Gravity gradient suppression in spaceborne atomic tests of the equivalence principle
GGravity gradient suppression in spaceborne atomic tests of the equivalence principle
Sheng-wey Chiow, Jason Williams, Nan Yu ∗ Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109
Holger M¨uller † Department of Physics, University of California, Berkeley, CA 94720 (Dated: October 2, 2018)The gravity gradient is one of the most serious systematic effects in atomic tests of the equivalenceprinciple (EP). While differential acceleration measurements performed with different atomic speciesunder free fall test the validity of EP, minute displacements between the test masses in a gravitygradient produces a false EP-violating signal that limits the precision of the test. We show thatgravity inversion and modulation using a gimbal mount can suppress the systematics due to gravitygradients caused by both moving and stationary parts of the instrument as well as the environment,strongly reducing the need to overlap two species.
PACS numbers: 04.80.Cc, 04.60.-m, 03.75.Dg, 37.25.+k, 95.55.-n
The equivalence principle (EP) is one of the pillars ofthe general theory of relativity, and has become a touch-stone for quantum gravity, dark matter, and dark en-ergy [1]. Precision tests of the EP are powerful toolsto probe Planck-scale physics at low energy, and physicsbeyond the standard model [2], including the low-energylimit of a possible theory of quantum gravity [3–5] andsearches for dark matter [6] and dark energy [7]. The EPwas tested at a precision of ∼ − by dropping bulkmasses [8], and at ∼ − by using a torsion balance [9].Quantum tests of the Universality of Free Fall (UFF)by comparing accelerations of different atomic specieswith atom interferometry (AI) have been proposed [10]and conducted [11, 12] at ∼ − by several researchgroups, as summarized in Ref [1]. The extended obser-vation times and improved control of systematic effectsin spaceborne AI promise orders of magnitude improve-ments in sensitivity [1]. In such equivalence principletests the gravity gradient is the most serious systematiceffect, where species-dependent acceleration arises dueto imperfect spatial overlap of test masses [1, 10, 13, 14].Various measures are necessary to control this system-atic, e.g., extensive in-flight calibrations [13, 14] or en-vironmental control [10]. However, the effectiveness ofthese mitigations is not all obvious. There is even a re-cent claim asserting the existence of a fundamental mea-surement limit in the presence of gravity gradients, as aresult preventing a high precision EP measurement [15].In this letter, we show that inversion of gravity by ro-tating the AI apparatus on a gimbal can suppress thesensitivity to gravity gradients caused by both the rotat-ing parts of the instrument itself and the environment,strongly reducing the requirements on the overlap of thetwo species (this concept has been employed for bulktest masses [13, 14, 16]). An example of this scheme,utilizing the International Space Station (ISS) for dual-species AI experiments in microgravity [1], is illustratedin the instrument frame of reference in Fig. 1. The sup- pression is based on the fact that inversion of the in-strument leaves the gravity gradient tensor unchanged inits principal-axis frame, while the gravitational acceler-ation changes sign and can thus be measured indepen-dently, if the apparatus (in particular spatial mismatchbetween the atomic samples) is otherwise unaffected bythe inversion. This is the case in microgravity with goodmagnetic shielding. The gravity modulation thus greatlysuppresses systematics and their variations.The significance of the gravity gradient for AI-basedEP tests is briefly summarized as follows. The phase φ of an AI in the Mach-Zehnder configuration is (to theleading orders): φ = (cid:126)k eff · (cid:126)g T + (cid:126)k eff · ↔ γ · ( (cid:126)z i + (cid:126)v i T ) T + · · · , (1)where (cid:126)k eff is the effective wavevector of the AI laser pulses(direction fixed to the apparatus), (cid:126)g, ↔ γ are the first andthe second order derivatives of the gravitational potential(also commonly recognized as the gravitational acceler-ation vector and the gravity gradient tensor), T is thetime between AI laser pulses, and (cid:126)z i , (cid:126)v i are the initialposition and velocity of the atom [1, 10]. In practice,the effective location and velocity of an extended atomicsample are hard to control. Typically, (cid:126)k eff is chosen par-allel to (cid:126)g to maximize the sensitivity [17]. In general, ↔ γ = ↔ γ ⊕ + ↔ γ S consists of the Earth’s gravity gradient ↔ γ ⊕ and the gravity gradient of the instrument ↔ γ S , the selfgravity gradient (SGG).The E¨otv¨os parameter η is defined as η = 2 g A − g B g A + g B , (2)where g A ( g B ) is the gravitational acceleration measuredwith atomic species A ( B ). Due to the gravity gradient,a direct comparison of AI phases of different species willhave nonzero signal even if η = 0 and g A = g B : δφ θ =0 = η (cid:126)k eff · (cid:126)g T + (cid:126)k eff · ↔ γ · ( δ(cid:126)z i + δ(cid:126)v i T ) T + · · · , (3) a r X i v : . [ phy s i c s . a t o m - ph ] M a r r g 𝜽 = p ar Turntable modulated AI physics package onboard the ISS a r -g 𝜽 = 0 ra g z < 0𝛾 %% > 0 a g z > 0𝛾 %% > 0 Science ChamberDual-SpeciesCold-Atom Source 2 P i ez o - A c t u a t e d R e t r o r e f l ec t o r ImagingSensor 2
Dual-SpeciesCold-Atom Source 1
Bragg AI Laser
ImagingSensor 1
FIG. 1. Illustration of the gravity modulation scheme in the context of the QTEST concept [1]. The center figure is the atominterferometer EP experimental setup on a turntable which is modulated between 0 and π along the nadir direction ( r ). Inthe reference frame of the turntable (experiment), the projection of the Earth’s gravitational acceleration along z ( g z ), andtherefore the EP signal ( η k eff gT ), changes sign as the turntable is modulated from θ = 0 (left) to θ = π (right). On theother hand, the gravity gradient from the Earth, γ zz , remains the same under the modulation (by definition, the gravity andthe gravity gradient of the instrument remain constant in the experiment reference frame). Therefore, the gravity gradienteffects can be effectively suppressed in differential measurements. The insets zoom to the apparatus size, emphasizing thatthe modulation changes the relative sign between the gravitational acceleration ( g z ) and the gravity gradient ( γ zz ) seen by theexperiment. where δ(cid:126)z i ( δ(cid:126)v i ) is the initial position (velocity) differenceof the two species. With Earth’s nominal gravity gradi-ent γ ⊕ (cid:39) per meter in the radial direction,establishing a bound of 10 − on η would require over-lapping of the two atomic clouds with a high accuracy of δz i = 10 − g/γ ⊕ (cid:39) (cid:126)k eff is aligned vertically to (cid:126)g and therefore parallel to themajor principal axis of ↔ γ ⊕ , and the gimbal rotation isideally 180 ◦ about a second principal axis of ↔ γ ⊕ . In thisideal case, δφ θ =0 = η k eff g T + k eff γ ( z i + v i T ) T + · · · . As seen in the apparatus frame, g → − g when thegimbal is turned, while k eff , γ S , z i , v i remain the same.The gravity gradient γ ⊕ , too, remains the same: γ (cid:48)⊕ ≡ ∂ ∂z Φ ⊕ ( L t − z ) (cid:12)(cid:12)(cid:12)(cid:12) z = L t (4)= ∂ ∂ ( L t − z ) Φ ⊕ ( L t − z ) (cid:12)(cid:12)(cid:12)(cid:12) z = L t = ∂ ∂z Φ ⊕ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) z = L t = γ ⊕ , where Φ ⊕ is the Earth’s gravitational potential, and L t is the turning point of the gimbal and the origin of the coordinate system. The phase difference of two AIs mea-sured at the flipped orientation is δφ θ = π = − η k eff g T + k eff γ ( δz i + δv i T ) T + · · · . (5)The difference δφ θ =0 − δφ θ = π = 2 η k eff g T is immuneto γ . The result is still valid when considering only partof the instrument is rotating on the gimbal: the poten-tial of the non-rotating part of the instrument, includingthe spacecraft and housing, can be combined with theEarth’s potential, thus the gravity gradient remains thesame under flipping.Extending the analysis to three dimensions and con-sidering a finite inversion imperfection angle δθ (cid:28)
