Sensitivity limits of a Raman atom interferometer as a gravity gradiometer
F. Sorrentino, Q. Bodart, L. Cacciapuoti, Y.-H. Lien, M. Prevedelli, G. Rosi, L. Salvi, G. M. Tino
SSensitivity limits of a Raman atom interferometer as a gravity gradiometer
F. Sorrentino, Q. Bodart, L. Cacciapuoti, Y.-H. Lien, M. Prevedelli, G. Rosi, L. Salvi, and G. M. Tino ∗ Dipartimento di Fisica e Astronomia & LENS, Universit`a di Firenze,INFN Sezione di Firenze, via Sansone 1, I-50019 Sesto Fiorentino (FI), Italy European Space Agency, Research and Scientific Support Department,Keplerlaan 1, 2200 AG Noordwijk, The Netherlands Centre for Cold Matter, Department of Physics,Imperial College of London, London, SW7 2BW, UK Dipartimento di Fisica dell’Universit`a di Bologna, Via Irnerio 46, I-40126, Bologna, Italy (Dated: January 26, 2018)We evaluate the sensitivity of a dual cloud atom interferometer to the measurement of verticalgravity gradient. We study the influence of most relevant experimental parameters on noise andlong-term drifts. Results are also applied to the case of doubly differential measurements of thegravitational signal from local source masses. We achieve a short term sensitivity of 3 × − g/ √ Hzto differential gravity acceleration, limited by the quantum projection noise of the instrument. Activecontrol of the most critical parameters allows to reach a resolution of 5 × − g after 8000 s onthe measurement of differential gravity acceleration. The long term stability is compatible with ameasurement of the gravitational constant G at the level of 10 − after an integration time of about100 hours. I. INTRODUCTION
Atom interferometry provides extremely sensitive andaccurate tools for the measurement of inertial forces, find-ing important applications both in fundamental physicsand applied research [1, 2]. Quantum sensors based onatom interferometry had a rapid development during thelast two decades, and are expected to play a crucial rolefor science and technology in the next future.The performances of atom interferometry sensors havebeen already demonstrated in the measurements of grav-ity acceleration [3–6], Earth’s gravity gradient [7–9], androtations [10–13]. Experiments based on atom interfer-ometry are currently running to test the Einstein’s Equiv-alence Principle [14, 15], to measure the Newtonian grav-itational constant G [9, 16, 17] and the fine structureconstant α [18, 19], and to test fundamental physics ef-fects in atomic systems [20, 21], while experiments test-ing general relativity [5, 15, 22] and the 1 /r Newton ’slaw [23–26] or searching for quantum gravity effects [27]and for gravitational waves detection [28–30] have beenproposed. Accelerometers based on atom interferometryhave been developed for several applications includingmetrology, geodesy, geophysics, engineering prospectingand inertial navigation [8, 31–34].While the sensitivity of such quantum inertial sensorshas not yet reached its ultimate limits, recent progressesin atom optics are expected to yield further improve-ments by some orders of magnitude by increasing themomentum transfer during the interferometer sequence[35, 36]. These instruments are expected to reach theirultimate sensitivity in space where free fall conditionsallow very long interrogation times. [37–41]. ∗ guglielmo.tino@fi.infn.it One of the most interesting features of atom interfer-ometry sensors, besides their sensitivity, is the ability tocontrol systematic effects. This in turn follows from thepossibility to use the quantum nature of atom-light inter-actions as a tool to control several sources of biases. Thismakes atom interferometry sensors particularly suited forapplications requiring long term stability and accuracy.In this paper we analyze the influence of the most rel-evant experimental parameters on the stability and ac-curacy of our apparatus for gravity gradient measure-ments by atom interferometry. We also consider a spe-cific experimental configuration for the measurement ofthe gravitational signal from local source masses. In thisway, we show that the present state of our experimentis compatible with the measurement of the gravitationalconstant G with a precision of 10 − .The paper is organized in the following way: section IIdescribes the experimental apparatus and the measure-ment scheme of our gradiometer; section III describeshow to extract differential acceleration measurementsfrom the instrument raw data; in section IV we analyzethe effect of most relevant experimental parameters onthe gravity gradient and G measurements; finally, sec-tion V describes the sensitivity and long term stabilityperformance of our apparatus. II. EXPERIMENTAL APPARATUS
In the following we present the measurement principleof our apparatus. An extensive description is given in[17, 42, 43].Our experiment is based on a dual atom interferome-ter, to measure the differential acceleration between twoclouds of cold rubidium atoms in free fall, and on a wellcharacterized set of source masses, to produce a con-trolled gravity acceleration at the location of the atomic a r X i v : . [ qu a n t - ph ] D ec probes.The atom gravimeter is based on Raman light-pulseinterferometry [3]: atoms are first launched verticallyin a fountain configuration, and then illuminated by asequence of light pulses acting as beam splitters andmirrors for the atomic wave packets. The light pulsesare generated by two vertically aligned and counter-propagating laser beams, inducing two-photon Ramantransitions between the hyperfine levels of the Rb groundstate. An atom optics beam splitter consists in a π/ τ = π/ π -pulse with length τ = π/ Ω, swapping the atomic populations between the twohyperfine states. Since the two laser beams are counter-propagating, the Raman transitions result in a momen-tum exchange by an amount of (cid:126) k e = (cid:126) ( k + k ), where k and k are the wave numbers of the two Raman laserfields. The atom interferometer is composed of a se-quence of three Raman pulses separated by two equaltime intervals T , i.e. a π/ π -pulse toredirect, and a π/ P a = (1 − cos φ ) /
2, where φ is the phase dif-ference accumulated by the wave packets along the twointerferometer arms. In the presence of a uniform grav-ity field, the phase shift φ = k e gT is proportional to thegravitational acceleration g . The gravity gradiometer isobtained by operating two simultaneous gravimeters withtwo vertically separated atomic clouds illuminated by thesame Raman laser pulses. This configuration provides ameasurement of the differential acceleration between thetwo samples with an excellent common-mode rejection ofvibration noise.A scheme of our gravity gradiometer is shown in figure1 with the two typical configurations of the source masses( C and C ). We collect the Rb atoms in a magneto-optical trap (MOT) at the bottom of the apparatus. Welaunch the samples with the moving molasses techniquealong the symmetry axis of the vacuum tube, at a tem-perature of about 2 . µ K. For the gravity gradiometer weemploy two atomic clouds simultaneously reaching theapogees of their ballistic trajectories at about 60 cm and90 cm above the MOT. We cope with the short time de-lay between the two launches ( ∼
80 ms) by juggling theatoms loaded in the MOT [44]. In this way we are ableto launch about 10 atoms in each cloud. Shortly afterlaunch the atoms enter the magnetically shielded verticaltube shown in figure 1, where a uniform magnetic fieldof 29 µ T along the vertical direction defines the quan-tization axis. The field gradient along this axis is lowerthan 10 µ T/m (see section IV D 2). At this stage, the twoatomic samples are first simultaneously addressed with acombination of a Raman π pulses and resonant blow-away laser pulses to select a narrow velocity class and upper gravimeterlower gravimeterRaman beams C C source massesRaman beamsMOTpopulationdetection FIG. 1. Scheme of the gravity gradiometer (from [43]). Rbatoms are first loaded in the magneto-optical trap (MOT),and then launched vertically in the vacuum tube with themoving optical molasses method. Around the apogees of theatomic trajectories, the atoms are illuminated by a sequenceof laser pulses for the Raman interferometry scheme. Externalsource masses are typically positioned in two different config-urations ( C and C ) and the induced phase shift is measuredas a function of masses positions. to prepare the atoms in the ( F = 1 , m F = 0) state. TheRaman lasers propagate along the vertical direction fromthe bottom, and are retro-reflected on a mirror above thevacuum tube. The atom interferometry sequence takesplace around the apogee of the atomic trajectories, witha sligth asymmetry to avoid double resonance at the cen-tral π − pulse (see section IV D 2). We complete the ex-perimental cycle by measuring the normalized populationof the ground state F = 1 , φ u − φ l , which is proportional to the gravitygradient, is then obtained from the eccentricity and therotation angle of the ellipse best fitting the experimentaldata [45].For the measurement of the gravitational signal fromlocal source masses, as for the determination of G , thegravity gradient measurement is repeated in the twodifferent configurations of source masses shown in fig- FIG. 2. Two experimental ellipses obtained with 720 (left)and 79000 (right) experimental points; each point is acquiredin 1.9 s. ure 1. In this way, we are able to isolate the effect ofsource masses from other biases introduced by Earth’sgravity gradient, Coriolis forces, etc. The position ofsource masses is modulated between the two configura-tions shown in figure 1 with a period T mod (cid:39) ÷
30 min.
