GGeneralized gravity-gradient mitigation scheme (Published as Physical Review A , 023305 [2021])Christian Ufrecht ∗ Institut für Quantenphysik and Center for Integrated Quantum Science and Technology (IQ ST ),Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany A major challenge in high-precision light-pulse atom interferometric experiments such as in tests of theweak equivalence principle is the uncontrollable dependency of the phase on initial velocity and position of theatoms in the presence of inhomogeneous gravitational fields. To overcome this limitation, mitigation strategieshave been proposed, however, valid only for harmonic potentials or only for small branch separations in moregeneral situations. Here, we provide a mitigation formula for anharmonic perturbation potentials including localgravitational effects that vary on length scales much smaller than the spatial extent probed by the atoms andoriginate e.g. from buildings that surround the experiment. Furthermore, our results are applicable to generalinterferometer geometries with arbitrary branch separation and allow for compensation of Coriolis effects inrotating reference frames.
I. INTRODUCTION
The high sensitivity of light-pulse atom interferometry withpromising applications as inertial sensor in gravimetry [1, 2],gradiometry [3–5] and tests of the weak equivalence principle(WEP) [6–8] has led to ambitious proposals on ground [9–12]and in space [13, 14].A serious challenge for next-generation atom interferometrichigh-precision measurements is posed by non-linearities inthe gravitational potential and Coriolis forces which lead tonon-perfect overlap of the trajectories in both momentum andposition after the final laser pulse. As a consequence, thephase contains the initial position and velocity of the atomsand the contrast drops dramatically in long-time interferometry[13, 15]. Since control over the initial conditions is limited[16], mitigation strategies had to be developed. Interferometerschemes insensitive to initial kinematics in the presence ofhomogeneous gradients and rotations can be constructed byfolding the interferometer geometry symmetrically [17, 18].However, in these schemes also the dominant part of the phasefrom linear gravity cancels, including a possible violation signalin a test of the WEP.Similar to the mitigation strategies developed for Corioliseffects by using tip-tilt mirrors [19–21], the crucial insight toachieve compensation of initial-condition-dependent phasesin the presence of homogeneous gravity gradients was a mod-ification of the pulse timing [15] or the momentum transferof the central pulse e.g. in a Mach-Zehnder (MZ) interfer-ometer as a function of the gradients [22–24]. This methodreduces the mismatch of the trajectories at the end of theinterferometer sequence while leaving the phase from lineargravity unaffected. Besides other successive work to mimican inertial frame [25, 26], gravity-gradient compensation wasextended to spatially inhomogeneous gravity gradients [24]based on an additional modification of the momentum transferof the final laser pulse. However, the derivation in Ref. [24]is based on the midpoint theorem [27], which becomes less ∗ [email protected] accurate with increasing branch separation when applied toanharmonic potentials. Therefore, future proposals includinglarge-momentum-transfer techniques [28, 29] to increase thespace-time area of the interferometer or long-time interferome-try with large branch separation require a further generalizationof the method to such situations.In this article particular emphasis is put on the perturbativecharacter of the description which allows the application ofthe mitigation scheme to arbitrary anharmonic perturbationsin the gravitational potential as for example present in theexperimental setups of Refs. [24, 30]. The formula derived inthe present article is furthermore valid for general interferometergeometries and arbitrary branch separation. Our derivationwithin a full quantum-mechanical framework also allows toconsistently include contributions from wave-packet dynamics.In Sec. II we briefly review the perturbative formalism em-ployed to derive the compensation formula for inhomogeneousgravity gradients in Sec. III. In Sec. IV we generalize theformula to include rotations and finally discuss conditions forvalidity of our derivation in Sec. V. II. PATH-DEPENDENT DESCRIPTION
In this article, we rely on the perturbative formalism recentlydeveloped in Refs. [31, 32]. The Hamiltonianˆ 𝐻 ( 𝛼 ) = ˆ 𝐻 ( 𝛼 ) + 𝑉 ( ˆ 𝒓 , 𝑡 ) (1)describes the evolution through the interferometer along theupper ( 𝛼 = 𝑢 ) and the lower ( 𝛼 = 𝑙 ) branch. It consists of adominant part ˆ 𝐻 ( 𝛼 ) with respect to which the interferometeris closed (that is perfect overlap after the final pulse) and aperturbation 𝑉 ( ˆ 𝒓 , 𝑡 ) which slightly opens the interferometerand renders the phase dependent on initial position and velocityof the atoms. As an example we illustrate in Fig. 1 the caseof an MZ interferometer where a small cubic potential leadsto a slight mismatch of the trajectories at the final laser pulseand explain how custom-tailored laser pulses can mitigate thiseffect. a r X i v : . [ phy s i c s . a t o m - ph ] F e b Figure 1.