1, thephase difference becomes: δφ θ =0 − δφ θ = π (cid:39) η k eff g T (6) − k eff γ ⊕⊥ ( δz i + δv i T ) ⊥ δθ T + 2 k eff γ ⊕(cid:107) ( δz i + δv i T ) (cid:107) δθ T + · · · , where the subscript (cid:107) indicates the component in the (cid:126)k eff direction, and the subscript ⊥ indicates the component inthe direction orthogonal to both (cid:126)k eff and the turning axis.Even taking experimental imperfections into account, asuppression factor of 10000 can still be obtained with δθ = 0 . Phase term Relative magnitude k eff ( T zz + t zz )(( δz + δz ) + ( δv z + δv z ) T ) T cos θ cos ψ . × − C − k eff ( δv x + δv x ) δ Ω y T cos θ cos ψ . × − ∗ − k eff ( T xx + t xx )( δv x + δv x )Ω y T cos θ cos ψ . × − ∗ k eff ( T zz + t zz )( δv x + δv x )Ω y T cos θ cos ψ . × − ∗ k eff (( T xz + t xz )( δv x + δv x ) + ( T yz + t yz )( δv y + δv y )) T cos θ cos ψ . × − B − k eff ( δv y + δv y ) δ Ω z Ω y T cos θ cos ψ . × − A − k eff ( T xx + t xx )Ω y ( δx + δx ) T cos θ cos ψ . × − ∗ k eff (( T xz + t xz )( δx + δx ) + ( T yz + t yz )( δy + δy )) T cos θ cos ψ . × − B − k eff ( T xx + t xx )( δv x + δv x ) T sin θ . × − F k eff ( T yy + t yy )( δv y + δv y ) T cos θ sin ψ . × − ∗ − k eff ( T zz + t zz )( δv z + δv z )Ω y T cos θ cos ψ . × − ∗ − k eff ( δv z + δv z ) δ Ω y Ω y T cos θ cos ψ . × − ∗ k eff ( T xx + t xx )( δv z + δv z )Ω y T cos θ cos ψ . × − ∗ − k eff t xz ( δv z + δv z )Ω y T cos θ cos ψ . × − ∗ − k t zzz (cid:126) LT cos θ cos ψ (2 + cos θ cos ψ ) (cid:16) m − m (cid:17) . × − D k t zzz (cid:126) T cos θ cos ψ (1 + cos θ cos ψ ) (cid:16) m − m (cid:17) . × − E − k eff t xy ( δv y + δv y )Ω y T cos θ cos ψ . × − ∗ k eff ( T zz + t zz ) ( δv z + δv z ) T cos θ cos ψ . × − ∗ − k eff Ω y ( δ Ω z ( δy + δy ) + δ Ω y ( δz + δz )) T cos θ cos ψ . × − G − k eff ( δv z + δv z )Ω y T cos θ cos ψ . × − ∗ − k eff t xx ( δx + δx ) T sin θ . × − ∗ k eff t yy ( δy + δy ) T cos θ sin ψ . × − ∗ − k eff ( δv z + δv z ) δ Ω y T sin θ . × − ∗ k eff ( δv y + δv y ) δ Ω x Ω y T cos θ cos ψ . × − H k t zzz (cid:126) T cos θ cos ψ (cid:16) m − m (cid:17) . × − I TABLE I. Phase terms with relative magnitude > − after (cid:126)k -reversal and internal state modulation (operation of Bragg AIsequentially in two hyperfine states for suppressing magnetic field sensitivity [1]), without gravity modulation. Parameters aredefined in Table III. Letter superscipts of the relative magnitudes indicate the corresponding phase terms in Table II, whileasterisk superscripts show < − contribution of each after modulation.Phase term Relative magnitude − k eff ( δv y + δv y ) δ Ω z Ω y T cos θ cos ψ . × − A k eff T yz (( δy + δy ) + ( δv y + δv y ) T ) T cos θ cos ψ . × − B − k eff t zz (( δv x + δv x ) (cid:15) + ( δv y + δv y ) (cid:15) ) T cos θ cos ψ . × − C − k t zzz (cid:126) LT cos θ cos ψ (2 + cos θ cos ψ ) (cid:16) m − m (cid:17) . × − D k t zzz (cid:126) T cos θ cos ψ (cid:16) m − m (cid:17) . × − E k eff t xx ( δv x + δv x ) (cid:15) T cos θ cos ψ . × − F − k eff δ Ω z Ω y ( δy + δy ) T cos θ cos ψ . × − G k eff ( δv y + δv y ) δ Ω x Ω y T cos θ cos ψ . × − H k t zzz (cid:126) T cos θ cos ψ (cid:16) m − m (cid:17) . × − I TABLE II. Phase terms with relative magnitude > − after (cid:126)k -reversal and internal state modulation, with gravity modula-tion. Superscipts of the relative magnitudes indicate the corresponding phase terms in Table I. of QTEST [1], a concept of an apparatus on a turntableaboard ISS running simultaneous Rb and Rb AIs(each with 500:1 signal-to-noise ratio per shot governedby 10 atoms