III. SHORT-TERM SENSITIVITY IN GRAVITYGRADIENT MEASUREMENTS
The differential gravity acceleration is calculated fromthe phase angle of an ellipse whose points ( x, y ) are thefraction of atoms in the F = 1 state of each cloud as mea-sured from the fluorescence signals. Figure 2 shows twotypical elliptical plots for 720 and ∼ (cid:40) x ( t ) = A sin( t ) + By ( t ) = C sin( t + Φ) + D (1)where the A and C parameters represent the amplitudesof the interference fringes for the upper and lower in-terferometers, and ( B, D ) are the coordinates of the el-lipse center (see eq. (7) below). Although more sophisti-cated algorithms have been proposed to retrieve Φ withBayesian estimators [46], least-squares ellipse fitting isadequate for the analysis of sensitivity and long termstability [43].
A. Detection noise and quantum projection noise
The sensitivity to gravity gradient measurement canbe modeled by including noise terms in eq. (1), i.e. aterm δ Φ describing differential phase fluctuations, twoterms δA and δC describing fringe contrast fluctuations,and two terms δB and δD describing fringe bias fluctu-ations; moreover, we add two terms δx d ( t ) and δy d ( t )respectively to the lines of eq. (1), describing additive FIG. 3. Typical plot of detection signals after the atominterferometry sequence; the two curves are for the F = 1and F = 2 channels respectively; for each curve, the twopeaks are for the lower and upper cloud, respectively. detection noise. The total fluctuations δx ( t ) and δy ( t ) ofthe atom interferometry signals depend on the parameter t ; by taking the average over t (cid:40) (cid:104) δx (cid:105) = (cid:104) δA (cid:105) + (cid:104) δB (cid:105) + (cid:104) δx d (cid:105)(cid:104) δy (cid:105) = (cid:104) δC (cid:105) + (cid:104) δD (cid:105) + (cid:104) δy d (cid:105) + C (cid:104) δ Φ (cid:105) (2)The different contributions are not easily disentangledexperimentally; in this section we give a model for de-tection noise, and in the following section we discuss theeffect of contrast and bias fluctuations.The detection signals are obtained by collecting theatomic fluorescence from the two hyperfine states in twoseparate regions (see section IV B) using independentphotodiodes. Typical photodiode signals are shown infigure 3. The population n ij of the F = i state ( i = 1 , A ij of the corresponding peakin the detection signal, i.e. n ij = η i A ij , where j = 1 , n ij depend on t . Let usassume for simplicity that the detection efficiency is thesame for the two channels, i.e. η = η . Since the sig-nal x ( t ) in eq. (1) is given by the normalized population n / ( n + n ), the detection noise can be written as δx d ( t ) = x ( t ) δn + [1 − x ( t )] δn n x (3)where n x = n + n , and δn ( δn ) is the detectionnoise for F = 1 ( F = 2) atoms. A fundamental lowerlimit to δx d is given by the quantum projection noise(QPN) δn ij = n ij . In this case eq. (3) reads δx d ( t ) = x ( t )[1 − x ( t )] /n x ; applying eq. (1) and averaging over t we obtain (cid:104) δx QP N (cid:105) = (cid:113) B (1 − B ) − A n x (cid:104) δy QP N (cid:105) = (cid:113) D (1 − D ) − C n y (4)In our typical experimental conditions, n x (cid:39) n y (cid:39) × atoms, A (cid:39) C (cid:39) . B (cid:39) D (cid:39) .
5, thusthe noise per shot amounts to δx QPN (cid:39) δy QPN (cid:39) . t is uni-formly distributed in [0; π ] and adding Gaussian noiseto each line, with standard deviation 0.0011. We calcu-lated the Allan variance for Φ as resulting from least-square fitting of simulated ellipses with contrast and biasclose to our typical values. The Allan variance σ Φ ( N )drops as the square root of the number N of points, σ Φ ( N ) = 0 . / √ N ; we repeated the simulation for dif-ferent values of A (cid:39) C , and verified that σ Φ ( N ) scaleswith the inverse of the contrast.Detection noise can be larger than the QPN limit dueto technical noise sources such as intensity and frequencyfluctuations of probe laser beams, electronic noise, straylight etc. In our setup, an upper limit to technical detec-tion noise can be estimated from the fit of the detectionpeaks (see figure 3). After removing a small amount ofcrosstalk between channels, we fit each peak to the prod-uct of a Gaussian and a fourth order polynomial h (1 + αx + βx + γx + δx ) exp (cid:20) − ( x − x ) σ (cid:21) + B (5)in order to account for the signal distortion due to thefinite bandwidth of the photodiode, so that the area A ij of a peak is given by A ij = hσ √ π (1 + βσ + 3 δσ ) (6)We collect the atomic fluorescence with two large areaphotodiodes (Hamamatsu S7510, active area 11 × ),with 1 GΩ transimpedance amplifiers. The bandwidth isof the order of a few kHz. We optimize the noise andbandwidth by bootstrapping the large capacitance of thephotodiode with a low noise JFET, as described in [47].In this way we reach a current noise level of ∼ √ Hzlimited by the Johnson noise of the 1 GΩ resistor andthe photodiode dark current. In our typical conditions,RMS fluctuations on peak height and width are δh/h ∼ δσ/σ ∼ .
