Determination of the Newtonian Gravitational Constant Using Atom Interferometry

0.22 MB·English

by  G. Lamporesi

Determination of the Newtonian Gravitational Constant Using Atom Interferometry
G. Lamporesi1, A. Bertoldi1, L. Cacciapuoti2, M. Prevedelli 3, and G.M. Tino1∗
1Dipartimento di Fisica and LENS, Universit`a di Firenze - INFN
Sezione di Firenze, Via Sansone 1, 50019 Sesto Fiorentino, Italy
2ESA Research and Scientific Support Department, ESTEC,
Keplerlaan 1 - P.O. Box 299, 2200 AG Noordwijk ZH, The Netherlands
3Dipartimento di Chimica Fisica e Inorganica, Universit`a di Bologna, V.le del Risorgimento 4, 40136 Bologna, Italy
(Dated: February 2, 2008)
We present a new measurement of the Newtonian gravitational constant G based on cold atom
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0 interferometry. Freely falling samples of laser-cooled rubidium atoms are used in a gravity gra-
0 diometer to probe the field generated by nearby source masses. In addition to its potential sen-
2 sitivity, this method is intriguing as gravity is explored by a quantum system. We report a value
of G=6.667·10−11m3kg−1s−2, estimating a statistical uncertainty of ±0.011·10−11m3kg−1s−2
n and a systematic uncertainty of ±0.003·10−11m3kg−1s−2. The long-term stability of the instru-
a
ment and the signal-to-noise ratio demonstrated here open interesting perspectives for pushing the
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measurement accuracy below the100 ppm level.
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The Newtonian constant of gravity G is one of the terferometry. An atomic gravity gradiometer is used to
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h most measured fundamental physical constants and at measure the differential acceleration experienced by two
p the same time the least precisely known. Improving the freely falling samples oflaser-cooledrubidiumatoms un-
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knowledge of G has not only a pure metrological inter- der the influence of nearby tungsten masses. The mea-
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est, but is also important for the key role that it plays surement is repeated in two different configurations of
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in theories of gravitation, cosmology, particle physics, the source masses and modeled by a numerical simula-
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a in geophysical models, and astrophysical observations. tion. From the evolution of the atomic wavepackets and
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s However,theextremeweaknessofthegravitationalforce the distribution of the source masses, we evaluate the
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and the impossibility of shielding the effects of gravity expected differential acceleration, having G as unique
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s make it difficult to measure G, while keeping systematic free parameter. A value for the Newton’s constant of
y
effects well under control. Many of the measurements gravity is finally extracted by comparing experimental
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p performed to date are based on the traditional torsion data and numerical simulations. Proof-of-principle ex-
[ pendulum method [1], direct derivation of the historical periments with similar schemes using lead masses were

experiment performed by Cavendish in 1798. Recently, alreadypresentedin[15,16]. Inthepresentwork,specific
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many groups have set up new experiments based on dif- effortshavebeendevotedto the controlofsystematicef-
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0 ferent concepts and with completely different systemat- fects related to atomic trajectories,positioning of source
8 ics: a beam-balance system [2], a laser interferometry masses, and stray fields. In particular, the high density
5 measurement of the acceleration of a freely falling test of tungsten is crucial in our experiment to compensate
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mass[3],experimentsbasedonFabry-Perotormicrowave for the Earth gravity gradient. In both configurations of
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1 cavities [4, 5]. However, the most precise measurements thesourcemasses,atominterferometerscanthereforebe
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available today still show substantial discrepancies, lim- operatedinspatialregionswheretheoverallacceleration
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iting the accuracy of the 2006 CODATA recommended is slowly varying and the sensitivity of the measurement
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: value for G to 1 part in 104. From this point of view, to the initial position and velocity of the atoms strongly
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the realization of conceptually different experiments can reduced.
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X help to identify still hidden systematic effects and there-
r fore improve the confidence in the final result. Our atom interferometer uses light pulses to stimu-
a late 87Rb atoms on the two-photon Raman transition
Cold-atominterferometryhasdemonstratedoutstand-
between the hyperfine levels F = 1 and F = 2 of the
ing performances for the measurement of tiny rotations
ground state [17]. The light field is generated by two
andaccelerationsanditis widelyusedformanyapplica-
counter-propagating laser beams with wave vectors k1
tions: precisionmeasurementsofgravity[6],gravitygra- and k2 ≃ −k1 aligned along the vertical axis. Atoms
dient[7],androtationoftheEarth[8,9],butalsotestsof
interact with the Raman lasers on the three-pulse se-
theEinstein’sweakequivalenceprinciple[10],testsofthe quence π/2 − π − π/2 which splits, redirects, and re-
Newton’s law at short distances [11], and measurement
combines the atomic wavepackets. At the end of the
of fundamental physical constants [12, 13]. Applications
interferometer, the probability of detecting the atoms
of these techniques for fundamental physics experiments in the state F = 2 is given by P2 = 1/2·(1−cosΦ),
in space are under study [14].
