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Measurement of the Gravity-Field Curvature by Atom Interferometry G. Rosi1, L. Cacciapuoti2, F. Sorrentino1,∗ M. Menchetti1,† M. Prevedelli3, and G. M. Tino1‡ 1Dipartimento di Fisica e Astronomia and LENS, Universita` di Firenze, INFN Sezione di Firenze, via Sansone 1, I-50019 Sesto Fiorentino, Firenze, Italy 2European Space Agency, Keplerlaan 1, 2200 AG Noordwijk, Netherlands 3Dipartimento di Fisica e Astronomia, Universita` di Bologna, Via Berti-Pichat 6/2, I-40126 Bologna, Italy (Dated: January 8, 2015) Wepresent thefirst direct measurement of thegravity-fieldcurvaturebased on threeconjugated 5 atom interferometers. Three atomic clouds launched in the vertical direction are simultaneously 1 interrogated bythesameatom interferometry sequenceandusedtoprobethegravityfieldat three 0 2 equallyspacedpositions. Theverticalcomponentofthegravity-fieldcurvaturegeneratedbynearby sourcemasses ismeasured from thedifferencebetween adjacent gravitygradient values. Curvature n measurementsareofinterestingeodesystudiesandforthevalidationofgravitationalmodelsofthe a surrounding environment. The possibility of using such a scheme for a new determination of the J Newtonian constant of gravity is also discussed. 7 ] In the last two decades, atom interferometry [1] has the simultaneous determination of gravity acceleration h p profoundly changed precision inertial sensing, leading and its derivatives improves the inversion procedure by - to major advances in metrology and fundamental and introducingadditionalconstraintsforthevalidsolutions. m applied physics. The outstanding stability and accu- Gravitygradientsurveysarealreadyusedtodetectshort- o racy levels [2, 3] combined with the possibility of easily wavelength density anomalies or in situations where the t implementing new measurement schemes [4–7] are the vibrationnoiseseriouslylimitsabsolutegravitymeasure- a . mainreasonsforthe rapidprogressoftheseinstruments. ments. Thesecondderivativeofthegravityfieldcanvary s Matter-wave interferometry has been successfully used by several orders of magnitude when measured across c i to measure local gravity [8], gravity gradient [9–11], the shallowdensityanomalies,promisinghighspatialresolu- s y Sagnac effect [12], the Newtonian gravitational constant tions and sharp signals for their localization[27]. Simul- h [13–16], the fine structure constant [17], and for tests of taneousin situ measurementsofthe gravityacceleration p generalrelativity[18,19]. Accelerometersbasedonatom and its derivatives can also be used for remote sensing [ interferometry have been developed for many practical to estimate the evolution of the gravitational field along applications including geodesy, geophysics, engineering the direction of the local plumb line. Such a method 1 v prospecting,andinertialnavigation[20–22]. Instruments couldfind interesting applicationsin regionalheightsys- 0 for space-based research are being conceived for differ- temstomeasuredifferencesinthegravitationalpotential 0 entapplicationsrangingfromweakequivalenceprinciple with respect to a reference station, e.g., located on the 5 testsandgravitational-wavedetectiontogeodesy[23,24]. geoid [28]. Indeed, in the presence of shallow density 1 One of the most attractive features of atom interfer- anomalies, the knowledge of both the gravity gradient 0 ometry sensors is the ability to perform differential ac- and the curvature can provide centimeter-level resolu- . 1 celeration measurements by simultaneously interrogat- tion (∼ 0.1 m2/s2) in the measurement of differential 0 ing two separated atomic clouds with high rejection of geopotential heights by integrating the gravity field over 5 common-mode vibration noise, as demonstrated in grav- baselines of several hundreds of meters. The simulta- 1 ity gradiometry applications [3, 9]. In principle, such a neous measurement of gravity gradient and higher-order : v scheme can be extended to an arbitrary number of sam- derivativeswouldalsohelpwithcorrectingforNewtonian i X ples, thus, providinga measurementof higher-orderspa- noise in future gravitational-wavedetectors [29]. r tialderivativesofthegravityfield. Geophysicalmodelsof In this Letter, we report for the first time the direct a the Earth’s interior rely on the inversion of gravity and measurement of the gravity-fieldcurvature generated by gravity gradient data collected at or above the surface nearbysourcemasses,assuggestedinRef. [9]. Ouratom [25]. The solution to this problem, which is, in general, interferometer, which simultaneously probes three freely notunique,leadstoimagesofthesubsurfacemassdistri- fallingsamplesof87Rb,isabletoperformmeasurements bution over different scale lengths [26]. In this context, ofgravity,gravitygradient,and curvaturealong the