004 and the noise δA ij /A ij ∼ δn ij /n ij ∼ . ∼ B. Noise on ellipse contrast and bias
Sensitivity and long term stability of the gravity gradi-ent measurement can be limited by noise in the x and y signals, by fluctuations and/or drifts in the contrast andcenter of the ellipses, and by sources of instability of theΦ value itself. The main sources of instability in ellipsecontrast, bias and phase angle are discussed in sectionIV. Let us call t e the measurement time to acquire an el-lipse. As shown in [43], it is possible to obtain a reliable value for Φ with an ellipse containing a few hundreds ofpoints. We typically use 100 ÷
700 points per ellipse, cor-responding to a measurement time t e ∼ ÷ δ Φ ofthe differential phase on time scales shorter than t e arenegligible (see section V C). On the other hand, the slowchanges in the A , B , C and D parameters occurring ona time scale longer than t e , as visible on the right of fig-ure 2 , are efficiently rejected. The short term sensitivitywill be mainly determined by detection noise, and possi-bly by fast fluctuations of ellipse contrast and position,such as those caused by changes in the detection effi-ciency (see section IV B) or in the Raman laser power (seesection IV C). Contrast and bias fluctuations on timeslonger than t e do not affect the long term stability ofgravity gradient measurement, which is thus only lim-ited by slow distortions and rotations of the ellipse, suchas those from Coriolis acceleration (see section IV C 4) ordetection efficiency changes (see section IV B 1). The fol-lowing section provides a systematic characterization ofthe influence of most relevant experimental parameterson ellipse contrast, bias and rotation angle. IV. LONG-TERM STABILITY ANDACCURACY: IMPACT OF MOST RELEVANTEXPERIMENTAL PARAMETERS
Noise sources which equally affect the upper and loweratom interferometer (i.e. vibrations, tidal effects, rela-tive phase noise of Raman lasers, etc.) are rejected ascommon mode in the gravity gradient measurement. Inthe following subsections, we will investigate those exper-imental parameters which affect the two atom interferom-eters differently; such parameters can in principle limitthe sensitivity and long term stability of gravity gradientmeasurements. Due to the double differential scheme, themeasurement of gravity signal from local source massesis even more robust with respect to noise cancellationand control of systematic effects. Indeed, the only effectswhich can affect the measurement of local source massesare those which either depend on the position of sourcemasses, or change on a time scale shorter than the cyclingtime T mod of masses positions.We separately investigated the effect of various pa-rameters. We recorded the ellipse phase angle in thetwo configurations of source masses, Φ C and Φ C , fordifferent values of each parameter α ; for each value ofthe parameter, we calculated the average ellipse angle¯Φ = (Φ C + Φ C ) / C − Φ C .From ¯Φ( α ) we can deduce requirements on the stabilityof the parameter α on time scales shorter than T mod forthe measurement of local source masses, as well as onthe long term stability of α for gradient measurements;from ∆Φ we can deduce requirements on the long termstability of α for the measurement of local source masses. FIG. 4. Time fluctuations of MOT laser intensities; the up-per plot shows the relative reading of the two photodiodesat the input of the 1 → A. Intensity fluctuations of cooling laser
The total power and intensity ratio of the six MOTlaser beams affect the number of atoms as well as thetemperature and launching direction in the atomic foun-tain. Such effects may influence the upper and lowerinterferometers differently. With the chosen launch con-figuration in the atomic fountain, the six MOT beamsare produced in two independent triplets from the out-put of a single MOPA (Master Oscillator Power Ampli-fier). The MOPA output is split in two parts, the “up”and “down” beams, which are separately controlled infrequency and amplitude with two AOMs. Each beam isthen coupled into a polarization maintaining (PM) opti-cal fiber, and sent to a 1 → → → → FIG. 5. Average and differential ellipse angle for the twoconfiguration of source masses, versus the power ratio betweenupper and lower cooling laser beams. Solid lines are leastsquares parabolic (black points) and linear (white squares)fits to the data. beams, by keeping the total power constant. The re-sults are shown in figure 5. The average angle ¯Φ is verysensitive to the intensity ratio, while there is no clear ev-idence of any variation of the difference angle ∆Φ. Froma linear fit we find that changing the up/down inten-sity ratio by 1% induces a shift of 0 . ± .
06 mrad on¯Φ. A parabolic fit of the ∆Φ data provides an upperestimate of ∼ µ rad / % to a possible small quadraticdependence. Changing the intensity ratio also modifiesthe position of the ellipse center and the ellipse ampli-tude, with sensitivity ∼ − /% and ∼ . × − /%,respectively.By recording the time of flight (TOF) of atomic cloudsfrom launch to the detection region, we observe thatchanges in the up/down intensity ratio induce a verti-cal shift of the MOT position with a coefficient of about0.1 mm/%. However, this effect cannot explain the mea-sured shift in ¯Φ; in fact, we investigated the vertical grav-ity gradient and the magnetic gradient in the interferom-eter tube (see [48] and section IV D 2): the calculated ¯Φshift from both gradients is smaller than the measured ∼ ∼ µ rad/% for the linear coefficient of both ¯Φ and ∆Φ. B. Detection
We measure the normalized number of atoms in the F = 1 and F = 2 states by fluorescence spectroscopyafter the interferometry sequence. In the detection re-gion, the atomic clouds cross two horizontal laser beams,resonant with the F = 2 → F (cid:48) = 3 transition, having ahorizontal size of 15 mm, a vertical size of ∼ ∼
20 mm. Both lasers are retro-reflected only for the upper ∼ ∼ F = 2 atoms right after detection.The two beams are split from a single laser source witha polarizing beam splitter (PBS) close to the detectionchamber. A weak repumper beam, resonant with the F = 1 → F (cid:48) = 2 transition, propagates horizontally be-tween the two probe beams, to optically pump F = 1atoms before detecting them on the F = 2 transitionin the lower interaction region. The optical intensity ofprobe and repumper beams affects the photon scatteringrate of the atoms, and thus the signal at the two detec-tion channels. Any unbalance between the efficiencies η i of the two detection channels may result in principle ina shift of the ellipse phase angle. In the following wediscuss possible sources of detection unbalance, and themagnitude of the effect on the gravity gradient measure-ment.
1. Relative efficiency of detection channels
The differential gravity acceleration is calculated fromthe phase angle of an ellipse whose points ( x, y ) aregiven by the normalized number of atoms in the F = 1state for each cloud, i.e. x = n / ( n + n ) and y = n / ( n + n ). However, atomic populations n ij are measured from the areas A ij of detection peaks. if thedetection efficiencies of the two channels are not equal,i.e. ξ = η /η (cid:54) = 1, then the Lissajous plot obtained with (cid:40) x = A A + A = n n + ξn y = A A + A = n n + ξn (7)results in a distorted ellipse. The phase angle obtainedfrom least-squares ellipse fitting depends on the relativedetection efficiency ξ ; this bias is not efficiently removedin the doubly differential scheme for G measurement. Inorder to evaluate the effect of detection unbalance onnoise and systematic error, we calculated the phase an-gle Φ and rms error δ Φ of least squares ellipse fittingversus ξ using a set of synthetic data. Both Φ( ξ ) and δ Φ( ξ ) have a minimum around ξ = 1. In order to keepthe systematic error on Φ below 100 µ rad, the relative de-tection efficiency must be calibrated to better than 3%.The systematic error depends on the noise level on theellipses: in our simulations points we applied a noise levelcomparable to that of our typical experimental data. An-other consequence of detection unbalance is a shift of theellipse center: if ξ changes, the ellipse translates alongthe x = y direction.An efficiency unbalance between the two detectionchannels may arise from either differences of size andpower of probe beams, from limited repumping efficiencyof F = 1 atoms or from geometrical differences betweenthe two optical systems for fluorescence collection. FIG. 6. Ellipse phase angle versus detection efficiency ratiofor a typical data set.
In principle, it is possible to compensate for any de-tection unbalance originated from geometrical differencesby properly adjusting the intensity ratio of probe beams.However, if the probe beams have unequal intensities,the saturation parameter is different for the two detec-tion channels; as a result, even common mode intensityfluctuations will be converted to ξ changes.Absolute calibration of the relative detection efficiency ξ at the ∼
1% level is technically challenging, due to theunavoidable differences in the geometry of collection op-tics. However, it is possible to determine the detectionunbalance introducing ξ as a parameter in eq. (7): thevalue ¯ ξ corresponding to the minimum of Φ( ξ ) or δ Φ( ξ )as determined from ellipse fitting represent our best es-timate for the effective detection efficiency ratio. Figure6 shows the Φ( ξ ) values obtained from a set of experi-mental data. The location of ¯ ξ is the same as for thecorresponding δ Φ( ξ ) curve within the experimental er-ror. We checked the consistency of our method, which isequivalent to introduce an additional parameter ξ in theleast squares fitting, by a numerical simulation. We gen-erated a large number of ellipses with contrast, bias, noiseand detection efficiency ratio similar to our experimen-tal conditions. We verified that our algorithm returnsthe correct value of Φ within ∼ µ rad. In principle,the detection efficiency ratio ξ might be different for thetwo simultaneous interferometers, due to difference in thecloud size and velocity. However, in our typical experi-mental conditions, we verified that minimizing δ Φ withrespect to two independent ξ parameters does not changeour estimate of Φ by more than ∼ µ rad.
2. Frequency fluctuations of probe laser beams
Frequency jitter of the detection light changes the scat-tering rate. During the detection sequence the scatteringrate has to be constant to allow for normalization, sincethe interferometer ports are read out sequentially. Thespectral density of frequency noise of our probe laser is ∼ Hz/ √ Hz. Given our typical values for detuning
FIG. 7. Average and differential ellipse angle for the twoconfigurations of source masses, versus power of probe laserbeams. Solid lines are least squares linear (black points) andparabolic (white squares) fits to the data. and intensity for the probe laser, and a duration of thedetection sequence ∼
15 ms for each cloud, the contribu-tion of frequency noise is below the QPN limit for 10 atoms.
