Detecting inertial effects with airborne matter-wave interferometry

Abstract

Inertial sensors relying on atom interferometry offer a breakthrough advance in a variety of applications, such as inertial navigation, gravimetry or ground- and space-based tests of fundamental physics. These instruments require a quiet environment to reach their performance and using them outside the laboratory remains a challenge. Here we report the first operation of an airborne matter-wave accelerometer set up aboard a 0g plane and operating during the standard gravity (1g) and microgravity (0g) phases of the flight. At 1g, the sensor can detect inertial effects more than 300 times weaker than the typical acceleration fluctuations of the aircraft. We describe the improvement of the interferometer sensitivity in 0g, which reaches 2 x 10-4 ms-2 / [SQRT]Hz with our current setup. We finally discuss the extension of our method to airborne and spaceborne tests of the Universality of free fall with matter waves.

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Received 15 Apr 2011 | Accepted 15 Aug 2011 | Published 20 Sep 2011 DOI: 10.1038/ncomms1479

R. Geiger1,2, V. Mnoret1, G. Stern1,2,3, N. Zahzam4, P. Cheinet1, B. Battelier1,3, A. Villing1, F. Moron1, M. Lours3, Y. Bidel4, A. Bresson4, A. Landragin3 & P. Bouyer1,5

Inertial sensors relying on atom interferometry offer a breakthrough advance in a variety of applications, such as inertial navigation, gravimetry or ground- and space-based tests of fundamental physics. These instruments require a quiet environment to reach their performance and using them outside the laboratory remains a challenge. Here we report the rst operation of an airborne matter-wave accelerometer set up aboard a 0 g plane and operating during the standard gravity (1 g) and microgravity (0 g) phases of the ight. At 1 g, the sensor can detect inertial effects more than 300 times weaker than the typical acceleration uctuations of the aircraft. We describe the improvement of the interferometer sensitivity in 0 g, which reaches 2 10 m s / Hz

4 2

Detecting inertial effects with airborne matter-wave interferometry

with our current setup. We nally discuss the extension of our method to airborne and spaceborne tests of the Universality of free fall with matter waves.

1 Laboratoire Charles Fabry, UMR 8501, Institut d Optique, CNRS, Univ. Paris Sud 11 , 2, Avenue, Augustin Fresnel, 91127 Palaiseau, France.2 CNES,18 Avenue Edouard Belin, 31401 Toulouse, France.3 LNE-SYRTE, Observatoire de Paris, CNRS and UPMC, 61 avenue de l Observatoire , 75014 Paris , France. 4 ONERA, DMPH, Chemin de la Hunire, 91761 Palaiseau, France.5 Laboratoire Photonique Num rique et Nanosciences, Universit Bordeaux 1, IOGS and CNRS , 351 cours de la Lib ration, 33405 Talence , France . Correspondence and requests for materials should be addressed to P.B. (email: [email protected]).

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1479

Matter-wave inertial sensing relies on the capability of controlling the wave nature of matter to build an interferometer and accurately measure a phase dierence 1,2.

As the particle associated to the matter wave senses inertial or gravitational eects, the interferometer represents an accurate inertial probe. In particular, atom interferometers (AIs) have beneted from the outstanding developments of laser-cooling techniques and reached accuracies comparable to those of inertial sensors based on optical interferometry. Because of their long-term stability, AIs oer a breakthrough advance in accelerometry, gyroscopy and gravimetry, for applications to inertial guidance 3,

geoid determinations 4, geophysics5 and metrology 6.

In addition, AIs are excellent candidates for laboratory-based tests of general relativity that could compete with the current tests that consider astronomical or macroscopic bodies 7 . For example, AIs may provide new answers to the question of whether the free-fall acceleration of a particle is universal, that is, independent of its internal composition and quantum properties. Although this principle known as the Universality of free fall (UFF) has been tested experimentally 8,9 to a few parts in 10 13 , various extensions to the current theoretical physics framework predict its violation (for a review of these theories, see ref. 10). It is thus important to test experimentally these theoretical models with dierent types of particles. AIs also open perspectives for further tests of general relativity such as the detection of gravitational waves 11 . All these fundamental tests may benet from the long interrogation times accessible on microgravity platforms 7,12,13, or in space 14.

Because of its high sensitivity, running an AI has required, until now, low-vibration and high-thermal stability environments that can only be found in dedicated ground or underground platforms. We report here the rst operation of a matter-wave inertial sensor in an aircra , both at 1 g and in microgravity (0 g). Our matter-wave interferometer uses 87 Rb atoms and operates aboard the Novespace A3000g aircra taking ofrom Bordeaux airport, France ( http:// www.novespace.fr/). Th is plane carries out parabolic ights during which 22 s ballistic trajectories (0 g ) are followed by 2 min of standard gravity ight (1 g). Th e AI measures the local acceleration of the aircra with respect to the inertial frame attached to the interrogated atoms that are in free fall. In the rst part of this communication, we describe the inertial measurements performed by our instrument and show how the matter-wave sensor achieves a resolution level more than 300 times below the plane-acceleration level. We present the general method that is used to operate the AI over a wide acceleration range and to reach such a resolution.