where Φ represents the phase difference accumulated by
In this paper, we present a new determination of the the wavepackets along the two interferometer arms. In
Newtonian constant of gravity based on cold-atom in- the presence of a gravity field, atoms experience a phase
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eters take place at the center of the vertical tube shown
in Fig. 1. In this region, surrounded by a system of two
µ-metalshields(76dB attenuationfactorinthe axialdi-
rection),auniformmagneticfieldof∼250mGalongthe
verticaldirectiondefinesthe quantizationaxis. The field
gradientalongthisaxisislowerthan5·10−5G/mm. The
population of the groundstate is measuredin the cham-
ber placed just above the MOT by selectively exciting
the atoms on the F = 1, 2 hyperfine levels and detect-
ingthe light-inducedfluorescenceemission. We typically
detect 105 atoms on each rubidium sample after the in-
terferometer sequence.
The Raman beams are generated by two extended-
cavity diode lasers and amplified in a tapered amplifier.
An optical phase-locked loop keeps their frequency dif-
ference in resonancewith the transitionbetweenthe two
FIG.1: Schematicoftheexperimentshowingthegravitygra- hyperfinelevelsofthe87Rbgroundstate(∼6.8GHz)and
diometerset-upwiththeRamanbeamspropagatingalongthe
stabilizestheirrelativephasetoabout100mrad(rms)in
vertical direction. During the G measurement, the position
of the source masses is alternated between configuration C1 a5Hz−10MHzbandwidth[20]. Thetwoco-propagating
(left) and C2 (right). beams enter the vacuum system from the bottom of the
MOT cell with the same linear polarization,travelalong
theaxisoftheverticaltubeand,aftercrossingaquarter-
wave plate, are reflected back producing a lin⊥lin po-
shift Φ = (k1−k2)·g T2 depending on the local grav- larization scheme. Their frequency difference is ramped
itational acceleration g [18]. The gravity gradiometer down with continuous phase during the experiment cy-
consists of two absolute accelerometers operated in dif- cle to compensate for the Doppler effect and ensure that
ferential mode. Two spatially separated atomic clouds freely falling atoms remain resonantwith the laser light.
in free fall along the same vertical axis are simultane- In this way, only one pair of counterpropagating laser
ouslyinterrogatedby the sameRamanbeams to provide beams with crossed linear polarizations is able to stimu-
ameasurementofthedifferentialaccelerationinducedby late the atoms on the two-photon transition. The three-
gravity on the two samples. pulse interferometerhas a durationof2T =320ms. The
Figure 1 shows a schematic of the experiment. The πpulselasts48µsandoccurs5msaftertheatomicclouds
gravity gradiometer set-up and the configurations of the reachtheirapogees. Therefore,duringtheπ/2−π−π/2
source masses (C1 and C2) used for the G measurement pulsesequence,atomictrajectoriesarealmostsymmetric
are visible. Further details on the atom interferometer with respect to the apogees, allowing a better control of
apparatus and the source masses assembly can be found systematic effects induced by spurious magnetic fields.
in [15, 19]. A magneto-optical trap (MOT) with beams The source masses [19] are composed of 24 tungsten
orientedina1-1-1configurationcollectsrubidiumatoms alloy (INERMET IT180) cylinders, for a total mass of
fromthebackgroundvaporofthechamberatthebottom about516kg. Theyarepositionedontwotitaniumplat-
of the apparatus. Using the moving molasses technique, formsanddistributedinhexagonalsymmetryaroundthe
thesampleislaunchedverticallyalongthesymmetryaxis vertical axis of the tube (see Fig. 1). Each cylinder is
of the vacuum tube and cooled down to a temperature machined to a diameter of 100.00 mm and a height of
of 2.5µK. The gravity gradient is probed by two atomic 150.20 mm after a hot isostatic pressing treatment ap-
clouds moving in free flight along the verticalaxis of the plied to compress the material and reduce density inho-
apparatus and simultaneously reaching the apogees of mogeneities. The internal structure of the material has
their ballistic trajectories at 60 cm and 90 cm above the been analyzed using different methods: ultrasonic tests,
MOT. Such a geometry, requiring the preparation and microscope analysis, and surface studies. Finally, a de-
the launch of two samples with high atom numbers in structivetesthasbeenperformedtocharacterizetheden-
a time interval of about 100 ms, is achieved by juggling sitydistribution. We havemeasuredamaximumdensity
theatomsloadedinthe MOT:Afirstcloudof109 atoms variation of 2.6·10−3 and an average density inhomo-
is launched to 80 cm above the MOT; during its ballis- geneity of 6.6·10−4 over volume samples of about 1/40
tic flight, 5·108 atoms are loaded and launched to 90 of the whole cylinder volume.
