We develop a general framework for calculating the leadingorder fullyrelativistic contributions to the gravitational phase shift in singlephoton atom interferometers within the context of linearized gravity. We show that the atom gradiometer observable which only depends on the atom interferometer propagation phase can be written in terms of three distinct contributions the Doppler phase shift which accounts for the tidal displacement of atoms along the baseline the Shapiro phase shift which accounts for the delay in the arrival time of photons at atomlight interaction points and the Einstein phase shift which accounts for the gravitational redshift measured by the atoms. For specific atom gradiometer configurations we derive the signal and response functions for two physicallymotivated scenarios i transient gravitational waves in the transversetraceless gauge and for the first time in the proper detector frame and ii transient massive objects sourcing weak and slowvarying Newtonian potentials. We find that the Doppler contribution of realistic Newtonian noise sources e.g. a freight truck or a piece of space debris at proposed atom gradiometer experiments such as AION MAGIS and AEDGE can exceed the shot noise level and thus affect physics searches if not properly subtracted.
['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'High Energy Physics - Phenomenology', 'High Energy Physics - Theory', 'Physics - Atomic Physics']
Signatures of Linearized Gravity in Atom Interferometers a Simplified Computational Framework
{'Signatures of Linearized Gravity in Atom Interferometers: a Simplified Computational Framework': 'Leonardo Badurina, 1, ∗ Yufeng Du, 1, † Vincent S. H. Lee, 1, ‡ Yikun Wang, 1, § and Kathryn M. Zurek 1, ¶  \n1  \nWalter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA (Dated: September 9, 2024)  \nWe develop a general framework for calculating the leading-order, fully-relativistic contributions to the gravitational phase shift in single-photon atom interferometers within the context of linearized gravity. We show that the atom gradiometer observable, which only depends on the atom interferometer propagation phase, can be written in terms of three distinct contributions: the Doppler phase shift, which accounts for the tidal displacement of atoms along the baseline, the Shapiro phase shift, which accounts for the delay in the arrival time of photons at atom-light interaction points, and the Einstein phase shift, which accounts for the gravitational redshift measured by the atoms. For specific atom gradiometer configurations, we derive the signal and response functions for two physically-motivated scenarios: (i) transient gravitational waves in the transverse-traceless gauge and, for the first time, in the proper detector frame, and (ii) transient massive objects sourcing weak and slow-varying Newtonian potentials. We find that the Doppler contribution of realistic Newtonian noise sources ( e.g. , a freight truck or a piece of space debris) at proposed atom gradiometer experiments, such as AION, MAGIS and AEDGE, can exceed the shot noise level and thus affect physics searches if not properly subtracted.', 'I. INTRODUCTION': "Atom interferometry is a versatile and rapidly-developing experimental technique that can be used for a wide variety of precision measurements [1]. For instance, atom interferometers (AIs) have been used to measure fundamental constants [2-4], probe the foundational principles of general relativity [5-9] and quantum mechanics [10-12], and test models of dark energy and modified gravity [13-16]. Atom gradiometers (AGs), which consist of two spatially-separated AIs that are referenced by common lasers, have also been proposed to detect gravitational waves (GWs) in the unexplored 'mid-frequency band' [1721], search for violations of the universality of free-fall [22] and measure time-varying corrections to atomic transition energies induced by scalar ultralight dark matter [23-25].  \nIn recent years, a number of ambitious AG experiments have been proposed as quantum sensors for fundamental physics (see Ref. [26] for a recent review). These include large-scale terrestrial experiments, such as AION [27], MAGIS [28], MIGA [29], ELGAR [30], and ZAIGA [31], and futuristic space-based experiments, such as STE-QUEST [32] and AEDGE [33]. By overcoming a number of experimental systematics, these experiments are expected to operate at the shot-noise level. However, fluctuations in an atom's local gravitational field, e.g. due to seismic waves [34] or Newtonian noise (NN) [35], may significantly reduce the projected reach in the key (10 -3 -1) Hz frequency window. If left unmitigated, these effects will dramatically limit the physics potential of these ambitious experiments.  \nInstead of cutting large sections of an experiment's time series [35], it may be possible to subtract the phase shift from transient sources of NN and recover an experiment's shot-noise limited sensitivity provided that this phase shift is known with sufficient precision. Depending on an experiment's projected reach, such a strategy may require a fully relativistic calculation. For example, a massive object traveling with speed v s will induce a phase shift by accelerating the atoms, given by ∆ ϕ non -rel ∼ k eff aT 2 [36], with k eff being the maximum momentum difference between the two arms of an interferometer, T being the interrogation time, and a being the acceleration of the atoms. This is typically calculated utilizing only classical Newtonian mechanics. However, relativistic corrections are expected. Using dimensional analysis and the invariance of general relativity under parity and time-reversal, the dominant relativistic phase shift is expected to scale as ∆ ϕ rel ∼ v s ∆ ϕ non -rel . Although useful to estimate the size of the effect, ∆ ϕ rel does not inform us of: (i) the coefficient of this phase shift term, which may be sequencedependent, and (ii) the shape of the power-spectrum associated with this phase shift, both of which can only be inferred from a general-relativistic ( i.e. general coordinate-invariant) phase shift calculation.  \nSeveral coordinate-invariant formalisms have been proposed [7, 37-40] and have been used to calculate, e.g. , the phase shift induced by gravitational waves [17, 18, 41] and static potentials ( e.g. the Earth's gravitational field) [7, 37, 39, 40]. Notably, the formalism proposed in Ref. [37] defines all tunable experimental parameters in a frame-independent manner, solves for the geodesics of the freely falling  \natomic wavepackets and laser pulses, and determines the momentum transferred to the atoms as a result of atom-light interactions in the atom's local interial frame. Although crucial for correctly predicting the size of relativistic contributions to the phase shift, these formalisms are computationally cumbersome when the number of interaction points exceeds O (1). This is especially relevant in proposed atom gradiometer experiments employing large momentum transfer (LMT) such as AION and MAGIS, which plan up to O (10 4 ) atom-light interactions per cycle.  \nAside from these computational considerations, outstanding questions remain about the physical interpretation of existing gauge-invariant frameworks for computing gravitational phase shifts, such as the interpretation of AGs as gravitational antennas, or equivalently the mapping between AG and laser interferometer observables. For example, Ref. [37] decomposes the gauge-invariant phase shift for a single AI (and consequently for an AG) into three contributions: the phase shift associated with the free-evolution of atomic wavepackets in spacetime ( i.e. the propagation phase), the phase shift imprinted by the laser pulses during atom-light interactions ( i.e. the laser phase) and the phase shift associated with the degree to which the two spatially-separated wavepackets do not overlap at the application of the final beamsplitter pulse ( i.e. the separation phase). As shown for GWs in Ref. [42] and recently for more general metric perturbations via a proper time treatment in Ref. [43], the observable in laser interferometers can be written as a sum of three distinct (and separately not diffeomorphism-invariant) contributions: the time-delay caused by the tidal displacement of the mirrors along the baseline ( i.e. the Doppler time delay), the delay in the arrival time of photons at the mirrors ( i.e. the Shapiro time delay), and the time-delay due to the gravitational redshift measured by the beamsplitter ( i.e. the Einstein time delay). Since AGs have been proposed as gravitational wave interferometers, it should be possible to extract these three contributions from the gauge-invariant AG phase shift. This endeavor would clarify the interpretation of AGs as exquisite accelerometers [44] and time-keeping devices [45], and elucidate the origin of the relativistic phase shift contributions in different frames.  \nIn this work, we address these points by developing from first principles a simplified general coordinate-invariant framework for calculating phase shifts in single-photon AGs. Since the metric perturbation in the problems of interest is very small, we will work within the context of linearized gravity. Additionally, as the motion of the atoms relative to the laser sources is highly non-relativistic, we work to leading order in the atom velocity. Notably, for the case of AGs, we express the differential phase shift in terms of contributions which are in one-to-one correspondence with the time-delays that enter the laser interferometer observable: the phase shifts associated with the Doppler, Shapiro and Einstein time delays. Equipped with this formalism, we compute the signal and response function induced by two well-motivated physical scenarios: transient gravitational waves (GWs) and weak and slow-varying Newtonian potentials sourced by transient massive objects. Since the form of the response function depends on the pulse sequence, we will perform explicit calculations for gradiometers employing  \nMach-Zehnder and LMT configurations. Importantly, our examples highlight the gauge-invariance of our framework (as explicitly shown in the GW calculation, which is performed in both the transversetraceless and the proper detector frame) and the accuracy of our formalism in reproducing existing results in the literature in a more physically and computationally transparent fashion.  \nThis paper is structured as follows. After reviewing the basics of atom interferometry, in section II we introduce our general coordinate-invariant framework for computing gradiometer phase shifts in linearized gravity. After deriving the basis for our formalism in section II A, in section II B we introduce the gradiometer observable and provide expressions for Mach-Zehnder and LMT gradiometer configurations. As example applications, in section III we compute the phase shifts induced by gravitational waves and slow-varying weak Newtonian potentials. In section IV we summarize the key results of this paper. Appendices A-C support the calculations in sections II-III.", 'II. DERIVATION OF THE LEADING-ORDER GRADIOMETER PHASE SHIFT IN LINEARIZED GRAVITY': "Schematically, atom interferometers (AIs) utilize matter wave interference to detect the phase difference between two coherent atomic states in a spatial superposition. In order for an atom to be prepared in a spatial superposition and then measured via matter wave interference, the trajectories of the atomic wavepackets are manipulated using laser pulses. Most experiments rely on a two-level system where the external momentum and the internal energy state of the atoms can be manipulated via Rabi oscillations [46]. A laser pulse that interacts with the atom over a quarter of the Rabi cycle (a π/ 2-pulse) takes an atom in one state to an equal superposition of two states, thus acting as a 'beamsplitter'; a pulse over half of the Rabi cycle (a π -pulse) reverses the state of the atom, thus acting as a 'mirror' [47]. In this paper, we focus on experiments that rely on two spatially-separated AIs operating common lasers and single-photon transitions with energy separation ω a . We refer to these configurations as single-photon atom gradiometers (AGs).  \nIn order to correctly capture relativistic effects, which manifest through the dependence of dynamics on the curvature of spacetime, it is of paramount importance to describe the interferometer sequence and all experimental quantities in a frame-independent manner. Importantly, this guarantees that the observable is free of gauge artifacts. Let us consider the description of atomic fountains (which we assume in this work) in the language of general relativity ( e.g. Ref. [37]). In these AI experiments, the atoms are in free-fall. Provided that the radius of curvature is much larger than the wavepacket size, a semi-classical treatment is sufficient. In this regime, atom trajectories are described in terms of timelike geodesics, while photon trajectories are described in terms of null geodesics. The spacetime points at which atom-light interactions occur can be solved in coordinate time and position. Furthermore, for the  \nphase shift to be frame-independent, the coordinate times at which the laser pulses are emitted must be related to the established time difference measured by an observer traveling along the laser's worldline. Finally, the four momentum of a pulse must be related to the frequency of the pulse at emission; this four momentum is then evolved from emission to the designated atom-light interaction point, where the atom's recoil is computed in a local inertial frame.  \nInspired by this description, in the following sections we provide a detailed derivation from first principles of the gauge-invariant gradiometer phase shift. Importantly, our framework is valid to leading order in a generic metric perturbation and correctly predicts the coefficients of phase shift terms that are linear in the atom's recoil velocity. Starting from the phase shift for a single AI, we show that the single-photon AG observable can be reinterpreted in terms of coordinate time-delays that are well understood in the context of laser interferometers: the Doppler, Shapiro and Einstein time-delays.", 'A. Propagation Phase Shift': "Let us consider the semi-classical evolution of an atomic wavepacket | ψ ⟩ in spacetime. In this regime, the atom's dynamics can be described in terms of the evolution of the atom's center-of-mass (c.o.m.). Consequently, solving the Schrodinger equation, | ψ ⟩ ∝ exp(i S ), where S is the action of the atom's c.o.m. [48]. In general relativity, the action of a massive point-like particle can be expressed in terms of the proper time elapsed along particle's timelike worldline [49]. Therefore, the phase difference associated with the spacetime propagation of two spatially-separated wavepackets corresponds to the difference in the actions evaluated along the worldlines of the two wavepackets' c.o.m. from state initialization to measurement. For a spacetime with metric g µν , the propagation phase is  \n∆ ϕ = ∮ C mdτ = ∮ C m √ -g µν dx µ dt dx ν dt dt , (1)  \nwhere m is the path-dependent mass of the atom, with m = m o and m = m o + ω a for ground and excited states, respectively. 1 The proper time along the atom's worldline is defined as τ , which is not to be confused with the coordinate time x 0 = t , and the atom's path-dependent four coordinate-velocity is defined as dx µ /dt . We parameterize the atom's evolution with respect to t and perform the loop integral over the closed semi-classical path C . This path depends on: (i) the free evolution of the atomic wavepackets between atom-light interaction points, (ii) the arrival time of laser pulses at atom-light interaction points and (iii) the metric-dependent correction to the laser beam's wave-vector, which leads to a correction in the recoil of the atoms after atom-light interactions. Here, we assume that C is initiated when the superposition of states is created ( i.e. at the initial beamsplitter pulse) and closes when the  \nFIG. 1: Schematic spacetime diagram showing the atom trajectories used in deriving the framework. The dotted yellow lines denote hypersurfaces defined by the initial beam-splitter pulse, the final beamsplitted pulse, and the measurement. The unperturbed atom trajectory C o is schematically shown as the dashed black lines. In the presence of a metric perturbation, the atom follows a geodesic C (purple), which can be deformed into another geodesic ˜ C (blue) that closes at the final beam-splitter. ∆ x µ is the separation of path C at the final beam-splitter pulse. λ parameterizes the atom geodesics, and the deformation ξ ( λ ) is of order O ( h ) and only enters into the observable at O ( h 2 ). At O ( hv ), ˜ C is a good approximation for C in calculating the propagation phase shift.  \n<!-- image -->  \ninterference pattern is measured. The latter should not be confused with the final beamsplitter used to redirect the atomic states to the measurement ports.  \nNote that the full AI phase observable, ∆ ϕ ∣ ∣ AI , also depends on the laser phase ∆ ϕ laser , which is imprinted onto the atoms during atom-light interaction and arises from the linear coupling between the photon field and the atom's electric dipole [50]:  \n∆ ϕ ∣ ∣ AI = ∆ ϕ +∆ ϕ laser . (2)  \nIn gradiometer setups featuring more than one AI referenced by the same laser, the AIs experience a common laser phase which cancels out in a differential measurement. 2 Since we focus on gradiometer observables, we do not consider this effect further, though it is important to note that the laser phase contribution is generally non-zero in single-AI setups.  \nFor the relativistic effects considered in this work, it is sufficient to linearize the metric tensor, i.e. g µν = η µν + h µν , with h µν ≪ 1. By working at leading order in h µν and relating C to ancillary paths, which we schematically depict in Fig. 1, a number of simplifications will arise. First, in the absence of  \n̸  \nmetric perturbations, g µν = η µν , we assume that the atomic wavepackets travel along trajectories that recombine at the application of the final beamsplitter pulse; we will refer to this path as C o . Second, as alluded to above, we define C as the closed path traced by the atomic wavepackets in the presence of a non-zero metric perturbation h µν ; however, while C is closed at measurement, it is no longer closed at the final beamsplitter pulse. Finally, we define a third and fictitious path ˜ C , constructed from C assuming the same metric perturbation h µν ; the initial conditions ( e.g. atom velocity) are adjusted for the path to close at the final beamsplitter pulse. Hence, generally C = C .  \n˜ The advantage of introducing these ancillary paths is that, at leading order in the metric perturbation, the phase shift along the true path C is equivalent to the phase shift computed along the unphysical path ˜ C . This can be seen simply as follows. Let ξ µ ( λ ) be the coordinate separation between the atom geodesics in C and ˜ C , where λ ∈ [0 , 1] parameterizes the atom worldlines and ξ µ (0) = ξ µ (1) = 0, since both C and ˜ C are closed with respect to state initialization and state measurement. Expanding in the coordinate deviation ξ and defining L = √ g µν ˙ x µ ˙ x ν , where ˙ x µ = dx µ /dλ , the phase shift over the true path C takes the form 3  \n∆ ϕ = ∮ C mdτ = ∮ ˜ C mdτ + ∫ 1 0 ξ µ ( d dt ∂L ∂ ˙ x µ -∂L ∂x µ ) ∣ ∣ ∣ ∣ ∣ R dλ -∫ 1 0 ξ µ ( d dt ∂L ∂ ˙ x µ -∂L ∂x µ ) ∣ ∣ ∣ ∣ ∣ L dλ + O ( ξ 2 ) = ∮ ˜ C mdτ + O ( ξ 2 ) , (3)  \nwhere the left and right trajectories are marked by the L and R subscripts, respectively. The second and third terms in the first line vanish because the atom worldlines in ˜ C are geodesics, and hence extrema of the action.  \nWe next expand Eq. (3) to leading order in the metric perturbation, utilizing ˜ C = C o + δC with δC = O ( h ):  \n∆ ϕ = ∮ ˜ C m √ -η µν dx µ dt dx ν dt dt -1 2 ∮ C o m √ -η µν dx µ dt dx ν dt h µν dx µ dt dx ν dt dt + O ( h 2 ) . (4)  \nWriting the phase in Eq. (4) to leading order in O ( hv ), we find 4  \n∆ ϕ = ∮ ˜ C mdt -∮ C o mv i δv i dt -1 2 ∮ C o mh 00 dt -∮ C o mh 0 i v i dt + O ( h 2 , v 2 ) , (5)  \nwhere δv i is the O ( h ) correction to the atom three velocity arising from free-falling atoms, obtained by  \nsolving geodesic equations 5  \nδv i ( t, x ) = -∫ t t 0 dt ' Γ i 00 ( x, t ' ) = -η ij ∫ t t 0 dt ' ( ∂ 0 h j 0 -1 2 ∂ j h 00 ) + O ( h 2 , v ) , (6)  \nwhere Γ µ αβ is the Christoffel symbol in linearized gravity and t 0 is the time at which the experiment is initialized. We neglect the contribution to the atom's recoil velocity after atom-light interactions that arises from the O ( h ) correction to the photon's wave-vector, since the leading-order correction is proportional to v and therefore enters the observable at next-to-leading order.  \nThe first term on the RHS of Eq. (5) corresponds to phase shift due to the deformation of the atom trajectories from C 0 to ˜ C . We allow for multiple pulses to divide each interferometer arm into path segments labeled by k , so that this term can be written as  \n∮ ˜ C mdt ⊃ ∮ δC mdt = ( ∑ k ∈ E ω a ∆ t ( k ) )∣ ∣ ∣ ∣ ∣ R -( ∑ k ∈ E ω a ∆ t ( k ) )∣ ∣ ∣ ∣ ∣ L , (7)  \nwhere the leading order term is proportional to ω a since the rest mass contribution vanishes under a loop-integral, i.e. ∮ ˜ C m o dt = 0; 6 hence we only sum over the excited segments, denoted as the set E . As a result, only excited state path segments contribute to the phase shift in Eq. (7); this will be very important in the calculations below. In Eq. (7), ∆ t ( k ) is the O ( h ) coordinate time duration of the k -th path segment, and path segments on both the right ( R ) and left arm ( L ) are summed over. The overall minus sign in the left arm contributions originates from the loop integral. The coordinate time corrections can be computed by solving for the intersection of atom and photon worldlines. Importantly, these atom-light interaction points define the path segments. We label each path segment k by the laser pulse that starts the sequence. The k -th segment is ended by the laser pulse labeled as k + 1, which starts the subsequent segment k + 1. Denoting the perturbed initial and final times of the k -th path segment as ¯ t ( k ) + δt ( k ) and ¯ t ( k +1) + δt ( k +1) , the perturbed atom worldline as ¯ x i ( t ) + δx i ( t ), the worldlines of the photons as ¯ x i γ ( k ) ( t ) + δx i γ ( k ) ( t ) and ¯ x i γ ( k +1) ( t ) + δx i γ ( k +1) ( t ), with unperturbed quantities denoted by overbars, and n i as the unit vector pointing along the baseline (considering the motion of both the photons and atoms in a single spatial direction), the corresponding equation to solve is  \nn i [ ¯ x i ( ¯ t ( k ) + δt ( k ) ) + δx i ( ¯ t ( k ) )] = n i [ ¯ x i γ ( k ) ( ¯ t ( k ) + δt ( k ) ) + δx i γ ( k ) ( ¯ t ( k ) )] . (8)  \nExpanding Eq. (8) to O ( h ) and neglecting the unperturbed atom velocity (as the prefactor ω a in Eq. (7)  \nis parametrically O ( v )), we find  \nδt ( k ) = ( ± ) ( k ) n i [ δx i ( t ( k ) ) -δx i γ ( k ) ( t ( k ) )] , (9)  \nwhere ( ± ) ( k ) is taken to be +1 or -1 for outgoing or incoming photons ( i.e. parallel or anti-parallel to the baseline) interacting with the atom, respectively. Here we drop the overbars, which is valid to O ( h ). The perturbed atom positions, δx i ( t ), are given by integrating Eq. (6), i.e.  \nδx i ( t, x ) = ∫ t t 0 δv i ( t ' , x ) dt ' , (10)  \nand the perturbed photon trajectories, δx i γ ( k ) ( t ), are given by solving the null geodesic condition, ds 2 = g µν dx µ dx ν = 0, leading to  \nn i d dt δx i γ ( k ) ( t ) = ( ∓ ) ( k ) H ( ± ) ( k ) , (11)  \nwhere we defined  \nH ± ≡ 1 2 ( h 00 ± 2 h 0 i n i + h ij n i n j ) . (12)  \nPutting Eq. (10) and Eq. (11) into Eq. (9), the time shifts, ∆ t ( k ) , can thus be expressed as the sum of Doppler and Shapiro time delays,  \n∆ t ( k ) = δt ( k +1) -δt ( k ) ≡ ∆ t ( k ) D +∆ t ( k ) S . (13)  \nThe Doppler term is due to the atom's motion under the metric perturbation as derived in Eq. (10):  \n∆ t ( k ) D = n i [ ( ± ) ( k +1) δx i ( t ( k +1) , x ( k +1) ) -( ± ) ( k ) δx i ( t ( k ) , x ( k ) )] , (14)  \nwith x ( k ) and x ( k +1) being the unperturbed initial and final atom positions of the k -th and ( k +1)-th path segment. The Shapiro term, which corresponds to the time delay accrued by the photon along its geodesic as derived in Eq. (11), is given by  \n∆ t ( k ) S = ∆ T ( ± ) ( k +1) S ( t ( k +1) L , x ( k +1) L , x ( k +1) ) -∆ T ( ± ) ( k ) S ( t ( k ) L , x ( k ) L , x ( k ) ) +∆ t ( k ) laser , (15)  \nwhere we defined  \n∆ T ± S ( t, x 1 , x 2 ) ≡ ± ∫ x 2 x 1 H ± ( t ± ( x ' -x 1 ) , x ' ) dx ' . (16)  \nHere, the photons which interact with the atom at the k -th and ( k +1)-th intersection are emitted from the lasers at x ( k ) L and x ( k +1) L , and t ( k ) -t ( k ) L ≡ ( ± ) ( k ) ( x ( k ) -x ( k ) L ) with t ( k ) L the corresponding photon emission time. The quantity ∆ t ( k ) laser depends on the O ( h ) correction to the photon's emission spacetime points and cancels out in a differential measurement, as discussed in Appendix A.  \nThe remaining three terms on the RHS of Eq. (5) only depend on the unperturbed path C o . By using Eq. (6), these terms can be rewritten as  \n∆ ϕ E ≡ -∮ C o mv i δv i dt -1 2 ∮ C o mh 00 dt -∮ C o mh 0 i v i dt = -1 2 ∮ C o dt m ( 1 + v i ∫ t t 0 dt ' ∂ i ) h 00 . (17)  \nThese contributions only depend on h 00 , and are thus identified as the Einstein term ( i.e. time dilation as measured by the atoms). We note that this phase shift can be significantly simplified if the metric perturbation is spatially slow varying over C o , which allows it to be expanded as  \nh µν ( t, x AI + x a ) = h µν ( t, x AI ) + x i a ∂ i h µν ( t, x AI ) + O ( v 2 ) . (18)  \nHere, the distance traveled by the atoms from their initial position is x i a ( t ) ∝ v i and x AI is the unperturbed position of the AI at the start of the sequence. The expansion is valid when ( vT ) ∂ i h µν ≪ h µν , where T is the interrogation time, and holds for all examples considered in this work. 7 With this approximation and integrating by parts, Eq. (17) simplifies to  \n∆ ϕ E = ( ∑ k ∈ E ω a ∆ t ( k ) E )∣ ∣ ∣ ∣ ∣ R -( ∑ k ∈ E ω a ∆ t ( k ) E )∣ ∣ ∣ ∣ ∣ L , (19)  \nwhere the non-vanishing contribution comes from the excited state segments, as the ground state contribution sums to zero over the closed loop, and  \n∆ t ( k ) E = -1 2 ∫ C ( k ) o h 00 ( t, x AI ) dt + O ( v 2 ) (20)  \ndefines the Einstein time delay.  \nIn summary, the propagation phase shift in Eq. (5) can be written schematically as  \n∆ ϕ = ∆ ϕ D +∆ ϕ S +∆ ϕ E + O ( h 2 , v 2 ) , (21)  \nwhere the Doppler and Shapiro phase shifts originate from Eq. (7) with ∆ t ( k ) given in Eq. (13), while the Einstein phase shift originates from Eq. (17). The expressions of the Doppler, Shapiro and Einstein time  \ndelays are given in Eqs. (14), (15) and (20), respectively. Together with Eq. (19), Eq. (21) facilitates the direct comparison between atom interferometer experiments, such as AION and MAGIS, and laser interferometers, such as LIGO [52] and GQuEST [53]. Indeed, the time-delays that enter the atom interferometer propagation phase also appear in laser interferometer calculations. The only difference between the two propagation phase shifts is the physical origin of the frequency: in the case of atom interferometers, the phase shift is proportional to the energy difference between the ground and excited state; in the case of laser interferometers, the phase shift is proportional to the frequency of the laser pulse traveling along the baseline. More broadly, the appearance of the Doppler and Shapiro time-delays facilitates the analogy between the atom interferometers and the mirrors of a one-dimensional laser interferometer. Furthermore, the appearance of the Einstein time-delay clarifies the analogy between an atom interferometer and the beamsplitter in a laser interferometer. Finally, since we started with the action for massive point-like particles ( cf . Eq. (1)), the phase shift is manifestly invariant under diffeomorphisms.", 'B. Doppler, Shapiro, and Einstein Phase Shifts for Atom Gradiometers': "In an AG, the observable is the difference between the phase shifts measured by each AI, i.e. ∆ ϕ grad ≡ ∆ ϕ ∣ ∣ AI 1 -∆ ϕ ∣ ∣ AI 2 . Since the transitions in each AI are driven by common laser pulses, the gradiometer observable exclusively depends on the difference between the propagation phase shifts of each AI and, subsequently, on the difference between the coordinate time corrections between path segments, which we refer to as gradiometer time delays. Without loss of generality, we can write the gradiometer time delays for each segment k because the pair of laser pulses that start and end the path segment in a given AI also start and end a path segment in the other AI. Restricting our attention to such configurations, the O ( h ) correction to the laser's motion cancels, as we carefully show in Appendix A. Pulling all parts of the derivation together, the gradiometer phase shift for experiments using single-photon transitions can be schematically expressed as  \n∆ ϕ grad = ∆ ϕ grad , D +∆ ϕ grad , S +∆ ϕ grad , E + O ( h 2 , v 2 ) , ∆ ϕ grad , D , S , E = ∑ k ∆ ϕ ( k ) grad , D , S , E = ( ∑ k ∈ E ω a ∆ t ( k ) grad , D , S , E )∣ ∣ ∣ ∣ ∣ R -( ∑ k ∈ E ω a ∆ t ( k ) grad , D , S , E )∣ ∣ ∣ ∣ ∣ L , (22)  \nwhere the gradiometer time delay is expressed in terms of the single AI time delays ∆ t ( k ) D , S , E ( cf. Eqs. (14), (15) and (20)) as  \n∆ t ( k ) grad , D , S , E ≡ ∆ t ( k ) D , S , E ∣ ∣ ∣ AI 1 -∆ t ( k ) D , S , E ∣ ∣ ∣ AI 2 . (23)  \nSince the ground state contribution vanishes over the loop, we only sum over the path segments where the atoms are in the excited state.  \nExplicit expressions for the Doppler, Shapiro, and Einstein phase shifts can be extracted for particular configurations. In what follows, we provide analytical phase shift expressions for two popular single-photon gradiometer designs: the Mach-Zehnder (MZ) and the large-momentum-transfer (LMT) gradiometers. For convenience, we will describe both experiments in a frame where the lasers are at rest in the absence of a metric perturbation. 8 By convention, the right arm receives the initial momentum deposition.", '1. Mach-Zehnder Gradiometer': "As we showed in section II B 1, the Einstein, Doppler and Shapiro gradiometer phase shift contributions for a MZ gradiometer initiated at time t 0 may be expressed as  \n∆ ϕ MZ grad , E , D , S ( t 0 ) = ω a ( ∆ t MZ grad , E , D , S ( t 0 ) -∆ t MZ grad , E , D , S ( t 0 + T ) ) , (B1)  \nwhere the time-delays are defined as  \n∆ t MZ grad , E ( t ) = -1 2 [∫ t + T t h 00 ( t ' , x AI 1 ) dt ' -∫ t + T + L t + L h 00 ( t ' , x AI 2 ) dt ' ] , ∆ t MZ grad , D ( t ) = n i [ δx i ( t + T, x AI 1 ) -δx i ( t, x AI 1 ) + δx i ( t + L, x AI 2 ) -δx i ( t + T + L, x AI 2 ) ] , ∆ t MZ grad , S ( t ) = ∫ x AI 2 x AI 1 H + ( t ( x ' ) , x ' ) dx ' -∫ x AI 2 x AI 1 H + ( t ( x ' ) + T, x ' ) dx ' , (B2)  \nwith t ( x ' ) ≡ t + x ' -x AI 1 in the limit where the distance between the laser source and the first AI can be neglected. Using the Fourier transform convention  \n˜ x ( ω ) = ∫ ∞ -∞ dt x ( t ) e -i ωt , (B3)  \nand performing the time shift t 0 → t 0 -T , each gradiometer phase shift contribution may be expressed in the frequency domain as  \n∆ ˜ ϕ MZ grad , E , D , S ( ω ) = ω a ( 1 -e i ωT ) ∆ ˜ t MZ grad , E , D , S ( ω ) . (B4)  \nSimilarly, the Fourier transform of the Doppler gradiometer time delay can be expressed as  \n∆ ˜ t MZ grad , D ( ω ) = ( e i ωT -1 ) [ n i δ ˜ x i ( ω, x AI ) e i ω ( x AI -x AI 1 ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 . (B5)  \nFor convenience, let us rewrite the Shapiro gradiometer time-delay in terms of the Shapiro time-delay computed between the laser located at x L and a particular AI ( cf . Eq. (16)). Explicitly,  \n∆ t MZ grad , S ( t 0 ) = [ ∆ T + S ( t 0 + T, x AI 1 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 -[ ∆ T + S ( t 0 , x AI 1 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B6)  \nwith  \n∆ T + S ( t 0 , x AI 1 , x AI ) ≡ ∫ x AI x AI 1 H + ( t 0 +( x ' -x AI 1 ) , x ' ) dx ' . (B7)  \nEmploying this decomposition and adopting the procedure for deriving the MZ gradiometer phase shift, the Fourier transform of the Shapiro gradiometer time delay takes the following compact form:  \n∆ ˜ t MZ grad , S ( ω ) = ( e i ωT -1 ) [ ∆ ˜ T + S ( ω, x AI 1 , x AI ) ] x AI 1 x AI 2 . (B8)  \nTo express the Einstein gradiometer time delay in the frequency domain, we use the following identity:  \nFT t → ω {∫ t -∞ f ( t ' ) dt ' } = ˜ f ( ω ) i ω +2 π ˜ f (0) δ ( ω ) , (B9)  \nwhere δ ( ω ) is the Dirac delta function [62]. Consequently, for constant t A and t B , we find  \nFT t → ω {∫ t + t B t + t A f ( t ' ) dt ' } = ( ˜ f ( ω ) i ω +2 π ˜ f (0) δ ( ω ) ) ( e i ωt B -e i ωt A ) = ˜ f ( ω ) i ω ( e i ωt B -e i ωt A ) , (B10)  \nwhere the last equality follows from the definition of δ ( ω ). Using Eq. (B10), the Einstein gradiometer time delay takes the form  \n∆ ˜ t MZ grad , E ( ω ) = ( e i ωT -1 ) [ -1 2i ω ˜ h 00 ( ω, x AI ) e i ω ( x AI -x AI 1 ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 . (B11)  \nFinally, noting that ( e i ωT -1 ) 2 = -4 e i ωT sin 2 ( ωT/ 2) = -ω 2 T 2 e i ωT sinc 2 ( ωT/ 2) and using Eqs. (B1), (B5), (B8) and (B11), the three gradiometer phase shift contributions take the form  \n∆ ˜ ϕ MZ grad , E ( ω ) = ω a T 2 ω 2 K MZ ( ω ) [ -1 2i ω ˜ h 00 ( ω, x AI ) e i ω ( x AI -x AI 1 ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ ˜ ϕ MZ grad , D ( ω ) = ω a T 2 ω 2 K MZ ( ω ) [ n i δ ˜ x i ( ω, x AI ) e i ω ( x AI -x AI 1 ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ ˜ ϕ MZ grad , S ( ω ) = ω a T 2 ω 2 K MZ ( ω ) [ ∆ ˜ T + S ( ω, x AI 1 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B12)  \nwhere we defined the MZ response function as K MZ ( ω ) = e i ωT sinc 2 ( ωT/ 2).", '2. Large-momentum-transfer Gradiometer': "The derivation of the Einstein, Doppler and Shapiro gradiometer phase shift contributions for a LMT gradiometer follows in a similar fashion. As we showed in section II B 2, the gradiometer phase shift contributions take the form  \n∆ ϕ LMT grad , E , D , S ( t 0 ) = ω a n -2 ∑ k =0 even ( ∆ t ( k ) grad , E , D , S ( t 0 ) ∣ ∣ ∣ ∣ L 1 +∆ t ( k ) grad , E , D , S ( t 0 + T -( n -1) L ) ∣ ∣ ∣ ∣ L 2 -∆ t ( k ) grad , E , D , S ( t 0 + T ) ∣ ∣ ∣ ∣ L 1 -∆ t ( k ) grad , E , D , S ( t 0 +2 T -( n -1) L ) ∣ ∣ ∣ ∣ L 2 ) , (B13)  \nHere we sum over only even k because only excited states segments contribute to the observable phase shift. We quote Eq. (28)-(29) from Sec. B 2 for the expressions of ∆ t ( k ) grad , E , D , S ( t ) ∣ ∣ L 1 and ∆ t ( k ) grad , E , D , S ( t ) ∣ ∣ L 2 to start our derivation:  \n∆ t ( k ) grad , E ( t ) ∣ ∣ ∣ L 1 , L 2 = -1 2 [ ∫ t k +1 ( t,x AI ) t k ( t,x AI ) h 00 ( t ' , x AI ) dt ' ]∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ t ( k ) grad , D ( t ) ∣ ∣ ∣ L 1 , L 2 = n i [ ( ± ) ( k +1) δx i ( t k +1 ( t, x AI ) , x AI ) -( ± ) ( k ) δx i ( t k ( t, x AI ) , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ t ( k ) grad , S ( t ) ∣ ∣ ∣ L 1 , L 2 = [ ∆ T ( ± ) ( k +1) S ( t +( k +1) L, x ( k +1) L , x AI ) -∆ T ( ± ) ( k ) S ( t + kL, x ( k ) L , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B14)  \nwhere t k ( t, x ) is defined as  \nt k ( t, x ) = t + kL +( ± ) ( k ) ( x -x ( k ) L ) . (B15)  \nWe note that x ( k ) L is the location of the laser source that emits the k -th pulse, and ( ± ) ( k ) corresponds to the direction of the k -th laser pulse. An 'outgoing' pulse is defined to be parallel to the AG baseline ˆ n and takes the sign '+', while an 'incoming' pulse is anti-parallel to the baseline and takes the sign ' -'. In an LMT sequence, a pair of laser pulses that define a path segment always consists of two consecutive pulses from opposite ends of the baseline. There are two possible combinations of laser pulse pairs, ' > ' consists of an outgoing-incoming pair, and ' < ' consists of an incoming-outgoing pair.  \nFor the sake of clarity, in this section, we spell out all the x ( k ) L and ( ± ) ( k ) for different path segments in an LMT sequence, and write down explicit equations for the gradiometer time delays for the two kinds of segments. We define the gradiometer time delays for a path segment initiated by a pulse from x L 1 ( i.e. a ' > ' segment) to be ∆ t ( k ) grad , D , S , E ∣ ∣ L 1 . In this case, pulse k is emitted from x (0) L = x L 1 and is outgoing ( i.e. ( ± ) ( k ) = +1), while pulse k +1 is emitted from x ( k +1) L = x L 2 and is incoming ( i.e. ( ± ) ( k +1) = -1). We can write down the gradiometer time delays for excited state segments started by an outgoing pulse  \nfrom x L 1 :  \n∆ t ( k ) grad , E ( t ) ∣ ∣ ∣ L 1 = -1 2 [ ∫ t k +1 ( t,x AI ) t k ( t,x AI ) h 00 ( t ' , x AI ) dt ' ]∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ t ( k ) grad , D ( t ) ∣ ∣ ∣ L 1 = -n i [ δx i ( t k ( t, x AI ) , x AI ) + δx i ( t k +1 ( t, x AI ) , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ t ( k ) grad , S ( t ) ∣ ∣ ∣ L 1 = -[ ∫ x AI x L 1 H + ( t k ( t, x ' ) , x ' ) dx ' + ∫ x AI x L 2 H -( t k +1 ( t, x ' ) , x ' ) dx ' ]∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B16)  \nwhere we have for even k :  \nt k ( t, x ) = t + kL +( x -x L 1 ) , for | g ⟩ → | e ⟩ , t k +1 ( t, x ) = t +( k +1) L -( x -x L 2 ) , for | e ⟩ → | g ⟩ . (B17)  \nSimilarly, the gradiometer time delays for a path segment initiated by a pulse from x L 2 ( i.e. a ' < ' segment) is written as ∆ t ( k ) grad , D , S , E ∣ ∣ L 2 . In this case, pulse k is emitted from x ( k ) L = x L 2 and is incoming ( i.e. ( ± ) ( k ) = -1), while pulse k +1 is emitted from x ( k +1) L = x L 1 and is outgoing ( i.e. ( ± ) ( k +1) = +1). The gradiometer time delays for excited state segments started by an incoming pulse from x L 2 can be written as:  \n∆ t ( k ) grad , E ( t ) ∣ ∣ ∣ L 2 = -1 2 [ ∫ t k +1 ( t,x AI ) t k ( t,x AI ) h 00 ( t ' , x AI ) dt ' ]∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ t ( k ) grad , D ( t ) ∣ ∣ ∣ L 2 = n i [ δx i ( t k ( t, x AI ) , x AI ) + δx i ( t k +1 ( t, x AI ) , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ t ( k ) grad , S ( t ) ∣ ∣ ∣ L 2 = [ ∫ x AI x L 1 H + ( t k +1 ( t, x ' ) , x ' ) dx ' + ∫ x AI x L 2 H -( t k ( t, x ' ) , x ' ) dx ' ]∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B18)  \nwhere we have for even k :  \nt k ( t, x ) = t + kL -( x -x L 2 ) , for | g ⟩ → | e ⟩ , t k +1 ( t, x ) = t +( k +1) L +( x -x L 1 ) , for | e ⟩ → | g ⟩ . (B19)  \nIn the frequency domain, using the same procedure as before, Eq. (B13) can be rewritten as  \n∆ ˜ ϕ LMT grad , E , D , S ( ω ) = ω a ( 1 -e i ωT ) n -2 ∑ k =0 even ∆ ˜ t ( k ) grad , E , D , S ( ω ) ∣ ∣ ∣ ∣ L 1 + ω a e i ω ( T -( n -1) L ) ( 1 -e i ωT ) n -2 ∑ k =0 even ∆ ˜ t ( k ) grad , E , D , S ( ω ) ∣ ∣ ∣ ∣ L 2 . (B20)  \nUsing Eq. (B10) and the steps shown in Appendix B 1, and setting x L 2 = x L 1 + L , the Fourier transforms of the Einstein gradiometer time delays for a particular value of k take the form  \n∆ ˜ t ( k ) grad , E ( ω ) ∣ ∣ ∣ L 1 = e i ωkL [ -1 2i ω ˜ h 00 ( ω, x AI ) ( e i ωL e i ω ( x L 2 -x AI ) -e i ω ( x AI -x L 1 ) ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ ˜ t ( k ) grad , E ( ω ) ∣ ∣ ∣ L 2 = e i ωkL [ -1 2i ω ˜ h 00 ( ω, x AI ) ( e i ωL e i ω ( x AI -x L 1 ) -e i ω ( x L 2 -x AI ) ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B21)  \nwhile the Doppler contributions can be expressed as  \n∆ ˜ t ( k ) grad , D ( ω ) ∣ ∣ ∣ L 1 = e i ωkL [ -n i δ ˜ x i ( ω, x AI ) ( e i ω ( x AI -x L 1 ) + e i ωL e i ω ( x L 2 -x AI ) )] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ ˜ t ( k ) grad , D ( ω ) ∣ ∣ ∣ L 2 = e i ωkL [ n i δ ˜ x i ( ω, x AI ) ( e i ω ( x L 2 -x AI ) + e i ωL e i ω ( x AI -x L 1 ) )] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 . (B22)  \nTo achieve compact closed form expressions in the frequency domain, we express the Shapiro gradiometer time delay in terms of Eqs. (16). Explicitly, in terms of x L 1 and x L 2 , we find  \nwhere  \n∆ t ( k ) grad , S ( t ) ∣ ∣ ∣ L 1 = [ ∆ T -S ( t +( k +1) L, x L 2 , x AI ) -∆ T + S ( t + kL, x L 1 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ t ( k ) grad , S ( t ) ∣ ∣ ∣ L 2 = [ ∆ T + S ( t +( k +1) L, x L 1 , x AI ) -∆ T -S ( t + kL, x L 2 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B23)  \n∆ T + S ( t, x L 1 , x AI ) = ∫ x AI x L 1 H + ( t -x L 1 + x ' , x ' ) dx ' , ∆ T -S ( t, x L 2 , x AI ) = -∫ x AI x L 2 H -( t + x L 2 -x ' , x ' ) dx ' . (B24)  \nIn the frequency domain, Eqs. (B23) can be compactly expressed as  \n∆ ˜ t ( k ) grad , S ( ω ) ∣ ∣ ∣ L 1 = -e i ωkL [ ∆ ˜ T + S ( ω, x L 1 , x AI ) -e i ωL ∆ ˜ T -S ( ω, x L 2 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ ˜ t ( k ) grad , S ( ω ) ∣ ∣ ∣ L 2 = e i ωkL [ e i ωL ∆ ˜ T + S ( ω, x L 1 , x AI ) -∆ ˜ T -S ( ω, x L 2 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 . (B25)  \nNote that all of the gradiometer time-delay expressions in frequency space appear with a factor of exp(i ωkL ), where k is summed from zero to n -2 with only the even k 's. Since the k -dependence of all  \ncontributions can be isolated in this term, the gradiometer phase shifts appear with an overall factor  \nn -2 ∑ k =0 even e i ωkL = n 2 e i ω ( n -2) L/ 2 sinc( nωL/ 2) sinc( ωL ) . (B26)  \nMaking use of Eqs. (B20), (B21), (B22), (B25) and (B26), the gradiometer phase shift contributions in frequency space may be rewritten compactly as  \n∆ ˜ ϕ LMT grad , E ( ω ) = 1 2 k eff T 2 ω 2 K MZ ( ω ) × [ -1 2i ω ˜ h 00 ( ω, x AI ) ( K + LMT ( ω ) e i ω ( x AI -x L 1 ) -K -LMT ( ω ) e i ω ( x L 2 -x AI ) ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ ˜ ϕ LMT grad , D ( ω ) = 1 2 k eff T 2 ω 2 K MZ ( ω ) × [ n i δ ˜ x i ( ω, x AI ) ( K + LMT ( ω ) e i ω ( x AI -x L 1 ) + K -LMT ( ω ) e i ω ( x L 2 -x AI ) )] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , ∆ ˜ ϕ LMT grad , S ( ω ) = 1 2 k eff T 2 ω 2 K MZ ( ω ) × [ K + LMT ( ω )∆ ˜ T + S ( ω, x L 1 , x AI ) -K -LMT ( ω )∆ ˜ T -S ( ω, x L 2 , x AI ) ] ∣ ∣ ∣ ∣ ∣ x AI 1 x AI 2 , (B27)  \nwith k eff = nω a and the LMT-specific response functions defined as  \nK ± LMT ( ω ) = sinc( nωL/ 2) sinc( ωT/ 2)      sinc( ω ( T -( n -2) L ) / 2) sinc( ωL ) ( 1 -( n -2) L T ) for outgoing photons (+) , sinc( ω ( T -nL ) / 2) sinc( ωL ) ( 1 -nL T ) for incoming photons ( -) . (B28)", 'III. APPLICATIONS': 'Equipped with the formalism introduced in the previous section, we now compute the leading order phase shift in both MZ and LMT configurations for two well-motivated examples: a transient gravitational wave (GW) and a slow-varying weak Newtonian potential.', 'A. Gravitational Waves': 'The first application that we consider in this work is the signal induced by a transient GW. By working in different frames and with different single-photon gradiometer configurations, we demonstrate that our result: (i) is invariant under general coordinate transformations and (ii) accurately reproduces results from the literature. Here, we note that an analogous calculation in these two gauges, within the context of a laser interferometer, is well documented in the literature (see, e.g. , Ref. [56]).', '1. Transverse-traceless Gauge': "Firstly we compute the phase shift in the transverse-traceless (TT) gauge. Assuming that the GW propagates in the z direction, in TT-gauge the metric tensor perturbation takes the form  \nh TT ij ( t, z ) =      h + h × 0 h × -h + 0 0 0 0      ij cos ( ω g ( t -z ) + θ ) . (33)  \nHere, i, j ∈ { 1 , 2 , 3 } , h × and h + are the amplitudes of the two polarizations, ω g is the GW angular frequency, and θ is the phase of the GW at ( t, z ) = (0 , 0) [56]. Without loss of generality, we set the gradiometer at z = 0 and the baseline in the x -direction. To make contact with the literature, we work in the very long-baseline limit, i.e. L = x L 2 -x L 1 ≈ x AI 2 -x AI 1 . In this gauge, the Einstein phase shift, which only depends on h 00 , vanishes ( cf. Eq. (17)). The Doppler phase shift, which depends on both h i 0 and h 00 through the atom's geodesic equation, also vanishes ( cf. Eqs. (6), (10) and (14)), regardless of the gradiometer configuration. Hence, in this gauge, the phase shift can only depend on the Shapiro time delay.  \nLet us first focus on the simpler MZ gradiometer configuration. Making use of Eqs. (24)-(25), the leading order phase shift in this gauge takes the simple form  \n∆ ϕ MZ grad ( t 0 ) = -2 h + Lω a sinc ( ω g L 2 ) sin 2 ( ω g T 2 ) cos ( ω g t 0 + ω g ( T + L 2 ) + θ ) , (34)  \nwhose amplitude can be rewritten as  \n∣ ∣ ∆ ϕ MZ grad ∣ ∣ = 2 h + Lω a ∣ ∣ ∣ ∣ sinc ( ω g L 2 )∣ ∣ ∣ ∣ sin 2 ( ω g T 2 ) . (35)  \nSimilarly, using Eqs. (28)-(30), the phase shift induced by a transient GW in an LMT gradiometer takes the form  \n∆ ϕ LMT grad ( t 0 ) = -2 h + Lk eff sinc ( nω g L 2 ) sin ( ω g T 2 ) sin ( ω g ( T -( n -1) L ) 2 ) × cos ( ω g t 0 + ω g ( T + L 2 ) + θ ) , (36)  \nwhere k eff = nω a . The amplitude of the signal can be rewritten as  \n∣ ∣ ∆ ϕ LMT grad ∣ ∣ = 2 h + Lk eff ∣ ∣ ∣ ∣ sinc ( nω g L 2 ) sin ( ω g T 2 ) sin [ ω g ( T -( n -1) L ) 2 ]∣ ∣ ∣ ∣ . (37)  \nNotably, Eqs. (35) and (37) agree with previous results from the literature [18, 41]. Furthermore, our result highlights the origin of the effect in TT-gauge: since test masses in free-fall are unaffected by a transient GW in TT-gauge, the effect of interest appears solely through the delay in the arrival time of photons between the two AIs.  \nThis analysis can also be repeated in the frequency domain. Using the master equation for the MZ configuration ( cf . Eq. (26)), the Fourier transform of the Shapiro time delay ( cf . Eq. (16)) and the time-dependence of the GW's phase, we find that the MZ phase shift takes the form  \n∆ ˜ ϕ MZ grad ( ω ) = -ω a T 2 ω 2 K MZ ( ω )∆ ˜ T + S ( ω, 0 , L ) = -π 2 ω a h + Lω 2 T 2 K MZ ( ω ) sinc ( ωL 2 ) e i θ e i ωL/ 2 δ ( ω -ω g ) , (38)  \nwhere we restricted our attention to positive frequencies. Similarly, using the master equation for the LMT configuration ( cf . Eq. (31)) the gradiometer phase shifts in the frequency domain can be expressed as  \n∆ ˜ ϕ LMT grad ( ω ) = -1 2 k eff T 2 ω 2 K MZ ( ω ) ( K + LMT ( ω )∆ ˜ T + S ( ω, 0 , L ) + K -LMT ( ω )∆ ˜ T -S ( ω, L, 0) ) = -π 4 k eff h + Lω 2 T 2 sinc ( ωL 2 ) e i θ e i ωL/ 2 K MZ ( ω ) [ K + LMT ( ω ) + K -LMT ( ω ) ] δ ( ω -ω g ) . (39)  \nImportantly, after some algebra, one can show that Eqs. (38) and (39) agree exactly with the Fourier transform of Eqs. (34) and (36), respectively, thus validating the form of the master equations in section II B.  \nTo explicitly verify that our result is invariant under general coordinate transformations, we now repeat the GW calculation in the gradiometer's proper detector frame. The coordinate system in this frame is constructed along the worldline of a freely-falling ( i.e. non-accelerating) observer by extending spacelike vectors orthogonal to the observer's four-velocity [57]. In this frame, the metric takes the form  \nds 2 = ( -1 + ̂ h 00 ) dt 2 + ̂ h 0 i dtdx i +(1 + ̂ h ij ) dx i dx j , (40)  \nwhere the components of the metric perturbation tensor are defined as  \n̂ h 00 = -∞ ∑ r =0 2 ( r +2)! ̂ R 0 m 0 n,k 1 ··· k r x m x n x k 1 · · · x k r , ̂ h 0 i = -∞ ∑ r =0 2( r +2) ( r +3)! ̂ R 0 min,k 1 ··· k r x m x n x k 1 · · · x k r , ̂ h ij = -∞ ∑ r =0 2( r +1) ( r +3)! ̂ R imjn,k 1 ··· k r x m x n x k 1 · · · x k r . (41)  \nHere, the comma preceding the indices k 1 · · · k r denotes differentiation with respect to these coordinates and the hat above any function implies that this function is evaluated along the observer's worldline x obs e.g. ,  \nˆ R µνρσ,k 1 ...k r ≡ ∂ r R µνρσ ∂x k 1 . . . ∂x k r ∣ ∣ ∣ ∣ x = x obs , (42)  \nwith R µνρσ = ( ∂ ν ∂ ρ h µσ + ∂ µ ∂ σ h νρ -∂ µ ∂ ρ h νσ -∂ ν ∂ σ h µρ ) / 2 the Riemann tensor in linearized gravity [58]. In this frame, x i is the spacelike separation between the observer's worldline and an arbitrary spacetime point at fixed time; setting x i = 0 we recover the metric of flat spacetime. Importantly, the Riemann tensor is invariant (element-wise) under gauge transformations to leading order in the metric perturbation. Therefore, without loss of generality, we may evaluate it in TT-gauge. Since the gradiometer probes test mass dynamics along a single spatial component, and we assumed that the GW travels in the z -direction with the atoms at z = 0 and moving only along the x -direction ( i.e. i = 1), one can easily show that the metric tensor perturbation in this frame takes the especially simple form [59]  \n̂ h µν = ̂ h TT 00 , ̂ h TT 00 = -( x i ) 2 ̂ R TT 0 i 0 i = x 2 2 ∂ 2 0 h TT 11 = -ω 2 g x 2 2 h + cos( ω g ( t -z ) + θ ) . (43)  \nFrom the results of section B, we observe that in this frame all contributions to the gradiometer phase shift are non-zero.  \nUsing Eq. (43), we compute the phase shift for the MZ and LMT pulse sequences in the proper detector frame. Using our master equations and working in the limit x AI 2 -x AI 1 ≈ L , the Doppler, Shapiro and Einstein phase shift contributions to the MZ observable take the form  \n∆ ϕ MZ grad , D ( t 0 ) = -2 h + Lω a sin 2 ( ω g T 2 ) cos ( ω g t 0 + ω g ( T + L ) + θ ) , ∆ ϕ MZ grad , S ( t 0 ) = + 2 h + Lω a sin 2 ( ω g T 2 ) cos ( ω g t 0 + ω g ( T + L ) + θ ) -2 h + Lω a sinc ( ω g L 2 ) sin 2 ( ω g T 2 ) cos ( ω g t 0 + ω g ( T + L 2 ) + θ ) + h + L 2 ω a ω g sin 2 ( ω g T 2 ) sin ( ω g t 0 + ω g ( T + L ) + θ ) , ∆ ϕ MZ grad , E ( t 0 ) = -h + L 2 ω a ω g sin 2 ( ω g T 2 ) sin ( ω g t 0 + ω g ( T + L ) + θ ) . (44)  \nAdding the three contributions, we exactly recover Eq. (34), as desired. To gain a better understanding of the effects induced by a transient gravitational wave on the AG, it is advantageous to take the limit ω g L ≪ 1 and isolate the leading order phase shifts in Eq. (44). Explicitly,  \n∆ ϕ MZ grad , D ( t 0 ) = -2 h + Lω a sin 2 ( ω g T 2 ) cos ( ω g t 0 + ω g T + θ ) + O ( ω g L ) , ∆ ϕ MZ grad , S ( t 0 ) = 1 3 h + L 3 ω a ω 2 g sin 2 ( ω g T 2 ) cos( ω g t 0 + ω g T + θ ) + O ( ω 3 g L 3 ) , ∆ ϕ MZ grad , E ( t 0 ) = -h + L 2 ω a ω g sin 2 ( ω g T 2 ) sin ( ω g t 0 + ω g T + θ ) + O ( ω 2 g L 2 ) . (45)  \nFrom Eq. (45), the hierarchy of contributions is evident: in this frame, the Doppler phase shift dominates, while the Einstein and Shapiro phase shift are subdominant, with the latter being further suppressed with respect to the leading order Einstein phase shift. This can be understood as follows. In the PD frame, the effect of a GW can be described in terms of Newtonian forces. Indeed, ∂ 2 0 x i = h TT ij x j / 2, which follows from the geodesic deviation equation [56]; in turn this implies that ˆ h 00 = x i a i , where a i ≡ ∂ 2 0 x i is the acceleration induced by the gravitational wave on photons and atoms. In NR computational frameworks ( e.g. [60]), the leading order phase shift due to forces acting on atoms scales as k eff aT 2 . Hence, the Doppler phase shift contribution, which precisely accounts for the O ( h ) acceleration of the atoms in the AG, recovers this term at leading order. 11 The Einstein and Shapiro contributions are pure GR effects, since the latter depends on the rest mass correction of excited state path segments while the former depends on the finite speed of light. It is therefore unsurprising that these effects are parametrically suppressed with respect to the leading order Doppler phase shift.  \nThe calculation for the LMT pulse configuration follows analogously. Using the definitions in sec-  \ntion II A, the gradiometer phase shift contributions take the form  \n∆ ϕ LMT grad , D ( t 0 ) = 2 h + Lω a sin ( ω g L ) sin ( nω g L 2 ) sin ( ω g T 2 ) × { sin ( ω g t 0 + ω g ( nL + T 2 ) + θ ) -cos( ω g L ) sin ( ω g t 0 + ω g ( 3 T -( n -2) L 2 ) + θ )} , ∆ ϕ LMT grad , S ( t 0 ) = -2 h + Lk eff sinc ( nω g L 2 ) sin ( ω g T 2 ) sin ( ω g ( T -( n -1) L ) 2 ) cos ( ω g t 0 + ω g ( T + L 2 ) + θ ) + h + L 2 ω g ω a sin ( nω g L 2 ) sin ( ω g T 2 ) sin ( ω g t 0 + ω g ( 3 T -( n -2) L 2 ) + θ ) -2 h + Lω a sin ( ω g L ) sin ( nω g L 2 ) sin ( ω g T 2 ) × { sin ( ω g t 0 + ω g ( nL + T 2 ) + θ ) -cos( ω g L ) sin ( ω g t 0 + ω g ( 3 T -( n -2) L 2 ) + θ )} , ∆ ϕ LMT grad , E ( t 0 ) = -h + L 2 ω g ω a sin ( nω g L 2 ) sin ( ω g T 2 ) sin ( ω g t 0 + ω g ( 3 T -( n -2) L 2 ) + θ ) . (46)  \nAs for the MZ case, the sum of the terms in Eq. (46) exactly recovers the TT-gauge result in Eq. (36). In the long wavelength limit, the result exhibits interesting features. Explicitly,  \n∆ ϕ LMT grad , D ( t 0 ) = -2 h + Lk eff sin 2 ( ω g T 2 ) cos ( ω g t 0 + ω g T + θ ) + O ( ω g L ) , ∆ ϕ LMT grad , S ( t 0 ) = 1 3 h + L 3 ω 2 g k eff sin 2 ( ω g T 2 ) cos( ω g t 0 + ω g T + θ ) + O ( ω 3 g L 3 ) , ∆ ϕ LMT grad , E ( t 0 ) = -1 2 h + L 3 k eff ω 2 g sin ( ω g T 2 ) sin ( ω g t 0 + 3 ω g T 2 + θ ) + O ( ω 3 g L 3 ) . (47)  \nWhile the Shapiro and Doppler contributions are structurally equivalent to their respective contributions in Eq. (45), the leading-order LMT Einstein phase shift term is further suppressed by a factor of L/T in the limit ωT ≪ 1, as anticipated at the end of section II B for a generic metric perturbation.  \nIn summary, we explicitly showed that, for a transient GW, the phase shift obtained in TT-frame agrees with the observable computed in the PD frame. Furthermore, the results obtained using our simplified framework agree with previous work. This confirms the gauge-invariant nature of the gradiometer observable derived in section II.", 'B. Slow-varying Weak Newtonian Potential': "It is challenging to correctly account for the GR contributions to the gradiometer phase shift induced by a moving point mass, as a fully covariant treatment is often algebraically involved, and unfeasible for complicated configurations such as the LMT sequence. However, with the simplified formalism derived in Sec. II, we can obtain the leading order O ( hv ) contributions to the gradiometer phase shift arising from a slow-varying and weak Newtonian potential in a tractable manner, which can be used to model  \nNewtonian noises from passing planes or trains, as well as to study transient signals from new physics.  \nIn the presence of a weak Newtonian potential Φ and in the rest frame of the source, the metric in harmonic gauge takes the form  \nds 2 = -(1 + 2Φ) dt 2 +(1 -2Φ) dx i dx i , (48)  \nwhere i = 1 , 2 , 3 and we make use of the Einstein summation convention. In a frame where the source is moving at a constant velocity v s with its amplitude v s ≪ 1, to leading order the metric takes the form  \nds 2 = -(1 + 2Φ) dt 2 -8Φ v s,i dx i dt +(1 -2Φ) dx i dx i , (49)  \n̸  \nwhich differs from Eq. (48) in so far as h 0 i = h i 0 = -4Φ v s,i = 0. For a derivation, see Appendix C. In the weak field limit, potentials that are sourced by different bodies can be added linearly. This allows us to conveniently choose an inertial frame at rest in the Earth's gravitational field.  \nUnlike signals from periodic sources ( e.g. , GW, ultralight bosonic fields), transient signals from a moving Newtonian potential do not factorize into signal strength and response function in the time domain, making it challenging to understand the signal frequency dependence and analyze the detector sensitivity. Therefore, we directly work in the Fourier domain, where the factorization of the detector response is more straightforward. As the source of the Newtonian potential can move in a general direction, it is necessary to express the result in three spatial dimensions. For simplicity, in this section, we also assume that the two AIs are located at two ends of the baseline, i.e. x AI 1 = x L for the MZ sequence, and x AI 1 = x L 1 , x AI 2 = x L 2 for the LMT sequence. The direction of the baseline is defined by n .  \nLet us first focus on the MZ configuration. Following Eq. (26), the total phase shift in Fourier space reads  \n∆ ˜ ϕ MZ grad ( ω ) = ω a T 2 K MZ ( ω ) { -ω 2 ∆ ˜ T + S ( ω, x AI 1 , x AI 2 ) ︸ ︷︷ ︸ Shapiro + [ n · ( ∇-4 iω v s ) ︸ ︷︷ ︸ Doppler -iω ] ︸ ︷︷ ︸ Einstein [ ˜ Φ( ω, x AI 1 ) -e iωL ˜ Φ( ω, x AI 2 ) ] } . (50)  \nHere, the Shapiro time delay is expressed in terms of Eq. (16) with the metric perturbation introduced  \nin Eq. (49), i.e.  \n∆ T + S ( t, x AI 1 , x AI 2 ) = -2(1 + 2 v s · n ) ∫ L/ 2 -L/ 2 Φ ( t + L 2 + x ' , x mid + x ' n ) dx ' ∆ T -S ( t, x AI 2 , x AI 1 ) = -2(1 -2 v s · n ) ∫ L/ 2 -L/ 2 Φ ( t + L 2 -x ' , x mid + x ' n ) dx ' , (51)  \nwhere we have changed the integration variable and defined x mid ≡ ( x AI 1 + x AI 2 ) / 2 as the midpoint of the baseline. 12 Inspecting the frequency dependence in Eq. (50), at low frequencies ( i.e. ωT, ωL ≪ 1), the signal is dominated by the difference in the gradient of the potential evaluated at the location of the two AIs, i.e. the acceleration gradient. Recalling the asymptotic behavior of the MZ response function, the total phase shift is approximately  \n∆ ˜ ϕ MZ grad ( ω ) ≈ ω a T 2 n · [ ∇ ˜ Φ( ω, x AI1 ) -∇ ˜ Φ( ω, x AI2 ) ] , (52)  \nwhich agrees with the familiar leading order non-relativistic phase shift [36].  \nSimilarly, for an LMT sequence, following Eq. (31), the total phase shift in the Fourier space induced by a slow-varying Newtonian potential reads  \n∆ ˜ ϕ LMT grad ( ω ) = k eff T 2 K MZ ( ω ) 1 2 { -ω 2 [ K + LMT ∆ ˜ T + S ( ω, x AI 1 , x AI 2 ) + K -LMT ∆ ˜ T -S ( ω, x AI 2 , x AI 1 ) ] ︸ ︷︷ ︸ Shapiro -iω [ ( K + LMT -e iωL K -LMT ) ˜ Φ( ω, x AI 1 ) -( K + LMT -e -iωL K -LMT ) e iωL ˜ Φ( ω, x AI 2 ) ] ︸ ︷︷ ︸ Einstein + n · ( ∇-4 iω v s ) [ ( K + LMT + e iωL K -LMT ) ˜ Φ( ω, x AI 1 ) -( K + LMT + e -iωL K -LMT ) e iωL ˜ Φ( ω, x AI 2 ) ] ︸ ︷︷ ︸ Doppler } , (53)  \nwhere it is understood that K ± LMT are functions of frequency as given in Eq. (32). Notice that  \nlim ω → 0 K + LMT ( ω ) = 1 -( n -2) L T , lim ω → 0 K -LMT ( ω ) = 1 -nL T . (54)  \nAt low frequencies ( i.e. ωT, nωL ≪ 1), the phase shift is therefore dominated by  \n∆ ˜ ϕ LMT grad ( ω ) ≈ k eff T 2 ( 1 -( n -1) L T ) n · [ ∇ ˜ Φ( ω, x AI1 ) -∇ ˜ Φ( ω, x AI2 ) ] . (55)  \nNotice that this result differs from the MZ phase shift by a factor of (1 -( n -1) L/T ). This can be attributed to the effective spacetime area enclosed by a single AI. In an LMT sequence, because of  \nFIG. 3: The square root of the power spectral density, √ S n ( ω ) ≡ √ ω/ 2 π | ∆ ϕ grad ( ω ) | , induced by a moving Newtonian noise source. The left panel shows a freight truck with mass M = 10 3 kg, velocity v s = 25 km/h, and impact parameter b = 100 m, moving near a terrestrial, single-photon atom gradiometer (AG) utilizing the large-momentum-transfer (LMT) pulse sequence with baseline L = 1 km, interrogation time T = 1 . 7 s, and n = 2500. Experimental parameters align with advanced designs potentially used in future vertical gradiometers such as AION-100, MAGIS-100, and AION-km (see Table 1 in Ref. [34]). The right panel depicts a piece of space debris with mass M = 10 4 kg, velocity v s = 10 m/s, and impact parameter b = 100 m, passing by a space-based AG, also utilizing the LMT sequence, with baseline L = 4 . 