In the second part, we demonstrate the rst operation of a matter-wave inertial sensor in 0 g . Microgravity oers unique experimental conditions to carry out tests of fundamental physics. However, these experiments are conducted on platforms such as planes, sounding rockets or satellites, which are not perfectly free-falling, so that the residual cra vibrations might strongly limit the sensitivity of the tests. Overcoming this problem generally requires the simultaneous operation of two sensors to benet from a common mode vibration noise rejection. For example, conducting a matter-wave UFF test implies the simultaneous interrogation of two dierent atomic species by two AIs measuring their acceleration dierence 15 . In the present work, we investigate the 0 g operation of our one-species AI in a dierential conguration to illustrate a vibration noise rejection. Our achievements (0 g operation and noise rejection) constitute major steps towards a 0 g-plane-based test of the UFF with matter waves at the 10 11 level, and towards a space-based test 16 below1015 . Such an experiment in space has been selected for the next medium-class mission in ESA s Cosmic Vision 2020 22 in the frame of the STE QUEST project 17 (the

report describing the STE QUEST project is available in ref. 18 ).

Th is paper presents the rst airborne and microgravity operation of a matter-wave inertial sensor. We introduce a new and

original method that allows to use the full resolution of an atom interferometer in the presence of high levels of vibration. We also show how high-precision tests of the weak equivalence principle may be conducted with dierential atom interferometry.

ResultsDescription of the airborne atom interferometer. Our experiment relies on the coherent manipulation of atomic quantum states using light pulses 19,20 . We use telecom-based laser sources that provide high-frequency stability and power in a compact and integrated setup 21 . Starting from a 87 Rb vapour, we load in 400 ms a cloud of about 3 10 7 atoms laser cooled down to 10 K, and select the atoms in a magnetic eld insensitive ( mF=0) Zeeman sublevel. We then apply a velocity selective Raman light pulse 22 carrying two counterpropagating laser elds so as to keep 10 6 atoms that enter the AI with a longitudinal velocity distribution corresponding to a temperature of 300 nK. Th e Raman laser beams are aligned along the plane wings direction ( Y axis, Fig. 1 ) and are retroreected by a mirror attached to the aircra structure and following its motion. Th e AI consists of a sequence of three successive Raman light pulses to split, redirect and recombine the atomic wavepackets ( Fig. 1d ). Th e acceleration measurement process can be pictured as marking successive positions of the free-falling atoms with the pair of Raman lasers, and the resulting atomic phase shi

is the dierence between the phase of the two Raman lasers at the atom s successive classical positions, with respect to the retroreecting mirror 23.As

the Raman beam phase simply relates to the distance between the atoms and the reference retroreecting mirror, the AI provides a measurement of the relative mean acceleration am of the mirror during the interferometer duration, along the Raman beam axis. Th e information at the output of the AI is a two-wave interference

X Z

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Figure 1 | Description of the experiment in the plane. ( a) The parabolic manoeuvre consists of a 20 s pull-up hypergravity (1.8 g) phase, the 22 s ballistic trajectory (0 g) and a 20 s pull-out 1.8 g phase. This manoeuvre is alternated with standard gravity (1 g) phases of about 2 min and carriedout 31 times by the pilots during the ight. ( b) Picture of the experiment in the plane during a 0-g phase. ( c) Zoom in the science chamber where the atoms are laser cooled and then interrogated by the Raman laser beams (red) that are collinear to the Y axis and retroreected by a mirror (blue).( d) Space-time diagram of the AI consisting of three successive light pulses that split, reect and recombine the two matter waves represented bythe dashed and the solid lines. The blue and red arrows represent the two Raman laser beams.

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Figure 2 | AI revealing information on the plane acceleration.( a) Acceleration signal recorded by the MAs (red); the standard deviation am of the acceleration signal is about 0.5 m s 2 at 1 g and 0.2ms2 in 0 g.

( b) AI discrete measurements corresponding to the atomic uorescence of the 87Rb atoms in the F=2 state, normalized to the uorescence ofall the atoms; the total interrogation time is here 2 T=3ms. The blackand green points correspond to the 1 g and 0 g phases, respectively;we have removed the 1.8 g phases where the AI is not designed to operate. ( c,d) Atomic measurements plotted versus the signal stemming from the MAs at 1 g (c) and in 0 g (d); the sinusoidal correlations show that theAI contains information on the acceleration of the plane.