cm; finally,the first cloudis recapturedin the MOT and Toevaluatethestabilityofthegravitygradiometer,we
launchedagaintoaheightof60cm. Shortlyafterthisse- performed a 5-hour measurement run in which the dif-
quence,the twoatomicsamplesarevelocityselectedand ferential phase shift was continuously monitored. Each
prepared in the (F = 1,mF = 0) level. The interferom- atominterferometerinthe gravitygradiometermeasures
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al
gnd) 0.6
nterference si (upper clou 000...543
I 0.3 0.4 0.5 0.6 0.3 0.4 0.5 0.6
Interference signal Interference signal
(lower cloud) (lower cloud)
1.9
d) 1.7
a
ϕ (r
∆
1.5
FIG. 2: Acceleration along the symmetry axis of the system
1.3
in configuration C1 (left) and C2 (right). The simulation in-
0 5000 10000
cludes the effect of the source masses and the Earth gravity
gradient. Thespatial extension of theatomic trajectories for Time (s)
theupperand lower interferometers is indicated bythethick
FIG.3: Typicaldatasetshowingthemodulationofthediffer-
lines. The shaded regions show the positions of the source
entialphaseshiftmeasuredbytheatomicgravitygradiometer
masses.
when the distribution of the source masses is alternated be-
tween configuration C1 (red squares) and C2 (blue squares).
Eachphasemeasurementisobtainedbyfittinga24-pointscan
the local acceleration with respect to the common ref- of the atom interference fringes to an ellipse.
erence frame identified by the wavefronts of the Raman
lasers. Therefore, even if the phase noise induced by vi-
brationsontheretro-reflectingmirrorcompletelywashes
out the atom interference fringes, the signals simulta- ment is obtained by fitting a 24-point scan of the atom
neously detected on the upper and lower accelerometer interference fringes to an ellipse. The modulation on the
remaincoupledandpreserveafixedphaserelation. As a differentialphaseshiftmeasuredbythegravitygradiome-
consequence, when the trace of the upper accelerometer ter can be resolved with a signal-to-noise ratio of 180
is plotted as a function of the lower one, experimental after about one hour. Experimental data have been col-
points distribute along an ellipse. The differential phase lected during two separate measurementruns performed
shift is then obtained from the eccentricity and the ro- in February and May 2007.
tation angle of the ellipse fitting the data [21]. We have The differential nature of the measurement ensures
evaluatedtheAllanvarianceofthedifferentialphaseshift an efficient rejection of many systematic errors: Firstly,
andverifiedthatitdecaysastheinverseoftheintegration each gravity gradient measurement is highly insensitive
time, showing the typical behavior expected for white to phase shifts appearing in common mode on the two
noise. The instrument has a sensitivity of 140mrad at interferometers [7]; in addition, the differential measure-
1s of integration time, corresponding to a sensitivity to ment performed by alternating the source masses be-
differential accelerations of 3.5·10−8g in 1s. These per- tween configuration C1 and C2 removes effects not de-
formancesarecompatiblewithanumericalsimulationas- pending on the masses position. A budget of the sys-
suming quantum-projection-noise-limiteddetection with tematic errors affecting the value of G is presented in
105 atoms. Table I. Each effect is taken into account in our nu-
The Newtonian gravitational constant has been ob- merical simulation and its contribution evaluated on the
tained from a series of gravity gradient measurements basisofthe measurementsperformedonthe relevantpa-
performed by periodically changing the vertical position rameters. Positioning errors account for uncertainties in
of the source masses between configuration C1 and C2, the positionofthe 24tungstencylindersalongthe radial
with the atoms always launched along the same trajec- and vertical direction, both in configuration C1 and C2.
