Systematic Errors in High-Precision Gravity Measurements by Light-Pulse Atom Interferometry on the Ground and in Space

Systematic errors in high-precision gravity measurements by light-pulse atom interferometry on the ground and in space

Anna M. Nobili,1, 2 Alberto Anselmi,3 and Raffaello Pegna1 1Dept. of Physics “E. Fermi”, University of Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy 2INFN (Istituto Nazionale di Fisica Nucleare), Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy 3Thales Alenia Space Italia, Strada Antica di Collegno 253, 10146 Torino, Italy (Dated: February 18, 2020) We focus on the fact that light-pulse atom interferometers measure the atoms’ acceleration with only three data points per drop. As a result, the measured effect of the gravity gradient is systematically larger than the true one, an error linear with the gradient and quadratic in time almost unnoticed so far. We show how this error affects the absolute measurement of the gravitational acceleration g as well as ground and space experiments with gradiometers based on atom interferometry such as those designed for space geodesy, the measurement of the universal constant of gravity and the detection of gravitational waves. When atom interferometers test the universality of free fall and the weak equivalence principle by dropping different isotopes of the same atom one laser interrogates both isotopes and the error reported here cancels out. With atom clouds of different species and two lasers of different frequencies the phase shifts measured by the interferometer differ by a large amount even in absence of violation. Systematic errors, including common mode accelerations coupled to the gravity gradient with the reported error, lead to hard concurrent requirements –on the ground and in space– on several dimensionless parameters all of which must be smaller than the sought-for violation signal.

Light-pulse atom interferometers (AIs) are based on We focus on the fact that AIs measure the atoms’ po- quantum mechanics. As the atoms fall, the atomic wave sition along the trajectory only three times per drop (in packet is split, redirected, and finally recombined via correspondence with the three light pulses), unlike laser three atom-light interactions at times 0, T , and 2T . The interferometers in falling corner-cube gravimeters which phase that the atoms acquire during the interferometer make hundreds to a thousand measurements per drop [4]. sequence is proportional to the gravitational acceleration Hence, although predicted exactly, the gravitational ac- that they are subjected to. celeration measured by AIs is the true one only in a uniform field. Although one might think that the phase shift depends The predicted value of the measured acceleration first on quantum mechanical quantities “. . . this is merely an appeared in Ref. [5], and it has been confirmed ever illusion since we can write the scale factor [between the since[1, 6–9]. However, none of these works mentions phase shift and the gravitational acceleration] in terms that in the presence of gravity gradient this value is the of the parameters we control experimentally, i.e. Raman average free fall acceleration (at time T of the middle pulse vector k and pulse timing T . It then takes the form pulse) based on three position measurements. This is, of kT 2. . . . We can simply ignore the quantum nature of the course, only an approximation to the true acceleration, atom and model it as a classical point particle that car- expressed mathematically by the second time derivative ries an internal clock and can measure the local phase of of the position, or obtained experimentally with a suffi- the light field.” ([1], Sec. 2.1.3). The same reference also ciently large number of measurements per drop. demonstrates that, in the case of a gravitational field with a linear gradient, both the exact path integral approach Initially, the lack of precise measurements made the and the purely classical one lead to the same exact closed difference unimportant, and by the time the precision form for the phase shift and free fall acceleration mea- improved nobody went back to this issue. However, sured