Figure 3 | High resolution measurement of the plane acceleration. ( a)

In red, signal recorded by the MAs in the time window ( 2 T, 2T) around the measurement times t i =iTc (vertical green lines), with T c =500ms and2 T=3ms. The MAs signal has been ltered by the response function ofthe AI described in the Methods section, and the red points represent the value of the signal at t i . This value determines the reciprocity region where the AI operates, delimited by two horizontal dashed lines. In this way,the MAs provide the coarse acceleration measurement (black step-like signal). ( b) The AI is then used for the high-resolution measurement within its reciprocity region, bounded by the two blue dashed lines at aR/2,with aR=

/kT 20.087ms 2. The error bars represent the noise of the atom accelerometer, which equals 0.0065 m s 2 per shot in this example (SNR=4.3). (c) The total acceleration ( am) is the sum of the black step-like signal in ( a) and of the AI measurements in ( b). (d) Full signal measured by the hybrid Mas AI sensor aboard the A300 0 g aircraft during successive 1- g and 0- g phases of the ight. For this data set, where 2 TMA=3ms,

SNR=4.3 and

T c =500ms, the resolution of the sensor in one second is more than 100 times below the plane acceleration uctuations

am.

sinusoidal signal P=P0A cos

, where

P is the transition probability between the two 87 Rb ground states, and P0 (resp. A) is the oset (resp. the amplitude) of the interference fringes. Th is signal is modulated by the atomic phase shi

=amkT 2, where k=2 2

/

780 nm is the Raman lasers eective wave vector and T is the time between the light pulses (see the rst Methods subsection for the calculation of the phase shi).

In the aircra, the acceleration along Y (Fig. 2a) uctuates over time by

am~0.5ms2 (1 s.d.), and is at least three orders of magnitude greater than the typical signal variations recorded by laboratory-based matter-wave inertial sensors. For this reason, the signal recorded by the AI rst appears as random, as shown in Figure 2b . To quantify the information contained in the atomic measurements, we use mechanical accelerometers (MAs) xed on the retroreecting mirror and search for the correlation between the MAs and the AI 24 . We use the signal aMA(t) continuously recorded by the MAs to estimate the mean acceleration aE(t i ) which

is expected to be measured by the AI at time t i =iT c , with

T c =500ms being the experimental cycle time (see the rst subsection in Methods). Plotting the atomic measurements P(t i ) versus aE(t i )

reveals clear sinusoidal correlations between the mechanical sensors and the AI, both at 1 g (Fig.2c)andin0g ( Fig. 2d ). This demonstrates that the AI truly holds information on the mirror acceleration am. We note that this result stands for the rst demonstration of the operation of an atom accelerometer in an aircra and in microgravity.

Retrieving the plane acceleration with the AI resolution. We now consider the application of our matter-wave sensor to precise measurements of the plane acceleration, by operating the AI beyond its linear range. For that purpose, we determine the AI acceleration response, dened by P

AI(am)=P0Acos(kT 2am), independently from the mechanical devices. We have developed a method (Methods) to estimate this response (that is, the parameters P0 and

A), and the signal to noise ratio (SNR) of the interferometer, which determines the acceleration noise of the sensor,

a =1/(SNRkT 2).

Th e knowledge of P0 and A enables us to extract the acceleration am(t i ) from the atomic data P(t i ) by inverting the model P AI(am). In this way, the acceleration is known within the region where the interferometer model can be inverted unambiguously and corresponding to an acceleration interval of range aR=

/kT 2.To obtain the total acceleration, we need the information on the reciprocity region ( n(t i )aR, (n(t i )+1)aR ) where the AI operates at measurement time t i , with

n(t i ) being the interference fringe number where the measurement point is located (Methods). To determine n(t i ),

we use the MAs which have a reciprocal response over a wide acceleration range. Thus, our instrument consists in a hybrid sensor that is able to measure large accelerations due to the mechanical devices, and able to reach a high resolution because of the atom accelerometer. Figure 3a c illustrate the measurement process that we use to measure the plane acceleration during successive 1 g and 0 g phases of the ight ( Fig. 3d ), with high resolution.

Main error sources. Because of their limited performances, mainly their nonlinear response and intrinsic noise (Methods), the MAs provide a signal that is not perfectly proportional to the acceleration am. Th is leads to errors on the estimated acceleration aE that blur the MAs AI correlation function and might prevent from nding the fringe index n(t i ) where the AI operates. These errors increase with the AI sensitivity and with the acceleration signal am . For a given acceleration level, the good measurement strategy consists in increasing T up to TMA where the MAs can still resolve the correlation fringes. Th is will set the scale factor kTMA2 of the AI.

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1479

Th e sensitivity of the accelerometer is then determined by the SNR of the matter-wave sensor, which is estimated independently from the MAs with our method. For instance at 1 g, where

am~0.5ms2, the MAs enables us to increase the AI interrogation time up to 2 TMA=6ms and to resolve the correlation fringes. Future improvements will rely on the use of well-characterized MAs, in particular on a calibration of their scale factor at the 10 3 level over the sensitivity bandwidth of the AI, to achieve interrogation times 2 TMA>20ms.

