Precision measurement of gravity with cold atoms in an optical lattice and comparison with a classical gravimeter
N. Poli, F.-Y. Wang, M. G. Tarallo, A. Alberti, M. Prevedelli, G. M. Tino
aa r X i v : . [ phy s i c s . a t o m - ph ] O c t Precision measurement of gravity with cold atoms in an optical latticeand comparison with a classical gravimeter.
N. Poli, F.-Y. Wang, ∗ M. G. Tarallo, A. Alberti, † M. Prevedelli, ‡ and G. M. Tino § Dipartimento di Fisica e Astronomia and LENS, Universit`a di FirenzeINFN Sezione di Firenze, Via Sansone 1, 50019 Sesto Fiorentino, Italy (Dated: October 26, 2018)We report on a high precision measurement of gravitational acceleration using ultracold strontiumatoms trapped in a vertical optical lattice. Using amplitude modulation of the lattice intensity, anuncertainty ∆ g/g ≈ − was reached by measuring at the 5 th harmonic of the Bloch oscillationfrequency. After a careful analysis of systematic effects, the value obtained with this microscopicquantum system is consistent with the one we measured with a classical absolute gravimeter at thesame location. This result is of relevance for the recent interpretation of related experiments as testsof gravitational redshift and opens the way to new tests of gravity at micrometer scale. PACS numbers: 91.10.Pp, 03.75.Dg, 37.25.+k, 37.10.Jk
Atom interferometry, and in general methods basedon quantum interference of ultracold atoms, were largelyused in recent years for gravitational physics experimentsand new exciting prospects can be envisioned in the nearfuture [1]. For example, Raman interferometry was usedfor precise measurements of Earth’s gravitational acceler-ation g [2] and its gradient [3], for determining the valueof the gravitational constant [4, 5], for a possible redefi-nition of the kg [6], and for geophysical applications [7].Schemes based on Bloch oscillations of atoms trapped invertical optical lattices were also used to measure gravitywith the possibility of combining high sensitivity and mi-crometric spatial resolution [8–10]. The results of atominterferometry experiments were interpreted as tests ofthe isotropy of post-Newtonian gravity [11], of quantumgravity [12], and of gravitational redshift [13]. Prospectsinclude high precision tests of the weak equivalence prin-ciple [14, 15], the detection of gravitational waves [16, 17],and future experiments in space [18].So far, however, Bloch oscillation measurements hadlimited accuracy compared to Raman atom interferome-ters. Here, we present a precision measurement of grav-itational acceleration g with a new method based on ul-tracold Sr atoms trapped in an amplitude-modulatedvertical optical lattice [19] and compare the result withthe value obtained with a classical absolute gravimeterbased on a Michelson interferometer with one arm in-cluding a freely-falling corner-cube. We also improvedour previous observation of long-lived Bloch oscillations[9] and discuss the precision of the two methods for thedetermination of g . In addition to demonstrating thesensitivity and accuracy of this new method, our datacan be interpreted as a measurement of the gravitationalredshift to the Compton frequency of Sr matter waves,as suggested by H. M¨uller et al. [13]. Our data surpassesprevious Bloch oscillation measurements by one order ofmagnitude, making it the most precise test of the gravita-tional redshift based on Bloch oscillations at micrometricspatial scales. The interpretation of atom interferometer redshift tests is complicated by special relativistic timedilation since the atoms are moving [20, 21], but Blochoscillations experiments with stationary lattices providea measurement of the purely gravitational effect.The experimental setup is based on cooled and trapped Sr atoms [9] (Fig. 1). Atoms from a thermal beamare slowed in a Zeeman slower and trapped in a “blue”Magneto Optical Trap (MOT) operating on the S - P resonance transition at 461 nm. The temperature is fur-ther reduced by a second cooling stage in a “red” MOToperating on the S - P intercombination transition at689 nm. This produces about 10 atoms at a tempera-ture of 0.6 µ K. Since the force of gravity is comparableto the force produced by the red MOT on the atoms, thecloud of trapped atoms assumes a dish-like shape diskwith a vertical size of 27 µ m and a radial size of 180 µ m.The atoms are adiabatically loaded in an optical latticein 300 µ s. The lattice potential is generated by a single–mode frequency–doubled Nd:YVO laser ( λ L = 532 nm) FC PMF MPMF PD AM AOM
FM IDSM
FG5Iaser
FIG. 1: Experimental setup for the measurement of gravitywith Sr atoms trapped in a vertical optical lattice and thecomparison with a classical absolute gravimeter (FG5). M:lattice mirror; FC: frequency comb; FM: freely-falling mir-ror; SM: stationary mirror; ID: interference detector; AOM:acousto-optical modulator; PMF: polarization maintainingfiber; AM: RF generator for amplitude modulation of the laserproducing the lattice; PD: photodiode. C l oud s i z e ( m ) modulation frequency (Hz) =31 mHz FIG. 2: Spectrum recorded by modulation of the lattice depthat the 5 th harmonic of the Bloch frequency for 10.4 s with amodulation depth of 7%. The red line is a fit of experimentaldata with a sinc function. delivering up to 1 W on the atoms with a beam waistof 557(7) µ m. The beam is vertically aligned and retro-reflected by a mirror. The atomic sample in