June 1, 2002 / Vol. 27, No. 11 / OPTICS LETTERS 951

Method of phase extraction between coupled atominterferometers using ellipse-specific fitting

G. T. Foster, J. B. Fixler, J. M. McGuirk,* and M. A. Kasevich

Department of Physics, Yale University, 217 Prospect Street, New Haven, Connecticut 06520-8120

Received November 19, 2001

We present a method of analysis involving ellipse-specific fitting of sinusoidally coupled data from two gravime-ters in a gradiometer configuration. This method permits rapid extraction of induced gradient phase shiftsin the presence of common-mode vibrational phase noise. Gravity gradients can be accurately measured inthe presence of large vibrational accelerations. © 2002 Optical Society of America

OCIS codes: 020.0020, 120.5050, 000.2780.

A significant limitation of gravimetry is vibration ofthe reference platform, which appears as accelerationnoise as a consequence of the equivalence principle.A gradiometer rejects this noise by taking the differ-ence of two simultaneous acceleration measurementssharing a common reference frame. We developedand demonstrated an accurate and sensitive gravitygradiometer based upon atom interferometric tech-niques.1,2 Common-mode vibrational-noise rejectionhas been demonstrated to a high level, but accurateextraction of small gravitational gradient signals wasdifficult if the vibrationally induced phase noise waslarger than p�2 rad. We have now implemented atechnique that is inherently immune to common-modephase noise and allows extraction of accurate gravitygradient phase shifts without the necessity of activevibration isolation of the reference platform.

The key idea is that the signal from the two gravime-ters are sinusoids that parametrically describe anellipse. Common phase noise in the two sine signalsdistributes the data points around the ellipse but doesnot change the ellipticity. By fitting of an ellipseto the data sets, the phase shift can be rapidly andaccurately determined even in the presence of largecommon phase noise. In this Letter we detail theapplication of this technique to our experiment.

The data take the form of two sinusoidal signals witha relative phase difference, Df:

x � A0 sin�f� 1 B 0, y � C 0 sin�f 1 Df� 1 D 0.

(1)

In the limit of Df � p�2 and A0 � C 0, the data form acircle in the x y plane centered at �x, y� � �B 0,D 0�. Inthe extreme case of Df � 0 (or p), the data collapse toa line.

The general algebraic form of a conic is

a ? x � Ax2 1 Bxy 1 Cy2 1 Dx 1 Ey 1 F � 0 , (2)

where x � �x2,xy, y2,x, y, 1� and a � �A,B,C,D,E,F �T . We can express the phase difference, Df,in terms of the conic parameters:

Df � cos21�2B�2pAC � . (3)

0146-9592/02/110951-03$15.00/0

To fit the ellipse parameters, we employ an ellipse-specific f itting routine developed for pattern recogni-tion and vision simulation.3 This technique finds thealgebraic coefficients a that minimize the sum of thesquared algebraic distances to the conic for the datapoints:

minkDak2, (4)

where D � �x1x2 . . .xn�T is the n 3 6 design matrix forn data points. To avoid the trivial solution a � 0 andkeep the system determined, we constrain the f it to bean ellipse through the discriminant: B2 2 4AC , 0.We are free to scale the data so that the constraintbecomes an equality, B2 2 4AC � 21, allowing theminimization problem to be solved through the use ofLagrange multipliers. This reformulates the probleminto determining the solutions of a matrix eigenvalueequation with a constraint. The eigenvector with thepositive eigenvalue provides real parameters that sat-isfy the ellipse constraint.3

Our gravity gradiometer experiment is described indetail elsewhere.1,2 The gradiometer consists of twogravimeters separated by 1.4 m that share a commonvertical measurement axis. Each gravimeter consistsof laser-cooled Cs atoms that are trapped in a magneto-optical trap and then launched via light-induced forceson a ballistic trajectory. During the 320-ms atomicfountain, a series of Doppler-sensitive two-photonRaman laser pulses4 drives transitions between the Cs6S1/2 F � 3 and F � 4 hyperf ine ground states. Thelaser acts as a phase reference, while the atoms accruea gravitationally induced phase shift throughout thefountain. Following a p�2 p p�2 interferometersequence, the probability of finding an atom in theF � 4 state is given by5

Pj4� �12

�1 2 cos�f0 1 Dfg�� . (5)