High-frequency (>10 Hz) vibration damping could also be used to constrain the frequency range where the MAs are needed, and, therefore, push forward a particular sensor technology.

Second, parasitic inertial eects due to the rotation of the plane might be experienced by the matter-wave sensor and not by the MAs. At 1 g, the atoms fall down before interacting with the Raman beams (aligned along the Y axis), so that the interferometer has a physical area and is thus sensitive to the Sagnac eect. The resulting Coriolis acceleration, measured by the AI and not by the MAs, might impair the correlation and limit the performance of the hybrid sensor. For shot-to-shot uctuations of the plane rotation of the order of 10 3rads1 , we estimate a limit to the sensitivity at the 10 4ms 2 level at 1 g. Th is error source may be signicantly reduced in the future with the use of extra sensors to measure the rotation of the plane and to take it into account in the calculation of the estimated acceleration.

Finally, the atomic SNR limits the sensitivity of the inertial sensor. During the ight, we measure at 1 g a SNR of 3.1 for 2 T = 6 ms, which is in agreement with the value measured in our laboratory for the same interrogation time. With our experimental setup, the signal ( A~0.1) is essentially limited by the imperfections of the atomic beam splitters and mirror due to the temperature of the cloud and to the gaussian intensity prole of the Raman beams, whereas the noise is mainly due to detection noise. In these conditions, with Tc = 500 ms,theacceleration noiseof theAIequals 1 6 10 3 2

. /

+ .It results, in the linear range of the interferometer, from the quadratic sum of two independent contributions: the atomic phase noise sAI SNR

1/ , and the vibration noise

vib not measured by the

MAs. As we can estimate

AI independently from the correlation (see the second subsection in Methods), the comparison of

corr and

AI indicates whether the sensor sensitivity is limited by the atomic noise (

corr

AI ) or by the residual vibration noise (

corr >

AI).In

Figure 4e,f , we have represented

corr (error bars) and

AI (vertical spacing between the green lines) for the one-loop and the two-loop interferometers, respectively. Figure 4e shows that the sensitivity of the three-pulse sensor is limited by the vibration noise not measured by the MAs, as

corr >

AI . On the contrary, Figure 4f reveals that the atomic phase noise is the main limit to the sensitivity of the four-pulse sensor, as

corr

AI.Th e SNR in the four-pulse interferometer is less than in its three-pulse counterpart as the extra light pulse reduces the interference fringe contrast due to the Raman beam intensity inhomogeneities and the transverse temperature of the cloud. In spite of greater atomic noise, the quality of the correlation 4f is better than that of correlation 4e, which shows that the four-pulse sensor operates a vibration noise rejection.

In the four-pulse geometry, the two elementary interferometers operate one aer another and share accelerations of frequency below1/2T . In a UFF test, two AIs of equal scale factor ( kT 2)will interrogate two dierent atoms almost simultaneously, so that the noise rejection is expected to be much more effi cient (see ref. 16 and Supplementary Information ). In that case, the precision of the inertial sensor might be limited by the atomic phase noise, that is, by the SNR of the interferometer. For the 3-pulse AI data in Figure 4c, e (2 T=20ms), we estimate a SNR of 2.1, which corresponds to an acceleration sensitivity of the matter-wave sensor of 2 10 /

4 2

m s Hz , in the context of the high-vibration noise rejection expected for the UFF test. Th at sensitivity level may be greatly improved in the future by using a highly collimated atomic source in microgravity 13.

Discussion

We nally discuss possible improvements of our setup, both for inertial guidance and fundamental physics applications. In the former case, increasing the resolution of the accelerometer will be achieved by using well-characterized MAs to increase TMA, and by improving the SNR of the AI. Reaching an interrogation time 2 TMA=40ms in

=

2 2

m s Hz . At this sensitivity level, the hybrid sensor is able to measure inertial eects more than 300 times weaker than the typical acceleration uctuations of the aircra. We emphasize that reaching such a high resolution is possible because of the appropriate combination of MAs (Methods), the success of operating the AI in the plane, and the use of our method for the acceleration measurement. In the present conguration of the experiment, the SNR degrades at 1 g when 2T increases above 20 ms because the atoms fall down and escape the Raman and detection beams 12.Th is limitation could be overcome because of extra Raman beam collimators and by changing the orientation of the detection lasers. In 0 g , the experiment falls with the atoms and the SNR is not constrained by gravity any more. The SNR may also be improved signicantly in the future because of a better detection system (for example, more stable detection lasers) and the use of ultra-cold atoms.

