We report on a high precision measurement of gravitational acceleration using
ultracold strontium atoms trapped in a vertical optical lattice. Using
amplitude modulation of the lattice intensity, an uncertainty $\Delta g /g
\approx 10^{-7}$ was reached by measuring at the 5$^{th}$ harmonic of the Bloch
oscillation frequency. After a careful analysis of systematic effects, the
value obtained with this microscopic quantum system is consistent with the one
we measured with a classical absolute gravimeter at the same location. This
result is of relevance for the recent interpretation of related experiments as
tests of gravitational redshift and opens the way to new tests of gravity at
micrometer scale.Comment: 4 pages, 4 figure

Alberti, A.

Poli, N.

Prevedelli, M.

Tarallo, M. G.

Tino, G. M.

Wang, F. -Y.

English

arXiv

Wang, F. Y.

METRICA Archivio istituzionale della ricerca - INRIM

Precision Measurement of Gravity with Cold Atoms in an Optical Lattice and Comparison with a Classical Gravimeter

We report on a precision measurement of gravitational acceleration using ultracold strontium atoms confined in an amplitude-modulated vertical optical lattice. An uncertainty dg/g  ~ 10^-7 is reached by measuring at the 5th harmonic of the Bloch frequency. The value obtained with this microscopic quantum system is consistent with the one measured with a classical gravimeter. Using lattice modulation to prepare the atomic sample, we also achieve high visibility of Bloch oscillations for ~20 s. These results can be of relevance for testing gravitational redshift and Newtonian law at micrometer scale

N. Poli

F. Y. Wang

M. G. Tarallo

A. Alberti

G. M. Tino

PREVEDELLI, MARCO

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna

Precision Measurement of Gravity with Cold Atoms in an Optical Lattice
and Comparison with a Classical Gravimeter