3. Intensity fluctuations of probe laser beams
In our setup, the intensity ratio of probe beams is pas-sively stabilized to 0.1% with a high extinction polarizerplaced before the PBS. As a result, probe intensity fluc-tuations on time scales longer than the delay between F = 2 and F = 1 detection (i.e. ∼
15 ms) are essen-tially common mode between the two channels. However,fast fluctuations would yield noise on the measurement ofnormalized atomic population. Moreover, as seen in theprevious paragraph the ellipse phase angle, bias and con-trast can still depend on the total power of probe beams,as well as on the power of the repumping beam.We recorded the ellipse phase angle, contrast and bias,in the two configurations of source masses, for differentvalues of the total probe laser intensity I p and of theintensity ratio between the two probe beams. In bothcases, the change in detection efficiency ratio produces atranslation and a distortion of the ellipses, which mod-ify the center, amplitude and rotation angle of the bestfitting ellipse. As an example, the plot of ellipse phaseangle versus total intensity I p is shown in figure 7. Theslope of the ¯Φ( I p ) curve decreases when I p is above thesaturation intensity. Around our typical experimentalconditions, I p · ∂ ¯Φ /∂I p = − . ± .
04 mrad/%. For thesensitivity of the difference angle ∆Φ on I p we derive anupper limit of ∼ µ rad/%. The sensitivity of ¯Φ on in-tensity ratio is ∼ µ rad/%, while the sensitivity of ∆Φis lower than ∼ µ rad/%.The X and Y coordinates of ellipse center depend onthe total probe intensity at fixed ratio with a sensitivity I P · ∂B/∂I p ∼ I P · ∂D/∂I p ∼ − × − /%. The ellipse FIG. 8. Average and differential ellipse angle for the twoconfiguration of source masses, versus power of repumpinglight in the probe beam. Solid lines are least squares linearfits to the data. amplitude has a weaker sensitivity I P · ∂A/∂I p ∼ I P · ∂C/∂I p ∼ − . × − /%. The sensitivities of ellipsecenter and amplitude on intensity ratio are (1 . ± . × − /% and (0 . ± . × − /%, respectively.We also measured the effect of intensity changes inthe repumper beam. The results are shown in figure 8.Changes in the optical intensity of repumper I r modifythe detection efficiency in the F = 1 channel; around ourtypical experimental conditions, the phase angle ¯Φ( I r )decreases with repumper power with a slope I r · ∂ ¯Φ /∂I r = − . ± .
02 mrad/%. The sensitivity of the differentialphase angle ∆Φ is below 0.1 mrad/%, while ellipse centerand amplitude have sensitivities of ( − . ± . × − /%and ( − . ± . × − /%, respectively.
4. Noise and biases arising from detection of differentatomic velocity classes
Shortly before the atom interferometry sequence, weselect a narrow class of vertical velocity from the ther-mal clouds; an efficient elimination of the thermal back-ground from the velocity selected atoms is important toachieve high contrast ellipses and to control systematicshifts on the ellipse phase angle. In order to investigatethe effect of residual thermal atoms on the measurementaccuracy, we compared two different methods for veloc-ity selection. After launch in the atomic fountain, al-most all the atoms in the thermal cloud are pumped inthe F = 2 state. In the first method (single pulse selec-tion), we apply a velocity selective Raman pulse, tunedto the | F = 2 , m F = 0 (cid:105) → | F = 1 , m F = 0 (cid:105) transition,shortly after launch. Atoms in a narrow velocity class,which is Doppler shifted to resonance, are pumped intothe | F = 1 , m F = 0 (cid:105) state. The vertical velocity spread δv of selected atoms is determined by the duration τ ofthe Raman pulse, i.e. δv (cid:39) ( τ k e ) − ∼ . ∼
18 nK. Before start-
FIG. 9. Fit residuals of detection peaks after a single velocityselection. Upper plot: F = 2 state; lower plot: F = 1 state. ing the interferometry sequence, we then eliminate theresidual atoms in the F = 2 state with a 5 ms pulse(slightly divergent, circularly polarized) tuned to the cy-cling | F = 2 (cid:105) → | F = 3 (cid:105) transition. However, whenusing this method for state and velocity selection we al-ways find a non negligible fraction of thermal atoms inthe F = 1 detection, producing a wide pedestal belowthe detection peak. Figure 9 shows typical fit residualsfor F = 2 and F = 1 atoms (see section III A about peakfitting models). The thermal pedestal is indeed resultingfrom the off-resonant photon scattering from the Ramanbeams during the velocity selection pulse. In the pres-ence of a large fraction of thermal atoms, determinationof the F = 1 peak area and calculation of the normal-ized F = 1 population are not reliable. This is due toboth the large RMS error of the peak fitting, and to thefact that part of the thermal atoms do not interact withRaman lasers in the interferometry sequence, while stillbeing detected.The geometry of our apparatus prevents the possibil-ity to employ Zeeman state selection with microwavepumping. In order to eliminate the thermal pedestal inthe F = 1 channel, we implemented a different stateand velocity selection (triple pulse selection), based onthe application of three Raman pulses to transfer theatoms back and forth between the | F = 2 , m F = 0 (cid:105) and | F = 1 , m F = 0 (cid:105) states. After each Raman pulse, weapply a resonant laser pulse to blow away the remain-ing atoms in the initial state. The blow-away laser for F = 1 atoms is resonant to the | F = 1 (cid:105) → | F = 0 (cid:105) tran-sition. After the application of the triple pulse velocityselection, the F = 1 peaks show no detectable thermalpedestal, and no clear structures in the fit residuals. TheRMS of fit residuals is now the same for the two channels(see section III A). With the number of thermal atoms de-tected in the pedestal of TOF signals reduced by a factor >
30, the systematic effects on the gravity gradient mea-surements can be controlled to better than 100 µ rad.A drawback of the triple pulse velocity selection is a re-duction in the number of selected atoms by a factor ∼ ∼
70 %) efficiency of Raman π pulses. However, the reduced signal is well compensatedby a larger contrast of the interference fringes. As a re-sult, the sensitivity of gravity gradient measurement isimproved with respect to the use of single pulse velocityselection (see section V B). C. Influence of Raman lasers on noise and bias ofthe atom interferometer
Fluctuations in the frequency, intensity and alignmentof Raman laser beams may induce changes in the el-lipse phase angle. One of the two Raman lasers (mas-ter) is frequency locked, with a red detuning of 2 GHz, tothe reference laser, which is frequency stabilized to the | F = 2 (cid:105) → | F = 3 (cid:105) Rb line with the modulation trans-fer spectroscopy technique [49]. The absolute frequencyof the master laser is stable within 0.5 MHz. The otherRaman laser (slave) is phase locked to the master laser,with an offset of 6.8 GHz given by a RF synthesizer. Inour experimental conditions the effect of phase and fre-quency fluctuations of Raman lasers on the ellipse phaseangle is negligible.When the detuning of the Raman lasers is fixed, the in-tensities of Raman beams determine the Rabi frequency,i.e. the probability of the Raman transitions, as well asits spatial distribution through the inhomogeneous lightshift. We set the ratio between the optical intensity ofmaster and slave lasers, I M and I S , to the value whichcancels the first-order light-shift at the frequency detun-ing from the | F = 2 (cid:105) → | F = 3 (cid:105) resonance selected forthe Raman lasers (see next section). We fix the durationof Raman pulses, and adjust the total optical power ofthe Raman beams in order to maximize the efficiency of π pulses. Intensity fluctuations or drifts may change theellipse contrast; more importantly, they might change thevelocity class of selected atoms because of residual lightshift.