Dierential measurement in 0 g . We focus now on the micro-gravity operation of the matter-wave inertial sensor and on its possible application to fundamental physics tests such as that of the UFF. Such a test can be carried out with two AIs measuring the acceleration of two dierent atoms with respect to the same mirror that retroreects the Raman lasers. In airborne or space-borne experiments, the mirror constitutes an ill-dened inertial reference because of the cra s vibrations. This might degrade the sensitivity of the test, unless the vibrations impact the two interferometers in the same way.

Vibration noise rejection occurs when conducting dierential measurements such as in the operation of gradiometers 25 or

gyroscopes 26, and is expected in UFF tests based on atom interferometry 16.Th e rejection effi ciency depends on the two species used for the UFF test and is maximum for simultaneous interrogation of the two atoms ( Supplementary Information ). In the case of

a nite rejection, the impact of the acceleration noise can further be reduced by measuring the vibrations of the mirror with MAs to substract them from the dierential phase measurement. In this way, the MAs are used to release the requirements on the vibration damping of the cra as they measure the accelerations not rejected in the dierential operation of the two AIs. The effi ciency of that technique is limited by the performances of the MAs as their imperfections (for example, their nonlinearities) translate into residual vibration noise impairing the dierential acceleration measurement. In the following, we investigate vibration noise rejection by operating our 87Rb sensor in a dierential mode. We use a sequence of four light pulses to build a two-loop AI 25 that is equivalent to two successive one-loop interferometers head to tail ( Fig. 4b ). The four-pulse AI provides a signal resulting from the coherent substraction of two spatially and temporally separated inertial measurements and is therefore expected to be less sensitive to the low-frequency inertial eects.

To illustrate the noise rejection, we operate the three-pulse and four-pulse interferometers with a same total interrogation time of 20 ms, and compare the MAs AI correlation for each geometry ( Fig. 4 ). Figure 4c,d show that the quality of the correlation (ratio of the sinusoid contrast to the mean error bar) is clearly improved in the dierential geometry compared with the 3-pulse interferometer. To understand this dierence, we estimate the total noise of the correlation, dened as s s s

corr AI vib

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Figure 4 | Comparison of two AI geometries in 0g. ( a,b) 3-pulse and four-pulse interferometers considered in this work. ( c,d) Corresponding MAs AI correlation functions recorded during the 0 g phase of consecutive parabolas, with total interrogation times of 2 T=20ms and 4T=20ms, respectively. To obtain these plots, we have sorted the estimated phase data (200 and 180 points, respectively) and averaged the correlation points by packetsof 20. The error bars equal the standard deviation of each packet divided by 20 , and are transferred to the vertical axis in this averaging procedure. The red line is a sinusoidal t to the points. To make the comparison of the two correlations easier, we have scaled the vertical axis so as toobtain the same amplitude for the two sinusoids. ( e,f) Comparison of the total correlation noise

corr (error bars) and of the atomic phase noise

sAI SNR

= 1/( 20)

for the two interferometer geometries (the vertical spacing between the green lines is 2

AI). The atomic phase data {

(t i )} have

been obtained from the atomic measurements { P(t i )} by inverting the AI model P

AI(

), in the linear region of the interferometer ranging from

/4 to

3

/ 4. To facilitate the comparison of

corr and

AI, we have set

E=

(the points are thus aligned on the rst bisector). Figure ( e) (resp. ( f)) shows that the sensitivity of the sensor is limited by the vibration noise

vib not measured by the MAs (respectively by the atomic phase noise).

the plane (where the acceleration uctuations are

am~0.5ms2r.m.s. at 1 g ) would require MAs whose scale factor frequency response is determined with a relative accuracy of 2 10 4. We believe that such precision can be achieved with state-of-the-art MA technology, for example, capacitive MicroElectroMechanical Systems (MEMS) sensors, and we now work at implementing these sensors in our setup. Together with a shot noise limited AI with SNR ~ 200 (as in ref. 26), the resolution of the hybrid sensor would be 8 10 7ms 2 per shot, which would represent a major advance in inertial navigation as well as in airborne gravimetry. We note that the present analysis does not report the in-ight bias of the matter-wave sensor because no other airborne accelerometer of similar accuracy was available onboard to proceed to the comparison. However, it has been demonstrated in laboratories that atom accelerometers can reach biases of the order of 10 8ms 2 under appropriate conditions 27.