4 × 10 7 m, interrogation time T = 150 s, and n = 250. These parameters match the proposed AEDGE experiment [33]. Both setups employ the clock transition (5 s 21 S 0 ↔ 5s5p 3 P 0 ) in Sr87 with an angular transition frequency ω a = 2 . 70 × 10 15 Hz. The source's impact parameter spans an angle of 45 · with the baseline. Individual contributions from the Einstein (green), Doppler (blue), and Shapiro (orange) effects, as derived in Eq. (53), are illustrated. The estimated level of shot noise, √ S n ( f ) ∼ 10 -5 Hz -1 / 2 , is plotted in grey for comparison. The low-frequency behaviors of the dominant contributions are shown in sky blue. Finally, the characteristic frequencies of the noise spectrum and the response functions, ω = v s /b and ω = 2 π/T , are plotted in black for reference.  \n<!-- image -->  \nthe incremental velocity boost, the maximum separation between atom paths is ∆ x ∝ k eff ( T -( n -1) L ) instead of ∆ x ∝ k eff T . Hence, the enclosed spacetime area receives a correction proportional to ( n -1) L/T with respect to the MZ case.  \nTo better understand the impact of Newtonian noise on the spectrum of an AG experiment, we use the results from this section to compute the power spectral density of the gradiometer phase, which is defined as S n ( ω ) ≡ √ ω/ (2 π ) | ∆ ˜ ϕ grad ( ω ) | 2 [55]. As an example, in the left panel of Fig. 3, we plot the square root of the PSD induced by a freight truck with a mass M = 10 3 kg, an impact parameter b = 100m from the nearest AI, and a constant velocity v s = 25km/h. Experimental parameters are derived from Table 1 in [34] to align with advanced designs that are potentially achievable by future experiments employing the LMT pulse sequence, such as AION-km. The estimated shot noise level is shown for comparison.  \nThe contributions from Doppler, Shapiro and Einstein term are individually displayed. As expected, the spectrum is dominated by the Doppler term, which far exceeds the noise floor for ω ≲ 2 π × 10 -2 Hz and would therefore impact a terrestrial experiment's projected reach, e.g. , to mid-frequency GWs. At low frequencies, the dominant phase shift contribution of the Doppler phase shift originates from the first AI as | ∆ ϕ grad | ∼ k eff T 2 (2 GM/bv s ) | ˆ b · ˆ n | , where (2 GM/bv s ) is the characteristic amplitude of the acceleration caused by a transiting source with impact parameter b , as indicated in Fig. 3, where we show the analytic scaling with a cyan dotted curve. The Shapiro contribution is the most subdominant and is parametrically suppressed by v 2 s compared to the Doppler term due to its nature as a relativistic correction. Finally, the Einstein contribution is parametrically suppressed by ∼ v s ( L/T ) compared to the Doppler term. All contributions scale as power laws at low frequency, and are exponentially supressed at high frequency above ω ∼ v s /b , which is the (inverse) characteristic time-scale of the moving source.  \nIt is worth exploring whether the Einstein and Shapiro terms can surpass the shot noise floor in a more futuristic setup. In the right panel of Fig. 3, we show the PSD for a piece of space debris with mass M = 10 4 kg, an impact parameter b = 100 m from the nearest AI, and a constant velocity v s = 10 m/s relative to a space-based AG. The small relative velocity could arise from both the debris and the AG orbiting at similar altitudes above Earth's surface. Experimental parameters are chosen to match the AEDGE proposal [33]. As indicated by Eq. (53), the Einstein spectrum peaks at ω ∼ v s /b , with ∆ ϕ E ∼ 2 k eff LTω ( GM/v s ). An estimate with this formula naively suggests that the Einstein contribution might exceed the noise floor for certain realistic Newtonian noise parameters. However, the response functions, K MZ ( ω ) and K ± LMT ( ω ) as defined in Eq. (27) and Eq. (32), introduce additional suppression factors of ( ωT ) -1 for frequencies beyond ω ∼ 2 π/T , which suppresses all contributions at the naively expected peak spectrum frequency. The sin( ωT ) term in the response functions is also responsible for the choppiness in the shape of the spectrum for frequencies ω ≳ 2 π/T . Consequently, it is unlikely that the Einstein contribution to a realistic Newtonian noise will be significant in any experimental search. A similar conclusion can be drawn for the Shapiro contribution, which, although not suppressed by factors of L/T , is further suppressed by a factor of v s . It is important to note that while noise sources with a larger v s could be considered, the peak frequency of these sources is greater than 2 π/T , thereby further suppressing the noise amplitude, and thus can be neglected.", 'IV. DISCUSSION & CONCLUSIONS': "In this work, we have developed a simplified gauge-invariant formalism for calculating the leadingorder gravitational phase shift at O ( hv ) in a single-photon atom gradiometer under the influence of a generic metric perturbation. We have found that the leading-order contributions can be fully accounted for by the rest mass correction of the atom excited states and can be conveniently written in terms of  \nDoppler, Shapiro, and Einstein time delays in analogy with laser interferometers.  \nWe applied this formalism to MZ and LMT gradiometer configurations, and we calculated the gravitational signals from GWs and slow-varying weak Newtonian potentials in these experiments. Our calculations of the GW phase shift are performed in both the TT gauge and the proper detector frame. Our result confirms the gauge-invariance of the leading-order phase shift and agrees with the literature [18, 41]. In the proper detector frame, the Doppler phase shift gives the familiar ∆ ϕ ∝ k eff aT 2 with an acceleration induced by GW in the long wavelength limit ω g L ≪ 1. We also derived the signal of a slow-varying weak Newtonian potential in the frequency domain and calculated the response functions for both the MZ and LMT configurations. We found that in the laser's rest frame, the dominant effect comes from the Doppler phase shift, which gives the k eff aT 2 scaling in the low frequency regime; the Einstein contribution receives a v s L/T suppression.  \nThe formalism presented in this paper is valid to O ( hv ). Extending the range of validity to higher order in the atom velocity and the metric perturbation requires a punctilious analysis, which we leave to future work. For instance, to calculate effects at O ( v 2 ) or O ( h 2 ), one would need a more careful and precise treatment of the separation phase and atom paths, as well as the momentum transferred in atom-light interactions. In our derivation, we assumed that the physical path C is closed; to achieve this in the laboratory, prior knowledge of the metric is required. In practice, however, this is not possible for the type of searches envisaged in this work. Consequently, the atom's worldlines are not expected to overlap at the moment the measurement is performed. In this case, the phase shift for each AI in the gradiometer set-up receives a correction ∼ ¯ p µ ∆ X µ commonly known as the separation phase. Here, ¯ p µ is the average four momentum of the two wavepackets being recombined at a particular measurement port and ∆ X µ is the coordinate separation between the two wavepackets on the null hypersurface defining the measurement. Following the same arguments from section II A, ∆ X µ is zero in the absence of a metric perturbation, and both ∆ X µ and ¯ p µ vanish in the limit of zero recoils. Hence, the correction from this separation phase enters the observable at O ( hv 2 ). Furthermore, the corrections to the atom's initial conditions in ˜ C also enter the observable at O ( hv 2 ), as do the corrections to the atom recoil velocities due to the photons' propagation in curved spacetime and the initial launch velocity of the atoms. Finally, the equivalence between the phase shift evaluated over C and ˜ C only holds at leading order in the metric, so an extension of our formalism to O ( h 2 ) would require a careful treatment order by order in perturbation theory.  \nAlthough we have restricted our attention to vertical gradiometers employing single-photon transitions, our results can also be modified to predict the observable in AG experiments which utilize horizontal baselines and two-photon transitions ( e.g. MIGA, ELGAR and ZAIGA). In these types of experiments, the baseline is perpendicular to the initial launch velocity of the atoms and the transition from ground to excited state (and vice versa) is driven by two counter-propagating laser beams which in-  \nteract with a particular atomic wavepacket almost synchronously. As a result, the emission times of two pulses are interrelated. However, following the same arguments presented in Appendix A, the observable is not sensitive to O ( h ) corrections to the laser pulse emission times; therefore, only the unperturbed emission times would have to be carefully chosen.  \nIn conclusion, the formalism developed in this paper opens exciting new avenues for future studies. For example, our simplified framework, and especially the gradiometer phase shift expressions presented in this work, can assist with the development of Wiener filters in atom gradiometer experiments to mitigate Newtonian noise from nearby traffic (air crafts, trains, etc.) and well-monitored seismic activity. Additionally, because of the generality of our framework insofar as the form of the metric perturbation is concerned, the key results of our work can be used to study signatures of new physics, such as macroscopic dark matter, which we explore in a companion paper [61], and more exotic signatures, such as violations of general coordinate invariance.", 'ACKNOWLEDGMENTS': "We thank Albert Roura and Allic Sivaramakrishnan for useful discussions. L.B., Y.D., V.L., Y.W. and K.Z. are supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics under Award Number DE-SC0011632, and by the Walter Burke Institute for Theoretical Physics. KZ is also supported by a Simons Investigator award, and by Heising-Simons Foundation 'Observational Signatures of Quantum Gravity' collaboration grant 2021-2817.", 'Appendix A: Shapiro and Doppler Phase Shifts in Single-Photon Atom Gradiometry': 'In atom gradiometers utilizing single-photon transitions, the Shapiro and Doppler phase shifts are manifestly independent of the dynamics of the laser. This is due to the fact that a laser pulse that starts an excited state segment of one atom interferometer will also start the excited state of the other atom interferometer, and those excited state segments are ended by another common laser pulse. To see this explicitly, we consider the Doppler and Shapiro phase shifts for an arbitrary excited state path segment.  \nThroughout this appendix, we use overbars to identify unperturbed quantities, and denote O ( h ) corrections to coordinates by δt and δx . Let us first consider a gradiometer configuration with a single laser ( e.g. a Mach-Zehnder gradiometer). For simplicity, let us set the baseline along the x -direction ( i.e. i = 1). Let ( t I L , x I L ) = ( ¯ t I L + δt I L , ¯ x I L + δx I L ) be the spacetime point at which the laser pulse that initialises the excited state paths in both AIs is emitted. This pulse will drive the transition from ground to excited state in the first and second AI at ( t I 1 , x I 1 ) = ( ¯ t I 1 + δt I 1 , ¯ x I 1 + δx I 1 ) and ( t I 2 , x I 2 ) = ( ¯ t I 2 + δt I 2 , ¯ x I 2 + δx I 2 ), respectively. Let ( t F L , x F L ) = ( ¯ t F L + δt F L , ¯ x F L + δx F L ) be the spacetime point at which the laser pulse that drives the transition from excited to ground state is emitted. This second pulse will drive the transition from excited to ground state in the first and second AI at ( t F 1 , x F 1 ) = ( ¯ t F 1 + δt F 1 , ¯ x F 1 + δx F 1 ) and ( t F 2 , x F 2 ) = ( ¯ t F 2 + δt F 2 , ¯ x F 2 + δx F 2 ), respectively. We show this diagrammatically in the left panel of Fig. 4. Since photons travel on null geodesics, the O ( h ) corrections to the coordinate time at which the atom and laser worldlines intersect are given by  \nδt I,F 1 , 2 = δt I,F L + ∫ δx I,F 1 , 2 δx I,F L dx \' + ∫ ¯ x I,F 1 , 2 ¯ x I,F L H + ( ¯ t I,F L -¯ x I,F L + x \' , x \' ) dx \' + O ( h 2 ) , (A1)  \nwhere we introduced H + in Eq. (12) with n = ˆ x . Note the appearance of the O ( h ) correction to the spacetime points at which the photons are emitted, which in turn gives rise to the ∆ t ( k ) laser term in Eq. (15). From Eq. (22), the sum of the Shapiro and Doppler phase shifts for this particular excited state path segment is given by  \n∆ ϕ ( ⋆ ) grad , D +∆ ϕ ( ⋆ ) grad , S = ω a ( δt F 1 -δt I 1 ) -ω a ( δt F 2 -δt I 2 ) = ω a ( δt I 2 -δt I 1 ) -ω a ( δt F 2 -δt F 1 ) . (A2)  \nCombining Eq. (A1) with Eq. (A2), it immediately follows that the δt L and δx L terms cancel out (i.e. the gradiometer observable is insensitive to ∆ t ( ∗ ) laser ). Furthermore, from the linearity of integration, it also follows that the integration limits do not depend on δx L . Explicitly,  \nδt I,F 2 -δt I,F 1 = ∫ δx I,F 2 δx I,F 1 dx \' + ∫ ¯ x I,F 2 ¯ x I,F 1 H + ( ¯ t I,F L -¯ x I,F L + x \' , x \' ) dx \' . (A3)  \n<latexit sha1\\_base64="wwlTEqLfXHYnVEYj5S49pMifVgc=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PQi8eI5gHJEmYns8mQ2dllplcMSz7BiwdFvPpF3vwbJ8keNFrQUFR1090VJFIYdN0vp7C0vLK6VlwvbWxube+Ud/eaJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LR9dRvPXBtRKzucZxwP6IDJULBKFrp7pFgr1xxq+4M5C/xclKBHPVe+bPbj1kacYVMUmM6npugn1GNgkk+KXVTwxPKRnTAO5YqGnHjZ7NTJ+TIKn0SxtqWQjJTf05kNDJmHAW2M6I4NIveVPzP66QYXvqZUEmKXLH5ojCVBGMy/Zv0heYM5dgSyrSwtxI2pJoytOmUbAje4st/SfOk6p1Xz25PK7WrPI4iHMAhHIMHF1CDG6hDAxgM4Ale4NWRzrPz5rzPWwtOPrMPv+B8fAMUwY2u</latexit>  \nxt  \n<latexit sha1\\_base64="g2EL6HCcpJz8bLtJCSH2Ns283ak=">AAADFXichZJdS+NAFIYn8avbdbW6l3szbFlQXEomiHopCosFL1zYqtDUMJlO2sHJBzMnYgn9E974V7zxQln2Vti7/Tc7jfWjNeCBwMl73od5czJBKoUGx/ln2TOzc/MLlQ/Vj4uflpZrK6vHOskU4y2WyESdBlRzKWLeAgGSn6aK0yiQ/CQ43x/NTy640iKJf8Eg5Z2I9mIRCkbBSP6KteFJHsLape9FFPoqyg+HZz++w8Srp0SvD+vY86ol7uaku1nizl/mPhka/6TwDuFOE+4r4gkgRWpSGpYUGUlpNLfg3FLOLTj3mfNrdafhFIXfNmTc1NG4jvzaX6+bsCziMTBJtW4TJ4VOThUIJvmw6mWap5Sd0x5vmzamEdedvPirQ/zNKF0cJso8MeBCfU3kNNJ6EAXGOVqNnp6NxLJZO4Nwp5OLOM2Ax+zxoDCTGBI8uiK4KxRnIAemoUwJkxWzPlWUgblIVbMEMv3Jb5tjt0G2Gls/N+u7e+N1VNAX9BWtIYK20S46QEeohZh1Zd1Yd9a9fW3f2r/tP49W2xozn9FE2Q//Af7a+pQ=</latexit>  \n<latexit sha1\\_base64="g2EL6HCcpJz8bLtJCSH2Ns283ak=">AAADFXichZJdS+NAFIYn8avbdbW6l3szbFlQXEomiHopCosFL1zYqtDUMJlO2sHJBzMnYgn9E974V7zxQln2Vti7/Tc7jfWjNeCBwMl73od5czJBKoUGx/ln2TOzc/MLlQ/Vj4uflpZrK6vHOskU4y2WyESdBlRzKWLeAgGSn6aK0yiQ/CQ43x/NTy640iKJf8Eg5Z2I9mIRCkbBSP6KteFJHsLape9FFPoqyg+HZz++w8Srp0SvD+vY86ol7uaku1nizl/mPhka/6TwDuFOE+4r4gkgRWpSGpYUGUlpNLfg3FLOLTj3mfNrdafhFIXfNmTc1NG4jvzaX6+bsCziMTBJtW4TJ4VOThUIJvmw6mWap5Sd0x5vmzamEdedvPirQ/zNKF0cJso8MeBCfU3kNNJ6EAXGOVqNnp6NxLJZO4Nwp5OLOM2Ax+zxoDCTGBI8uiK4KxRnIAemoUwJkxWzPlWUgblIVbMEMv3Jb5tjt0G2Gls/N+u7e+N1VNAX9BWtIYK20S46QEeohZh1Zd1Yd9a9fW3f2r/tP49W2xozn9FE2Q//Af7a+pQ=</latexit>  \n<latexit sha1\\_base64="7NaxWW4epP/psj9P5Wnex1Glsbk=">AAAB83icbVDLSgMxFL1TX7W+qi7dBIvgqsyIVJdFNy4r2Ad0hpJJ0zY0kxmSO2IZ+htuXCji1p9x59+YtrPQ1gOBwzn3lRMmUhh03W+nsLa+sblV3C7t7O7tH5QPj1omTjXjTRbLWHdCargUijdRoOSdRHMahZK3w/HtzG8/cm1ErB5wkvAgokMlBoJRtJLvI3/CTNoBetorV9yqOwdZJV5OKpCj0St/+f2YpRFXyOwI0/XcBIOMahRM8mnJTw1PKBvTIe9aqmjETZDNb56SM6v0ySDW9ikkc/V3R0YjYyZRaCsjiiOz7M3E/7xuioPrIBMqSZErtlg0SCXBmMwCIH2hOUM5sYQyLeythI2opgxtTCUbgrf85VXSuqh6tWrt/rJSv8njKMIJnMI5eHAFdbiDBjSBQQLP8ApvTuq8OO/Ox6K04OQ9x/AHzucPw3iSLQ==</latexit>  \n(  \n)  \n( x L 2 ,t L 2 ) ( x F 1 ,t F 1 ) ( x I 1 ,t I 1 ) ( x F 2 ,t F 2 ) ( x I 2 ,t I 2 ) 2 2 ( x I 1 ,t I 1 ) ( x F 2 ,t F 2 ) ( x I 2 ,t I 2 ) FIG. 4: Schematic representation of an arbitrary excited state path segment (dashed red line) in a singlephoton atom gradiometer. On the left, we show the relevant spacetime diagram for a MZ gradiometer sequence, where the transitions are driven by a single laser source. On the right, we show the relevant spacetime diagram for a LMT gradiometer sequence, where the transitions are driven by two laser sources  \n(  \nx  \n,t  \n)  \n)  \n)  \n)  \n(  \n)  \n(  \nx  \nF  \nL  \nx  \nx  \nI  \nF  \nL  \nI  \nF  \n(  \nF  \n<!-- image -->  \nI  \n2  \n,t  \nxt  \nI  \n2  \n)  \n(  \n)  \n(  \n)  \nAfter some algebra, the terms of the LHS of Eq. (A2) may be rewritten using Eq. (A3) as  \n∆ ϕ ( ⋆ ) grad , D = ω a ( ∫ δx I 2 δx I 1 dx \' -∫ δx F 2 δx F 1 dx \' ) = ω a ( δx I 2 + δx F 1 -δx I 1 -δx F 2 ) , ∆ ϕ ( ⋆ ) grad , S = ω a ( ∫ ¯ x I 2 ¯ x I 1 H + ( ¯ t I L -¯ x I L + x \' , x \' ) dx \' -∫ ¯ x F 2 ¯ x F 1 H + ( ¯ t F L -¯ x F L + x \' , x \' ) dx \' ) . (A4)  \nIn this example, we assumed that the excited state path segment is on the right arm of the interferometer. If this segment were on the left arm of the interferometer, one can easily show that the Doppler and Shapiro would gain an overall minus sign.  \nLet us now show the same result for a configuration employing two laser sources ( e.g. a LMT gradiometer configuration). Let ( t L 1 , x L 1 ) = ( ¯ t L 1 + δt L 1 , ¯ x L 1 + δx L 1 ) be the spacetime point at which the first laser pulse that initialises the excited state paths in both AIs is emitted. This pulse will drive the transition from ground to excited state in the first and second AIs at spacetime points ( t I 1 , x I 1 ) = ( ¯ t I 1 + δt I 1 , ¯ x I 1 + δx I 1 ) and ( t I 2 , x I 2 ) = ( ¯ t I 2 + δt I 2 , ¯ x I 2 + δx I 2 ), respectively. Similarly, let ( t L 2 , x L 2 ) = ( ¯ t L 2 + δt L 2 , ¯ x L 2 + δx L 2 ) be the spacetime point at which the second laser pulse that drives the transition from excited to ground state is emitted, so that the spacetime endpoints of the excited state paths of the AIs are ( t F 1 , x F 1 ) = ( ¯ t F 1 + δt F 1 , ¯ x F 1 + δx F 1 ) and ( t F 2 , x F 2 ) = ( ¯ t F 2 + δt F 2 , ¯ x F 2 + δx F 2 ). We show this diagrammatically in the right panel of Fig. 4. The O ( h ) corrections to the coordinate time at which the atom and laser worldlines intersect now depend on whether the pulse was emitted from the first or second laser pulse.  \nx  \n(  \n<latexit sha1\\_base64="g2EL6HCcpJz8bLtJCSH2Ns283ak=">AAADFXichZJdS+NAFIYn8avbdbW6l3szbFlQXEomiHopCosFL1zYqtDUMJlO2sHJBzMnYgn9E974V7zxQln2Vti7/Tc7jfWjNeCBwMl73od5czJBKoUGx/ln2TOzc/MLlQ/Vj4uflpZrK6vHOskU4y2WyESdBlRzKWLeAgGSn6aK0yiQ/CQ43x/NTy640iKJf8Eg5Z2I9mIRCkbBSP6KteFJHsLape9FFPoqyg+HZz++w8Srp0SvD+vY86ol7uaku1nizl/mPhka/6TwDuFOE+4r4gkgRWpSGpYUGUlpNLfg3FLOLTj3mfNrdafhFIXfNmTc1NG4jvzaX6+bsCziMTBJtW4TJ4VOThUIJvmw6mWap5Sd0x5vmzamEdedvPirQ/zNKF0cJso8MeBCfU3kNNJ6EAXGOVqNnp6NxLJZO4Nwp5OLOM2Ax+zxoDCTGBI8uiK4KxRnIAemoUwJkxWzPlWUgblIVbMEMv3Jb5tjt0G2Gls/N+u7e+N1VNAX9BWtIYK20S46QEeohZh1Zd1Yd9a9fW3f2r/tP49W2xozn9FE2Q//Af7a+pQ=</latexit>  \n<latexit sha1\\_base64="g2EL6HCcpJz8bLtJCSH2Ns283ak=">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</latexit>  \n(  \nx  \nF  \nL  \nx  \nx  \nL  \n1  \nI  \nL  \n,t  \nF  \nL  \nI  \nL  \n,t  \n,t  \n)  \n)  \n<!-- image -->  \nF  \nL  \n(  \nx  \n1  \n(  \n(  \n<latexit sha1\\_base64="g2EL6HCcpJz8bLtJCSH2Ns283ak=">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</latexit>  \n<latexit 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sha1\\_base64="wwlTEqLfXHYnVEYj5S49pMifVgc=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PQi8eI5gHJEmYns8mQ2dllplcMSz7BiwdFvPpF3vwbJ8keNFrQUFR1090VJFIYdN0vp7C0vLK6VlwvbWxube+Ud/eaJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LR9dRvPXBtRKzucZxwP6IDJULBKFrp7pFgr1xxq+4M5C/xclKBHPVe+bPbj1kacYVMUmM6npugn1GNgkk+KXVTwxPKRnTAO5YqGnHjZ7NTJ+TIKn0SxtqWQjJTf05kNDJmHAW2M6I4NIveVPzP66QYXvqZUEmKXLH5ojCVBGMy/Zv0heYM5dgSyrSwtxI2pJoytOmUbAje4st/SfOk6p1Xz25PK7WrPI4iHMAhHIMHF1CDG6hDAxgM4Ale4NWRzrPz5rzPWwtOPrMPv+B8fAMUwY2u</latexit>  \nExplicitly,  \nδt I 1 , 2 = δt L 1 + ∫ δx I 1 , 2 δx L 1 dx \' + ∫ ¯ x I 1 , 2 ¯ x L 1 H + ( ¯ t L 1 -¯ x L 1 + x \' , x \' ) dx \' + O ( h 2 ) , δt F 1 , 2 = δt L 2 -∫ δx F 1 , 2 δx L 2 dx \' -∫ ¯ x F 1 , 2 ¯ x L 2 H -( ¯ t L 2 + ¯ x L 2 -x \' , x \' ) dx \' + O ( h 2 ) . (A5)  \nMaking use of Eq. (A5) to write  \nδt I 2 -δt I 1 = ∫ δx I 2 δx I 1 dx \' + ∫ ¯ x I 2 ¯ x I 1 H + ( ¯ t L 1 -¯ x L 1 + x \' , x \' ) dx \' , δt F 2 -δt F 1 = -∫ δx F 2 δx F 1 dx \' -∫ ¯ x F 2 ¯ x F 1 H -( ¯ t L 2 + ¯ x L 2 -x \' , x \' ) dx \' , (A6)  \nthe Doppler and Shapiro gradiometer phase shift terms ( cf . Eq. (A2)) for this path segment can be rewritten as  \n∆ ϕ ( ⋆ ) grad , D = ω a ( δx I 2 + δx F 2 -δx I 1 -δx F 1 ) , ∆ ϕ ( ⋆ ) grad , S = ω a ( ∫ ¯ x I 2 ¯ x I 1 H + ( ¯ t L 1 -¯ x L 1 + x \' , x \' ) dx \' + ∫ ¯ x F 2 ¯ x F 1 H -( ¯ t L 2 + ¯ x L 2 -x \' , x \' ) dx \' ) . (A7)  \nHere, we assumed that the excited state path segment is on the right arm of the AI. The Doppler and Shapiro contributions from left-arm paths gain an overall minus sign. Furthermore, we assumed that the first pulse originated from the end of the baseline closest to the first AI. When the first pulse originates from the opposite end of the baseline, the Doppler and Shapiro phase shifts are given by  \n∆ ϕ ( ⋆ ) grad , D = -ω a ( δx I 2 + δx F 2 -δx I 1 -δx F 1 ) , ∆ ϕ ( ⋆ ) grad , S = -ω a ( ∫ ¯ x I 2 ¯ x I 1 H -( ¯ t L 2 + ¯ x L 2 -x \' , x \' ) dx \' + ∫ ¯ x F 2 ¯ x F 1 H + ( ¯ t L 1 -¯ x L 1 + x \' , x \' ) dx \' ) , (A8)  \nfor right-arm paths. The contribution from left-arm segments differs by a relative sign. Note, also, that Eqs. (A4), (A7) and (A8) can be expressed in terms of the gradiometer time-delays introduced in sections II. Indeed, the ω a -independent piece in Eq. (A4) is the Doppler/Shapiro MZ gradiometer time delay, while the ω a -independent piece in Eqs. (A7) and (A8) is the Doppler and Shapiro LMT gradiometer time-delay of \' > \' or \' < \' excited state path segments, respectively ( cf . section II B).', 'Appendix B: Derivation of the Phase Shift Formulae in the Frequency Domain': 'In this appendix, we derive the form of the gradiometer phase shift inthe frequency domain. We specialise to the MZ and LMT configurations defined in section II B.', 'Appendix C: Tools for Computing the Gradiometer Phase Shift Induced by a Slow-varying Weak Newtonian Potential': "In this appendix we provide tools for computing the single-photon gradiometer phase shift induced by a slow-varying weak Newtonian potential. We first derive the metric sourced by a massive point-like object moving at constant velocity v s ≡ v i s . Finally, we collect key results which are necessary for computing the gradiometer phase shift induced by such an object ( e.g. expressions for the potential in the frequency-domain, geometric factors, etc.).  \nIn the rest frame of the source, spacetime can be described by the Schwarzschild metric. In the weak-field limit, the Schwarzschild metric in isotropic coordinates can be written as [63]:  \nds 2 = ( -1 -2Φ) dt 2 +(1 -2Φ) dx i dx i , (C1)  \nwhere Φ( r ) ≡ -GM/r = -GM/ √ x i x i is the gravitational potential sourced by the massive point-like object. Boosting to the laboratory frame, the metric becomes  \nds 2 = [ -1 -2 ( 1 + 2 v 2 s 1 -v 2 s Φ v s )] dt 2 + [ 1 -2 ( 1 + 2 v 2 s,i 1 -v 2 s ) Φ v s ] dx i dx i -8 v s,i 1 -v 2 s Φ v s dtdx i -8 v s,i v s,j 1 -v 2 s Φ v s dx i dx j , (C2)  \n̸  \nwhere Φ v s = -GM/ | γ ( r -v t ) -( γ -1)( r -( r · n ) n ) | is the boosted potential, with γ = 1 / √ 1 -v 2 s , and the last term is evaluated for i = j . For a slow moving source, we can express the metric to O ( v s ), i.e.  \nds 2 = -(1 + 2Φ) dt 2 +(1 -2Φ) dx i dx i -8Φ v s,i dtdx i . (C3)  \nWe note that the diagonal terms reproduce the static weak field metric in the isotropic coordinates, and the leading order relativistic corrections appears in g 0 i .  \nWe now derive the Fourier transform of the Newtonian potential due to a moving point source, which enters the Fourier-transformed signal in Eqs. (50)-(55). The trajectory of a point mass moving with a constant velocity v s can be written as r s = b + v s ( t -t s ), where b is its impact parameter (defined with respect to the origin), and t s is the time when it is closest to the origin. The potential evalauted at ( t, x ) is then given by  \nΦ( t, x ) = -GM | b + v s ( t -t s ) -x | , (C4)  \nwhere M is the mass of the object. The Fourier transform of the potential is  \n˜ Φ( ω, x ) = -2 GM v s e -iωt s e iω ˆ v s · x vs K 0 ( r ⊥ v s ω ) , (C5)  \nwhere we have defined the transverse component r ⊥ and its magnitude to be  \nr ⊥ ( x ) ≡ b -x +( ˆ v s · x ) ˆ v s , (C6)  \nr ⊥ ( x ) ≡ √ | b -x | 2 -( ˆ v s · x ) 2 , (C7)  \nand v s ≡ | v s | . The gradient of the potential, which induces accelerations to test masses, takes the following form in Fourier space  \n∇ ˜ Φ( ω, x ) = -2 GM v 2 s e -iωt s e iω ˆ v s · x vs ω [ ˆr ⊥ K 1 ( r ⊥ v s ω ) + i ˆ v s K 0 ( r ⊥ v s ω )] . (C8)  \nThe argument x is the location of the AI that measures the phase shift. For gradiometer configurations, these functions should be evaluated at the two baseline-separated AIs, with the difference evaluated according to, e.g. , Eq. (50) and Eq. (53). Lastly, the Shapiro phase shift is related to the integrated potential along the baseline, given by  \n∫ ∞ -∞ dt e -iωt ∫ L/ 2 -L/ 2 dx ' Φ( t ± x ' , r + x ' ˆ n ) = -GM n ' e -iωt s      1 ω exp [ -| n ' × ( b -r ) | v s ω ] , for b ≪ L/ 2 2 L v s K 0 ( | b -r | v s ω ) , for b ≫ L/ 2 , (C9)  \nwhere n ' ≡ ˆ n ∓ v s with n ' ≡ | n ' | . Here argument r is understood as the mid-point of the two baselineseparated AIs, i.e. r = ( x 1 + x 2 ) / 2.  \n- [1] K. Bongs, M. Holynski, J. Vovrosh, P. Bouyer, G. Condon, E.M. Rasel et al., Taking atom interferometric quantum sensors from the laboratory to real-world applications, Nature Reviews Physics 1 (2019) 731 .\n- [2] L. Morel, Z. Yao, P. Clad'e and S. Guellati-Kh'elifa, Determination of the fine-structure constant with an accuracy of 81 parts per trillion, Nature 588 (2020) 61.\n- [3] B. Estey, C. Yu, H. Muller, P.-C. Kuan and S.-Y. Lan, High-resolution atom interferometers with suppressed diffraction phases, Physical Review Letters 115 (2015) [ 1410.8486 ].\n- [4] R.H. Parker, C. Yu, W. Zhong, B. Estey and H. Muller, Measurement of the fine-structure constant as a test of the standard model, Science 360 (2018) 191 [ https://www.science.org/doi/pdf/10.1126/science.aap7706 ].\n- [5] M. Zych, F. Costa, I. Pikovski and C. Brukner, Quantum interferometric visibility as a witness of general relativistic proper time, Nature Commun. 2 (2011) 505 [ 1105.4531 ].\n- [6] G. Rosi, G. D'Amico, L. Cacciapuoti, F. Sorrentino, M. Prevedelli, M. Zych et al., Quantum test of the equivalence principle for atoms in superpositions of internal energy eigenstates, Nature Commun. 8 (2017) 5529 [ 1704.02296 ].\n- [7] A. Roura, Gravitational redshift in quantum-clock interferometry, Phys. Rev. X 10 (2020) 021014 [ 1810.06744 ].\n- [8] V. Xu, M. Jaffe, C.D. Panda, S.L. Kristensen, L.W. Clark and H. Muller, Probing gravity by holding atoms for 20 seconds, Science 366 (2019) 745 [ https://www.science.org/doi/pdf/10.1126/science.aay6428 ].\n- [9] P. Asenbaum, C. Overstreet, M. Kim, J. Curti and M.A. Kasevich, Atom-Interferometric Test of the Equivalence Principle at the 10 -12 Level, Phys. Rev. Lett. 125 (2020) 191101 [ 2005.11624 ].\n- [10] A. Bassi, K. Lochan, S. Satin, T.P. Singh and H. Ulbricht, Models of Wave-function Collapse, Underlying Theories, and Experimental Tests, Rev. Mod. Phys. 85 (2013) 471 [ 1204.4325 ].\n- [11] A.G. Manning, R.I. Khakimov, R.G. Dall and A.G. Truscott, Wheeler's delayed-choice gedanken experiment with a single atom, Nature Physics 11 (2015) 539.\n- [12] M. Arndt and K. Hornberger, Testing the limits of quantum mechanical superpositions, Nature Physics 10 (2014) 271.\n- [13] C. Burrage, E.J. Copeland and E.A. Hinds, Probing Dark Energy with Atom Interferometry, JCAP 03 (2015) 042 [ 1408.1409 ].\n- [14] P. Hamilton, M. Jaffe, P. Haslinger, Q. Simmons, H. Muller and J. Khoury, Atom-interferometry constraints on dark energy, Science 349 (2015) 849 [ 1502.03888 ].\n- [15] B. Elder, J. Khoury, P. Haslinger, M. Jaffe, H. Muller and P. Hamilton, Chameleon Dark Energy and Atom Interferometry, Phys. Rev. D 94 (2016) 044051 [ 1603.06587\n- ].\n- [16] D.O. Sabulsky, I. Dutta, E.A. Hinds, B. Elder, C. Burrage and E.J. Copeland, Experiment to detect dark energy forces using atom interferometry, Phys. Rev. Lett. 123 (2019) 061102 [ 1812.08244 ].\n- [17] S. Dimopoulos, P.W. Graham, J.M. Hogan, M.A. Kasevich and S. Rajendran, Gravitational Wave Detection with Atom Interferometry, Phys. Lett. B 678 (2009) 37 [ 0712.1250 ].\n- [18] P.W. Graham, J.M. Hogan, M.A. Kasevich and S. Rajendran, A New Method for Gravitational Wave Detection with Atomic Sensors, Phys. Rev. Lett. 110 (2013) 171102 [ 1206.0818 ].\n- [19] J. Ellis and V. Vaskonen, Probes of gravitational waves with atom interferometers, Phys. Rev. D 101 (2020) 124013 [ 2003.13480 ].\n- [20] L. Badurina, O. Buchmueller, J. Ellis, M. Lewicki, C. McCabe and V. Vaskonen, Prospective sensitivities of atom interferometers to gravitational waves and ultralight dark matter, Phil. Trans. A. Math. Phys. Eng. Sci. 380 (2021) 20210060 [ 2108.02468 ].\n- [21] H. Banks, D.M. Grabowska and M. McCullough, Gravitational wave backgrounds from colliding exotic compact objects, Phys. Rev. D 108 (2023) 035017 [ 2302.07887 ].\n- [22] P.W. Graham, D.E. Kaplan, J. Mardon, S. Rajendran and W.A. Terrano, Dark Matter Direct Detection with Accelerometers, Phys. Rev. D 93 (2016) 075029 [ 1512.06165 ].\n- [23] A. Arvanitaki, P.W. Graham, J.M. Hogan, S. Rajendran and K. Van Tilburg, Search for light scalar dark matter with atomic gravitational wave detectors, Phys. Rev. D 97 (2018) 075020 [ 1606.04541 ].  \n[24]  \nL. Badurina, D. Blas and C. McCabe,  \nRefined ultralight scalar dark matter searches with compact atom gradiometers, Phys. Rev. D  \n105  \n(2022)  \n023006 [  \n2109.10965  \n].  \n- [25] L. Badurina, A. Beniwal and C. McCabe, Super-Nyquist ultralight dark matter searches with broadband atom gradiometers, Phys. Rev. D 108 (2023) 083016 [ 2306.16477 ].\n- [26] Terrestrial Very-Long-Baseline Atom Interferometry: Workshop Summary, 10, 2023.\n- [27] L. Badurina et al., AION: An Atom Interferometer Observatory and Network, JCAP 05 (2020) 011 [ 1911.11755 ].\n- [28] MAGIS-100 collaboration, Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100), Quantum Sci. Technol. 6 (2021) 044003 [ 2104.02835 ].\n- [29] B. Canuel et al., Exploring gravity with the MIGA large scale atom interferometer, Sci. Rep. 8 (2018) 14064 [ 1703.02490 ].  \n- [30] B. Canuel et al., ELGAR-a European Laboratory for Gravitation and Atom-interferometric Research, Class. Quant. Grav. 37 (2020) 225017 [ 1911.03701 ].\n- [31] M.-S. Zhan et al., ZAIGA: Zhaoshan Long-baseline Atom Interferometer Gravitation Antenna, Int. J. Mod. Phys. D 29 (2019) 1940005 [ 1903.09288 ].  \n[32]  \nSTE-QUEST  \ncollaboration,  \nSTE-QUEST: Space Time Explorer and QUantum Equivalence principle Space Test,  \n2211.15412  \n.  \n- [33] AEDGE collaboration, AEDGE: Atomic Experiment for Dark Matter and Gravity Exploration in Space, EPJ Quant. Technol. 7 (2020) 6 [ 1908.00802 ].\n- [34] L. Badurina, V. Gibson, C. McCabe and J. Mitchell, Ultralight dark matter searches at the sub-Hz frontier with atom multigradiometry, Phys. Rev. D 107 (2023) 055002 [ 2211.01854 ].\n- [35] J. Carlton and C. McCabe, Mitigating anthropogenic and synanthropic noise in atom interferometer searches for ultralight dark matter, Phys. Rev. D 108 (2023) 123004 [ 2308.10731 ].\n- [36] M. Kasevich and S. Chu, Atomic interferometry using stimulated raman transitions, Phys. Rev. Lett. 67 (1991) 181.\n- [37] S. Dimopoulos, P.W. Graham, J.M. Hogan and M.A. Kasevich, Testing general relativity with atom interferometry, Phys. Rev. Lett. 98 (2007) 111102 [ gr-qc/0610047 ].\n- [38] S. Dimopoulos, P.W. Graham, J.M. Hogan and M.A. Kasevich, General Relativistic Effects in Atom Interferometry, Phys. Rev. D 78 (2008) 042003 [ 0802.4098 ].\n- [39] M. Werner, P.K. Schwartz, J.-N. Kirsten-Siemß, N. Gaaloul, D. Giulini and K. Hammerer, Atom interferometers in weakly curved spacetimes using Bragg diffraction and Bloch oscillations, Phys. Rev. D 109 (2024) 022008 [ 2310.03719 ].\n- [40] A. Roura, Atom interferometer as a freely falling clock for time-dilation measurements, 2402.11065 .\n- [41] P.W. Graham, J.M. Hogan, M.A. Kasevich and S. Rajendran, Resonant mode for gravitational wave detectors based on atom interferometry, Phys. Rev. D 94 (2016) 104022 [ 1606.01860 ].\n- [42] M. Rakhmanov, Response of test masses to gravitational waves in the local Lorentz gauge, Phys. Rev. D 71 (2005) 084003 [ gr-qc/0406009 ].\n- [43] V.S.H. Lee and K.M. Zurek, Proper time observables of general gravitational perturbations in laser interferometry-based gravitational wave detectors, 2408.03363 .\n- [44] A.D. Cronin, J. Schmiedmayer and D.E. Pritchard, Optics and interferometry with atoms and molecules, Rev. Mod. Phys. 81 (2009) 1051.\n- [45] A.D. Ludlow, M.M. Boyd, J. Ye, E. Peik and P.O. Schmidt, Optical atomic clocks, Rev. Mod. Phys. 87 (2015) 637 [ 1407.3493 ].\n- [46] S. Abend, M. Gersemann, C. Schubert, D. Schlippert, E.M. Rasel, M. Zimmermann et al., Atom interferometry and its applications, in International School of Physics 'Enrico Fermi' Course 197 'Foundations of Quantum Theory', E.M. Rasel, W.P. Schleich and S. Wolk, eds., pp. 345-392, IOS, Amsterdam; SIF, Bologna, 2019 [ 2001.10976 ].\n- [47] C. Foot, Atomic physics, Oxford University Press, USA (2005).\n- [48] J.J. Sakurai and J. Napolitano, Modern Quantum Mechanics, Quantum physics, quantum information and quantum computation, Cambridge University Press (10, 2020), 10.1017/9781108587280.\n- [49] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, W. H. Freeman, San Francisco (1973).\n- [50] C. Foot, Atomic Physics, Oxford Master Series in Physics, OUP Oxford (2005).\n- [51] R.M. Wald, General Relativity, Chicago Univ. Pr., Chicago, USA (1984), 10.7208/chicago/9780226870373.001.0001.\n- [52] LIGO Scientific collaboration, LIGO: The Laser interferometer gravitational-wave observatory, Rept. Prog. Phys. 72 (2009) 076901 [ 0711.3041 ].\n- [53] S.M. Vermeulen et al., Photon Counting Interferometry to Detect Geontropic Space-Time Fluctuations with GQuEST, 2404.07524 .\n- [54] C. Overstreet, P. Asenbaum and M.A. Kasevich, Physically significant phase shifts in matter-wave interferometry, Am. J. Phys. 89 (2021) 324 [ 2008.05609 ].\n- [55] C.J. Moore, R.H. Cole and C.P.L. Berry, Gravitational-wave sensitivity curves, Class. Quant. Grav. 32 (2015) 015014 [ 1408.0740 ].\n- [56] M. Maggiore, Gravitational Waves. Vol. 1: Theory and Experiments, Oxford University Press (2007), 10.1093/acprof:oso/9780198570745.001.0001.\n- [57] M. Rakhmanov, Fermi-normal, optical, and wave-synchronous coordinates for spacetime with a plane gravitational wave, Class. Quant. Grav. 31 (2014) 085006 [ 1409.4648 ].\n- [58] K.-P. Marzlin, Fermi coordinates for weak gravitational fields, Phys. Rev. D 50 (1994) 888 [ gr-qc/9403044 ].\n- [59] A. Berlin, D. Blas, R. Tito D'Agnolo, S.A.R. Ellis, R. Harnik, Y. Kahn et al., Detecting high-frequency gravitational waves with microwave cavities, Phys. Rev. D 105 (2022) 116011 [ 2112.11465 ].\n- [60] P. Storey and C. Cohen-Tannoudji, The Feynman path integral approach to atomic interferometry: A tutorial, J. Phys. II 4 (1994) 1999.\n- [61] L. Badurina, Y. Du, V.S. Lee, Y. Wang and K.M. Zurek, Atom interferometer searches of macroscopic dark matter, To appear (2024) .  \n- [62] R. Bracewell, The Fourier Transform and its Applications, McGraw-Hill Kogakusha, Ltd., Tokyo, second ed. (1978).\n- [63] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley and Sons, New York (1972)."}