Abstract
Quantum technology based on coldatom interferometers is showing great promise for fields such as inertial sensing and fundamental physics. However, the finite freefall time of the atoms limits the precision achievable on Earth, while in space interrogation times of many seconds will lead to unprecedented sensitivity. Here we realize simultaneous ^{87}Rb–^{39}K interferometers capable of operating in the weightless environment produced during parabolic flight. Large vibration levels (10^{−2} g Hz^{−1/2}), variations in acceleration (0–1.8 g) and rotation rates (5° s^{−1}) onboard the aircraft present significant challenges. We demonstrate the capability of our correlated quantum system by measuring the Eötvös parameter with systematiclimited uncertainties of 1.1 × 10^{−3} and 3.0 × 10^{−4} during standard and microgravity, respectively. This constitutes a fundamental test of the equivalence principle using quantum sensors in a freefalling vehicle. Our results are applicable to inertial navigation, and can be extended to the trajectory of a satellite for future space missions.
Introduction
The field of quantum physics and atom optics is promising major leaps forward in technology for many applications, including communication, computation, memory and storage, positioning and guidance, geodesy, and tests of fundamental physics. Among these developments, the coherent manipulation of atoms with light, which exploits the particle–wave duality of matter, has led to the development of matterwave interferometers exhibiting groundbreaking precision^{1,2,3,4}—particularly for measuring inertial effects such as rotations^{5,6,7} and accelerations^{3,4,8,9}. However, the exquisite sensitivity of these quantum inertial sensors often limits their applicability to very quiet and wellcontrolled laboratory settings—despite recent efforts that have led to major technological simplifications and the emergence of portable devices^{3,10,11}. The precision of these instruments becomes particularly relevant when it comes to fundamental tests of general relativity. For instance, the universality of free fall (UFF), a cornerstone of general relativity, which states that a body will undergo an acceleration in a gravitational field that is independent of its internal structure or composition, can be probed at the quantum scale^{12,13}. Tests of the UFF generally involve measuring the relative acceleration between two different test masses in free fall with the same gravitational field, and are characterized by the Eötvös parameter
where a_{1} and a_{2} are the gravitational accelerations of the two masses. Presently, the most precise measurement of η using atom interferometry has been carried out with the two isotopes of rubidium at the level of a few 10^{−8} (ref. 14)—five orders of magnitude less precise than the best tests with classical bodies^{15,16}. This has motivated increasing the sensitivity of matterwave interferometers (which scales as the square of the freefall time) by circumventing the limits set by the gravitational free fall on Earth, either by building a largescale vertical apparatus^{1,17,18} or by letting the entire setup fall in an evacuated tower^{19}. This is also one of the main goals for spaceborne experiments^{20,21}, where the satellite can be viewed as an ideal ‘Einstein elevator’.
Our experiment, where two matterwave sensors composed of rubidium87 and potassium39 operate simultaneously in the weightless environment produced by parabolic flight (Fig. 1), represents an atominterferometric test of the UFF in microgravity. We demonstrate measurements of η with precisions of 10^{−3} during steady flight and a few 10^{−4} in weightlessness using a new interferometer geometry optimized for microgravity operation. Since the aircraft’s trajectory during parabolic flight closely mimics that of a satellite in an elliptical orbit, but with residual accelerations of ∼1% terrestrial gravity, a precise analysis of its trajectory was necessary to compute the systematic effects on the interferometer phase. This enabled us to quantify the present performance of our atomic sensors onboard the aircraft, and has direct consequences for future implementations of the strapdown inertial navigation algorithm with matterwave interferometers^{10,22,23}. This analysis has also allowed us to put strict requirements on the satellite trajectory in future space missions that target precisions of δη≃10^{−15} (refs 20, 21).