Gravity-gradient compensation in a small cubic potential.
In an MZ interferometer a 𝜋 / 𝑡 = ℏ 𝑘 on oneof them. The atoms are then redirected by a 𝜋 pulse at 𝑡 = 𝑇 and finallythe wave function is recombined by a second 𝜋 / 𝑡 = 𝑇 .In absence of a non-linear gravitational potential the atoms followthe unperturbed trajectories (thin solid lines). These trajectories aremodified (dashed lines) in presence of a small perturbation potentialso that the branches do not overlap perfectly in both momentum andposition at the final laser pulse, the interferometer is then referred to asopen. Note that this deviation is displayed strongly exaggerated in thefigure. If the wave vector of the second and the final pulse is modifiedappropriately by Δ 𝒌 ℓ (decreased in the example shown in the figure),the interferometer can be closed (thick solid lines) and dependence ofthe phase on the initial conditions is eliminated to first order. In a gravimeter configuration the natural choice for theunperturbed Hamiltonian isˆ 𝐻 ( 𝛼 ) = ˆ 𝒑 𝑚 − 𝑚 𝒈 ˆ 𝒓 + 𝑉 ( 𝛼 ) em ( ˆ 𝒓 , 𝑡 ) (2)where 𝑚 is the mass of the atoms and 𝒈 is the local linearacceleration. Furthermore, 𝑉 ( 𝛼 ) em ( ˆ 𝒓 , 𝑡 ) = − ℏ ∑︁ ℓ [ 𝒌 ( 𝛼 ) ℓ ˆ 𝒓 + 𝜑 ( 𝛼 ) ℓ ] 𝛿 ( 𝑡 − 𝑡 ℓ ) (3)is the effective laser interaction potential, imprinting the mo-mentum ℏ 𝒌 ℓ and the laser phase 𝜑 ( 𝛼 ) ℓ at time 𝑡 ℓ on branch 𝛼 .Additional contributions to the gravitational potential such asgradients of Earth’s gravitational potential, gravitational fieldsof the local environment such as from buildings surroundingthe experiment [30] etc., are incorporated into the perturbationpotential 𝑉 ( ˆ 𝒓 , 𝑡 ) and treated perturbatively.The phase 𝜑 and contrast 𝐶 of a matter-wave interferometeris defined as (cid:104) ˆ 𝑈 ( 𝑙 )† ˆ 𝑈 ( 𝑢 ) (cid:105) = (cid:104) e i ˆ 𝜙 (cid:105) = 𝐶 e i 𝜑 (4)where ˆ 𝑈 ( 𝛼 ) is the time-evolution operator with respect toHamiltonian (1) for the respective branch and the expectationvalue is taken with respect to the initial wave function. In Refs. [31, 32] we merged the two time-evolution operators onthe left-hand side in favour of the operator ˆ 𝜙 , which reads tofirst order in the perturbationˆ 𝜙 = 𝜙 − ℏ ∮ d 𝑡 𝑉 ( ˆ 𝒓 ( 𝑡 )) (5)and where 𝜙 is the interferometer phase for vanishing pertur-bation. The integral runs along the upper branch and returnsalong the lower. It is taken over the perturbation potential eval-uated at the branch-dependent and operator-valued Heisenbergtrajectories ˆ 𝒓 ( 𝛼 ) ( 𝑡 ) generated by the unperturbed Hamiltonian(2).The Heisenberg trajectories can be decomposed into the sumof the classical trajectory with the initial conditions given bythe initial mean position and velocity of the wave packet and afluctuation operator of the form ˆ 𝒓 ( 𝑡 ) = ˆ 𝒓 − (cid:104) ˆ 𝒓 (cid:105) + [ ˆ 𝒑 − (cid:104) ˆ 𝒑 (cid:105)] 𝑡 / 𝑚 where the expectation values are taken with respect to theinitial wave function [32]. This decomposition is valid asthe unperturbed Hamiltonian is only linear in the positionoperator. The standard deviation of the fluctuation operatoris a measure for the size of the expanding wave packet andcharacterizes wave-packet effects. In a modification to the formfrom