Tests of fundamental physics would also require perfor mant MAs to remove the residual aircra s or satellites acceleration noise not rejected in the dierential measurement. For a UFF test in the 0 g plane and a vibration rejection effi ciency of 300 (explained in ref. 16 and in the Supplementary Information ), a dierential acceleration sensitivity of 3 10 10 m s 2 per shot

(SNR = 200, 2T = 2 s) could be achieved with MAs of 2104 relative accuracy, if the vibrations during the 0 g phase are damped to the5104 m s 2 level. In space, high-performance MAs such as the sensors developed for the GOCE mission could be used to determine the residual accelerations of the satellite ( ~ 10 6 m s 2)

with a resolution 28 of the order of 10 12 m s 2. Th e vibration rejection effi ciency of 300 would thus limit the impact of the acceleration noise on the interferometric measurement to 3 10 15 m s 2

per shot, which would stand for a minor contribution in the error budget. Th erefore, high-precision test of the equivalence principle could be conducted in space without the strong drag-free constraints on the satellite that represent a major challenge in current space mission proposals.

To conclude, we have demonstrated the rst airborne operation of a cold-atom inertial sensor, both at 1 g and in micro gravity. We have shown how the matter-wave sensor can measure the

cra acceleration with high resolutions. Our approach proposes to use mechanical devices that probe the coarse inertial eects and allow us to enter the ne measurement regime provided by the atom accelerometer. In the future, instruments based on the combination of better characterized mechanical sensors and a shot noise limited AI could reach sensitivities of the order of few 10 7 m s 2 in one second aboard aircras. Th us, our investigations indicate that sensors relying on cold-atom technology may be able to detect inertial eects with resolutions unreached so far by instruments aboard moving cra s characterized by high accele ration levels. Cold-atom sensors oer new perspectives in inertial navigation because of their long-term stability as they could be used to correct the bias ( ~ few 10 5 m s 2 ) of the traditional sensors monitoring the cra s motion, below the 107 m s 2

level. In geophysics, airborne gravity surveys may also benet from the accuracy of AIs 29.

Moreover, we have operated the rst matter-wave sensor in microgravity. We have shown how dierential interferometer geometries enable to reject vibration noise of the experimental platform where new types of fundamental physics tests will be carried out. Our result in 0 g suggests that the high sensitivity level of matter-wave interferometers may be reached on such platforms, and support the promising future of AIs to test fundamental physics laws aboard aircras, sounding rockets or satellites where long interrogation times can be achieved 17 . While many quantum gravity theories predict violations of the UFF, AIs may investigate its validity at the atomic scale, with accuracies comparable to those of ongoing or future experiments monitoring macroscopic or astronomical bodies 30.

MethodsAI response function and MAs AI correlation. For a time-varying acceleration a(t) of the retroreecting mirror, the phase of the interferometer at time t i =iTc is given by

( ) ( , ) ( ) ,

t k f t t a t t

i i

= [tildenosp] d (1)(1)

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Retroreflecting mirror

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P(ti)

Figure 5 | Estimation of the AI response and SNR. PDF of the AI measurements P(t i ) (normalized atomic uorescence) for the data of

Figures 2 and 3 . P0 and A are, respectively, the offset and the amplitude of the interference fringes, and

P is the standard deviation of the atomic

noise.

P0 A

2

Acceleration am ([afii9843]/kT 2)

Figure 6 | Schematic of the two-step acceleration measurement process. The signal of the MAs is ltered by the response function f(t) of the AI (described in the rst paragraph of the Methods section) and informs on the reciprocity region where the AI operates at the measurement timet i =iT c . In this example, the reciprocity region corresponds to the 0

/kT

2

where f is the acceleration response of the 3-pulse AI. It is a triangle-like function 16 that reads:

f t t t t if t t t T

T t t if t t T t T

i

interval, that is, to n(t i )=0. The value provided by the AI,

P(t i ), is then used to rene the acceleration measurement within the reciprocity region. The acceleration [tildenosp]

a ti

( ) is obtained from P(t i ) by inverting the AI response P

AI(am)

(red curve).

( , ) [ , ]

( ) [ , ],

[tildenosp] [tildenosp][tildenosp] [tildenosp]

[tildenosp][tildenosp] [tildenosp]

= +

+ +

2 2

(2)(2)

(3)(3)

(4)(4)

(5)(5)

i i ii i i

with [tildenosp]ti being the time of the rst Raman pulse at the ith measurement. Th e mean acceleration that we infer is dened as am=

illustrates the method for the data used in Figures 2, 3 , corresponding to the interrogation time 2 T=3ms. The tted parameters are P0=0.50, A=0.074 and SNR=4.3.

Th is method is independent on the MAs signal and informs both on the response of the AI, and on its one-shot acceleration sensitivity

a = 1/(SNRkT 2). The characteristics of the interferometer are thus known without any extra calibration procedure. In particular, we can estimate the noise level of the matter-wave sensor, given by Tc a

s for a white atomic phase noise, and which equals 4.6 10 m s / Hz

-3 -2

/kT

2.