Poli, Nicola

Tino, Guglielmo Maria

Florence Research

Precision Measurement of Gravity with Cold Atoms in an Optical Latticeand Comparison with a Classical GravimeterN. Poli, F.-Y. Wang,* M.G. Tarallo, A. Alberti,† M. Prevedelli,‡ and G.M. TinoxDipartimento di Fisica e Astronomia and LENS, Universita` di FirenzeINFN Sezione di Firenze, Via Sansone 1, 50019 Sesto Fiorentino, Italy(Received 1 October 2010; published 18 January 2011)We report on a precision measurement of gravitational acceleration using ultracold strontium atomsconfined in an amplitude-modulated vertical optical lattice. An uncertainty g=g  107 is reached bymeasuring at the 5th harmonic of the Bloch frequency. The value obtained with this microscopic quantumsystem is consistent with the one measured with a classical gravimeter. Using lattice modulation toprepare the atomic sample, we also achieve high visibility of Bloch oscillations for 20 s. These resultscan be of relevance for testing gravitational redshift and Newtonian law at micrometer scale.DOI: 10.1103/PhysRevLett.106.038501 PACS numbers: 91.10.Pp, 03.75.Dg, 37.10.Jk, 37.25.+kAtom interferometry, and in general methods based onquantum interference of ultracold atoms, were largely usedin recent years for gravitational physics experiments andnew exciting prospects can be envisioned in the near future[1]. For example, Raman interferometry was used forprecise measurements of Earth’s gravitational accelerationg [2] and its gradient [3], for determining the value of thegravitational constant [4,5], for a possible redefinition ofthe kg [6], and for geophysical applications [7]. Schemesbased on Bloch oscillations of atoms trapped in verticaloptical lattices were also used to measure gravity with thepossibility of combining high sensitivity and micrometricspatial resolution [8–10]. The results of atom interferom-etry experiments were interpreted as tests of the isotropyof post-Newtonian gravity [11], of quantum gravity [12],and of gravitational redshift [13]. Prospects include highprecision tests of the weak equivalence principle [14,15],the detection of gravitational waves [16,17], and futureexperiments in space [18].Here, we present a precision measurement of gravita-tional acceleration g with a method based on ultracold 88Sratoms confined in an amplitude-modulated vertical opticallattice [19] and compare the result with the value obtainedwith a classical absolute gravimeter based on an opticalinterferometer with one arm including a freely fallingcorner cube. By careful analysis of systematics effects,we improved the accuracy of our previous gravity deter-minations [9,20] by more than 2 orders of magnitude.Using amplitude modulation of the lattice to prepare theatomic sample, we also achieved high visibility of theBloch oscillations for 20 s. We discuss the precision ofthe two methods for the determination of g. The use ofatoms confined in a volume of a few micrometers sizemakes this experiment qualitatively different from similar,higher precision experiments performed with freely fallingatoms [2,21]. For example, our data can be interpretedas a measurement of the gravitational redshift to theCompton frequency of Sr matter waves, as suggested byMu¨ller et al. [13]. While the interpretation of atom inter-ferometer redshift tests is complicated by special relativ-istic time dilation if the atoms are moving [22,23], Blochoscillations experiments with stationary lattices provide ameasurement of the purely gravitational effect.The experimental setup is based on cooled and trapped88Sr atoms [9] (Fig. 1). Atoms from a thermal beam areslowed in a Zeeman slower and trapped in a ‘‘blue’’magneto-optical trap (MOT) operating on the 1S0-1P1resonance transition at 461 nm. The temperature is furtherreduced by a second cooling stage in a ‘‘red’’ MOT oper-ating on the 1S0-3P1 intercombination transition at 689 nm.This produces about 106 atoms at a temperature of 0:6 K.Since the force of gravity is comparable to the forceproduced by the red MOT on the atoms, the cloud oftrapped atoms assumes a disklike shape with a verticalsize of 27 m and a radial size of 180 m. The atomsare adiabatically loaded in an optical lattice in 300 s. Thelattice is generated by a single-mode frequency-doubledNd:YVO4 laser (L ¼ 532 nm) delivering up to 1Won thePMFPMF PDAMAOM FCCIDRCCFG5IaserFIG. 