by the AI, which is then expanded in power series its physical consequences do deserve to be carefully ad- of the local gravity gradient γ for convenience [1]. The re- dressed. Using the classical approach [1] we point them arXiv:1806.04397v5 [physics.atom-ph] 17 Feb 2020 coil velocity is the only part of the atom-light interaction out when AIs are used to measure the absolute value of which is not found in the classical model; however, it does the gravitational acceleration g, for gravity gradiometry not appear in the phase shift actually measured by AIs, and for testing the universality of free fall (UFF), both because they are operated symmetrically so as to cancel on the ground and in space. it out [2, 3]. Thus, the classical approach gives excellent Since UFF tests are included, we allow from the start predictions of the phase shift measured by the interfer- the possibility that the equivalence of inertial and grav- ometer, while including the quantum mechanical details itational mass may be violated for atoms of different related to the internal degrees of freedom is needed to species A, B in the field of Earth (violation of the weak account for smaller effects, such as the finite length of equivalence principle, WEP) hence violating UFF [10]. g i the light pulses. We therefore write the masses as mA,B = mA,B(1+ηA,B), 2

g i M⊕ = M⊕(1 + η⊕), where superscripts i, g refer to in- If ηA,B = 0 (WEP and UFF hold) this is the same as ertial or gravitational mass and the Eötvösparameters in [1]; it is the expansion to order γ of an exact result ηA, ηB , η⊕ may not be exactly zero (although they must which can be obtained in closed form by an exact path be smaller than 1 by many orders of magnitude [11, 12]). integral treatment or within a purely classical descrip- The equation of motion for atoms A or B reads tion. In a gravitational field with a linear gradient the free GM i ⊕ fall acceleration of the atoms at time T is obtained from zÄ,B = − 2 (1 + η⊕ + ηA,B) (1) (R⊕ + zA,B) (3) while the AI measurement gives, at the same time T , the value (7), which is systematically larger (in modulus) where R⊕ is the Earth’s radius and the z axis points upward. UFF is tested by measuring the differential ac- than the true one by the amount: celerationz ¨B − zÄ, then η⊕ cancels out, and there is a 1 2 violation if, with identical initial conditions and no noise, ∆a = γg◦T (8) 12 the ratio with a relative error ∆a = 1 γT 2. The discrepancy was z¨B − zÄ g◦ 12 = ηB − ηA ≡ η (2) (¨zA +z ¨B)/2 pointed out in Ref. [17] where it was explained with the simple algebra involved in computing (6) from (5) and differs from zero; thus, what matters is the different com- (4). In physical terms, it is due to the limitation, intrin- position of the atoms under test, which should be maxi- sic to the AI instrument, of making only three position g i mized [13–15]. Hence we assume M⊕ = M⊕ ≡ M⊕. measurements per drop. Whether it can be neglected or Using a perturbative approach for a gravity field with not will depend on the specific experiment. If not, appro- a linear gradient γ (see, e.g., Ref. [16]) the equation of priate systematic checks are required in order to partially motion reads model this term, while any remaining unknown fraction 1 of it must be too small to matter. z¨ ' −g (1 + η ) − γ g t2 − v◦ t − z◦ , (3) A,B ◦ A,B 2 ◦ A,B A,B With the same perturbative approach the calculation 2 2 −2 can be extended to order γ by using the solution zA,B(t) where g◦ = GM⊕/R⊕ ' 9.8 ms , γ = 2g◦/R⊕ ' 3.1 × 10−6 s−2, and z◦ and v◦ are the initial position and to first order in γ as given by (4), rather than to order A,B A,B zero (i.e. z | = − 1 g t2 + v◦ t + z◦ ), as used in velocity errors of the atoms at release (the exact values A,B γ=0 2 ◦ A,B A,B are assumed to be zero). The solution is (3). The new equation of motion reads

1 h 1 2 ◦ ◦ ◦ ◦ 2 z¨A,B ' − g◦(1 + ηA,B) + γ g◦t − v t − z zA,B(t) ' zA,B + vA,Bt − g◦(1 + ηA,B)t A,B A,B 2 2 (9) (4) 1 1 1 i 2 1 2 1 ◦ 1 ◦ 2 4 ◦ 3 ◦ 2 −γt g t − v t − z . +γ g◦t − vA,Bt − zA,Bt . 24 ◦ 6 A,B 2 A,B 24 6 2