Results
Operation during steady flight
When the aircraft is in steady flight, each of the matterwave inertial sensors acts as an atombased gravimeter^{3,8,9,24}, where counterpropagating light pulses drive Dopplersensitive singlediffraction Raman transitions between two hyperfine ground states and , where p is the momentum of the atoms resonant with the Raman transition. This creates a superposition of two internal states separated by the twophoton momentum ħk^{eff}, where ħ is the reduced Planck’s constant and k^{eff}≃(4π/λ) is the effective wavevector of the Raman light (λ=780 nm for rubidium and 767 nm for potassium). Because the Raman beams are retroreflected, this transfer can occur along either the upward (−k^{eff}) or downward (+k^{eff}) directions, with an efficiency determined by the vertical velocity v of the atoms. If the velocity is large enough (for example, the Doppler shift k^{eff}·v is larger than the spectral width k^{eff}σ_{v} associated with sample’s velocity spread σ_{v}), a specific momentum transfer direction can be selected by an appropriate choice of the Raman laser frequency difference. Changing the sign of the transfer direction allows the rejection of directionindependent systematics by summing two consecutive, alternated measurements^{13,24}. For each transfer direction, the output of the interferometer is given by
where P^{0} is the mean probability of finding the atom in one interferometer output port, C is the fringe contrast and Φ^{±} is the total interferometer phase corresponding to a particular momentum transfer direction (±ħk^{eff}). This phase has contributions from the gravitational acceleration ϕ^{acc}=k^{eff}·aT^{2} (where a is the relative acceleration between the reference mirror and the atoms, and T is the freefall time between light pulses), vibrations of the reference mirror ϕ^{vib}, the total laser phase imprinted on the atoms by the Raman beams ϕ^{las}, systematic effects ϕ^{sys} and a phase corresponding to a potential violation of the equivalence principle for either atomic species.
Operation during parabolic flight
To operate in weightlessness, we introduced a new interferometer geometry consisting of two simultaneous singlediffraction Raman transitions in opposite directions, which we refer to as double single diffraction (DSD). In microgravity, the residual Doppler shift is small and the two opposite Raman transitions are degenerate. Thus, we choose a fixed Raman detuning δ within the spectral width defined by the atomic velocity distribution that simultaneously selects two velocity classes of opposite sign: ±v. This results in two symmetric interferometers of opposite area (Fig. 2), which sum to yield the output signal for a particular internal state
where 2P^{0}, C≤1/2 since the sample is initially split into two velocity classes by the first π/2pulse. The DSD interferometer signal given by equation (3) is a product of two cosines—one containing the halfsum ΣΦ=(Φ^{+}+Φ^{−}), which exhibits only noninertial contributions (ϕ^{las}, and directionindependent systematics), and one with the halfdifference ΔΦ=(Φ^{+}−Φ^{−}), which contains all inertial contributions (ϕ^{acc}, ϕ^{vib} and , and directiondependent systematics). Since noninertial and inertial contributions are now separated, we fix ϕ^{las} such that the contrast (2C cos(ΣΦ)) is maximized, and the fringes are scanned by the inertially sensitive phase ΔΦ. The DSD interferometer has the advantage of simultaneously rejecting directionindependent systematics during each shot of the experiment, since they affect only the fringe contrast. Hence, the systematic phase shift per shot is greatly reduced compared to the singlediffraction configuration.
Correlated atomic sensor measurements
Onboard the aircraft the dominant source of interferometer phase noise is caused by vibrations of the reference mirror, which serves as the inertial phase reference for both ^{87}Rb and ^{39}K sensors. Hence, the atomic signal caused by its motion is indistinguishable from motion of the atoms. To make this distinction, we measured the mirror motion with a mechanical accelerometer from which we compute the vibrationinduced phase ϕ^{vib} and correlate it with the normalized output population of each species. We refer to this process as the fringe reconstruction by accelerometer correlation (FRAC) method^{10,24,25}. Furthermore, since the two pairs of Raman beams follow the same optical pathway and operate simultaneously, the vibration noise is common mode and can be highly suppressed from the differential phase between interference fringes.