1 (color online). Experimental setup for the measurementof gravity with 88Sr atoms trapped in a vertical optical lattice andthe comparison with a classical absolute gravimeter (FG5). AM:amplitude-modulation signal; AOM: acousto-optical modulator;FCC: falling corner cube; ID: interference detector; PD: photo-diode; PMF: polarization-maintaining fiber; RCC: referencecorner cube.PRL 106, 038501 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending21 JANUARY 20110031-9007=11=106(3)=038501(4) 038501-1  2011 American Physical Societyatoms with a beam waist of 557ð7Þ m. The beam isvertically aligned and retroreflected by a mirror. Theatomic sample in the lattice has a vertical rms size of14 m and a horizontal size of 100 m. The Bloch fre-quency is B ¼ mSrgL=2h ’ 574:3 Hz where mSr is themass of 88Sr atoms and h is Planck constant. The corre-sponding lattice photon recoil energy is ER ’ 8 kHz h.In typical conditions the lattice depth ranges from 2.3 to 3ER, while the energy gap EG at the recoil momentum kL isEG ’ ER. The width of the first energy band in the latticepotential is about 0:5 EG. Landau-Zener tunneling isnegligible in these conditions. The lattice depth is stabi-lized by a servo loop acting on the rf signal driving anacousto-optical modulator (AOM). The same AOM is usedto add an amplitude modulation to the lattice potential.The atomic cloud can be imaged either in situ or withthe usual time-of-flight technique using resonant absorp-tion imaging on a CCD camera with a spatial resolutionof 5 m.After loading 88Sr atoms in the vertical lattice, the trapdepth is modulated sinusoidally. When the modulationfrequency matches an integer harmonic of the Bloch fre-quency, atoms start to tunnel in neighboring lattice sitesgiving rise to a net increase in the vertical spatial atomicdistribution which is observed in situ by resonant absorp-tion imaging (Fig. 2). In this experiment, the modulationat a frequency m ¼ 5 B is applied for a time 10 swhich is experimentally set as a compromise between thelinear increase of resolution with time and the exponentialdecay of the signal due to the loss of atoms from the lattice.Because of the extremely small scattering length of 88Sr,the effects of cold atoms collisions are negligible over thistime scale. The amplitude-modulation depth is varied inthe range 4%–10% in order to optimize the signal-to-noiseratio; the corresponding estimated tunneling rate is3–6 s1 [19]. In this interval, we do not observe anydependence of the resonance position and shape on themodulation depth. The recording time for a whole reso-nance spectrum is about 1 h and leads to a maximumresolution of 1:5 107 for B.A number of sources of systematic error must beconsidered. Since the lattice laser is not frequency stabi-lized, a calibration of its frequency is required. To thispurpose, part of the lattice laser light is sent to an opticalfrequency comb. The measured fluctuation of the laserfrequency is ’ 100 MHz on the relevant time scale of theexperiment. The correction for the index of refraction ofthe Sr cloud and the background gas in the vacuum cham-ber is negligible. The vertical alignment of the lattice ischecked with a precision of 0.2 mrad, corresponding to arelative uncertainty of 2 108 on g, by overlapping thedownward laser beam with the reflection from the surfaceof water in a glass container inserted in the beam. Atiltmeter with a resolution of 1:7 rad attached to theoptical table is employed to check the alignment stabilityduring the measurements. The potential resulting from theinterference of the two Gaussian beams depends on theirlongitudinal intensity profile and their wave front curva-tures, producing an inhomogeneity of the optical lattice inthe axial direction [24]. This gives rise to two correctionterms on g, namely, gU for the spatial derivative of thelattice potential and gk for the spatial derivative of thedifference between the Gouy phase of the two counter-propagating beams. The two terms are estimated by aprecise determination of the geometry of the incomingand the reflected trapping beams and the position of thecloud with respect to the beam waist, with a relativeuncertainty of 1%. An independent determination of thetransverse beam size at the position of the atomic cloudis obtained by measuring the axial and radial atomic traposcillation frequencies through parametric heating[25,26]. For typical experimental parameters, the twoterms are gU ¼ 1:53ð3Þ  105 m=s2 and gk ¼1:0ð2Þ  108 m=s2. Tidal effects are evaluated and re-moved from the raw data using the same algorithm andpotential model used for the absolute gravimeter dataprocessing. The peak-to-peak effect of tides at our siteis 2 106 m=s2. Since each measurement lasts about1 h the variation of g during a single measurement due totides is below 107 m=s2. The values ofmSr and h are bothknown with a relative uncertainty of 5 108. All theother potential sources of systematic shifts that we eval-uated (spurious higher harmonics of amplitude modula-tion, Bloch-Siegert shift [27], gravity due to surroundingmasses, magnetic field gradients) are below the currentaccuracy level. Table I summarizes the main systematicshifts for the gravity measurements using the amplitude-modulation method. The values of individual shifts dependon the experimental conditions; the quoted uncertaintiesare typical values.Figure 3 presents a set of 21 determinations of g with88Sr atoms. The error bars result from the quadraturesum of the statistical errors coming from the fit of theamplitude-modulation resonance signal and the uncer-tainty on systematic corrections. The weighted mean ofFIG. 