We compute the phase shift δφA,B measured by the AI Its solution leads to the phase shift, hence to the accel- following the step-by-step algorithm outlined in Ref. [1]. eration measured by the AI to order γ2: Assuming the same k for all three pulses (~k is the mo- g (T ) ' g (1 + η ) mentum transfer, with } the reduced Planck constant) A,B meas ◦ A,B and the same time interval T between subsequent pulses, 7 +γ g T 2 − v◦ T − z◦ it is 12 ◦ A,B A,B (10) 2 31 4 1 ◦ 3 7 ◦ 2 δφA,B = φA,B(2T ) − 2φA,B(T ) + φA,B(0) +γ g T − v T − z T . (5) 360 ◦ 4 A,B 12 A,B = k[zA,B(2T ) − 2zA,B(T ) + zA,B(0)] The result is the same as Eq. (2.19) in Ref. [1] (where and, using (4) it was obtained by expanding to second order the exact h result in closed form) and it is generally accepted. How- δφ (T ) ' −kT 2 g (1 + η ) A,B ◦ A,B ever, it differs from the true acceleration given by (9) (at (6) 7 i the same time T ) with a relative systematic error: +γ g T 2 − v◦ T − z◦ . 12 ◦ A,B A,B ∆a 1 2 1 v◦ T 3 1 z◦ T 2 With the scale factor kT 2 (k and T measured experimen- A,B = γT 2 + γ2 T 4 − A,B − A,B , g 12 45 12 g 12 g tally), this gives the free fall acceleration g (T ) ◦ ◦ ◦ A,B meas (11) that the AI is predicted to measure at time T of the middle pulse. In modulus, though we limit our analysis to order γ. 7 Let us now consider an AI experiment in space, inside g (T ) ' g (1 + η ) + γ g T 2 − v◦ T − z◦ . A,B meas ◦ A,B 12 ◦ A,B A,B a spacecraft in low Earth orbit such as the International (7) Space Station (ISS). The ISS is Earth pointing, the AI 3 axis is aligned with the radial direction and the nominal is of order γorb. We are led to the measured acceleration point O of atoms’ release (origin of the radial axis ζ point- (in modulus): ing away from Earth) is at distance h from the center of mass of the spacecraft (e.g., closer to Earth than the cen- aA,B meas(T ) ' atide + gorbηA,B ter of mass itself). When testing UFF with atom species 7 (15) +γ γ hT 2 − Υ◦ T − ζ◦ A and B we can assume ηspacecraft = η⊕ = 0 since they orb 12 orb A,B A,B cancel out anyway [18]. The equation of motion reads GM mi which, by comparison with its theoretical counterpart i ¨ ⊕ A,B mA,BζA,B = − 2 (1 + ηA,B) (14) at the same time, shows a systematic relative er- [r − h + ζA,B] (12) ∆a 1 2 ror = γorbT , similar to the ground experiment. i 2 atide 12 +mA,Bn (r − h + ζA,B) When testing UFF release errors result in position and with r being the orbital radius of the spacecraft (constant velocity offsets between the two atom clouds which – for simplicity) and n being its orbital angular velocity because of gravity gradient– give rise to a systematic 2 3 obeying Kepler’s third law n r = GM⊕. Since (h − differential acceleration error that mimics a violation sig- ζA,B)/r 1, we can write nal [17]. The effect of release errors is known to be a major issue in all UFF experiments based on “mass drop- ¨ ζA,B ' −(atide + gorbηA,B) + γorbζA,B (13) ping”, while it does not occur if the test masses oscil- where atide = γorbh is the tidal acceleration at the nom- late around an equilibrium position, as in torsion bal- 2 −2 inal release point and gorb = GM⊕/r ' 8.7 ms , ance tests or in the proposed Galileo Galilei (GG) ex- −6 −2 γorb = 3gorb/r ' 3.8 × 10 s are the gravitational periment in space [21]. As proposed by Roura [22], the acceleration and gravity gradient of Earth (the numer- effect of release errors coupled to the local gradient can ical values refer to an orbiting altitude of ' 400 km). be eliminated if the momentum transfer of the second This equation shows that in orbit the largest accelera- laser pulse is modified by a small quantity of order γ such atide h that the atoms fall as if they were moving in a uniform tion is the tidal one, with g ' 3 r 1 while the orb field. On the ground the nominal value k to be applied driving acceleration of UFF violation is gorb (slightly 2 1 2 weaker than in ground drop tests), meaning that when at the second pulse is k2 = k + ∆k2 = k + k 2 γT .A ◦ ◦ the free fall accelerations of two atom species are sub- residual acceleration γres(zA,B + vA,BT ) remains if this tracted a composition dependent violation signal would value is not implemented exactly (a successful reduction γ /γ ' 10−2 has been reported [3]): be gorbη, with η = ηB − ηA. The violation signal, if res any, is