vertical direction at the same time, opening new perspec- tives for geodesy studies and Earth monitoring applications. Using this scheme, we also demonstrate a new ∗ Present address: INFN Sezione di Genova, Via Dodecaneso 33, method to measure the Newtonian constant of gravity. I-16146Genova, Italy † Presentaddress: SchoolofPhysicsandAstronomy,Universityof Thedetailsoftheexperimentalapparatuscanbefound Birmingham, Edgbaston, Birmingham, B15 2TT, United King- in Refs. [16, 30]. In the following, a description of the dom measurement sequence and data analysis will be pro- ‡ tino@fi.infn.it vided, with particular emphasis on the new features in- 2 troducedbythethirdatomicsampleandthegravitycur- vature determination. A magneto-optical trap (MOT) with beams oriented ina1-1-1configurationcollects87Rbatomsandlaunches them vertically at a temperature of about 4 µK. A high- fluxsourcebasedona2DMOTprovideslargeatomnum- bers (∼ 109) in short loading times (∼ 40 ms). Larger atom numbers could be obtained by using the juggling technique[30];however,onlyadirectlaunchcanberead- ily implemented in our measurementcycle and extended tothreeormoresamples. Welaunchthreeatomicclouds alongthe verticaldirectionseparatedby ∼30 cm,which reach the apogees of their atomic trajectories simultaneously. A series of velocity selection and blow-away pulses prepares the samples in the magnetically insen- sitive |F = 2,m = 0i sublevel of the 87Rb ground F state. The Mach-Zehnder interferometer simultaneously addressesthe threecloudswithaπ/2−π−π/2sequence of verticalvelocity-selective Raman pulses [31]. The Ra- man lasers, with effective wave vector keff ≃ 16× 106 FIG. 1. (color online) a) Scheme of the experiment. 87Rb m−1, are resonant with the 6.8 GHz two-photon transi- atomsaretrappedandcooledinaMOT.Threeatomicclouds tion |F = 2,m = 0i → |F = 1,m = 0i of the 87Rb are launched in rapid sequence along the vertical direction F F groundstateandhavea2GHzreddetuningwithrespect with a moving optical molasses. Near the apogees of the to the 52S |F = 2i → 52P |F = 3i transition to the atomic trajectories, a measurement of the vertical accelera- 1/2 3/2 tionsensedbythethreecloudsisperformedbyRamaninter- excited state. The sequence has a duration of 2T = 320 ferometry. External source masses are positioned in order to ms. The π pulse lasts 24 µs and occurs 5 ms after the maximize the average gravity curvature at the three clouds’ atomic clouds have reached their apogees. The interfer- positions. b) Gravitational acceleration along the symme- ence fringes are obtained by measuring the normalized try axis (az) produced by the source masses and the Earth’s population in one of the two hyperfine levels of the 87Rb gravity gradient; a constant value accounting for the Earth’s groundstate. Weuseasetofhigh-densitysourcemasses, gravitationalaccelerationwassubtracted. Thespatialregions for a total of 516 kg, to enhance the gravity-field curva- of the three atom interferometers are indicated by the thick ture sensed by the three atomic samples. The source red lines. masses are composed of 24 tungsten alloy (INERMET IT180) cylinders [32]. They are positioned on two tita- eters are distributed around a Lissajous ellipse lying in nium platforms and distributed in hexagonal symmetry 3D space. around the axis of the interferometer tube. The verti- Theellipsebestfittingthedatapointsin3Dspacecan cal position of the platforms is accurately controlled by be expressed with the parametric equations precisionscrewssynchronouslydrivenbystepper motors and measured by an optical readout system. The ex- x(θ)=A+Bsinθ, perimental setup is shown in Fig. 1, together with the  axial acceleration profile due to the source masses and y(θ)=C+Dsin(θ+ϕ1), (1) the Earth’s gravity. z(θ)=E+Fsin(θ+ϕ +ϕ ). 1 2 Atomic gravity gradiometers use the same Raman  lasers to simultaneously probe two spatially separated Here A, B, C, D, E, and F representthe amplitude and atom clouds on the same interferometric sequence. In offset of the fringes of the three atom interferometers, θ this configuration, vibration noise that couples into the isthe phaseangleparameter,whichvariesrandomlydue phase of the Raman lasers is seen as common mode to common-mode vibrationnoise, andϕ1 andϕ2 are the and can be efficiently rejected. As a consequence, when phaseanglesproportionaltothedifferentialaccelerations the normalized atomic populations (x,y) simultaneously between adjacent interferometers. Fitting an ellipse to measured at the two spatially separated interferometers points in 3D space can be recast as a 2D problem [34]. areplottedin2Dspace,anellipseisobtained. Common- Given a set of n data points (xi,yi,zi), the χ2 function modephasenoiseaffectingthefringesofthetwoatomin- can be written as terferometersdistributestheexperimentalpointsaround