Figure 3 displays interferometer fringes for both ^{87}Rb and ^{39}K, recorded during steady flight (1 g) and in weightlessness (0 g) while undergoing parabolic manoeuvres, for interrogation times T=1 and 2 ms. Owing to the large Doppler shift induced by the gravitational acceleration, fringes recorded in 1 g were obtained with the singlediffraction interferometer along the +k^{eff} direction. Matterwave interference in 0 g was realized using the DSD configuration along both ±k^{eff} simultaneously, which requires a Doppler shift close to zero. Leastsquares fits to these fringes yield the FRAC phases , which are related to the gravitational acceleration of each species. From these fits we measure a maximum signaltonoise ratio of SNR≃8.9, and infer an acceleration sensitivity of (k^{eff}T^{2} SNR)^{−1}≃1.8 × 10^{−4} g per shot. The best performance onboard the aircraft was achieved with the Rb interferometer at T=5 ms (SNR≃7.6), which yielded 3.4 × 10^{−5} g per shot—more than 1,600 times below the level of vibration noise during steady flight (∼0.055 g).
Correlation between the potassium and rubidium interferometers is clearly visible when the same data are presented in parametric form (Fig. 3c,f). We obtain general Lissajous figures when the acceleration sensitivity of the two species are not equal^{25}, as shown in Fig. 3c. These shapes collapse into an ellipse (with an ellipticity determined by the differential phase) only when the interferometer scale factor ratio κ≃1 (Fig. 3f). This configuration is advantageous because both interferometers respond identically to lowfrequency mirror vibrations (that is, frequencies ≲1/2T), and the Lissajous shape remains fixed regardless of the commonmode phase span. We achieve this condition by ensuring the interrogation times satisfy .
Tests of the UFF
Using the sensitivity to gravitational acceleration along the z axis of the aircraft, we made a direct test of the UFF in both standard gravity and in weightlessness. The relative acceleration between potassium and rubidium atoms is measured by correcting the relative FRAC phase shift for systematic effects (see Methods), and isolating the differential phase due to a possible UFF violation
where is the ratio of interferometer scale factors when T is much larger than the Raman pulse durations^{25}. The Eötvös parameter was then obtained from , where a^{eff} is the average projection of the gravitational acceleration vector a along the z axis over the duration of the measurements. This quantity depends strongly on the trajectory of the aircraft. For our experiments, we estimate m s^{−2} and m s^{−2} during 1 g and 0 g, respectively, where the uncertainty is the 1σ variation of the projection resulting from the aircraft’s orientation. We used the Earth gravitational model EGM2008 to estimate changes in local gravity over the range of latitude, longitude and elevation during the flight and found these effects to be negligible compared with those caused by the variation in the aircraft’s roll and slope angles. The fact that is less than g originates from the large variation in the aircraft’s slope angle over a parabola (±45°). From the data shown in Fig. 3d–f, we measure an Eötvös parameter of η_{1 g}=(−0.5±1.1) × 10^{−3} during steady flight. Here the uncertainty is the combined statistical (=4.9 × 10^{−5}) and systematic (=1.1 × 10^{−3}) error—which was limited primarily by directionindependent phase shifts due to the quadratic Zeeman effect. Similarly, in microgravity we measure η_{0 g}=(0.9±3.0) × 10^{−4}, with corresponding statistical (=1.9 × 10^{−4}) and systematic (=2.3 × 10^{−4}) errors. Here the increased statistical error is a result of fewer data available in 0 g. However, the systematic uncertainty improves by a factor of ∼5 compared with measurements in standard gravity. This is a direct result of the reduced sensitivity of the DSD interferometer to directionindependent systematic effects. Both measurements are consistent with η=0.
Discussion
Although the systematic uncertainty was dominated by technical issues related to timevarying magnetic fields, the sensitivity of our measurements was primarily limited by two effects related to the motion of the aircraft—vibrational noise on the retroreflection mirror and rotations of the interferometer beams. These effects inhibited access to large interrogation times due to a loss of interference contrast, and are particularly important for future satellite missions targeting high sensitivities with freefall times of many seconds.