Ref. [32] we define the fluctuation operator 𝛿 ˆ 𝒓 ( 𝑡 ) = 𝛿 𝒓 ( 𝑡 ) + ˆ 𝒓 − (cid:104) ˆ 𝒓 (cid:105) + ˆ 𝒑 − (cid:104) ˆ 𝒑 (cid:105) 𝑚 𝑡 (6)which includes the deviation of the trajectories 𝛿 𝒓 ( 𝑡 ) due touncertainties in the initial conditions. Thus, the Heisenbergtrajectory reads ˆ 𝒓 ( 𝛼 ) ( 𝑡 ) = 𝒓 ( 𝛼 ) ( 𝑡 ) + 𝛿 ˆ 𝒓 ( 𝑡 ) (7)where 𝒓 ( 𝛼 ) ( 𝑡 ) is the classical unperturbed trajectory withoutthis deviation. Consequently, we find for the expectation valuethat (cid:104) 𝛿 ˆ 𝒓 ( 𝑡 )(cid:105) = 𝛿 𝒓 ( 𝑡 ) . Note that all expectation values aretaken with respect to the initial wave function. In summary,in the real classical unperturbed trajectories 𝒓 ( 𝛼 ) ( 𝑡 ) + 𝛿 𝒓 ( 𝑡 ) we choose 𝒓 ( 𝛼 ) ( 𝑡 ) as a fixed reference while 𝛿 𝒓 ( 𝑡 ) describestheir uncertainty due to e.g. limited initial characterization time[16].Making use of the decomposition from Eq. (7), a small valueof 𝛿 𝒓 ( 𝑡 ) and a small wave-packet size will allow for a Taylorexpansion of the perturbation potential around the classicalunperturbed trajectories in the next section. The dominant linearcontribution of 𝛿 ˆ 𝒓 ( 𝑡 ) in ˆ 𝜙 not only introduces a dependence onthe initial conditions but also leads to a loss of contrast [13, 15]when calculating the expectation value in Eq. (4). FollowingRefs. [22], slightly modifying the momentum transfer of thelaser pulses will eliminate 𝛿 ˆ 𝒓 ( 𝑡 ) to leading order and thereforestrongly mitigate these two effects.In our fully quantum-mechanical treatment phase contri-bution that arise from the square and higher powers of 𝛿 ˆ 𝒓 ( 𝑡 ) include both the residual, strongly suppressed dependence onthe initial conditions and wave-packet effects due to e.g. differ-ent dynamics along the interferometer branches. III. GRAVITY GRADIENTS
As anticipated in the previous section, we start by replacing 𝒌 ( 𝛼 ) ℓ → 𝒌 ( 𝛼 ) ℓ + Δ 𝒌 ( 𝛼 ) ℓ for each laser pulse and modify the effective laser-atom inter-action potential in Eq. (3) accordingly. As the validity of theperturbative approach requires a closed unperturbed interferom-eter [32], we keep the unperturbed Hamiltonian (2) unchangedand therefore consider Δ 𝑉 ( 𝛼 ) em ( ˆ 𝒓 , 𝑡 ) = − ℏ ∑︁ ℓ Δ 𝒌 ( 𝛼 ) ℓ ˆ 𝒓 𝛿 ( 𝑡 − 𝑡 ℓ ) as part of the perturbation potential 𝑉 ( ˆ 𝒓 , 𝑡 ) . In the following wewill omit the branch index 𝛼 whenever possible. Furthermore,quantities without operator hat are understood to be evaluated atthe unperturbed trajectories 𝒓 ( 𝑡 ) and we will omit the explicittime dependence.Inserting Eq. (7) into the perturbation potential followedby Taylor expansion about the classical trajectories, we find 𝑉 ( ˆ 𝒓 ) = 𝑉 − 𝑚 𝒂 𝛿 ˆ 𝒓 + ... with the acceleration 𝒂 = −∇ 𝑉 / 𝑚 andconsequently to first order in 𝛿 ˆ 𝒓 ˆ 𝜙 = 𝜙 − ℏ ∮ d 𝑡 (cid:16) 𝑉 − 𝑚 𝒂 𝛿 ˆ 𝒓 (cid:17) + Δ 𝜙 𝑘 + ∑︁ ℓ Δ 𝒌 ℓ 𝛿 ˆ 𝒓 ( 𝑡 ℓ ) . (8)The last contribution in Eq. (8) and the phase Δ 𝜙 𝑘 = ∑︁ ℓ Δ 𝒌 ( 𝑢 ) ℓ 𝒓 ( 𝑢 ) ( 𝑡 ℓ ) − Δ 𝒌 ( 𝑙 ) ℓ 𝒓 ( 𝑙 ) ( 𝑡 ℓ ) originate from the perturbation Δ 𝑉 em evaluated at Heisenbergtrajectories and we abbreviated Δ 𝒌 ℓ = Δ 𝒌 ( 𝑢 ) ℓ − Δ 𝒌 ( 𝑙 ) ℓ to alleviatenotation.Thanks to the simple form of the unperturbed Hamiltonianwe find 𝛿 𝒓 = 𝛿 𝒓 i + 𝛿 𝒗 i 𝑡 where 𝛿 𝒓 i and 𝛿 𝒗 i are the uncertaintiesof initial