and 50% contrast) for an EP test, shownschematically in Fig. 1. QTEST is designed to achieve η ≤ − in 1 year of continuous operation, thanks tothe insensitivity to vibrations and thus not interferingwith ISS operations or astronaut activities. In QTEST,there is a dual species source at each end of a magneti-cally shielded science chamber, where the sources and the chamber are mounted on a turntable orienting the cham-ber toward or away from the Earth. At each turntableorientation, dual AIs are launched from each source to-ward the other end, driven by common Bragg pulsesretroreflected along the science chamber to facilitate si-multaneous (cid:126)k -reversal measurements. The experimentalsequence is operated at 70 s duty cycle, and the orienta-tion of the turntable is changed every 10 runs [1]. Thesimulation is conducted in the ISS frame, and four sce-narios are included to account for dual source regions and Parameter Description Value T AI interrogation time 10 sΩ y ISS angular velocity 1.13 mrad/s δ Ω x , δ Ω y , δ Ω z rotation compensation error 1.13 µ rad/s T xx , T yy , T zz Earth’s gravity gradient (diagonal) ( − . , − . , × /m T xy , T xz , T yz Earth’s gravity gradient (off-diagonal) 0.001 T zz t xx , t yy , t zz gravity gradient of the rotating part (diagonal) 3500 nm/s /m t xy , t xz , t yz gravity gradient of the rotating part (off-diagonal) 0.01 t zz T zzz rd order derivative of Earth’s potential − . × − / s / m t zzz rd order derivative of the rotating potential − × − / s / m θ, φ misalignment angle of (cid:126)k eff to vertical in the x -, y -direction 1 mrad δx i , δy i , δz i initial position mismatch at source location i ( i = 1 ,
2) 1 µ m δv xi , δv yi , δv zi initial velocity mismatch at source location i ( i = 1 ,
2) 1 µ m/s x t , y t , z t location of the center of mass of the rotating part 0.2 m L separation of the two source locations 0.5 m (cid:15) , (cid:15) angular error of turntable flipping in the x -, y -direction 0.1 mrad ∗ TABLE III. Definition of parameters in the simulation. The Earth’s gravity potential is expanded at 400 km altitude as: − (cid:0) gz + T zz z + T zzz z + T xx x + T yy y + T xy xy + · · · (cid:1) . The gravitational potential of the rotating part is modeled as: − (cid:0) t zz ( z − z t ) + t zzz ( z − z t ) + t xy ( x − x t )( y − y t ) + · · · (cid:1) . When the turntable points in the + z -direction, source location1 is near (0 , ,
0) and source location 2 is near (0 , , L ). A stationary dual species cloud at each source location is first launchedwith momentum ± (cid:126) (cid:126)k eff toward the other source location; after T / T . Flipping of the turntable is about the y -axis at (0 , , L t ) with L t = 0 . ↔ γ ⊕ and 10 mrad to the principal axesof ↔ γ S , respectively. Note that the contribution of shot-to-shot fluctuations of initial conditions, e.g. due to the cloud positionand velocity profiles, is below atom shot noise as detailed in Ref. [1]. ∗ Space-qualifiable turntables with < µ rad repeatabilityand accuracy are available, e.g., the Aerotech ALAR series [18]. two turntable positions. In each scenario, the phase dif-ference ∆Φ between Rb and Rb is calculated and ex-pressed in terms of initial conditions, parameters of grav-itational and magnetic potentials, and pointing change of (cid:126)k eff for rotation compensation [1, 10]. A combination ofphase differences features both (cid:126)k -reversal and turntablemodulation cancellation.The simulation result is summarized in Tables I and II,where relative magnitude is defined as the size of the cor-responding phase term over k eff gT for T = 10 s. Table Ilists residual phase terms up to 10 − relative magnitudeafter (cid:126)k -reversal and internal state modulation [1], with-out gravity modulation, and Table II lists residuals afteradditional gravity modulation with flipping angle errors (cid:15) , (cid:15) . The error terms induced by SGG are all