In addition to phase noise on the interferometer, large levels of mirror vibrations cause a loss of interference contrast due to a Doppler shift of the twophoton resonance. To avoid significant losses, the Doppler shift must be well bounded by the spectral width of the Raman transition Ω^{eff} during each light pulse. A model of this effect (Supplementary Note 1) confirms that it is most significant when the s.d. of mirror vibrations is . Figure 4a shows the mean power spectral density of vibrations onboard the aircraft during 1 g and 0 g. We use these data to estimate upper limits on T corresponding to a relative contrast loss of ∼60%. For our experimental parameters, we find T^{max}≃20 and 30 ms for 1 g and 0 g, respectively. Conversely, for future space missions planning interrogation times of order T=5 s and Ω^{eff}≃2π × 5 kHz (ref. 20), our model predicts an upper limit on the vibration noise of μg. One strategy to mitigate this effect is to suppress highfrequency vibrations using an active isolation system modified to operate in microgravity^{26}. However, for inertial navigation applications, measuring the vibrations is critical to accurate positioning, thus a hybrid classicalquantum solution may be more viable^{27}. Onboard the aircraft, a combination of these two solutions will give access to freefall times up to ∼1 s, above which the jerk of the aircraft will be too large to keep the atoms in the interrogation region defined by the Raman beams.
During parabolic manoeuvres, the aircraft’s trajectory is analogous to a Nadirpointing satellite in an elliptical orbit. The rotation of the experiment during a parabola causes a loss of contrast due to the separation of wavepacket trajectories (Fig. 4c) and the resulting imperfect overlap during the final π/2pulse^{1,28,29,30}. For a rotation vector Ω_{T} transverse to k^{eff} and a velocity spread σ_{v}, the wavepacket displacement can be shown to produce a relative contrast loss of (Supplementary Note 2). Hence, during a parabola where Ω_{T}≃5° s^{−1}, the loss of contrast reaches 60% by T=5 ms for our ^{87}Rb sample and by T=2.8 ms for ^{39}K. Figure 4d,e shows the measured contrast loss as a function of T for each species during both steady and parabolic flight. We fit a model to these data, which includes effects due to both vibrations and rotations. Using only a vertical scale factor as a free parameter, we find good agreement with the data. This loss of contrast can be compensated by counterrotating the retroreflection mirror during the interferometer sequence^{1,28}. In addition, imaging the atoms on a camera can mitigate this effect, since the rotationinduced spatial fringes in the atomic density profile can be measured directly^{1,31}. Using the model we validated with our experiment, we estimate the rotation limitations of a highly elliptical orbit such as in STEQUEST^{20}. In the case of a Nadirpointing satellite with an orbital rotation rate near perigee (700 km) of ∼2.7° s^{−1}, we estimate a 60% loss of contrast by T≃73 ms for the experimental parameters proposed in ref. 20. This justifies the choice of inertial pointing, where the rotation of the satellite counteracts that of the orbit, to reach the target sensitivity of 3 × 10^{−12} m s^{−2} per shot at T=5 s. We estimate a loss of <1% at T=5 s can be achieved if the residual rotation rate is <6 × 10^{−5}° s^{−1}.
We have realized simultaneous dual matterwave inertial sensors capable of operating onboard a moving vehicle—enabling us to observe correlated quantum interference between two chemical species in a weightless environment, and to demonstrate a UFF test in microgravity at a precision two orders of magnitude below the level of ambient vibration noise. With the upcoming launch of experiments in the International Space Station^{32,33}, and in a sounding rocket^{34}, this work provides another important test bed for future coldatom experiments in weightlessness. In the ZeroG aircraft, even if the limit set by its motion cannot be overcome, an improvement of more than four orders of magnitude is expected by cooling the samples to ultracold temperatures, and actively compensating the vibrations and rotations of the inertial reference mirror. This will approach the desired conditions for nextgeneration atom interferometry experiments, such as those designed for advanced tests of gravitation^{35}, gradiometry^{36} or the detection of gravitational waves^{37}.