The induced gravitational phase shift is given byDfg � 2keffgT 2, where keff is the effective laser wavevector, g is the gravity over the interferometer path,and T is the time between interferometer pulses.f0 � f1 2 2f2 1 f3 contains the laser phase at eachpulse. A p pulse denotes a pulse envelope area such

© 2002 Optical Society of America

952 OPTICS LETTERS / Vol. 27, No. 11 / June 1, 2002

that the Cs atoms undergo complete population rever-sal from one hyperfine state to the other. A p�2 pulsecreates a superposition of states. We use balanceddetection with modulation transfer spectroscopy6 todetect the F � 4 population. We acousto-opticallyscan the phase of the f irst interferometer p�2 pulse,f1, to map the sinusoidal interference fringe givenby Eq. (5). In the absence of vibrational noise in f0,the laser phase is precisely known, and the phaseof the interference fringe yields the gravitationalacceleration.

We previously2 showed that the difference phase sup-presses common-mode vibrations at a level of betterthan 140 dB for Df � 0. In this case, both sinusoidsare in phase, and the common-mode performance of thesystem can be characterized through direct subtractionof appropriately normalized signals by use of Gaussianelimination. In the absence of vibrations, the relativephase can be directly determined by fitting of sinusoidsto the data sets. We previously showed that this tech-nique is also effective at suppressing small commonphase noise. In a noisier environment, an active servois required for keeping the system noise in a regime inwhich sinusoids can be fitted to the data sets.

Figure 1(a) shows typical signals from the twogravimeters plotted individually for a T � 150 msDoppler-sensitive interferometer in which commonphase noise dominates (uncorrelated amplitude noiseis estimated at the 100:1 level of the detected atomsignal). The large phase noise prevents a sinusoidalleast-squares fit from producing accurate results.Figure 1(b) reveals the underlying elliptic constraintof the two data sets. The spread of the data aroundthe ellipse is due to the presence of the large amountof phase noise. If no phase noise were present, thedata would be bunched closer to the scan phases aboutthe ellipse, as shown with simulated data in Fig. 1(c).

To validate the accuracy of phase extraction throughellipse f itting, we apply a phase shift that can beindependently measured. We use a magnetic biaspulse that is applied for 66.7 ms during the f irst halfof the fountain in the lower chamber via Helmholtzcoils. The atoms in the fountain are state selected tobe mf � 0, so they are insensitive to the first-orderZeeman shift but do experience a second-order Zeemanshift. This phase shift can be measured indepen-dently with a p�2 p�2 microwave clock experimentduring the fountain instead of the Doppler-sensitiveinterferometer. This exhibits the same sensitivityto the magnetic bias phase shift but is insensitive toaccelerations. We least-squares fit a sinusoid to themicrowave fringes and calibrate the phase shift for agiven applied coil current with the microwave clock.The ellipse-fitting algorithm is used to extract themagnetic bias shift applied during Doppler-sensitiveinterferometer sequences. Figure 2 compares the ex-tracted magnetic phase shift with the known appliedshift. Over more than a 2p rad phase shift, there is aclose correspondence to at least a part in 100, limitedby the statistical uncertainty of the measurements.We have also found the ellipse and least-squaresfitting to produce statistically consistent results forthe low-noise microwave clock fringes.

A small deviation is evident at the applied phases ofp�2 and 3p�2. The inset of Fig. 2 expands the viewof the deviation near p�2. We plot the measuredbias shift, which we determine by subtracting out theEarth’s �p�2 rad gravity gradient phase shift as mea-sured with the Doppler-sensitive interferometer fromDf. At an applied bias phase of p�2 rad, Df � 0,and the data from the two chambers are in phase andnearly describe a line. The ellipse-f itting methodcannot accurately fit the phase near Df � 0 �mod p�when amplitude noise is present. This deviation

Fig. 1. (a) Signals from the two gravimeters exhibitinga high level of vibrationally induced phase noise. Thephase of each interferometer is scanned over 2p every 16data points. The top data plot is offset by one amplitudeunit for clarity. (b) Upper trace �y� versus the lower trace�x�, demonstrating the elliptic constraint of the data. Thesolid curve is an ellipse fit to the data. (c) Simulatedscanned data without vibrational phase noise. We haveincluded amplitude noise at the same level as the data.