propor-tional to the flipping error and < − with the simula-tion parameters listed in Table III, where the relevanceand justification of the parameter values are discussed indetail in [1]. The largest two terms in Table II, both ofwhich are aligned to the turning axis and not modulated,can be suppressed with the imaging technique outlinedin Ref. [1] by at least 10 , and thus will not limit theperformance of QTEST at 10 − .The above simulation assumes that the initial condi-tion mismatches are 100% fixed to the apparatus. Prac-tical imperfections, e.g., assuming that 0.01% of δx and δv are not fixed to the apparatus due to gravitational andmagnetic fields outside the turning apparatus, will effec-tively leave the same proportion in amplitude of phase terms in Table I unmodulated and not suppressed. Thisis equivalent to additional errors of 10 − of Table I, re-sulting in additional error contribution < − . Since Rb and Rb have the same linear Zeeman shift, themagnetic field bias itself doesn’t contribute to displace-ment. The differential gravitational sag δz sag , which isthe weight difference over the spring constant mω in theproposed quadrupole-Ioffee configuration (QUIC) trap [1]for QTEST, is: δz sag = δmm a res ω , (7)where δm/m (cid:39) .
02 for Rb isotopes, a res is the residualacceleration in microgravity, and ω (cid:39) π ×
25 Hz will bethe trap frequency. To constrain δz sag ≤ . a res ≤ .
12 mm/s , corresponding to an altitudedifference of the apparatus to the center of mass of theISS of a res / (Ω + γ ⊕ ) ≤ . − for 10 atoms) than the atom-shot-noise-limiteduncertainty of the ensemble centroid in each measure-ment [15]. The uncertainty in determining the centroidof a source, governed by the central limit theorem, de-creases as the number of measurements increases. Thisstatistical nature of both uncertainty limits suggests thatthe atom shot noise will always dominate for the thermalclouds considered in this letter.Our result is consistent with the gradient effect sup-pression scheme adopted in the MICROSCOPE missionwhere bulk test masses are used with a target sensitiv-ity of 10 − [13, 14], despite that MICROSCOPE has afixed and characterized overlap mismatch while QTESThas random but specified mismatch tolerances. The EPsignal in MICROSCOPE is modulated at the orbitingfrequency (or the orbiting plus the spinning frequenciesin the spinning mode), while the gravity gradient signalis mostly at twice the frequency. We interpret the resultin this letter a direct consequence of the fact that grav-ity gradients don’t change sign under flipping as shownin Eq. (4).In summary, we report that the gravity gradient depen-dent systematics, including those due to the self gravitygradient, can be totally suppressed in AI-based EP testswhen the apparatus is inverted by a gimbal. This sup-pression is based on the fixation of initial condition mis-match of two species to the apparatus under micrograv-ity, and on the fact that gravity gradient is the secondorder derivative of a scalar function so that the sign re-mains the same when inverted. We discuss the physicsin both the Earth frame and the apparatus frame reach-ing the same conclusion, which is supported by a moreelaborated simulation. We conclude that, thanks to thissuppression scheme, QTEST can reach the targeted EPsensitivity of 10 − .We thank Brian Estey for his contributions on theoryverification. This work was carried out at the Jet Propul-sion Laboratory, California Institute of Technology, un-der a contract with the National Aeronautics and SpaceAdministration. Government sponsorship acknowledged. ∗ [email protected] † [email protected][1] Jason Williams, Sheng-wey Chiow, Nan Yu, and HolgerM¨uller. Quantum test of the equivalence principle andspace-time aboard the international space station. NewJournal of Physics , 18(2):025018, 2016.[2] Thibault Damour. Theoretical aspects of the equivalenceprinciple.