the lat-tice has a vertical RMS size of about 14 µ m and a hor-izontal size of about 100 µ m. The Bloch frequency is ν B ≃ . E R ≃ × h . In typical conditions thelattice depth ranges from 2.3 to 3 E R , while the energygap E G at the recoil momentum k L is E G ≃ E R . Thewidth of the first energy band in the lattice potential isabout 0 . × E G . Landau-Zener tunneling is negligiblein these conditions. The lattice depth is stabilized by aservo loop acting on the RF signal driving an acousto–optical modulator (AOM). The same AOM is also usedto add an amplitude modulation to the lattice potential.The atomic cloud can be imaged either in situ or withusual time-of-flight technique using resonant absorptionimaging on a CCD camera with a spatial resolution of5 µ m. The commercial lattice laser (Coherent V–5) em-ployed in the measurement is not frequency stabilizedand a precise calibration of its frequency is then required.To this purpose, part of the lattice laser light is sent to ahome–built self–referenced Ti:Sa optical frequency comb.Due to residual frequency instabilities on the time scaleof the Bloch frequency measurement, the uncertainty inthe laser frequency is ≃
100 MHz. The correction for theindex of refraction of the Sr cloud and the backgroundgas in the vacuum chamber is negligible. The verticalalignment of the lattice was checked with a precision of0.5 mrad, corresponding to a relative uncertainty of 10ppb on g , by overlapping the downward laser beam withthe reflection from the surface of water in a glass con-tainer inserted in the beam. A tiltmeter with a resolutionof 1.7 µ rad attached to the optical table was employed tocheck the alignment stability during the measurements.After loading Sr atoms in the vertical lattice, thetrap depth is modulated sinusoidally. When the mod-ulation frequency matches an integer harmonic of the Bloch frequency, atoms start to tunnel in neighbor latticesites giving rise to a net increase in the spatial verticalatomic distribution which is observed in situ by reso-nant absorption imaging (see Fig.2). The lifetime in thevertical lattice is about 20 s. This allows us to applyamplitude modulation to the lattice for ∼
10 s resultingin an increased quality factor observed on the resonanceat ν m = 5 × ν B . With a typical amplitude modulationdepth of the order of 7% we estimate a tunneling rate J l / ¯ h = 0 .
75 [19]. The recording time for a whole reso-nance spectrum is about 1 hour and leads to a maximumresolution of 150 ppb in ν B . The value of g is given by g = 2 h ν B / ( m Sr λ L ), where m Sr is the mass of Sr atomsand h the Planck constant which are both known with arelative uncertainty of ∼ × − .An important contribution to systematic shifts in grav-ity measurements with trapped neutral atoms is dueto the lattice light itself. Both the intensity and thewavevector of the lattice beam, which results from theinterference of two counter-propagating Gaussian beams,yield space-dependent terms to the potential U tot ( z ) = U s ( z ) + U l ( z ) cos(2 k ( z ) z ) − mgz , where U s ( z ) and U t ( z )depend on the squared sum and on the product of thetwo beam field amplitudes, respectively. This additionaldependence of the potential along the vertical directiongives rise to two correction terms in the typical Blochformula g = 2 h ν B / ( m Sr λ L ) + ∆ g U + ∆ g k given by thespatial derivative of the potential and the spatial deriva-tive of the difference between the Gouy phase [22] for thetwo beams at the position of the atomic cloud z at . Theshift introduced by these two extra terms is estimatedby a precise determination of the geometry of the incom-ing and the reflected trapping beams and the position ofthe cloud with respect to the beam waist, with a relativeuncertainty of 1%. Moreover, an independent determina-tion of the transverse beam size at z at has been done bymeasuring the axial and radial atomic trap oscillation fre-quencies through parametric heating technique [23, 24].For typical experimental parameters, the two terms are Effect Correction UncertaintyLattice wavelength 0 2Lattice beam vertical align. 0 0.1Stark shift (beam geometry) 14.3 ÷ ÷ < < ÷ × − ) for the gravity measurement with Sr atomsin the amplitude modulated optical lattice. th Apr, 30 th May, 28 th g atom =9.8049232(14) m/s g FG5 =9.804921609(84) m/s g ( m / s ^ ) measurement FIG. 3: (color online). Measurements of g using the am-plitude modulation technique. Each experimental point iscorrected for the systematic effects presented in Tab. I. Thered dashed line represents the weighted mean of the 21 mea-surements. The blue solid line is the value obtained with theclassical absolute FG5 gravimeter. ∆ g U = 1.53(3)x10 − m/s and ∆ g k = 1.0(2)x10 − m/s .Tidal effects were evaluated and removed from the rawdata using the same algorithm and potential model usedfor the absolute gravimeter data processing. The peak–to–peak effect of tides at our site is of the order of 2 × − m / s . Since each measurement lasts about 1 hourthe variation of g during a single measurement due totides is below 10 − m / s (i.e. below 10 ppb).We checked also for a possible dependence from mag-netic field gradients by performing a set of measurementsapplying a quadrupole magnetic field (from the MOTcoils) up to B =40 gauss/cm. The effect on ν B is smallerthan the statistical error. All the other sources of sys-tematic shift in the