an anomalous acceleration in the same direction h (and unknown sign) as the monopole gravitational at- ∆k2 2 δφA,B(T ) ' −kT g◦(1 + ηA,B) traction from the source body, in this case the radial (16) ◦ ◦ 1 2 i direction to the Earth’s center of mass. Hence, the equa- −γres(zA,B + vA,BT ) + γg◦T ) . tion of motion (13) contains the tidal acceleration (due 12 to gravity gradient) in the radial direction and not its The acceleration term (8) remains too, in which the transversal component [19, 20]. The ratio of the variable gradient is unaffected by whatever reduction has been acceleration γ ζ relative to the constant term a is orb A,B tide achieved for the previous one, as pointed out in the Com- γorbζA,B ζA,B 1 2 = ' γorbT , in analogy to the correspond- atide h 2 ment [23] and acknowledged by Roura [24]. This is in- γzA,B 1 2 ing ratio on the ground ' γT . Note that γorb g◦ 2 evitable because ∆k2 has been computed in order to nul- is only slightly larger than γ while it is expected that T lify the effect of the local gradient on the atoms whose can be several times larger in space than on the ground, motion is governed by (3). Instead, the acceleration mea- because of near weightlessness conditions. This is con- sured by the AI and used for tuning the change ∆k2, sidered the key motivation for moving the experiment to is affected by the error (8) which cannot therefore be space, since it means, for a given free fall acceleration, compensated. Indeed, attempts to compensate it [25] are a larger phase shift and hence higher sensitivity (as T 2). questionable because compensation would alter the free However, it also means a larger gradient effect (also as fall acceleration of the atoms and force it to equal a mea- T 2). With this warning we proceed with a perturbative sured value which (already to first order in γ) is not fully approach as on ground. To order γorb, it is correct. ¨ h The very fact that in proposing the gravity gradient ζA,B ' − atide + gorbηA,B (14) compensation scheme Roura did not address the acceler- 1 2 ◦ ◦ i ation term (8) indicates that the systematic error made +γorb atidet − Υ t − ζ 2 A,B A,B by taking only three measurements per drop has not been ◦ ◦ where ζA,B and ΥA,B are position and velocity errors at recognized. 2 release and the last term is of order γorb but cannot be ne- A similar approach in space leads to a residual gradi- glected because the free fall acceleration to be measured ent γorb−res < γorb after applying [22], and to the phase 4 difference: absolute measurement of g to be improved. h In gravity gradiometers, two spatially separated AIs ∆k2 2 δφA,B orb(T ) ' −kT atide + gorbηA,B with atoms of the same species interrogated by the same (17) laser (hence ∆T =0 and ∆k = 0) measure their individ- ◦ ◦ 1 2i −γorb−res(ζ + Υ T ) + γorbatideT ual free fall accelerations at their specific locations and A,B A,B 12 compute their difference. The advantage is that the dif- 2 where the error given by the last term contains γorb, but ferential (tidal) acceleration is less affected than g by 1 2 disturbances mostly in common mode, such as vibration amounts to 12 γorbT relative to atide = γorbh, which is the quantity to be measured, and therefore cannot be noise. They are used for geodesy applications, but also ignored as hinted by Refs. [23, 24]. for the measurement of the universal constant of grav- A previous approach to reducing gravity gradient and ity G and the detection of gravitational waves. On the initial offset errors in a proposed test of UFF on the ground, if the release points A and B are separated ver- ISS was based on the idea of rotating the interferome- tically by ∆h (A at the reference level and B higher by ter axis [26]. For a dedicated mission the idea of rotating ∆h), the differential acceleration is the whole spacecraft has been proposed by Rasel’s group as the key to reduce tidal effects [27]. In both cases the |gB theory − gA theory| ' γ∆h authors invoke a similarity with MICROSCOPE space 7 (18) +γ γ∆hT 2 + (v◦ − v◦ )T + z◦ − z◦ experiment [12]. 