Because of the MAs, we estimate the phase

E(t i ) which is expected to be measured by the interferometer by averaging in the time domain, the signal aMA(t) by the AI response function:

E MA d

( ) ( , ) ( ) .

t k f t t a t t

i i

= [tildenosp]

Th e MAs AI correlation function can be written as

P P A E

=

0 cos ,

and expresses the probability to measure the atomic signal P(t i )

at time t i , given the acceleration signal aMA(t ) recorded by the MAs. Th e estimated acceleration used in Figure 2 is dened by aE = E/kT2.

For simplicity, we have neglected, in equation (1), the Raman pulse duration

=20

s with respect to the interrogation time 2 T.

Th e exact formula can be found in ref. 31 and has been used in the data analysis to estimate the phase

E.

Estimation of the AI response and signal to noise ratio. We calculate the probability density function (PDF) of the AI measurements P(t i ) and t it with the PDF of a pure sine (a twin-horned distribution) convolved with a gaussian of standard deviation

P .The t

function reads:

F N

( ) [ ( ) ] exp( ( ) ),

for the data in Figure 5 , where T c = 500 ms.

Our analysis estimates the SNR by taking into account only the atomic noise due to detection noise or uctuations of the fringe oset and contrast. It does not account for the laser phase noise that could impact the sensitivity of the acceleration measurement for long interrogation times 32 . However, we demonstrated the low phase noise of our laser system during previous parabolic ight campaigns 12 , which is at least one order of magnitude below the estimated atomic phase noise for the interrogation times we consider in this work ( T 10ms). Th erefore, we have neglected in this communication the laser phase noise contribution when evaluating the sensitivity of the sensor.

Determination of the acceleration signal. Because of the MAs signal, we rst determine the fringe number n(t i )=oor[ aE(t i )/aR]

where the interferometer operates at time t i =iT c , with

aR=

0 2 1 2 2

2

/ s p s

where N is a normalization factor. In this way, we estimate the amplitude A and the oset value P0 of the interference fringes, that is, we estimate the AI response P

AI(

)=P0Acos

/kT 2 being the reciprocity interval of the matter-wave sensor. The values n(t i ) are represented by the black step-like curve in Figure 3a .

Second, we use the atomic measurements P(t i ) to deduce the acceleration [tildenosp]

a ti

( ) measured by the AI in its reciprocity region [ n(t i )aR, ( n(t i )+1)aR], and given by:

a t kT

P P t A

x x P x

A

x x P P

=

+

d 1

1

2 2

. The tted

parameters A and

P provide an estimate of the in-ight signal-to-noise ratio of the interferometer, given by SNR =

A/

P . Figure 5

[tildenosp][notdef]( i i

) arccos( ( )).

=

1

2

0 (6)(6)

6

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ARTICLE

Th e total acceleration a ti

m( ) is nally computed as:

a t n t a a t if n t

i n t a a t if n t

15. Fray, S., Diez, C. A., Hnsch, T. W. & Weitz, M. Atomic interferometer with amplitude gratings of light and its applications to atom based tests of the equivalence principle.Phys. Rev. Lett. 93, 240404 (2004).

16. Varoquaux, G. et al. How to estimate the dierential acceleration in a two-species atom interferometer to test the equivalence principle . New. J. Phys. 11, 113010 (2009).

17. Th e space missions selected by ESAs Cosmic Vision 2020 22 are listed here: http:// sci.esa.int/science-e/www/object/index.cfm?fobjectid=48467 (visited July 2011).

18.http://www.exphy.uni-duesseldorf.de/Publikationen/2010/STE-QUEST_nal. pdf (visited July 2011).

19.Bord, C. J. Atomic interferometry with internal state labeling.Phys. Lett. A 184, 12 (1989).

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21. Lienhart, F. et al. Compact and robust laser system for rubidium laser cooling based on the frequency doubling of a ber bench at 1560nm.Appl. Phys. B: Lasers Opt. 89, 177180 (2007).

22. Kasevich, M. et al. Atomic velocity selection using stimulated raman transitions.Phys. Rev. Lett. 66, 22972300 (1991).

23.Storey, P. & Cohen-Tannoudji, C. Th e feynman path integral approach to atomic interferometry.J. Phys. II France 4, 19992027 (1994).

24. Merlet, S. et al. Operating an atom interferometer beyond its linear range . Metrologia 46, 8794 (2009).

25.McGuirk, J. M., Foster, G. T., Fixler, J. B., Snadden, M. J. & Kasevich, M. A. Sensitive absolute-gravity gradiometry using atom interferometry . Phys. Rev. A 65, 033608 (2002).

26.Gauguet, A., Canuel, B., Lvque, T., Chaibi, W. & Landragin, A.

Characterization and limits of a cold-atom sagnac interferometer . Phys. Rev. A 80, 063604 (2009).27. Merlet, S. et al. Comparison between two mobile absolute gravimeters: optical versus atomic interferometers . Metrologia 47, L9L11 (2010).