position and velocity. Thus, making additionally useof the explicit form of the fluctuation operator shown in Eq. (6),all terms in Eq. (8) linear in 𝛿 ˆ 𝒓 vanish if we require that 𝑱 = − ∑︁ ℓ Δ 𝒌 ℓ and 𝑱 = − ∑︁ ℓ Δ 𝒌 ℓ 𝑡 ℓ (9)with the abbreviations 𝑱 = 𝑚 ℏ ∮ d 𝑡 𝒂 ( 𝑡 ) and 𝑱 = 𝑚 ℏ ∮ d 𝑡 𝒂 ( 𝑡 ) 𝑡 . (10)After eliminating the operator-valued terms in Eq. (8), theoperator ˆ 𝜙 has become a 𝑐 -number (to the order consideredhere) and we find 𝜑 = 𝜙 − ℏ ∮ d 𝑡 𝑉 + Δ 𝜙 𝑘 (11)where we stress again that 𝑉 is evaluated at the classicalunperturbed trajectories 𝒓 ( 𝛼 ) . The linear set of equations inEq. (9) can be solved in general if we slightly change the wave vectors of the laser at two different times, say 𝑡 and 𝑡 , so thatwe find Δ 𝒌 = − 𝑱 − 𝑱 𝑡 𝑡 − 𝑡 and Δ 𝒌 = 𝑱 − 𝑱 𝑡 𝑡 − 𝑡 . (12)The functions 𝑱 and 𝑱 in Eq. (10) allow an intuitive interpre-tation. The former is proportional to the integrated differentialacceleration between the branches, while the latter correspondsto its average over time. As shown in Appendix A and B, thedependence of Δ 𝒌 and Δ 𝒌 on these quantities allows to de-sign interferometer geometries for which the mitigation schemesimplifies. Furthermore, in Eq. (12) any two distinct laserpulses can be chosen that not necessarily have to correspond tothe second and final laser pulse.Once Δ 𝒌 and Δ 𝒌 are known, the shifts in momentumtransfer can be distributed between the two branches satisfying Δ 𝒌 ℓ = Δ 𝒌 ( 𝑢 ) ℓ − Δ 𝒌 ( 𝑙 ) ℓ . For example we find in case of a laserpulse imprinting opposite momentum on the two branches that Δ 𝒌 ( 𝑢 ) ℓ = − Δ 𝒌 ( 𝑙 ) ℓ = Δ 𝒌 ℓ / 𝑇 ) before taking the difference. Inground-based tests this procedure is only meaningful as longas the uncertainty in the wave vectors is smaller than the targetaccuracy of the WEP violation parameter. In microgravity,however, this constraint is significantly relaxed. If the miti-gation scheme is applied, remaining gradient-induced phaseshifts independent of the initial kinematics cancel differentiallyin case of homogeneous gradients. In case of locally varyinggravitational potentials, however, the atoms feel different lo-cal potentials along the species-dependent trajectories. As aconsequence, these phase contributions are only suppressedin the differential rescaled phase but not cancelled. Whilethis remaining differential phase is small, it might neverthelessimpose limits on future tests of the WEP on ground if thegravitational background is not known precisely.In Eq. (8) the Taylor expansion is truncated at first order in 𝛿 ˆ 𝒓 . Corrections to the phase from higher powers in the fluctua-tion operator can be calculated with the cumulant expansion[31, 32, 37], however, are often negligible [32]. Correctionsof this kind will be discussed in more detail in Sec. V. Asanharmonic potentials are treated quantum mechanically inthis work rather than within a semiclassical approximation, ourresults also cover the application of large-momentum transfertechniques where the branch separation can become compara-ble to the spatial extent probed by the atoms. In Appendix Awe explain the approximations needed to obtain the expressionsderived in the Supplemental material of