Methods
Experimental setup
Experiments were carried out onboard the Novespace A310 ZeroG aircraft, where the interferometers operated during more than 100 parabolic manoeuvres, each consisting of ∼20 s of weightlessness (0 g) and 2–5 min of standard gravity (1 g). Two lasercooled atomic samples (^{87}Rb at 4 μK, and ^{39}K at 18 μK) were simultaneously interrogated by a π/2−π−π/2 sequence of coherent velocitysensitive Raman pulses, separated by freefall times T_{Rb} and T_{K}, respectively (Fig. 1a), which set the acceleration response of each interferometer. The laser light used for this manipulation is aligned through the atoms and retroreflected along the yaw axis (z axis) of the aircraft (Fig. 1b). A detailed description of our experimental apparatus, fibrebased laser system and fluorescence detection scheme can be found in ref. 35. When the aircraft is in steady flight, the Raman beams are vertical—maximizing the sensitivity to gravitational acceleration. Owing to the high vibration levels onboard the aircraft, the interferometer fringes are reconstructed using a correlative method^{10,24,25} with measurements from a threeaxis mechanical accelerometer (Colibrys SF3600) fixed to the rear of the retroreflecting mirror. These acceleration measurements were also combined with software to discriminate between the 0 g, 1 g and 2 g phases of a parabola (Fig. 1a), and to automatically switch the interferometers between two different operating modes (single diffraction and DSD) during each manoeuvre. A frequency chirp is applied to the Raman frequency during 1 g to cancel the gravityinduced Doppler shift. The chirp is disabled by software during parabolic manoeuvres. Interferometer measurements taken during the 2 g phase were rejected during the data analysis process. Finally, the rotation rates Ω_{x}(t) and Ω_{y}(t) are continuously monitored during the flight using a twoaxis fibreoptic gyroscope (KVH DSP1750). Combined with continuous acceleration measurements, we integrate the equations of motion in the rotating frame to obtain the trajectory of the two atomic clouds with respect to the reference mirror for each shot of the experiment. These trajectories are used to estimate systematic shifts on the measurement of η due to the Coriolis effect and the magnetic gradient.
Evaluation of systematic effects
To evaluate the systematic effects on the measurement of η, we begin by separating the total interferometer phase (corresponding to atom j and momentum transfer direction ) into five contributions
where is the phase due to the relative gravitational acceleration a between the reference mirror and the atoms with scale factor and πpulse duration , is a phase shift from a possible UFF violation, is a random phase caused by mirror vibrations with corresponding timedependent acceleration a^{vib}(t) and interferometer response function f_{j}(t)^{25,38}, is the contribution from the Raman laser phase ϕ_{j}(t) at each interferometer pulse and represents the total systematic phase shift. We express the total systematic phase as the following sum
where i is an index corresponding to a given systematic effect. In general, these phases can depend on both the magnitude and the sign of . To simplify the analysis, we divide into two separate phases labelled for the directionindependent phase shifts and to denote the directiondependent shifts (that is, those proportional to the sign of ). We isolate these components by evaluating the sum and the difference between systematics corresponding to each momentum transfer direction
For the specific case of the singlediffraction interferometer used in 1 g along +, the systematic phase shift is given by . In comparison, for the DSD interferometer, only directiondependent systematic effects can shift the phase of the fringes measured as a function of . In the ideal case, the sum of is the sole contribution to the systematic shift of the DSD fringes, since is directionindependent and thus contributes only to the fringe contrast (Fig. 2). However, in the more general case, these two phases can indirectly affect the phase of the DSD interferometer when the two pairs of Raman beams do not excite the selected velocity classes with the same probability. We denote this contribution , thus the total systematic phase for the interferometers used in 0 g is
Table 1 displays a list of the systematic phase shifts affecting the interferometers operated at T≃2 ms onboard the aircraft (Fig. 3d–f).