28. Marque, J.- P. et al. Th e ultra sensitive accelerometers of the esa goce mission . In 59th IAC Congress, Glasgow, Scotland, 29 September-3 October 2008. Available at http://www.onera.fr/dmph/goce/IAC-08-B1.3.7.pdf.

29. Wu, X. Gravity Gradient Survey with a Mobile Atom Interferometer. Ph.D. thesis, Stanford University. http://atom.stanford.edu/WuTh esis.pdf (2009).

30.Touboul, P., Rodrigues, M., Mtris, G. & Tatry, B. Microscope, testing the equivalence principle in space C.R . Acad. Sci. Paris 4, 12711286 (2001).

31. Cheinet, P. PhD Th esis, Universit Pierre et Marie Curie, page 37, http://tel.archives-ouvertes.fr/tel-00070861/en/ (2006).

32. Nyman, R. et al. I.c.e.: a transportable atomic inertial sensor for test in microgravity.Appl. Phys. B: Lasers Opt. 84, 673681 (2006).

33. Cheinet, P. et al. Measurement of the sensitivity function in time-domain atomic interferometer.IEEE Trans. Instrum. Meas 57, 11411148 (2008).

Acknowledgements

Th is work is supported by CNES, DGA, RTRA-Triangle de la Physique, ANR, ESA,ESF program EUROQUASAR and FP7 program iSENSE. Laboratoire Charles Fabry and LNE-SYRTE are part of IFRAF. We are grateful to Alain Aspect and Mark Kasevich for discussions and careful reading of the manuscript, and to Linda Mondin for her productive implication in the ICE project. We thank Andr Guilbaud and the Novespace sta for useful technical advice.

Author contributions

R.G. and V.M. built and operated the experiment, took the data and analysed them. G.S. designed and operated the experiment and took the data. N.Z. and B.B designed the experiment and took part in the data-taking runs during parabolic ight campaigns. P.C. took part in the data-taking runs and in the data analysis. A.V., F.M. and M.L. contributed to the electronics used in the experiment. Y.B. contributed to the design of the laser system used in the experiment. A.B., A.L. and P.B. designed the experiment, coordinated the I.C.E. collaboration and took part in the data-taking runs. R.G. wrote the manuscript, which was improved by V.M., G.S., N.Z., B.B., P.C., A.L. and P.B. P.B. is the principal investigator of the project.

Additional information

Supplementary Information accompanies this paper at http://www.nature.com/ naturecommunications

Competing nancial interests: Th e authors declare no competing nancial interests.

Reprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/

How to cite this article: Geiger, R. et al. Detecting inertial eects with airborne matter-wave interferometry. Nat. Commun. 2:474 doi: 10.1038 / ncomms1479 (2011).

License: Th is work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivative Works 3.0 Unported License. To view a copy of this license, visit http:// creativecommons.org/licenses/by-nc-nd/3.0/

R

[tildenosp]

is even

(7)(7)

m

( ) ( ) ( ) ( )

= ( ( ) ) ( ) (

i i i i i i

++

1 )) .

is odd

R

[tildenosp]

Th e acceleration measurement process is illustrated in Figure 6 .

Limitations due to the Mechanical Accelerometers. Th e main limitation of the MAs comes from the nonlinearities in their frequency response (phase and gain), which means that the signal aMA(t) is not exactly proportional to the acceleration of the retro reecting mirror. Th is results in errors in the estimation of the phase

E that impair theMAsAIcorrelation.Th e nonlinearities typically reach amplitudes

nl ~ 5 % within the AI bandwidth that equals 1 / 2 T 500Hz. (Th e acceleration frequency response H(

) of an AI has been measured in ref. 33 and corresponds to the response of a second-order low pass lter of cut-o frequency 1/2T). To reduce them, we combine two mechanical devices of relatively at frequency response within two complementary frequency bands: a capacitive accelerometer (Sensorex SX46020) sensing the low-frequency accelerations (01Hz), and a piezoelectric sensor (IMI 626A03) measuring the rapid uctuations (1 500 Hz). In this way, we achieve

nl~2%.

For 2 TMA=6ms, these nonlinearities correspond to errors on

E of about 0.5 rad r.m.s. Further details on the errors in the estimation of the phase due to the MAs nonlinearities are given in the Supplementary Information .

Another limitation comes from the MAs internal noise, especially that of the capacitive one whose noise level integrated in (01 Hz) equals 3104 m s 2. Th is noise is about one order of magnitude lower than this due to the nonlinearities of the MAs, and is independent of the acceleration level in the plane. Axis cross-talk of the MAs of the order of 2 % is taken into account for the estimation of the phase, so that the errors on

E due to the MAs-axis coupling are negligible. Finally, the bias of the capacitive accelerometer is specied at the 0.05 m s2 level, so that the mean estimated acceleration might dri during the ight (4 h). Th is results in a displacement of the dark-fringe position in the MAs AI correlation plots.

References

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7

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