2 (color online). Spectrum recorded by modulation ofthe lattice depth at the 5th harmonic of the Bloch frequencyfor 10.4 s with a modulation depth of 8%. The red line is a fit ofexperimental data with a sinc function.PRL 106, 038501 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending21 JANUARY 2011038501-2our data is gatom ¼ 9:804 923 2ð14Þ m=s2 where the uncer-tainty corresponds to 1 standard deviation.The reference value for local gravitational accelerationis provided by an absolute gravimeter based on an opticalinterferometer with one arm including a freely fallingcorner-cube (FG5, Micro-g LaCoste). The measurementis performed in the same laboratory at a distance of 1.15 mfrom the atomic probe position. The difference in heightof 14(5) cm together with the estimated vertical gravitygradient value gzz ¼ 3:09 106 s2 at the laboratorysite is taken into account in the data analysis. The resultis gFG5 ¼ 9:804 921 609ð84Þ m=s2The comparison of the value obtained with the quantummechanical atomic sensor and the one obtained with theclassical gravimeter shows that they agree within the ex-perimental errors.With minor modifications of the experimental proce-dure, in this work we also determine g by measuring thefrequency of the Bloch oscillations of the atoms in thevertical optical lattice. Because of a better vacuum andtaking advantage of the lattice modulation method to re-duce the initial momentum distribution of the atoms in thelattice [19], we considerably improve the visibility of theoscillations and, as a consequence, the frequency resolu-tion compared with previous experiments [9]. After thetransfer of the atoms in the vertical optical lattice, anamplitude-modulation burst with typical duration of120 cycles at m ’ B is applied. The quantum phase ofthe atomic wave function induced by the amplitude modu-lation gives rise to an interference effect which results in anenhanced visibility of the Bloch oscillations peaks in thetime-of-flight image of the atomic cloud [28]. After turningoff the modulation, we let the atomic cloud evolve for atime T. Finally, we switch off the optical lattice within5 s to measure the momentum distribution of the atomsin ballistic expansion by taking an absorption picture witha CCD camera. In order to optimize the visibility throughthis quantum interference effect, we set the time of flight to14 ms. As shown in Fig. 4, we observe Bloch oscillationswith high visibility for 20 s. From the fit of the meanatomic momentum we can estimate the Bloch frequencyB with 1:7 107 statistical uncertainty. In comparisonwith the determination of B obtained with the resonantamplitude-modulation technique, however, we find a con-siderably larger scattering in repeated measurements,mainly due to the initial position instability of the atomictrap and to a higher sensitivity to the timing of the experi-ment. The value for g obtained with the Bloch oscillationtechnique is gBloch ¼ 9:804 88ð6Þ m=s2, which is consis-tent with the measurement presented above but is affectedby a larger relative uncertainty of 6 106.In conclusion, we have performed an accurate measure-ment of gravitational acceleration using ultracold 88Sratoms confined in a vertical optical lattice. The resultagrees within 140 ppb with the value obtained with aclassical FG5 gravimeter. This result improves by 1 orderof magnitude in sensitivity and by more than 2 inTABLE I. Systematic corrections and their associated uncer-tainties ( 107) for the gravity measurement with 88Sr atomsin the amplitude-modulated optical lattice.Effect Correction UncertaintyLattice wavelength 0 2Lattice beam vertical align. 