Phase corrections and η measurements
The raw interferometer phase for each species is measured directly from fits to the fringes reconstructed using the FRAC method (Fig. 3). We refer to this quantity as the FRAC phase . For the interferometers used in standard gravity, the measured fringes follow equation (2) with total phase . Since the vibration phase is the quantity used to scan , the FRAC phase is related to the sum of all other phase contributions through
where is an integer representing a certain fringe. Assuming that , and provided the total uncertainty from all other phases is much less than π, the UFF phase can be isolated from equation (10) by computing the fringe number from , where the square brackets indicate rounding to the nearest integer. A similar procedure can be carried out for the DSD fringes obtained in weightlessness, where the total phase is replaced with the halfdifference ΔΦ_{j}=(−). We point out that the laser phase does not contribute to the DSD interferometer because it is independent of the momentum transfer direction. Furthermore, since we are interested in only the differential UFF phase given by equation (4), the contribution due to the gravitational acceleration cancels .
The Eötvös parameter is obtained from
where a^{eff} is the effective gravitational acceleration to which the atom interferometer is sensitive over the duration of a measurement. To estimate a^{eff}, we first compute the gravitational acceleration along the vertical z′ axis, , over a twodimensional grid of latitude () and longitude (λ) coordinates at a fixed altitude h using the Earth gravitational model EGM2008 (ref. 39). From these values, we calculate the average projection of the gravitational acceleration vector on the axis of the Raman beams (). In the Earth frame, the interferometer axis is defined as after rotations about the x′ and y′ axes by roll angle θ_{x} and slope angle θ_{y}, respectively. It follows that the effective gravitational acceleration is given by
where 〈⋯〉 denotes an average. Table 2 contains the list of corrections applied to the raw data to obtain η. We now describe some of the dominant systematic effects that were specific to our experiment onboard the aircraft.
Coriolis phase shift
During steady flight, if the aircraft is tilted by angles θ_{x} and θ_{y} about the x′ and y′ axes, respectively (Fig. 1a), a component of the gravitational acceleration lies along the axes perpendicular to k^{eff}=k^{eff}, thus a rotation about these axes will cause a phase shift, ϕ^{Ω}, due to the Coriolis effect. To first order in the rotation rate Ω, this shift can be split into two main parts
where first term is due to an atomic velocity v_{0} at the start of the interferometer and the second originates from a constant acceleration a_{0}= g+δa. For small angles θ_{x} and θ_{y}, it is straightforward to show that
where δa is a small shottoshot variation due to the motion of the aircraft of order (Fig. 4a), and the initial velocity is related to δa via . Here is the selected atomic velocity determined by the frequency difference between Raman beams, and Δt represents the freefall time between cloud release and the first π/2pulse (Δt≃3 ms in our case).
Table 3 displays the mean value and range of variation of some inertial parameters during each flight configuration. These data imply that the dominant contribution to the Coriolis phase during steady flight is the instability in the roll angle. The corresponding phase shift at T=2 ms is estimated to be ≃0.1(3) mrad for both ^{87}Rb and ^{39}K. In comparison, during a parabolic trajectory the atoms are in freefall and the acceleration relative to the mirror is close to zero, hence the Coriolis phase shift is much less sensitive to the orientation of the aircraft relative to g. However, during this phase the aircraft can reach rotation rates of Ω>5° s^{−1} (Fig. 4b), which occurs primarily about the y axis (Fig. 1a). This causes small atomic velocities perpendicular to the direction of k^{eff} to produce significant phase shifts. We estimate ≃−3.7(3) mrad at T=2 ms for a mean rotation rate of Ω_{y}≃4.1° s^{−1}.
These simple estimates, although useful to give an intuitive understanding, do not include effects due to finite Raman pulse lengths , timevarying rotation rates Ω(t) or timevarying accelerations a(t). Since these effects are significant in our case, it was necessary to develop a new expression to accurately estimate the associated phase shift. The result of these calculations, which were based on the sensitivity function formalism^{40}, is the following expression
which describes the Coriolis phase shift due to an atomic trajectory undergoing a timedependent rotation Ω(t) and acceleration a(t). Here w^{Ω}(t) is a weight function
which contains the interferometer sensitivity function g^{s}(t). In the limit of short pulse lengths, and constant accelerations and rotations, equation (16) reduces to equation (13).
