ ) − sin( kR E − ωT + δ ) (cid:33) + ( α E + 2 α A ) g k eff ω ϕ (cid:32) cos( kR E + δ ) − kR E − ωT + δ ) + cos( kR E − ωT + δ ) (cid:33) − α A ( k eff ( v L + v R ) ω ) ϕ (cid:32) sin( kR E + δ ) + 2 sin( kR E − ωT + δ ) − sin( kR E − ωT + δ ) (cid:33) − k eff kϕ ω (cid:34)(cid:0) g T − v L − v R (cid:1)(cid:18)(cid:0) g T − v L T − v R T (cid:1) ω α A − g α A − g α E (cid:19) cos( kR E − ωT + δ ) − (cid:0) g T − v L − v R (cid:1)(cid:18)(cid:0) g T − v L T − v R T (cid:1) ω α A − g α A − g α E (cid:19) cos( kR E − ωT + δ ) + g (2 α E + 92 α A ) · (2 v L + v R ) cos( kR E + δ ) (cid:35) + k eff kϕ ω (cid:34)(cid:32)(cid:18) g T (2 g T − v L − v R ) ( α E + 3 α A ) + 32 v L ( v L + v R ) α A + 12 v R α A (cid:19) ω − g (cid:16) α A + α E (cid:17) (cid:33) sin( kR E − ωT + δ ) + (cid:32)(cid:18) g T (2 v L + v R − g T ) ( α E + 3 α A )3 − v L ( v L + v R ) α A − v R α A (cid:19) ω + 24 g (cid:0) α A + α E (cid:1)(cid:33) sin( kR E − ωT + δ ) + (cid:32)(cid:0) v L ( v L + v R ) + 12 v R (cid:1) ω α A − g ( α A + α E (cid:33) sin( kR E + δ ) (cid:35) − k eff T k ˜ dϕ c (cid:34) (cid:0) T g − (8 c + 4 v L + 2 v R ) g T + 4 c ( c + v L + v R ) (cid:1) (cid:18) T g − v L − v R (cid:19) sin ( kR E − T ω + δ )+ (cid:32) − g T + 14 (cid:0) c + 3 v L + 32 v R (cid:1) g T − g T (cid:18) v L + (cid:0) c + 32 v R (cid:1)(cid:0) v L + v R (cid:1)(cid:19) + (cid:18) v L + v L v R + 12 v R (cid:19) c + 12 v L + 12 ( v L + v R ) (cid:33) ( g T − c ) sin( kR E − ωT + δ ) (cid:35) − k eff T ˜ dϕ ωc (cid:34) c (cid:16) kg T − (cid:16) v L + v R (cid:17) g kT + c ω (cid:17) (2 T g − v L − v R ) cos ( kR E − ω T + δ )+ (cid:32) − kg T + (cid:16) v L + v R (cid:17) g kT − (cid:18)(cid:0) v L + v L v R + 12 v R (cid:1) k + c ω (cid:19) g T − (2 v L + v R ) c ω (cid:33) · cos ( kR E − T ω + δ ) (cid:35) (B8) [1] P. 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