tories. Figure 2 shows the acceleration along the sym- Density inhomogeneities of the source masses have been
metry axis for both configurations. Because of the high modeled. An upper bound of the systematic uncertainty
density of tungsten, the gravitational field produced by on G has been evaluated on the basis of the maximum
the source masses is able to compensate for the Earth density variation measured during the characterization
gravity gradient. Operating the interferometers close to of the tungsten cylinders. The contribution of the plat-
thesestationarypointsreducestheuncertaintyonGdue forms holding the source masses has been included as
to the knowledge of the atomic positions by two orders well. The velocity of the atomic clouds and their po-
ofmagnitude. Atypicaldatasequence usedforthe mea- sition at the time of the first interferometer pulse have
surement of G is shown in Fig. 3. Each phase measure- been calibrated performing time-of-flight measurements
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Systematic effect ∆G/G (×10−4)
VCCSRVueeyyaprrdllttiipnniiiaccoddlaareelltprrppssopsoolmdiassteiitaittnofsiiosnoosrinntmyiisnninmCChao12smsogeneity 001022......2928711 -113-1-210 m kg s) 6666....77764208 Bagley & Luther (1997) Karagioz (1998) Luo (1999; 2003) Gundlach & Merkovitz (2000; 2002) Quinn (2001) Kleinevoss (2002) Schlamminger (2002) Armstrong & Fitzgerald (2003) CODATA-2002 Schlamminger (2006) CODATA-2006 This work
G (
Initial position of the atomic clouds 0.18
6.66
Initial velocity of theatomic clouds 2.3
Gravity gradient 1
Stability of theon-axis B-field 0.3
FIG. 4: Our result of the Newtonian gravitational constant
Stability of thelaunch direction 0.6
compared to the most precise G measurements recently ob-
Total 4.6 tained and to CODATArecommended values.
TABLE I: Error budget of thesystematic shifts affecting the
G measurement.
ing the confidence in the knowledge of G. Presently, our
statistical error averages down reaching a value compa-
rable to the systematic uncertainty in about one day of
anddetecting the atomswhen crossingahorizontallight
integration time. As expected, the main contribution to
sheet twice, once on the way up and once on the way
the systematic erroron the Gmeasurementderives from
down. The knowledge of the gravity gradient is also im-
the positioning accuracy of the source masses. This er-
portant to identify the best positions for the apogees of
rorwillbereducedbyaboutoneorderofmagnitudeonce
the two atomic clouds with respect to the source masses
the position of the tungsten cylinders will be measured
(seeFig.2). Finally,specialcarehastobetakenofallthe
byalasertracker. Fromtheanalysisofsystematiceffects
systematicerrorsthatcoulddependonthesourcemasses
and the characterization of the stability of our appara-
distribution. For instance, the presence of an inhomoge-
tus, we foresee interesting perspectives for pushing the
neousmagneticfieldalongtheverticalaxisdependingon
measurement accuracy below the 100 ppm level.
thepositionofthesourcemassescanintroduceasystem-
This work was supported by INFN (MAGIA experi-
atic phase shift affecting the G measurement. However,
ment), MIUR, ESA, and EU (under contract RII3-CT-
the low magneticsusceptibility ofthe tungstenalloyand
2003-506350). G.M.T. acknowledges seminal discussions
the two µ-metal shields surrounding the interferometer
with M.A. Kasevich and J. Faller and useful suggestions
region ensure an excellent control of stray fields and re-
by A. Peters. M. Fattori, T. Petelski, and J. Stuhler
duce possible systematic errors below the measurement
contributed to the setting up of the apparatus. We are
sensitivity of our apparatus. Using a tiltmeter, we have
gratefulto A.CecchettiandB.DulachofINFN-LNF for
verified that the direction of the Raman lasers, which
thedesignofthesourcemassessupportandtoA.Peuto,
defines the sensitive axis of our interferometer, is stable
A.Malengo,andS.PettorrusoofINRIMfordensitytests
within1µradanddoesnotdependonthepositionofthe
on W masses.
source masses. At the same time, the stability of the
launchdirectionhasbeenevaluatedandanupperbound
on possible phase shifts induced by the Coriolis acceler-
ation on the differential measurement estimated.
After an analysis of the error sources affecting our ∗ Guglielmo.Tino@fi.infn.it
measurement, we obtain a value of G = 6.667 · [1] J. H. Gundlach and S. M. Merkovitz, Phys. Rev. Lett.
10−11m3kg−1s−2, with a statistical uncertainty of 85, 2869 (2000); T.J.Quinnetal., Phys.Rev.Lett.87,
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