During the flight, we measure the acceleration of the Raman mirror in the rotating frame (Fig. 1) using a threeaxis mechanical accelerometer, and the rotation rates Ω_{x}(t) and Ω_{y}(t) are measured using a twoaxis fibreoptic gyroscope. The rotation rate about the z axis of the aircraft, Ω_{z}, does not contribute significantly to the Coriolis phase (since it is parallel to k^{eff}), hence precise measurements of this quantity using a third gyroscope were not required. We then integrate the equations of motion in the rotating frame to obtain v(t) relative to the Raman mirror, and we use equation (16) to obtain the Coriolis phase shift for each shot of the experiment. The values reported in the third row of Table 1 represent the average of this phase taken over the coarse of all measurements during a given flight configuration. For the DSD interferometer used in 0 g, we computed the Coriolis shift for both upward and downward atomic trajectories and combined the results as in equation (8).
DSD phase shift
The DSD interferometer that we use in microgravity is sensitive to an additional systematic shift that is not present in the singlediffraction interferometer. This phase shift arises from the fact that we cannot distinguish between the atoms that are diffracted upwards and downwards. For instance, if there is an asymmetry in the number of atoms diffracted along these two directions, and the directionindependent phase ΣΦ^{ind} is nonzero, this will produce two phaseshifted fringe patterns with different contrasts. Since we measure the sum of these two fringe patterns, there is an additional phase shift that depends on the relative contrast ɛ=C^{−}/C^{+}−1 between the ±k^{eff} interferometers and ΣΦ^{ind} as follows
For the T≃2 ms fringes shown in Fig. 3e, we estimate ɛ≃0.05 for both rubidium and potassium interferometers. Hence, using the total directionindependent systematics listed in columns 5 and 7 of Table 1, we obtain DSD phase shifts of ≃−39(3) mrad and ≃−29(30) mrad.
Quadratic Zeeman effect and magnetic gradient
The primary source of systematic phase shift in this work originated from a timevarying Bfield during the interferometer produced by a large aluminium breadboard near the coils used to produce a magnetic bias field for the interferometers. Owing to the relatively large pulsed fields (∼1.5 G) required to sufficiently split the magnetically sensitive transitions in ^{39}K, Eddy currents produced in the aluminium breadboard during the interferometer significantly shift the resonance frequency of the clock transition via the quadratic Zeeman effect. We recorded the field just outside the vacuum system with a flux gate magnetometer (Bartington MAG03MCTPB500) and used these data, in conjunction with spectroscopic calibrations of the field at the location of the atoms, to compute the associated systematic phase shift for each shot of the experiment.
The secondorder (quadratic) Zeeman effect shifts the frequency of the clock transition as , where K_{Rb}=575.15 Hz G^{−2} for ^{87}Rb and K_{K}=8513.75 Hz G^{−2} for ^{39}K (ref. 41). This effect can shift the phase of the interferometers in three ways: (i) due to a Bfield that is nonconstant in time ; (ii) from a field that is nonconstant in space ; or (iii) via the force on the atoms from a spatial magnetic gradient . The total systematic shift due to magnetic field effects is the sum of these three phases
We model the local magnetic field experienced by the atoms as follows
where β_{0} is a magnetic bias field, β_{1}=∂B/∂z is a magnetic gradient and ξ(t) is a unitless envelope function that can describe the field turnon, as well as residual Eddy currents.
The phase shift due to a temporal variation of the Bfield can be computed using^{40}
where is the interferometer sensitivity function^{25,38} and is the clock shift at the initial position of the atoms. Similarly, the phase due to the clock shift from a spatially nonuniform field can be expressed as
Here is the centreofmass trajectory of atom j along the interferometer pathways, and are the initial atomic position and selected velocity, respectively, is the corresponding recoil velocity and a is a constant acceleration along the direction of z. We have ignored the influence of the magnetic gradient force on the atomic trajectory since it is small compared that of gravity. In equation (22), we have used the difference between the field experienced by a falling atom and that of a stationary atom at z= to separate the spatial effect of the field from the temporal one.