Ref. [24] and discusstheir validity. We furthermore show how Eq. (12) reduces to theresult originally derived in Ref. [22] for an MZ interferometerin presence of homogeneous gravity gradients. In Appendix Bwe investigate simplifications of our general results in case oftrajectories symmetric in time. We stress the importance oftreating the perturbation potential locally [38]. For instancethe gravitational profile reported in Ref. [30] cannot be Tay-lor expanded over the extent of the interferometer due to itsvariations on short lengths scales. A numerical integration ofEq. (10) for this example shows that these local perturbationsinfluence the value of Δ 𝒌 at the ten-percent level and above. IV. ROTATIONS
For experiments in a rotating reference frame Coriolis andcentrifugal forces need to be considered additionally. Fortu-nately, it is straightforward to extend our result to such situationsas will be shown next. The Hamiltonian in a rotating frame isobtained by adding ˆ 𝐻 Ω = 𝛀 · ( ˆ 𝒑 × ˆ 𝒓 ) to Hamiltonian (1) where 𝛀 is the rotation frequency. Ad-ditional centrifugal forces present for example in a referenceframe fixed on Earth’s surface only redefine the direction andabsolute value of 𝒈 . In complete analogy to Sec. III we inte-grate ˆ 𝐻 Ω along the Heisenberg trajectories shown in Eq. (7)and recall that ˆ 𝒑 ( 𝑡 ) = 𝑚 d / d 𝑡 ˆ 𝒓 ( 𝑡 ) . Consequently, Eq. (8) isextended by the term − ℏ ∮ d 𝑡 𝛀 ·[ ˆ 𝒑 ( 𝑡 )× ˆ 𝒓 ( 𝑡 )] = 𝜙 Ω + 𝑚 ℏ ∮ d 𝑡 [ 𝒗 ( 𝑡 ) × 𝛀 ]· 𝛿 ˆ 𝒓 ( 𝑡 ) where we neglected terms quadratic in the fluctuation operatorand made use of partial integration for which we appreciatedthat the unperturbed interferometer is closed. Furthermore, 𝒗 ( 𝑡 ) is the velocity of the atoms on the unperturbed trajectoriesand we abbreviated 𝜙 Ω = − ℏ ∮ d 𝑡 𝛀 · [ 𝒑 ( 𝑡 ) × 𝒓 ( 𝑡 )] . Consequently, by comparing to Eq. (8), the mitigation schemescan be generalized to rotating reference frames with the re-placement 𝒂 ( 𝑡 ) → 𝒂 ( 𝑡 ) + 𝒗 ( 𝑡 ) × 𝛀 in Eq. (10) and by adding 𝜙 Ω to Eq. (11). Alternatively, theeffects of rotations can be analyzed in a non-rotating frame,where the laser is rotating instead [39]. V. VALIDITY OF PERTURBATIVE TREATMENT
In the previous section we developed a general mitigationscheme based on a perturbative treatment, covering both ro-tations and gravity gradients. In the following we discuss thevalidity of this approach and the approximations made in thederivation.In a perturbative calculation of the phase in powers of the po-tential 𝑉 subsequent orders are suppresses by 𝜖 = Δ 𝑉𝑇 /( 𝑚𝜉 ) [32] where Δ 𝑉 is the characteristic change of the potential overthe interferometer size, 𝜉 is the typical length scale on whichthe potential changes and 𝑇 the characteristic interferometertime. The parameter 𝜖 can be understood as the deviation of thetrajectories caused by the perturbation compared to the length 𝜉 . For gravity gradients on Earth’s surface corresponding tothe potential 𝑉 = 𝑚 𝒓 T Γ 𝒓 /