n the ellipse, whose shape carries informationon the grav- χ2 ∝ [e2 (x ,y ,z )+h2], (2) ity gradientbetweenthe twoclouds [33]. We extend this X i,ellipse i i i i i=1 idea by introducing a third atom interferometer. In this case, the normalized atomic populations (x,y,z) mea- where e denotes the Euclidean 2D distance be- i,ellipse sured at the output ports of the three atom interferom- tween the ellipse and the projection of the point on the 3 plane of the ellipse, and h is the 3D point-plane dis- i tance. We assume equal uncertainties on all experimen- tal points. The χ2 function is then evaluated and min- imized with respect to the eight parameters of Eq. 1. This approach can be easily generalized to N interferometers (x ,...,x ) for the measurementof higher-order 1 N derivatives of the gravity field along the vertical axis. Itisworthpointingoutthataddinganextradimension (i.e., a third atom interferometer) opens the possibility of accurately measuring small gradiometric phase shifts introducing negligible bias on the fit results. In a two- cloudconfigurationandinthepresenceofasmallgravity gradient,the two output fringes are almostin phase and the ellipse degenerates to a line. On the other hand, in the presence of a third cloud, even if ϕ ∼ 0, ϕ can 2 1 be made quite large, e.g., by pulsing a magnetic field at the location of the third interferometer. In this case, |ϕ −ϕ |≫0and,eveninpresenceofnoisydata,ϕ can 2 1 2 FIG. 2. (color online) Bias error in the differential phase be reliably extracted from the fit in 3D space. Figure from the 2D elliptical fit (red circles), the new 3D fit rou- 2 compares the bias errors introduced by three different tine (black squares), and the Bayesian method for different methods: aleast-squarefitin2D,aleast-squarefitin3D ϕ2 angle values; ϕ1 is kept fixed at π/2. Synthetic data are space, and a Bayesian analysis, all as a function of ϕ2, generated with a Gaussian detection noise (σd =0.01) and a when ϕ1 is kept at π/2. Simulated data are affected by fringes contrast of 0.3. In the Bayesian analysis, we feed the significant Gaussian noise at detection (σ = 0.01) and algorithm both with the exact Gaussian noise as used in the d present a fringe contrast of 0.3. The plot clearly shows simulation (blue triangles) and with the value obtained after how the third interferometer becomes instrumental for introducing a 10% error on σd (green rhombi). For ϕ2 <0.1 rad, the χ2 numerical minimization fails in the 2D fit. The precisiongravitygradiometry. Thebiaserrorsintroduced knowledgeofthenoisemodelaffectingthedatabecomescrit- by the 2D fit are non-negligible. In addition, for φ < 2 icalintheBayesiananalysiswhenthephaseangleapproaches 0.1, the 2D fit routine fails to converge. The Bayesian zero. analysisperformsbetterthanthe2Dfit,butitintroduces significantbiasesatsmallphaseangleswhenthe a priori knowledge of the noise affecting the data varies by 10%. mentsperformedwithoppositek . Submillimetricverti- eff On the other hand, the 3D fit is very robust and, in caloverlapoftheinterferometer’sarmshasbeenachieved contrastwiththe Bayesianmethod,doesnotrequireany byproperlyadjustingtheRamanfrequencyrampandthe apriori knowledgeofthenoiseontheexperimentaldata. timing of the velocity selection. Their transverseoverlap is ensuredby using the same launchsequence. The hori- Our setup has been used to perform a direct mea- zontalvelocityspreaddue tothe finitetransverseatomic surement of the gravity-field curvature generated by the temperature is expected to introduce noise and system- source masses. One of the most critical aspects of the aticshiftsontheellipsephaseangleviatheCoriolisaccel- measurement is the presence of spurious and nonhomo- eration. Because of the double differential nature of the geneousmagneticfields inthe interferometerregion. Be- gravity-field curvature measurement, the effect of Cori- cause of the spatial separationbetween the three atomic olis accelerations depends on the difference between the clouds,primarilyimposedbytheMOTloadingtime,the relative initial velocities of the atomic clouds at the two gravity curvature measurement is averaged over a total adjacentgravitygradiometers. To further reduce this ef- distance ofabout 60 cm. In this configuration,the lower fect, the mirrorretroreflectingthe Ramanlaser beams is andtheupperatomicsamplesareclosetotheedgesofthe rotated to compensate for the Earth’s rotation [36, 37]. µ-metal shield surrounding the vertical tube, where the Two data sets of 720 points (2.5 s of measurement time passive attenuation of external magnetic fields is lower perpoint),oneforeachofthetwok oppositedirections and the internal bias field is less homogeneous. To re- eff (↑ and ↓ ), have been collected and analyzed. Figure 3 duce this source of systematic errors, the sign of