8 B A B A In MICROSCOPE, the offset vectors between the cen- ters of mass of the macroscopic test bodies –being due while the gradiometer measures: to construction and mounting errors– are fixed with the |g − g | ' γ∆h apparatus and therefore follow its rotation at all time, B meas A meas (19) allowing the main tidal effect to be distinguished from 49 2 ◦ ◦ ◦ ◦ +γ γ∆hT + (vB − vA)T + zB − zA a violation signal during the offline data analysis of a 48 sufficiently long run –this is not a mass dropping exper- with a systematic acceleration error proportional to γ2 iment (Ref. [21], Sec. 7). Instead, mass dropping tests which cannot be neglected relative to the tidal acceler- with AIs require a huge number of drops to reduce sin- ation measured by the gradiometer, the fractional error gle shot noise, each one with its own initial conditions being 7 γT 2. In space, with the release point A as in and mismatch vector between different atom clouds, and 48 (12), and B at a radial distance ∆h (farther away from the assumption that all these vectors are fixed with the Earth), the gradiometer would measure apparatus cannot be taken for granted. The argument presented in Ref. [26] that the proposed instrument “has 2h random but specified mismatch tolerances” is a weak one. δφB − δφA ' kT γorb∆h Being systematic, this error must be below the target ac- 7 i (20) +γ γ ∆hT 2 + (Υ◦ − Υ◦ )T + ζ◦ − ζ◦ celeration of the test in all drops; otherwise –should mis- orb 12 orb B A B A match reversal not occur even in a small number of drops 1 2 during the entire run (which is hard to rule out by direct with a fractional systematic error 12 γorbT . The error measurement)– the resulting average acceleration will be –like the physical quantity to be measured– contains the larger than the target, thus questioning the significance gradient. Therefore, depending on the target precision of a possible “violation” detection. and accuracy of the experiment, ad hoc independent mea- Roura’s proposal [22] is therefore to be preferred, as surements are needed in order to model and reduce it long as the acceleration term (8) is recognized and dealt below the target. with, if necessary. In tests of UFF with AIs in which different atoms On the ground the error (8) affects the absolute mea- A and B are dropped “simultaneously”, the individual surement of g. The best such measurement has achieved phase shifts are measured and their difference δφB − δφA ∆g/g ' 3 × 10−9 [7, 28], only about three times worse is computed, to yield zero if no composition dependent z¨B −zÄ effect is detected (i. e., η = ηB − ηA = = 0, than obtained by the absolute gravimeter with free falling (¨zA+¨zB )/2 corner-cube and laser interferometry [29]. With T = UFF and WEP hold). 7 2 160 ms, the acceleration 12 γg◦T in (7) exceeds the tar- Different isotopes of the same atom can be interro- get error and has required a series of ad hoc measure- gated with the same laser. In this case T is the same and 1 7 ments (drops from different heights) to be modelled and the gradient term with 12 or 12 coefficient cancels out. reduced below the target. Should it be possible to im- Different atom species need different lasers and a require- prove the sensitivity of the instrument by increasing T , ment arises, for a given target η of the UFF test, on the and to reduce the gradient and its effect coupled to ini- time difference ∆T , as pointed out by Refs. [23, 24]. How- tial condition errors as proposed by Ref. [22], the error ever, the main problem with different lasers is that differ- (8) would still remain and should be taken care of for the ent frequencies (kA 6= kB) result in widely different phase 5 shifts measured by the interferometer even in case of per- small as possible. Gravity gradient is relevant because it fect synchronization (∆T = 0), zero gradient (γ = 0), no couples also to common mode accelerations which would noise and no violation. For instance, using 87Rb and otherwise cancel out (also to higher order). Thus, vi- 39K, the fractional difference of the phase shifts is of the bration noise must meet the tight requirements (24) in kK −kRb −2 −1 order of ' 1.67 × 10 (kK = 4π/767 nm , differential and in common mode. Note that after ap- kK −1 kRb = 4π/780 nm ). When seeking a violation signal plying [22], the requirement on acm is relaxed only