Fig. 2. Phase extracted by ellipse fitting of Doppler-sensi-tive data (high phase noise) with a series of increasing ap-plied magnetic bias pulse phase shifts. The inset showsthe deviation from linearity of the ellipse-fit phase shiftaround the applied magnetic shift of p�2 rad. The boldcurve in the inset shows this behavior exhibited in simu-lations with amplitude noise of a percent. The line is thefit from the entire data set in the main plot.

June 1, 2002 / Vol. 27, No. 11 / OPTICS LETTERS 953

is well characterized in simulations, and we do notoperate near these phases. The magnetic bias pulsecan be applied to shift away from these positions ifnecessary.

The ellipse-analysis method has been proved to beimmune to the presence of common-mode phase noise.In addition to measurement of the Earth’s gravita-tional gradient and magnetic phase shifts, we are usingthe ellipse technique to measure gravitational phaseshifts from a 600-kg Pb test mass that produces asignal of �150 mrad when moved near the lowergravimeter. We have calibrated the accuracy of theellipse-fitting technique for detecting small gravita-tional phase shifts by applying a known magnetic biaspulse phase shift to null the signal from the test mass.We then measure the phase shift produced by the testmass without the nulling bias f ield and compare thedirectly measured phase shift with the inferred shiftfrom the magnetic bias used to null the gravitationalsignal. The phase shifts agree within the statisticaluncertainty of our measurements.

We tested the ellipse-f itting routine with simulateddata for a range of amplitude and phase noise levels,including the noise environment of the gravity gra-diometer. The results indicated no dependence oncommon-mode phase noise. Simulations show thatthe phase uncertainty is determined by the level ofamplitude noise. The standard deviation of the ex-tracted phase is proportional to the level of amplitudenoise present up to noise at a rms level of 20% ofthe signal amplitude. Uncorrelated phase noise sig-nificantly affects the extraction of an accurate phasedifference by spreading the data points within thelimits of the box determined by the ellipse amplitudes.At a noise level of 0.05 rad rms, the accuracy of thefit begins to deviate at a level larger than the meanuncertainty. This deviation is expected since theellipse method is contingent upon data that share adefinite phase relation. Uncorrelated phase noise inour gradiometer is far below this level.

Ellipse fitting by minimizing the squared algebraicdistance to each point is more sensitive to outlyingpoints outside of the ellipse than to outlying pointsinside the ellipse. The sensitivity to points outsidethe ellipse permits easy rejection of these points. Fornoisy data lying inside the ellipse, a rejection cut canbe made based on the ellipse parameters. By requir-ing minimization of the sum of the squares of the dis-tances to the ellipse, i.e., the geometric distance, di

2 ��kz 2 xik�2, from points �xi� to the ellipse �z�, one mayreduce this sensitivity. The fast algebraic f it is thenuseful for providing the initial estimates to be used in

an iterative, geometric f it. For typical interferometerdata, we found that the phase extracted by algebraicfits [Eq. (2)] versus various iterative geometric f its7

with initial parameters determined from an ellipse-specific fit agreed to within the statistical uncertaintyof the data. The ellipse fit can also be biased awayfrom the appropriate phase value if the data do notcover the entire ellipse. One can avoid this bias byfitting larger number of data points or by f ilteringout data sets that do not meet a distribution require-ment. Finally, the geometric criteria provide an effec-tive means for rejecting outlying data points.

In conclusion, we have implemented a method of ana-lyzing data from our gravity gradiometer that exhibitsimmunity to common-mode phase noise. This methodallows fast, accurate extraction of phase shifts with-out additional vibration isolation. Such a techniquemay be applicable to a variety of experiments outsidethe area of gradiometry. For example, systems withquadrature outputs that share common phase noise,such as in optical interferometers or homodyne de-tection, could benef it. Another possibility is in pre-cision measurements used to search for changes in thefine structure constant a via interspecies clock com-parisons.8 The method also might be useful in otheratom interferometry experiments, such as photon re-coil measurements for measurement of h̄�m.9

This work was supported by the U.S. Off ice of NavalResearch and NASA. G. T. Foster’s e-mail address isgreg@amo.physics.yale.edu.

*Present address, JILA, National Institute of Stan-dards and Technology, Boulder, Colorado 80309.

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