the shows a typical plot of the data points measured at the effective wave vector k of the Raman lasers is peri- eff threeconjugatedatominterferometers,togetherwiththe odically reversed during data acquisition [35]. This is ellipse in 3D space best fitting the data. The values for achieved by selecting a different Raman counterpropa- ϕ and ϕ are given by gating beam pair by properly adjusting the frequency 1 2 detuning tocompensateforthe Dopplershiftinducedby ϕ =(ϕ −ϕ )/2 , ϕ =(ϕ −ϕ )/2 . (3) 1 1,↑ 1,↓ 2 2,↑ 2,↓ theatomicmotioninthegravityfield. Inthisway,phase shifts that do not depend upon the effective wave vec- From the measurement of the clouds’ separation, d = tor, e.g., second-order Zeeman shifts or ac Stark shifts, (0.3098± 0.0002) m, it is possible to evaluate the av- arerejectedwhentakingthedifferencebetweenmeasure- erage gravity gradients γ = ϕ /(dk T2), obtaining 1,2 1,2 eff 4 such an experiment, it becomes important to optimize the distribution of the source masses to generate three quasistationary regions to host the conjugated atom interferometers,thus,reducingthesystematicsarisingfrom the positioning errors of the atomic clouds. Even if not specifically designed for this purpose, we used our appa- ratus to perform a proof-of-principle experiment. With thesourcemassesandthe atomiccloudspositionedasin Fig. 1, we measured Φ = ϕ −ϕ and compared it meas 2 1 with Φ obtained fromour single-particle Monte Carlo sim simulation. We obtained Φ =(0.5533±0.0006) rad, meas whichis ingoodagreementwithΦ =0.5528rad. The sim short-termsensitivityof3.8×10−2Gat1siscomparable with the one obtained in Refs. [3, 16, 38] by alternating the source masses position. An extensive evaluation of the systematic error sources that are affecting the measurement is beyond the scope of this work. In conclusion, by using three simultaneous atom interferometers, we have measured for the first time the component of the gravity curvature produced by nearby FIG. 3. (color online) Typical three-dimensional Lissajous source masses along one axis. The new analysis method figure obtained by plotting the output signal of the upper based on an elliptical fit in 3D space has proven to be atom interferometer as a function of the lower and central veryrobustwithrespecttoamplitude noiseandimmune one(redcircles) andellipsein3Dbestfittingthedata(black from noise-induced systematic shifts. The scheme has line). Orthogonal projections on the three Cartesian planes are also shown. also been used to perform a proof-of-principle measurement of the Newtonian gravitational constant based on two simultaneous gravity gradient measurements. Sensi- tivity and long-term stability of the G measurement are γ = (−4.112± 0.008)× 10−6 s−2 and γ = (0.223± 1 2 comparable with our previous work, opening the possi- 0.003)×10−6 s−2, and, thus, the average gravity curva- bility for further improvements after optimization of the ture ζ = (1.399±0.003)×10−5 s−2m−1. This measure- distributionofthe massesandthe positionofthe atomic ment is consistent with the value ζ = 1.397×10−5 sim clouds. This method can be extended to multiple inter- s−2m−1 obtained from our Monte Carlo model [38], ferometerswithsmallspatialseparation(∼5−10cm)in whichaccountsforthesourcemassesandadditionalcon- order to reconstruct acceleration profiles with high reso- tributionsintheimmediatevicinityoftheatomicclouds. lution and measure higher-order derivatives of the grav- Themeasurementofthe secondderivativeofthe gravitational acceleration. ityaccelerationisalsoaninterestingtoolfordetermining the Newtonian gravitational constant G, as proposed in ACKNOWLEDGMENTS Ref. [39]. The method consists of performing two simultaneous gravity gradient measurements in the presence of heavy source masses. The Earth’s gravity gradi- This work was supported by INFN (MAGIA exper- ent contribution is rejected when calculating the differ- iment), EC (FINAQS STREP/NEST project Contract ence between the two measurements without any need No. 012986) and ESA (SAI project Contract No. for modulating the position of the masses. In this way, 20578/07/NL/VJ).The authorsacknowledgeM.De An- systematic effects introduced by deformations and tilts gelis,R.delAguila,J.Flury,C.Rothleitner,andG.Sac- of the structure holding the masses can be removed. For corotti for a critical reading of the manuscript. [1] Atom Interferometry, Proceedings of the International [4] H.Mu¨ller,S.-w.Chiow,S.Herrmann, andS.Chu,Phys. School of Physics “Enrico Fermi”, Course CLXXXVIII, Rev. Lett.102, 240403 (2009). edited by G. M. Tino and M. A. Kasevich (Società Ital- [5] J. E. Debs, P. A. Altin, T. H. Barter, D. Döring, G. R. iana di Fisica and IOS Press, Amsterdam, 2014). Dennis,G.McDonald,R.P.Anderson,J.D.Close, and [2] A. 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