To measure B(z, t), we used velocityinsensitive Raman spectroscopy of magnetically sensitive twophoton transitions and we extracted the resonance frequency as a function of the time in free fall in standard gravity—yielding a map of B(z, t). However, this method cannot distinguish between the temporally and spatially varying components of the field. To isolate the spatial gradient β_{1}, we performed the same spectroscopy experiment with the bias field on continuously to eliminate the turnon envelope and to minimize Eddy currents. The difference between these measurements yielded the temporally varying component of the field. For typical experimental parameters during the flight (T_{j}∼2 ms, β_{0}∼1.5 G), we find ∼2.1 rad and ∼30.5 rad, as listed in the first row of Table 1. These relatively large phase shifts are produced by the large bias required to separate the states from , and a significant variation in the envelope during the interferometer (the field changes by ∼0.5 G in 2 ms) produced by the Eddy currents. Similarly, we estimate ∼0.032 rad and ∼0.96 rad during 1 g, which arises from a measured gradient of β_{1}≃13 G m^{−1}, as listed in the second row of Table 1.
The phase shift arising from the force on the atoms due to the magnetic gradient can be computed by evaluating the statedependent atomic trajectories and following the formalism of ref. 42. Up to order and =hK_{j}/M_{j}, this phase can be shown to be
where the ∓ sign convention corresponds to . We emphasize that this phase scales (Λ_{K}/Λ_{Rb})^{2}∼33 times more strongly for potassium than rubidium due to its lighter mass and smaller hyperfine splitting . However, since the magnetic gradient force is opposite in sign for and , and the states are exchanged halfway through the interferometer, this phase shift is generally much smaller than those produced by shifts of the clock transition. For typical experimental parameters during steady flight (T_{j}≃2 ms, ≃5 cm s^{−1}, a≃9.8 m s^{−2} and β_{1}≃13 G m^{−1}), the phase shift is ≃−0.12 mrad for ^{39}K and ≃−3.4 μrad for ^{87}Rb. We sum this phase with in the second row of Table 1.
Data availability
The authors declare that the primary data supporting the findings of this study are available within the article and its Supplementary Information file. Additional data are available from the corresponding author on request.
Additional information
How to cite this article: Barrett, B. et al. Dual matterwave inertial sensors in weightlessness. Nat. Commun. 7, 13786 doi: 10.1038/ncomms13786 (2016).
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Acknowledgements
This work is supported by the French national agencies CNES, ANR, DGA, IFRAF, action spécifique GRAM, RTRA ‘Triangle de la Physique’ and the European Space Agency. B. Barrett and L. AntoniMicollier thank CNES and IOGS for financial support. P. Bouyer thanks Conseil Régional d’Aquitaine for the Excellence Chair. Finally, the ICE team thank A. Bertoldi of IOGS for his assistance during the ZeroG flight campaign in May 2015; V. Ménoret of MuQuans for helpful discussions; D. Holleville, B. Venon, F. Cornu of SYRTE and J.P. Aoustin of the laboratory GEPI for their technical assistance building vacuum and optical components; and the staff of Novespace.
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Affiliations
LP2N, IOGS, CNRS and Université de Bordeaux, rue François Mitterrand, 33400 Talence, France
 Brynle Barrett
 , Laura AntoniMicollier
 , Laure Chichet
 , Baptiste Battelier
 & Philippe Bouyer
CNES, 18 avenue Edouard Belin, 31400 Toulouse, France
 Thomas Lévèque
LNESYRTE, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, 61 avenue de l’Observatoire, 75014 Paris, France
 Arnaud Landragin
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Contributions
P.B. and A.L. conceived the experiment and directed research progress; B. Battelier contributed to construction of the firstgeneration apparatus, helped to direct research progress and provided technical support; B. Barrett led upgrades to the second generation apparatus, performed experiments, carried out the data analysis and wrote the article; L.A.M. and L.C. helped upgrade the potassium interferometer and carried out experiments; T.L. provided technical support during flight campaigns. All authors provided comments and feedback during the writing of this manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Brynle Barrett.
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