measurement we evaluated (spurioushigher harmonics of amplitude modulation, Bloch-Siegertshift [25]) are far below the current accuracy level. Ta-ble I summarizes the systematic shifts for the gravitymeasurements with the amplitude modulation technique.The values of individual shifts depend on the experimen-tal conditions; the quoted uncertainties are typical val-ues.The reference value for local gravitational acceleration g was provided by an absolute gravimeter based on aMichelson interferometer with one arm including a freely-falling corner-cube (FG5, Micro-g LaCoste). The mea-surement was performed in the same laboratory at a dis-tance of 1.15 m from the atomic probe position. The dif-ference in height of 14(5) cm together with the estimatedvertical gravity gradient value g zz = − . × − s − atthe laboratory site was taken into account in the dataanalysis. The result is g F G = 9 . [26].Fig. 3 presents a set of 21 determinations of g with Sr FIG. 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 thetime of release from the lattice. Displayed are the first (a), the2900th (b), the 7500th (c), and the 9800th (d) Bloch cycle. atoms. The error bars are given by the quadrature sum ofthe statistical errors coming from the fit of the amplitudemodulation resonance and the uncertainty on systematiccorrections. The standard deviation is σ = 110 ppb witha χ =30. The resulting statistical uncertainty is σ × p χ / ( n − g Sr = 9 . , in good agreement with thevalue obtained using the FG5 gravimeter.With minor modifications of the experimental proce-dure, in this work we also determined g by measuringthe frequency of the Bloch oscillations of the atoms inthe vertical optical lattice. Thanks to a better vacuumand taking advantage of the lattice modulation methodto reduce the initial momentum distribution of the atomsin the lattice, we considerably improved the visibility andduration of the oscillations and, as a consequence, the fre-quency resolution compared with previous experiments[9]. After the transfer of ultracold atoms in the verti-cal optical lattice, an amplitude modulation burst withtypical duration of 120 cycles at ν m ≃ ν B is applied.The quantum phase of the atomic wavefunction inducedby the amplitude modulation gives rise to an interfer-ence effect in time of flight image of the atomic cloudwhich results in an enhanced visibility of the Bloch oscil-lations peaks [27]. After the modulation has turned off,we let the atomic cloud evolve for a time T. Finally, weswitch off the confinement within 5 µ s to measure the mo-mentum distribution taking an absorption picture on theCCD camera of the atoms in ballistic expansion. In orderto optimize the visibility through this quantum interfer-ence effect we set the time-of-flight to 14 ms. As shownin Fig. 4, we observe Bloch oscillations lasting up to 17 s.From the the fit of the mean atomic momentum we canestimate the Bloch frequency with 170 ppb statistical un-certainty. In comparison with the determination of Blochfrequency obtained with the resonant amplitude modu-lation technique, however, we observed a considerablylarger scattering in repeated measurements, mainly dueto initial position instability of the atomic trap and alsohigher dependence on the timing of the experiment. Thevalue for g obtained with the Bloch oscillation techniqueis indeed g Bloch = 9 . , which is consistentwith the measurement discussed above but is affected bya larger uncertainty of 6 ppm.It is important to notice that the amplitude modu-lation technique for gravity measurement allows furtherimprovements of both accuracy and sensitivity. In factour result is mainly limited by the lattice wavelengthstability and by the Stark shift. The first effect couldbe lowered by using a tunable laser and locking it to anatomic line. For instance, if the wavelength is stabilizedwithin 1 MHz, the uncertainty of this effect might bereduced by two orders of magnitude. The second mainsystematic effect could be reduced either by using a blue-detuned trapping laser [22] or by increasing the latticebeam waist. Also, the sensitivity could be increased byusing higher harmonic amplitude modulation frequenciesor by modulating for a longer time.In conclusion, we performed an accurate measure-ment of gravitational acceleration using ultracold atomstrapped in a vertical optical lattice. The result is inagreement at 140 ppb level with an independent deter-mination obtained with a classical FG5 gravimeter. Thisresult represents an improvement by an order of mag-nitude over previous results [9, 28] and is of interest asa test of the gravitational redshift [13]. Moreover wedemonstrated the validity of the amplitude modulationtechnique [19] for the measurement of forces with highspatial resolution [10]. We also observed persistent Blochoscillation up to 17 s which represents the longest coher-ence time observed to date. This result might also haveimportant applications in precision measurements in con-junction with nondestructive cavity QED techniques toprobe atomic momentum oscillations [29].We thank ENI and INGV for the measurement withthe FG5 gravimeter. We also thank M. Schioppo for hiscontribution in the early stage of the experiment, D. Su-tyrin for help with absolute frequency measurements andR. Ballerini, M. De Pas, M. Giuntini, A. Hajeb, A. Mon-tori for technical assistance. 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