1. Effect of light shift
To precisely cancel the first order light shift, we mea-sure the vertical velocity of the atomic clouds after asingle π pulse versus the power of Raman beams. Veloc-ity changes are detected with the time of flight method,i.e. by measuring the arrival time of the atomic cloudsin the detection region.At fixed detuning of the Raman lasers, the resonantfrequency of the Raman transition can be written as f ( I M , I S ) = f + C M I M + C S I S + O ( I ) (8)where f is the unperturbed resonance, and I M and I S are the intensities of master and slave laser beams, re-spectively. For a given value of I M , we measured theposition of the upper cloud versus I S , and vice versa.Figure 10 shows the results. We determine the C M and FIG. 10. Vertical displacement of velocity selected cloudsversus power of master and slave Raman beams. Solid linesare least squares linear fits to the data. C S coefficients by a linear fit to experimental data. Byproperly setting the intensity ratio I M I S = − C S C M (9)we can cancel the first order light shift independently onthe total Raman power. We determine this optimal ratioto be C S /C M = − . ± .
2. Intensity fluctuations of Raman lasers
In order to estimate the effect of Raman laser inten-sities on the gravity gradient measurement, we recordedthe ellipse phase angle for different values of the total Ra-man intensity I M + I S and of the intensity ratio I M /I S in the two configurations of source masses. The behaviorof ¯Φ and ∆Φ versus the intensity ratio of Raman laserbeams is shown in figure 11. The shift of ¯Φ is maxi-mum for an intensity ratio around 0 . ± .
05. From aparabolic fit we determine a curvature of 20 ± µ rad/% around the maximum. From the ∆Φ plot we extract alimit of 0.1 µ rad/% for the sensitivity of the differentialangle on the Raman intensity ratio.In a similar way we determine the influence of totalRaman intensity when I M /I S = 0 .
45. We measure a ¯Φsensitivity of 0 . ± .
04 mrad/%, which can be attributedto the combination of residual first-order light shift andsecond order light shift. The sensitivity of ∆Φ on totalintensity ratio is below 0.1 µ rad/%.Changing the intensity ratio and overall intensity ofRaman beams also modifies the position of the ellipsecenter, with sensitivity ∼ − /%, and the ellipse ampli-tude, with sensitivity ∼ . × − /%. FIG. 11. Average and differential ellipse angle for the twoconfiguration of source masses, versus intensity ratio of Ra-man lasers. Solid lines are least squares linear (black points)and parabolic (white squares) fits to the data.
3. Alignment fluctuations of Raman beams
We align the Raman beams along the vertical directionwith sub-mrad precision with the aid of a liquid mirror.However, small fluctuation in the propagation directionof Raman beams would couple with the gravity gradientmeasurement through the Coriolis effect.Assuming a small inclination θ of the k e vector alongthe East-West direction, and assuming that the upperand lower atomic clouds are launched vertically with ini-tial velocities v u (cid:39) . v l (cid:39) . φ C = − E k e T ( v u − v l ) cos α l sin θ (10)where Ω E is the Earth’s rotation rate and α l is the lat-itude angle at the location of our laboratory. In ourcase, with T = 160 ms and α l (cid:39) ◦ , the Coriolis shift is φ C (cid:39) − θ . All of the mountings for the optics deliveringthe Raman beams are chosen to be extremely rigid, andfluctuations in the propagation direction are essentiallydominated by the tilt of the retro-reflection mirror whichis mounted on the top of the structure holding the sourcemasses. We directly observed the effect of mirror tilt onthe ellipse phase angle. The Raman retro-flection mir-ror is mounted on a precision, dual-axis tiltmeter, thatmeasures the inclination θ x and θ y along two axes. The y axis is oriented along the West-East direction withina few degrees. We recorded several ellipses for differentvalues of the mirror tilt θ y , by keeping θ x constant, andvice versa. The results of average and differential ellipsephase angle versus θ y are shown in figure 12.From a linear fit to the ¯Φ data, we derive a sensi-tivity of − ± − ± θ x , which0 FIG. 12. Average and differential ellipse angle for the twoconfiguration of source masses, versus tilt of the Raman mir-ror along the y direction. Solid lines are linear fits to thedata. is compatible with eq. (10) assuming an angle of ∼ ◦ between the x axis and the North-South direction.From the ∆Φ data, there is no evidence of any di-rect effect of mirror tilt on the differential phase angle.Nevertheless, the influence of Coriolis shift on G mea-surement is not negligible, because of a tiny deformationof the mechanical structure which is induced by sourcemasses. In fact we observe that the vertical translation ofsource masses induces a tilt of the Raman mirror, so that θ y changes by ∼ µ rad between the two configurations.This results in a bias of ∼ µ rad on ∆Φ, correspondingto a systematic error of 6 × − on G . This bias can beeasily reduced by either correcting the ∆Φ data for themeasured mirror tilt in post-processing, or by activelystabilizing the angle of the mirror with PZT actuators.
4. Earth’s rotation compensation
As long as the atoms are launched with some resid-ual horizontal velocity along the East-West direction,the Coriolis force yields a phase shift on the atom in-terferometer output. This represents a source of bothsystematic errors and noise. The systematic error onthe gravity gradient measurement is proportional to theEast-West component of the average velocity differencebetween the two atomic clouds (see eq. (10)). Such effectwould cancel out in the doubly differential measurementof local masses, provided that the atomic velocities donot change when moving the source masses. Accordingto eq. (10), a change ∆ θ in the launching direction of theatomic fountain would produce a change ∆ φ ∼ − θ in the differential ellipse angle. In order to keep thesystematic effect on G measurement within ∼
50 ppm,i.e. ∆ φ C < µ rad, it is necessary to control possiblechanges ∆ θ in the launching direction within ∼ µ rad,i.e. to measure the shift in the center of atomic distribu-tion with micrometer precision. FIG. 13. Effect of Raman mirror rotation on the atom inter-ferometer sensitivity. The plot shows the fit error on ellipsephase angle and the fringe contrast versus the Raman mir-ror rotation rate along the North-South direction. The fringecontrast is defined as 2 A with reference to eq. (1). Lines areparabolic fits to experimental data. On the other hand, the horizontal velocity spread cor-responding to the ∼ µ K transverse atomic temperatureis expected to contribute to the noise on the ellipse phaseangle via Coriolis effect. This is shown in figure 13; weapply a uniform rotation rate to the retro-reflecting Ra-man mirror during the atom interferometry sequence bymeans of PZT actuators, as suggested in [50, 51]. Figure13 shows the rms error of ellipse phase angle versus themirror rotation rate; the rotation axis is roughly orientedalong the North-South direction. The optimal rotationrate, corresponding to the maximum contrast, is equiva-lent to the opposite of the local projection of the Earthrotation rate on the horizontal plane. In such conditions,the rms noise on ellipse fitting is minimum. As a result,the compensated error on ellipse angle is ∼
50 % lowerthan without compensation, while the contrast increasesby ∼ D. Effect of magnetic fields
Magnetic fields affect the atom interferometry mea-surement mainly in two ways: through the impact onatomic trajectories, and through the Zeeman shift of en-ergy levels along the cloud’s trajectories. We use severalcoils to separately create well controlled bias fields in theMOT region and in the fountain tube. In our experiment,the interferometer tube is surrounded by two concentriccylindrical µ -metal layers, that attenuate external mag-netic fields by more than 60 dB in the region of the atominterferometry sequence. The MOT and detection cham-bers, on the contrary, are not shielded.1 FIG. 14. Average and differential ellipse angle for the twoconfiguration of source masses, versus current in vertical com-pensation coils. Solid lines are parabolic fit to experimentaldata: however the black points with their error bars are alsoconsistent with a constant.
1. MOT compensation coils
The launching direction of the atoms in the fountainis sensitive to the magnetic field in the MOT region. Infact, we perform a fine tuning of the fountain alignmentby acting on the current of three pairs of Helmholtz coils,which are oriented along three orthogonal axes to createa uniform bias field at the position of the MOT. In orderto investigate the sensitivity of the gravity gradient mea-surements on the magnetic fields in the MOT region, werecorded several ellipses for different values of the cur-rent in the compensation coils. As an example, figure 14shows the plot of average and difference ellipse angle forthe two configurations of source masses, versus the biasfield produced by the vertical compensation coils. The¯Φ data clearly show the presence of maximum around31 µ T, with a curvature of 1 . ± .