Classical and Quantum Gravity , 29(18):184001,2012.[3] V Alan Kosteleck`y. Gravity, Lorentz violation, and thestandard model.
Physical Review D , 69(10):105009, 2004.[4] V Alan Kosteleck`y and Jay D Tasson. Matter-gravitycouplings and Lorentz violation.
Physical Review D , 83(1):016013, 2011.[5] Quentin G Bailey, V A Kostelecky, and R Xu. Short-range gravity and lorentz violation.
Physical Review D ,91(2):022006, 2015.[6] Peter W. Graham, David E. Kaplan, Jeremy Mardon,Surjeet Rajendran, and William A. Terrano. Dark mat-ter direct detection with accelerometers.
Phys. Rev. D ,93:075029, Apr 2016.[7] Paul Hamilton, Matt Jaffe, Philipp Haslinger, QuinnSimmons, Holger M¨uller, and Justin Khoury. Atom-interferometry constraints on dark energy.
Science ,349(6250):849–851, 2015.[8] S Carusotto, V Cavasinni, A Mordacci, F Perrone, E Po-lacco, E Iacopini, and G Stefanini. Test of g universalitywith a galileo type experiment.
Physical review letters ,69(12):1722, 1992.[9] Todd A Wagner, S Schlamminger, JH Gundlach, andEG Adelberger. Torsion-balance tests of the weakequivalence principle.
Classical and Quantum Gravity ,29(18):184002, 2012.[10] Jason M Hogan, David Johnson, and Mark A Kase-vich. Light-pulse atom interferometry. arXiv preprintarXiv:0806.3261 , 2008.[11] Lin Zhou, Shitong Long, Biao Tang, Xi Chen, Fen Gao,Wencui Peng, Weitao Duan, Jiaqi Zhong, ZongyuanXiong, Jin Wang, and et al. Test of equivalence prin-ciple at 10 − level by a dual-species double-diffractionRaman atom interferometer. Physical Review Letters ,115(1):013004, Jul 2015.[12] A Bonnin, N Zahzam, Y Bidel, and A Bresson. Char-acterization of a simultaneous dual-species atom inter-ferometer for a quantum test of the weak equivalenceprinciple.
Physical Review A , 92(2):023626, 2015.[13] ´Emilie Hardy, Agn`es Levy, Manuel Rodrigues, PierreTouboul, and Gilles M´etris. Validation of the in-flightcalibration procedures for the MICROSCOPE space mis-sion.
Advances in Space Research , 52(9):1634–1646, 2013.[14] P Touboul, G M´etris, V Lebat, and A Robert. TheMICROSCOPE experiment, ready for the in-orbit test ofthe equivalence principle.
Classical and Quantum Grav-ity , 29(18):184010, 2012.[15] Anna M. Nobili. Fundamental limitations to high-precision tests of the universality of free fall by droppingatoms.
Physical Review A , 93(2):023617, 2016.[16] Robert D Reasenberg. A new class of equivalence prin-ciple test masses, with application to sr-poem.
Classicaland Quantum Gravity , 31(17):175013, 2014.[17] In microgravity, where (cid:126)g is balanced by the centrifugalforce in the moving frame, (cid:126)k eff is chosen to point verti-cally so that it is parallel to (cid:126)g(cid:126)g