2, one would choose 𝜉 as the extentof the interferometer, given approximately by 𝜉 = 𝑔𝑇 / Δ 𝑉 ∼ 𝑚 Γ 𝜉 / 𝜖 = Γ 𝑇 , leading to a value 𝜖 < − for typicalinterferometer times. Local variations as in the gravitationalpotential of Ref. [30], in contrast, can lead to values of 𝜉 muchsmaller than the spatial extent probed by the atoms. A similarsuppression factor for rotations takes the form 𝜖 Ω = Ω 𝑇 with 𝜖 Ω < − for the rotation of Earth. Consequently, the relativeuncertainty in the phase achieved by a first-order calculationalready is of the order of 𝜖 .The term 𝑚 ∮ d 𝑡 𝒂 𝛿 ˆ 𝒓 / ℏ in Eq. (8) introduces the dominantdependence on the initial conditions. Estimating ∇ 𝑉 ∼ 𝛿𝑉 / 𝜉 [32] where 𝛿𝑉 is the change of the potential over the branchseparation and introducing the abbreviation 𝜂 = 𝛿𝑉𝑇 / ℏ , thisphase contribution scales as 𝜂𝛿𝑟 / 𝜉 .Application of the mitigation scheme requires prior knowl-edge of the gravitational background which can be obtainedby measurement, numerical simulation of the gravitationalsources surrounding the experiment, or a combination of both.However, determination of deviations from linear gravity willonly be possible to some relative uncertainty 𝜅 , which thenalso constitutes the suppression factor for initial-condition de-pendent phases. Estimations suggest that at least 𝜅 = − seems plausible [14, 24], thereby considerably relaxing therequirements on determination of initial position and velocityof the atoms. As described in the beginning of this section,further terms linear in the initial conditions which would resultfrom the second-order calculation in the perturbation potentialare suppressed by 𝜖 compared to the first-order terms. Con-sequently, an extension of the mitigation scheme to secondorder in the perturbation [31] is only necessary if 𝜅 < 𝜖 sinceotherwise initial-condition-dependent phases that are compen-sated only partially are still larger than contributions fromthe second-order calculation. Higher-order corrections to thedominant phase in Eq. (11), in contrast, can be obtained asshown in detail in Ref. [32].Moreover, Taylor expansion of Eq. (5) around the classicalunperturbed trajectories to first order neglects terms scaling as 𝜂 ( 𝛿 ˆ 𝒓 / 𝜉 ) . Comparing to the residual contribution 𝜅 𝜂𝛿 ˆ 𝒓 / 𝜉 fromthe first-order calculation, these terms and further correctionscan be disregarded if 𝛿𝑟 i / 𝜉 < 𝜅 and 𝛿𝑣 i 𝑇 / 𝜉 < 𝜅 . For 𝛿𝑟 i ∼ µ mas well as 𝛿𝑣 i ∼ µ m/s and 𝜉 ∼ VI. DISCUSSION
Finally, we conclude by the following remarks. The accelera-tion in Eq. (10) not necessarily points in direction of momentumtransfer. Consequently, gravity-gradient compensation mightalso require a tilt of the mirrors in order to adapt the directionof 𝒌 appropriately. However, generally for experiments onEarth’s surface the required modification of momentum trans-fer orthogonal to the sensitive axis ( 𝒌 pointing in direction of 𝒈 ) often is much smaller than the parallel component due tosymmetry in the mass distribution surrounding the apparatus[30].Obviously, the compensation method is equally applicableto perturbations of non-gravitational origin. However, initialcondition-dependent phases from e.g. magnetic field gradients[40], black-body radiation [41], etc. are generally much smallerthan those from gravity and can be neglected.Moreover, to avoid the necessity of a precise characterizationof the gravitational background, the compensation scheme canbe