06 mrad/ µ T .At our typical operating conditions (i.e. around 29 µ T)the linear sensitivity is ∼ . µ T. Since the verticalcompensation coils produce a field of 0.22 mT/A, thisconverts into a sensitivity of ∼
2. Bias field in the interferometer tube
A 1 m long solenoid inside the µ -metal tube creates auniform magnetic field B (cid:39) µ T to define the quan-tization axis during the atom interferometry sequence.The interferometer sequence is applied to atoms in the m F = 0 state; a uniform magnetic field produces a con-stant energy shift, yielding no extra phase shift due tothe symmetry of the interferometer. However, a mag-netic gradient would induce a phase shift on each of thetwo interferometers through the second order Zeeman ef-fect. The magnitude of the phase shift depends on the FIG. 15. Ellipse phase angle versus current in vertical biassolenoid, without magnetic pulse from the short coil. Thesolid line is a linear fit to experimental data. initial velocity v = gT + gt a at the first π -pulse, where t a is the time mismatch between the time at which theunperturbed cloud reaches the apogee and the π -pulse(see section II). We typically use t a (cid:39) γ , the differential phase shift inthe gravity gradiometer is δ Φ γ = παγ ( v r + 2 gt a ) T ∆ z (11)where α (cid:39) . is the differential coefficient ofquadratic Zeeman shift, v r is the recoil velocity and ∆ z is the vertical separation of the atomic clouds.We investigated the presence of magnetic gradients byrecording the ellipse phase angle versus the current i s inthe solenoid. The results are shown in figure 15. Thedata are reasonably consistent with a parabola with thevertex at i s = 0 and curvature 22 . ± . µ rad/mA ; sincethe solenoid produces a field ∂B ∂i s ∼ .
445 mT/A, thiscorresponds to ∼ µ rad/ µ T . The bias coil does notgenerate a perfectly uniform field: the magnetic gradientis proportional to the current i s in the solenoid, yieldingthe quadratic scaling. In an ideal solenoid the magneticfield would have a parabolic shape, and the theoreticaldifferential phase shift δ Φ γ would have a quadratic de-pendence of the order of ∼ , which cannotexplain the observed dependence in figure 15. We assumethat the solenoid is not ideal and derive from eq. (11) anestimate for the linear gradient in the solenoid; at ourtypical working point, i.e. 20 mA, we obtain γ (cid:39) µ T/m.On the other hand, there is no detectable stray magneticfield in the tube: an upper limit is obtained by fittingthe data with a parabola with a linear term. The vertexis at i s = 2 µ A which corresponds to 3 nT. In our typicalworking conditions, i.e. with i s (cid:39)
20 mA, the sensitivityto i s is below 0.9 mrad/mA.For the measurement of gravity gradient, it is neces-sary to extrapolate the angle at i s = 0. We obtain anangle of 580 ± . ± . × − s − for the gravity gradient.For the measurement of local source masses, a short2 FIG. 16. Ellipse phase angle for the two configurations ofsource masses versus current in vertical bias solenoid, withapplied magnetic pulse from the short coil (see text). Errorbars are not visible on this scale. Solid line are linear fits toexperimental data. (20 cm) coil creates a square pulse of length τ ∼
10 msof magnetic field, around the apogee of the lower atomiccloud, during the second half of the interferometry se-quence. During the pulse a field difference ∆ B ∼ µ Tis induced between the two clouds. The time τ is soshort that the clouds do not move by a distance overwhich ∆ B changes significantly. The corresponding ex-tra phase shift δ Φ B = 2 πατ [( B + ∆ B ) − B ] (12)is used to make the eccentricity of ellipses low enough andsymmetric in the two configurations of source masses.The magnetic gradient induced by the short coilstrongly enhances the sensitivity of ellipse phase an-gle to solenoid bias current. This fact can be used todetect possible effects of source masses on the staticmagnetic field in the interferometer tube. This ideais illustrated in figure 16, where the ellipse phase an-gle for the two configurations of source masses is plot-ted versus i s . The two plots show a linear depen-dence, in agreement with eq. (12) which predicts aslope 4 πατ ∆ B ∂B ∂i s (cid:39)
72 mrad/mA. The measured slope ∂ Φ /∂i s = 69 ± k -reversal technique (see section V C). V. HIGH PRECISION MEASUREMENT OFDIFFERENTIAL GRAVITY
The data presented in section IV allow to identify themain limits to the stability of Φ measurements, once the typical fluctuations of the parameters are known. Weconstantly monitor the value of most relevant experimen-tal parameters: the power of MOT, probe, repumper andRaman laser beams, the current in MOT compensationcoils, in pulse coil and in bias solenoid, the tilt of Ramanmirror, as well as the temperature in different points ofthe apparatus with a high resolution data logger. In thefollowing, we show how the active control of such param-eters allows to improve the precision on gravity gradientand G measurements. A. Active control of main experimental parameters
Table I summarizes the results of our characteriza-tion measurements about the influence of most relevantparameters on average and differential ellipse phase an-gle, respectively. The last two columns give the typicalRMS fluctuations of the parameters on two relevant timescales, i.e. over t e ∼ . G measurements are discussed in section V C.Table I shows that the main contributions will arisefrom instability of MOT laser beams intensity ratio,probe beams total power, current in the bias solenoidand MOT compensation coils, and tilt of the Raman mir-ror. However, noise in the coils current is fairly white,and would not entail the long term stability, while fluc-tuations in laser powers and mirror tilt exhibit a lowfrequency flickering.In order to improve the long-term stability, we activelystabilize the main experimental parameters, i.e. the op-tical intensity of cooling, Raman and probe laser beams,acting on the RF power driving acousto-optical modu-lators, and the Raman mirror tilt, acting on the piezotip/tilt system.The servo on cooling and Raman lasers intensity, aswell as on Raman mirror tilt, is implemented by meansof a slow digital loop: we sample the four powers (up anddown cooling beams, master and slave Raman beams)and the two components of mirror tilt every 72 experi-mental cycles (about 2 minutes); then we drive the RFpower of the corresponding AOMs, and the PZTs on Ra-man mirror, through a numerical loop filter. Residualfluctuations are below 0 . TABLE I. Sensitivity of average and differential phase angle, contrast and bias of ellipses to most relevant parameters.Parameter α ¯Φ( α ) slope ∆Φ( α ) slope Contrast sensitivity Bias sensitivity (cid:113) (cid:104) δα (cid:105) t e (cid:113) (cid:104) δα (cid:105) day Probe power ratio 40 µ rad/% < µ rad/% 0 . × − /% − . × − /% 0.1% 0.1%Probe power − . ± .
04 mrad/% < .
09 mrad/% − . × − /% − × − /% 0.5% 2%Repumper power − . ± .
02 mrad/% < .
01 mrad/% − . × − /% − . × − /% 0.5% 2%Raman intensity ratio 20 ± µ rad/% < . . × − /% 1 × − /% 0.5% 2%Raman total intensity 0 . ± .
04 mrad/% < . . × − /% 1 × − /% 0.5% 2%MOT total power < µ rad/% < µ rad/% < . × − /% < × − /% 0.5% 2%MOT power ratio 0 . ± .