0 0.2Stark shift (beam geometry) 14.3–17.3 0.4Experiment timing 0 0.2Tides 1:4–0:9 <0:1Height difference 4.3 0.2Refraction index 0 <0:01Fundamental constants 0 0.7Systematics total 17.2–22.5 2.2FIG. 4 (color online). Long-lived Bloch oscillations for Sratoms in the vertical lattice under the influence of gravity.Each picture shows one Bloch cycle in successive time-of-flightabsorption images giving the momentum distribution at the timeof release from the lattice. Displayed are the first (a), the 2900th(b), the 7500th (c), and the 9800th (d) Bloch cycles.FIG. 3 (color online). Measurements of g using the amplitude-modulation technique. Each experimental point is corrected forthe systematic effects presented in Table I. The red dashed linerepresents the weighted mean of the 21 measurements. The bluesolid line is the value obtained with the classical absolute FG5gravimeter.PRL 106, 038501 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending21 JANUARY 2011038501-3accuracy over previous results [9,20]. This provides acorrespondingly improved test of the gravitational redshiftin the frame of the model proposed in Ref. [13]. In thisexperiment, we also observe persistent Bloch oscillationswith high visibility for 20 s but we find that as a methodfor the measurement of g it is affected by a larger uncer-tainty. We believe that the amplitude-modulation methodfor gravity measurements has the potential for furtherimprovement. The present results are indeed mainly lim-ited by frequency stability and wave front curvature ofthe lattice laser beam. The first effect can be reduced byusing a frequency stabilized laser. The second could bereduced either by increasing the lattice beam waist or byusing a blue-detuned lattice [24]. Application of Ramsey-like schemes with short separated modulation pulses [19]might increase the resolution and reduce systematic effectsproduced by the lattice. The results of this work are ofinterest also for experiments on Casimir-Polder effect [29],tests of Newtonian gravitational law at micrometer dis-tances [10], and for precision measurements with nondes-tructive cavity QED techniques [30].We thank ENI and INGV for the measurement withthe FG5 gravimeter, M. Schioppo and D.V. Sutyrin fortheir help in the experiment, and R. Ballerini, M. De Pas,M. Giuntini, A. Hajeb, A. Montori for technical assistance.G.M. T. acknowledges useful discussions with H. Mu¨ller.This work is supported by INFN and LENS (ContractNo. RII3 CT 2003 506350).*Also ICTP, Trieste, Italy.†Present address: Institut fu¨r Angewandte Physik derUniversita¨t Bonn, Wegelerstrasse 8, 53115 Bonn,Germany.‡Permanent address: Dipartimento di Fisica, Universita` diBologna, Via Irnerio 46, 40126 Bologna, Italy.xGuglielmo.Tino@fi.infn.it[1] A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, Rev.Mod. Phys. 81, 1051 (2009).[2] A. Peters, C. K. Yeow, and S. Chu, Nature (London) 400,849 (1999).[3] J.M. McGuirk et al., Phys. Rev. A 65, 033608 (2002).[4] J. B. Fixler et al., Science 315, 74 (2007).[5] M. Fattori et al., Phys. Lett. A 318, 184 (2003); G.Lamporesi et al., Phys. Rev. Lett. 100, 050801 (2008).[6] S. Merlet et al., Metrologia 45, 265 (2008).[7] M.D. Angelis et al., Meas. Sci. Technol. 20, 022001(2009).[8] P. Clade´ et al., Europhys. Lett. 71, 730 (2005).[9] G. Ferrari et al., Phys. Rev. Lett. 97, 060402 (2006).[10] F. Sorrentino et al., Phys. Rev. A 79, 013409 (2009).[11] H. Mu¨ller et al., Phys. Rev. Lett. 100, 031101 (2008).[12] G. Amelino-Camelia et al., Phys. Rev. Lett. 103, 171302(2009).[13] H. Mu¨ller, A. Peters, and S. Chu, Nature (London) 463,926 (2010).[14] S. Fray et al., Phys. Rev. Lett. 93, 240404 (2004).[15] S. Dimopoulos et al., Phys. Rev. D 78, 042003 (2008).[16] G.M. Tino and F. Vetrano, Classical Quantum Gravity 24,2167 (2007), and references therein.[17] S. Dimopoulos et al., Phys. Rev. D 78, 122002 (2008).[18] G.M. Tino et al., Nucl. Phys. B, Proc. Suppl. 166, 159(2007).[19] A. Alberti et al., New J. Phys. 12, 065037 (2010).[20] V. V. Ivanov et al., Phys. Rev. Lett. 100, 043602 (2008).[21] S. Merlet et al., Metrologia 47, L9 (2010).[22] P. Wolf et al., Nature (London) 467, E1 (2010).[23] H. Mu¨ller, A. Peters, and S. Chu, Nature (London) 467, E2(2010).[24] P. Clade´ et al., Phys. Rev. A 74, 052109 (2006).[25] T.A. Savard, K.M. O’Hara, and J. E. Thomas, Phys. Rev.A 56, R1095 (1997).[26] R. Ja´uregui et al., Phys. Rev. A 64, 033403 (2001).[27] F. Bloch and A. Siegert, Phys. Rev. 57, 522 (1940).[28] A. Alberti et al., Nature Phys. 5, 547 (2009).[29] H. B. G. Casimir and D. Polder, Phys. Rev. 73, 360(1948).[30] B.M. Peden et al., Phys. Rev. A 80, 043803 (2009).PRL 106, 038501 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending21 JANUARY 2011038501-4

F.-Y. Wang

M. Prevedelli

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Physical Review Letters

Precision measurement of gravity with cold atoms in an optical lattice
  and comparison with a classical gravimeter

Precision measurement of gravity with cold atoms in an optical lattice and comparison with a classical gravimeter

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