implemented experimentally through calibration prior to themeasurement by introducing artificial large deviations of theinitial conditions [8, 24].In the reference frame of an inertial-pointing satellite orbitingEarth the gravitational potential is time dependent. In a WEPtest the varying projection of a possible violation signal on thesensitive axis can be utilized to demodulate systematic effects[14]. This technique also might relax the required accuracy[42] to which the gravity gradients have to be measured forapplication of the mitigation scheme. Note that formula (12)also applies to time-dependent gravitational potentials as inthis situation. VII. ACKNOWLEDGEMENTS
The author thanks É. Wodey and S. Loriani for fruitfuldiscussions and W. P. Schleich, A. Roura, A. Friedrich, F. DiPumpo and E. Giese for a careful reading of the manuscript.This work is supported by the German Aerospace Center(Deutsches Zentrum für Luft- und Raumfahrt, DLR) with funds provided by the Federal Ministry for Economic Affairsand Energy (Bundesministerium für Wirtschaft und Energie,BMWi) due to an enactment of the German Bundestag underGrant Nos. DLR 50WM1556 and 50WM1956. The authorthanks the Ministry of Science, Research and Art Baden-Württemberg (Ministerium für Wissenschaft, Forschung undKunst Baden-Württemberg) for financially supporting the workof IQ ST . APPENDIXAppendix A: Weakly varying potential
In this appendix we start from the general result in Eq. (12)and rederive the result of Ref. [24] in case of small branchseparation. The modified wave vector in Eq. (12) is a functionof the atom’s mass as the gravitational potential Φ with 𝑉 = 𝑚 Φ is evaluated at the mass-dependent trajectories. However, ifthe local acceleration varies only moderately over the branchseparation (of the order of centimeter for a few ℏ 𝑘 momentumtransfer and a 10-m baseline) this dependence cancels out andwe will find the result of Ref. [24] for an MZ interferometer.
1. Local gravity gradients
To prove this statement, we first decompose the trajectories 𝒓 ( 𝛼 ) ( 𝑡 ) = 𝒓 ( 𝑡 ) + 𝒓 ( 𝛼 )∗ ( 𝑡 ) into a suitably chosen branch-independent mean trajectory 𝒓 ( 𝑡 ) and the deviation 𝒓 ( 𝛼 )∗ ( 𝑡 ) . Thus, we Taylor expand 𝒂 ( 𝛼 ) ( 𝒓 ) = 𝒂 ( 𝒓 ) − Γ ( 𝒓 ) 𝒓 ( 𝛼 )∗ + ... (A1)where the gradient tensor is defined as Γ 𝑖 𝑗 = 𝜕 𝑖 𝜕 𝑗 𝑉 / 𝑚 . Substi-tuting Eq. (A1) into Eq. (10), we find 𝑱 = − 𝑚 ℏ ∮ d 𝑡 Γ ( 𝒓 ) 𝒓 ∗ and 𝑱 = − 𝑚 ℏ ∮ d 𝑡 Γ ( 𝒓 ) 𝒓 ∗ 𝑡 (A2)since 𝒂 ( 𝒓 ) is independent of the branch and therefore cancelsin the looped integrals. If the mean trajectory 𝒓 only containsthe mass-independent part of the trajectory generated by lineargravity while 𝒓 ∗ is the additional contribution from the laserpulses, Eq. (A2) becomes mass independent since 𝒓 ∗ is inverselyproportional to the mass through 𝒗 𝑟 = ℏ 𝒌 / 𝑚 . To connect withprevious results, we specify the case of an MZ gravimeterwhere the atoms are launched initially in 𝑧 direction so that 𝒓 = 𝑔𝑡 ( 𝑇 − 𝑡 / ) e 𝑧 . Consequently, with 𝒓 ( 𝑢 )∗ = 𝒗 𝑟 𝑡 , 𝒓 ( 𝑙 )∗ = ≤ 𝑡 < 𝑇 𝒓 ( 𝑢 )∗ = 𝒗 𝑟 𝑇 , 𝒓 ( 𝑙 )∗ = 𝒗 𝑟 ( 𝑡 − 𝑇 ) 𝑇 ≤ 𝑡 ≤ 𝑇 we find from Eq. (A2) the expressions 𝑱 = − 𝑇 ∫ d 𝑡 𝑡 Γ ( 𝒓 ) 𝒌 − 𝑇 ∫ 𝑇 d 𝑡 ( 𝑇 − 𝑡 ) Γ ( 𝒓 ) 𝒌 (A3) Figure 2.