06 mrad/% 20 ± µ rad/% . × − /% 1 × − /% 0.5% 2%vert. MOT comp. coil 56 ± µ rad/mA < µ rad/mA < . × − / mA < × − / mA 10 µ A 20 µ Abias solenoid (no pulse) 22 ± µ rad/mA n.a. n.a. n.a. 10 µ A 20 µ Abias solenoid (with pulse) 69 ± < µ rad/mA < . × − / mA < × − / mA 10 µ A 20 µ ARaman mirror E-W tilt 37 ± < < × − / mrad < × − / mrad 1 µ rad 10 µ radFIG. 17. (Color online) Allan deviation of the ellipse phaseangle in different configurations of the experiment. Data ina) correspond to the experiment status described in [17]; datain b) correspond to the experiment status described in [43],where a larger number of atoms and a faster repetition rateresulted from the implementation of a 2D-MOT and morepowerful Raman laser sources; c) resulted after minimizingthe stray light at detection photodiodes; in d) we improvedthe contrast by implementing the triple-pulse velocity selec-tion, and we reduced the technical noise on photodiodes withan improved readout electronics; in e) we further improvedthe number of atoms and applied the active stabilization ofcooling, detection and Raman laser beams intensity, and ofthe Raman mirror tilt; in f) Earth rotation was compensatedwith a piezo-driven tip tilt mirror. B. Sensitivity
In order to evaluate the sensitivity of our gradiome-ter, we split the atom interferometer data into groups of72 consecutive points, and obtain a value for Φ with itsestimated error from each group by ellipse fitting. Wethen evaluate the Allan deviation of Φ. Figure 17 showsthe Allan deviation of ellipse phase angle in different con-ditions. Several improvements of the apparatus have al-lowed to increase the number of atoms and the repetitionrate of the experiment, and also to reduce the technicalnoise at detection and increase the ellipse contrast. We currently achieve a sensitivity of 13 mrad at 1 s, in agree-ment with the calculated QPN limit for 2 × atoms,and corresponding to a sensitivity to differential accel-erations of 3 × − g at 1 s, about a factor seven betterthan in [43]. We can estimate the contribution of contrastand center fluctuations from the observed sensitivity tothe most relevant experimental parameters, as obtainedwith the same method as for the Φ sensitivity describedin section IV, and from the typical fluctuations of suchparameters on the time scale of t e . We find that δA and δB are smaller than δx d , which is in agreement with thefact that the observed sensitivity is close to the QPNlimit. Also noise δ ∆Φ on the differential phase appearsto be negligible at this stage. C. Reproducibility and long term stability
As a first test of the long term stability of our appa-ratus, we observe the statistical fluctuations of the gra-diometer measurements over about 20 hours, keeping thesource masses in a fixed position, and without active sta-bilizations of laser intensities and Raman mirror tilt. Atthe same time we monitor the value of most relevantexperimental parameters: the power of MOT, probe, re-pumper and Raman laser beams, the current in MOTcompensation coils, in pulse coil and in bias solenoid,the tilt of Raman mirror, as well as the temperature indifferent points of the apparatus. Figure 18 shows a typ-ical Allan deviation plot for a 20 hrs long measurement.For integration times τ lower than ∼
30 min the Allandeviation scales as the inverse of the square root of τ .For longer times we observe a bump, indicating a slowfluctuation of φ with a period of a few hours. The Φdata are well correlated with the measured temperatureof the laboratory. All the laser powers, as well as theRaman mirror tilt, are well correlated with the temper-ature with absolute values of the correlation coefficientsranging from ∼ . ÷ . FIG. 18. Allan deviation plots of the ellipse phase angle intwo different conditions; (upper points) without active stabi-lization of main experimental parameters; (lower points) withactive intensity stabilization of cooling and detection lasers,and Coriolis compensation.FIG. 19. Elliptical plots for configuration C of the sourcemasses; data in a) correspond to the experiment status de-scribed in [43]; data in b) resulted after reduction of technicaldetection noise, active intensity stabilization of cooling, Ra-man and detection lasers, and Coriolis compensation. in figure 18; the active control of cooling, Raman andprobe laser intensities, together with Coriolis compensa-tion, considerably improves both the short and long termstability. We reach a resolution of ∼ . ∼ × − g afteran integration time of about two hours.We tested the long term stability of the measurementof the gravitational field generated by our source massesby modulating their position as shown in figure 1. A typi-cal elliptical plot is shown in figure 19, together with thecorresponding ellipse of [43] for comparison. We movethe masses from the close ( C ) to the far ( C ) config-uration and viceversa every ∼
27 minutes, correspond-ing to 720 measurement cycles of 1.9 s each plus a dead
FIG. 20. Differential phase ∆Φ measured over 14 hours; theupper plot corresponds to the experiment status describedin [43]; lower plot corresponds to the present state of theapparatus. time of ∼ k e vector after each launch, in orderto cancel possible k e -independent systematic errors, suchas those arising from II order Zeeman shift and I orderlight shift [53]. We thus obtain two ellipses of 360 pointseach, corresponding to direct and reverse k e . We fit eachset of 360 points to an ellipse, and from each pair of el-lipses we determine the angle Φ n ( i ) = Φ dirn ( i ) − Φ revn ( i ) asthe difference between direct and reverse angles, and thestandard error δ Φ n ( i ) = (cid:113) δ Φ dirn ( i ) + δ Φ revn ( i ) . Here n = 1 , { Φ ( i ) , Φ ( i ) } a value for therotation angle ∆Φ( i ) = Φ ( i ) − Φ ( i ) due to the positionof the source masses can be obtained.Figure 20 shows two measurements of the differentialinterferometric phase ∆Φ( i ) on a period of 14 hours. Theupper plot corresponds to the experimental status de-scribed in [43]; the lower plot corresponds to the presentstate of the apparatus. The average values are not com-parable, since the positions C and C of source masseswere modified between the two measurements. The erroron each point is δ ∆Φ( i ) (cid:39) .
74 mrad. The weighted av-erage of data has a statistical error of 200 µ rad with a χ of 15. This corresponds to an uncertainty of 3 . × − after an integration time of 14 hours, expecting to reachthe 10 − level in about one week of continuous measure-ment. VI. CONCLUSIONS
We studied the sensitivity and long term stability of agravity gradiometer based on Raman atom interferome-try. We discussed the influence of the most relevant ex-perimental parameters, in particular for a measurement5of the Newtonian gravitational constant. Our experi-ment can run continuously for several days, showing areproducibility of the gravity gradient measurement atthe level of ∼ × − s − on the time scale of a fewweeks. Our measurement of the differential gravity sig-nal of source masses can reach a statistical uncertaintyof 3 . × − after ∼
10 hours of integration time.