by many orders of magnitude smaller this is a major prob- a factor 7 because the residual (8) contains the actual lem, which never occurred before in the long history of gradient γ and not the reduced one γres. these experiments that goes back to Galileo. Taking into Concerning synchronization, the requirement (24) 2 account only kA 6= kB the difference of phase shifts reads comes from the fact that each phase shift grows as T times the leading free fall acceleration, hence the relative δφ − δφ ' −g T 2[(k − k ) + (k η − k η )] , ∆T B A ◦ B A B B A A error 2 T competes with η. The issue has been faced (21) by [31] in a WEP test with 87Rb and 39K, though only to a few parts in 107. By chirping the lasers at a par- 2 or −g◦T [(kB −kA)(1+ηA)+kBη]. As mentioned earlier, ticular rate a wave acceleration was applied in order to there is a large term even if WEP holds, which makes compensate to some extent (for each species) the leading this quantity hardly suitable to detect a tiny violation. acceleration of the atoms which gives a phase shift pro- 2 Moreover, the Eötvösparameters ηA or ηB appear in portional to T . Raman pulse errors must meet similar addition to η = ηB − ηA [see Eq. (2)] mixed with the tight requirements, and it is envisaged to make use of Raman vectors. This mixing disappears and violation is frequency comb technology [30]. correctly expressed by η if we use the ratio of phase shifts In Ref. [30] the authors propose a data analysis that instead: would allow a violation term containing η to be separated from vibration noise. However, this is not the only vio- δφ k B ' B (1 + η) . (22) lation term in their equations, due to the mixing shown δφA kA by (21). Violation appears only as η in the ratio of the phase shifts, and (23) shows beyond question that vi- By defining k = k + ∆k , k = k + ∆k with A A A B B B bration noise, both in common and differential modes, kA, kB being the exact values and ∆kA, ∆kB being the cannot be separated from η; ∆adm and 1 γT 2 acm could respective experimental errors we get g◦ 6 g◦ be misinterpreted as a violation and therefore must be ( below the tight bounds (24). δφ k γ h i B ' B 1 + η − res (v◦ − v◦ )T + (z◦ − z◦ ) For a violation at level η to be detected, it is necessary δφ g B A B A A kA ◦ (i) that all errors are negligible with respect to η and ) (ii) that the absolute value of the ratio of the Raman ∆T ∆kA ∆kB ∆adm 1 2 acm +2 − + + − 2 × γT , vectors kB is measured to both precision and accuracy T kA kB g◦ 12 g◦ kA (23) better than η, so as to distinguish a deviation from the measured value at this tiny level. Instead, in UFF tests where we have included gravity gradient and initial condi- with macroscopic test masses, the physical quantity of tion errors, the synchronization and Raman vector errors, interest is zero if η = 0 (“null experiments” [21]). and also perturbations resulting in common mode accel- With η already established at levels below 10−13, −14 erations acm with inevitable differential residuals ∆adm. 10 [11, 12], the requirements (24) are challenging, and The most relevant and best studied is vibration noise [30]; each one needs a specific challenging technology, all to by using the same mirror it is ideally common mode, but be implemented together. Every error could be a WEP a differential residual remains due to imperfect rejection. violation and needs specific systematic checks in order to Systematic errors must obey the conditions be distinguished from it; all checks must have the target sensitivity and therefore require a total integration time γres ◦ ◦ ◦ ◦ each[17, 22]. (vB − vA)T + (zB − zA) < η, g◦ In space the ratio of phase shifts reads ∆a a 6 dm < η, cm < η, ( 2 (24) δφ k g g◦ g◦ γT B−orb ' B 1 + orb η ∆T η ∆k ∆k δφA−orb kA atide < , A < η B < η. h i T 2 kA kB γorb−res ◦ ◦ ◦ ◦ − (ΥB − ΥA)T + (ζB − ζA) atide The requirement on initial condition errors is very se- ) ∆T ∆kA ∆kB ∆adm 1 2 acm vere [17] and must be relaxed by applying, for each +2 − + + − γorbT ; species, an appropriate frequency shift at the second laser T kA kB atide 6 atide pulse [22] in order to make the residual gradient γres as (25) 6

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