Compensation for uniform gravity gradients. a) In anMZ interferometer the position of the geometric center 𝑡 𝑐 on thetime axis of the space-time area 𝑨 enclosed by the two branchescoincides with the position of the central pulse. For this reason, asshown in the main text, global gravity gradients can be compensatedby adapting the momentum transfer of the central pulse only. Therequired modification is proportional to the space-time area enclosedby the trajectories. b) In a Ramsey-Bordé interferometer the geometriccenter is situated exactly in between the two central pulses at 𝑡 = 𝑇 and 𝑡 = 𝑇 + 𝑇 . In this case compensation can be achieved by modifyingthe momentum transfer of these pulses equally. and 𝑱 = − 𝑇 ∫ d 𝑡 𝑡 Γ ( 𝒓 ) 𝒌 − 𝑇 ∫ 𝑇 d 𝑡 ( 𝑇 − 𝑡 ) 𝑡 Γ ( 𝒓 ) 𝒌 (A4)derived in the Supplemental material of Ref. [24] after appro-priate resummation of the integrals.In Eq. (A1) corrections from the next order of the Taylorexpansion are suppressed by 𝑣 𝑟 𝑇 / 𝜉 where 𝜉 , the characteristiclength scale on which the potential changes. For the localvariations in the gravitational profile from Ref. [30] the factor 𝑣 𝑟 𝑇 / 𝜉 might approach unity in future experiments involvinglarge-momentum transfer techniques and therefore limits thevalidity of Eqs. (A3) and (A4). Instead, using the midpointtheorem [27] without the approximation in Eq. (A1), the resultstill deviates from the exact expressions in Eq. (10) but only bya factor ( 𝑣 𝑟 𝑇 / 𝜉 ) which justifies its application in many casesbut care has to be taken when employing LMT techniques or thepotential changes on short length scales. This deviation resultsfrom the semiclassical approximation in the derivation of themidpoint theorem which limits its application to anharmonicpotentials.
2. Global gravity gradients
So far we have discussed general anharmonic perturbationswhich might even change rapidly over the branch separation.In this paragraph we assume that the deviations from lineargravity are smooth enough to be accurately described overthe extent of the whole interferometer by a global gradient.Correspondingly, 𝑉 = 𝑚 𝒓 T Γ 𝒓 where the position-independent gradient tensor Γ is chosen fullysymmetric. In case all laser pulses are aligned, we define thevector 𝑨 = ∮ d 𝑡 𝒓 , whose modulus corresponds to the space-time area enclosed by the two branches of the unperturbedinterferometer. With 𝒂 = − Γ 𝒓 the expression 𝑱 𝑡 𝑐 = 𝑱 (A5)defines the position of the geometric center 𝑡 𝑐 of this area onthe time axis and we can distinguish two different situationscorresponding to the two classes of interferometer geometriesdisplayed in Fig. 2: a) Suppose the interferometer exhibits alaser pulse at 𝑡 = 𝑡 𝑐 as for example in the MZ interferometervisualized in Fig. 2 a). From Eq. (12) together with Eq. (A5)we find Δ 𝒌 = 𝑚 ℏ Γ 𝑨 , and Δ 𝒌 = 𝑨 = 𝒗 𝑟 𝑇 where the modification of momentumtransfer is distributed equally over both branches. Thus, in caseof uniform gradients, compensation is particularly simple ifthe interferometer exhibits a laser pulse at the geometric centerof the space-time area enclosed by the branches. b) In contrastif 𝑡 𝑐 is located exactly in between two pulses at 𝑡 and 𝑡 , thatis 𝑡 𝑐 = ( 𝑡 + 𝑡 )/ = 𝑇 + 𝑇 /
2, we find Δ 𝒌 = Δ 𝒌 = 𝑚 ℏ Γ 𝑨 . This situation is for example found in a Ramsey-Bordé interfer-ometer shown in Fig. 2 b) for which we find 𝑨 = 𝒗 𝑟 𝑇 ( 𝑇 + 𝑇 ) . Appendix B: Symmetric trajectories
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