ACKNOWLEDGMENTS
This work was supported by INFN (MAGIA exper-iment) and EU (iSense STREP project Contract No. 250072).The authors acknowledge M. Depas, M. Giun-tini, A. Montori, R. Ballerini, M. Falorsi for technicalsupport. [1] A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard,Rev. Mod. Phys. , 1051 (2009).[2] G. M. Tino and M. A. Kasevich, eds., Atom Interferom-etry,
Proceedings of the International School of Physycs”Enrico Fermi”, Course CLXXXVIII, Varenna 2013 (So-ciet`a Italiana di Fisica and IOS Press, to be published).[3] M. Kasevich and S. Chu, Appl. Phys. B , 321 (1992).[4] A. Peters, K. Y. Chung, and S. Chu, Nature , 849(1999).[5] H. M¨uller, S.-W. Chiow, S. Herrmann, S. Chu, and K.-Y.Chung, Phys. Rev. Lett. , 031101 (2008).[6] J. L. Gou¨et, T. Mehlst¨aubler, J. Kim, S. Melet, A. Cla-iron, A. Landragin, and F. P. D. Santos, Appl. Phys. B , 133 (2008).[7] M. J. Snadden, J. M. McGuirk, P. Bouyer, K. G. Haritos,and M. A. Kasevich, Phys. Rev. Lett. , 971 (1998).[8] J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden,and M. A. Kasevich, Phys. Rev. A , 033608 (2002).[9] A. Bertoldi, G. Lamporesi, L. Cacciapuoti, M. de An-gelis, M. Fattori, T. Petelski, A. Peters, M. Prevedelli,J. Stuhler, and G. M. Tino, Eur. Phys. J. D , 271(2006).[10] T. L. Gustavson, P. Bouyer, and M. A. Kasevich, Phys.Rev. Lett. , 2046 (1997).[11] T. L. Gustavson, A. Landragin, and M. Kasevich, Class.Quantum Grav. , 2385 (2000).[12] B. Canuel, F. Leduc, D. Holleville, A. Gauguet, J. Fils,A. Virdis, A. Clairon, N. Dimarcq, C. J. Bord´e, A. Lan-dragin, and P. Bouyer, Phys. Rev. Lett. , 010402(2006).[13] A. Gauguet, B. Canuel, T. L´ev`eque, W. Chaibi, andA. Landragin, Phys. Rev. A , 063604 (2009).[14] S. Fray, C. A. Diez, T. W. H¨ansch, and M. Weitz, Phys.Rev. Lett. , 240404 (2004).[15] S. Dimopoulos, P. W. Graham, J. M. Hogan, M. A. Kase-vich, and S. Rajendran, Phys. Rev. D , 122002 (2008).[16] J. B. Fixler, G. T. Foster, J. M. McGuirk, and M. Ka-sevich, Science , 74 (2007).[17] G. Lamporesi, A. Bertoldi, L. Cacciapuoti, M. Prevedelli,and G. M. Tino, Phys. Rev. Lett. , 050801 (2008).[18] R. Bouchendira, P. Clad´e, S. Guellati-Kh´elifa, F. Nez, ,and F. Biraben, Phys. Rev. Lett. , 080801 (2011).[19] S. Y. Lan, P. C. Kuan, B. Estey, D. English, J. M. Brown,M. A. Hohensee, and H. M¨uller, Science , 554 (2013). [20] M. Jacquey, M. B¨uchner, G. Tr´enec, and J. Vigu´e, Phys.Rev. Lett. , 240405 (2007).[21] J. Gillot, S. Lepoutre, A. Gauguet, M. B¨uchner, andJ. Vigu´e, Physical review letters , 030401 (2013).[22] H. M¨uller, A. Peters, and S. Chu, Nature , 926(2010).[23] G. M. Tino, Nuclear Physics B-Proceedings Supplements , 289 (2002).[24] P. Wolf, P. Lemonde, A. Lambrecht, S. Bize, A. Landra-gin, and A. Clairon, Phys. Rev. A , 063608 (2007).[25] G. Ferrari, N. Poli, F. Sorrentino, and G. M. Tino, Phys.Rev. Lett. , 060402 (2006).[26] F. Sorrentino, A. Alberti, G. Ferrari, V. V. Ivanov,N. Poli, M. Schioppo, and G. M. Tino, Phys. Rev. A , 013409 (2009).[27] G. Amelino-Camelia, C. L¨ammerzahl, F. Mercati, andG. M. Tino, Phys. Rev. Lett. , 171302 (2009).[28] G. M. Tino and F. Vetrano, Class. Quantum Grav. ,2167 (2007).[29] S. Dimopoulos, P. Graham, J. Hogan, M. Kasevich, andS. Rajendran, Physics Letters B (2009).[30] in Gen. Relativ. Gravit. , Vol. 43, edited by G. M. Tino,F. Vetrano, and C. L¨ammerzahl (2011) p. 1901.[31] A. Peters, K. Y. Chung, and S. Chu, Metrologia , 25(2001).[32] A. Bresson, Y. Bidel, P. Bouyer, B. Leone, E. Murphy,and P. Silvestrin, Appl. Phys. B , 545 (2006).[33] M. de Angelis, A. Bertoldi, L. Cacciapuoti, A. Giorgini,G. Lamporesi, M. Prevedelli, G. Saccorotti, F. Sor-rentino, and G. M. Tino, Meas. Sci. Technol. , 022001(2009).[34] G. Geneves, IEEE Trans. on Instrum. and Meas. , 850(2005).[35] H. M¨uller, S.-W. Chiow, S. Herrmann, and S. Chu, Phys.Rev. Lett. , 240403 (2009).[36] S.-W. Chiow, T. Kovachy, H.-C. Chien, and M. A. Ka-sevich, Phys. Rev. Lett. , 130403 (2011).[37] G. M. Tino, L. Cacciapuoti, K. Bongs, C. J. Bord´e,P. Bouyer, H. Dittus, W. Ertmer, A. G¨orlitz, M. Inguscio,A. Landragin, P. Lemonde, C. Lammerzahl, A. Peters,E. Rasel, J. Reichel, C. Salomon, S. Schiller, W. Schle-ich, K. Sengstock, U. Sterr, and M. Wilkens, NuclearPhysics B (Proc. Suppl.) , 159 (2007).[38] S. G. Turyshev, U. E. Israelsson, M. Shao, N. Yu,A. Kusenko, E. L. Wright, C. W. F. Everitt, M. A. Ka- sevich, J. A. Lipa, J. C. Mester, R. D. Reasenberg, R. L.Walsworth, N. Ashby, H. Gould, and H.-J. Paik, Int. J.Mod. Phys. D , 1879 (2007).[39] R. Geiger, V. Menoret, G. Stern, N. Zahzam, P. Cheinet,B. Battelier, A. Villing, F. Moron, M. Lours, Y. Bidel,A. Bresson, A. Landragin, and P. Bouyer, Nat. Commun. , 474 (2011).[40] H. M¨untinga, H. Ahlers, M. Krutzik, A. Wenzlawski,S. Arnold, D. Becker, K. Bongs, H. Dittus, H. Duncker,N. Gaaloul, C. Gherasim, E. Giese, C. Grzeschik, T. W.H¨ansch, O. Hellmig, W. Herr, S. Herrmann, E. Kajari,S. Kleinert, C. L¨ammerzahl, W. Lewoczko-Adamczyk,J. Malcolm, N. Meyer, R. Nolte, A. Peters, M. Popp,J. Reichel, A. Roura, J. Rudolph, M. Schiemangk,M. Schneider, S. T. Seidel, K. Sengstock, V. Tamma,T. Valenzuela, A. Vogel, R. Walser, T. Wendrich,P. Windpassinger, W. Zeller, T. van Zoest, W. Ertmer,W. P. Schleich, and E. M. Rasel, Phys. Rev. Lett. ,093602 (2013).[41] G. M. Tino, F. Sorrentino, D. Aguilera, B. Batte-lier, A. Bertoldi, Q. Bodart, K. Bongs, P. Bouyer,C. Braxmaier, L. Cacciapuoti, N. Gaaloul, N. G¨urlebeck,M. Hauth, S. Herrmann, M. Krutzik, A. Kubelka,A. Landragin, A. Milke, A. Peters, E. M. Rasel, E. Rocco,C. Schubert, T. Schuldt, K. Sengstock, and A. Wicht,Nuclear Physics B-Proceedings Supplements , 203(2013).[42] M. Fattori, G. Lamporesi, T. Petelski, J. Stuhler, andG. M. Tino, Phys. Lett. A , 184 (2003).[43] F. Sorrentino, Y.-H. Lien, G. Rosi, G. M. Tino, L. Cac-ciapuoti, and M. Prevedelli, New J. Phys. , 095009 (2010).[44] R. Legere and K. Gibble, Phys. Rev. Lett. , 5780(1998).[45] G. T. Foster, J. B. Fixler, J. M. McGuirk, and M. A.Kasevich, Opt. Lett. (2002).[46] J. K. Stockton, X. Wu, and M. A. Kasevich, Phys. Rev.A (2007).[47] G. Brisebois, “Low noise amplifiers for small and largearea photodiodes,” (2006).[48] G. R. et al., “Effect of atomic motion in atom interferom-etry gravity gradient measurements,” (to be published).[49] G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Or-tiz, Appl. Phys. B , 145 (1983).[50] J. M. Hogan, D. M. S. Johnson, and M. A. Kasevich,in Atom Optics and Space Physics, Proceedings of theInternational School of Physics “Enrico Fermi” ; course168 , edited by E. Arimondo, W. Ertmer, W. P. Schleich,and E. Rasel (IOS Press Amsterdam, Washington, DC,2009) p. 411.[51] S.-Y. Lan, P.-C. Kuan, B. Estey, P. Haslinger, andH. M¨uller, Phys. Rev. Lett. , 090402 (2012).[52] F. Sorrentino, A. Bertoldi, Q. Bodart, L. Cacciapuoti,M. de Angelis, Y. Lien, M. Prevedelli, G. Rosi, and G. M.Tino, Applied Physics Letters , 114106 (2012).[53] A. Louchet-Chauvet, T. Farah, Q. Bodart, A. Clairon,A. Landragin, S. Merlet, and F. Pereira dos Santos, NewJournal of Physics13