 over a lone single-photon atom interferometer relies on the possibility of effectively attenuating laser noise. By operating the same laser pulse between both interferometers, the laser noise affecting the experiment's sensitivity cancels in a differential measurement [8].

Atom multi-gradiometry
We define an experiment consisting of N ≥ 2 atom interferometers that are referenced by the same set of lasers and located within the same baseline of length L as a an atom multigradiometer. This is to be contrasted with two atom gradiometers operating in different baselines, irrespective of their synchronisation. A schematic representation of this setup is shown in the right panel of Fig. 1, where the black squares represent the N AIs along the baseline. We will assume that the AIs can be situated at any point along the baseline and in particular, that they do not have to be equally spaced.
An atom gradiometer (AG) measurement consists of taking the difference between the phases collected by two AIs. Figure 1 shows an example of three AG measurements: using the first and second AIs; the (N − 1) th and N th AIs; and the second and (N − 1) th AIs. In total there are N (N − 1)/2 unique gradiometer measurements that can be performed. It then follows that a multi-gradiometer experiment employing two atom interferometers corresponds to a single atom gradiometer, as considered in previous works (see e.g., [12][13][14]40]). 3 Since the interferometers are located within the same baseline and are referenced by the same laser, the AIs will be characterised by the same interrogation time and number of LMT kicks. To simplify the subsequent analysis, we will assume that the launch velocity and atom number are identical for all atoms in all of the AIs.

Characterising signal and noise in an atom gradiometer
Interactions between scalar ULDM and electrons or photons can lead to oscillations in fundamental constants (see e.g. [42][43][44]). These ULDM-induced oscillations in turn alter atomic transition energies, which can be searched for with atom interferometer experiments [12,13]. In this section, we characterise the properties of the ULDM signal and the backgrounds considered in this work, namely GGN from seismic effects and atom shot noise, as recorded by an atom multi-gradiometer operating in a vertical configuration. While an interferometer's atom shot noise is an irreducible noise source and is uncorrelated between detectors, the ULDM signal and the GGN background will induce non-trivial cross-correlations between pairs of interferometers that can be exploited to improve the sensitivity to low frequency ULDM signals.
An atom interferometer measures a time-dependent signal by collecting phase differences over a campaign of duration T int : we refer to this as the integration time and it is zero at the time of the first phase measurement. Measurements are made at a sampling rate 1/∆t, where ∆t is the temporal separation between successive measurements. We assume that ∆t takes the same value over the measurement campaign. The integration time satisfies the relation T int = (N − 1)∆t, where N is the number of measurements, and we will use the approximation N ≈ T int /∆t since N 1. We define the phase measured by the i th interferometer (AI-i) at the end of a sequence that starts at time m∆t as Noise,m is the noise or background contribution. The gradiometer phase between the i th and j th AI, which will be referred to as AG-(i, j), is defined as where i, j ∈ {1, 2,..., N }, j > i, and the gradiometer signal and noise contributions are respectively defined as Signal,m would correspond to the differential ULDM phase recorded by the i th and j th AI. Instead, for GW searches, the phase would correspond to the GW gradiometer signal. In either case, the phase noise Φ (i,j) Noise,m captures all background contributions, including static effects, such as the Earth's gravitational field experienced by the atoms in free-fall, and time-dependent effects, such as GGN and atom shot noise. Because of the nature of the signal of interest, in this paper we shall focus on atom shot noise and GGN, as these are expected to be the dominant noise sources for long-baseline interferometry at high and low frequencies, respectively.

Ultralight dark matter signal
ULDM can be modeled as a temporally and spatially oscillating non-relativistic classical field. This follows from its high occupation number and its small mean velocity and velocity dispersion, which is characteristic of dark matter in the Milky Way [45]. We assume that the ULDM field can be decomposed into Fourier modes with frequencies sampled from the ULDM speed distribution, and random phases sampled from a uniform distribution between 0 and 2π. We assume a Maxwellian distribution of ULDM speeds according to the Standard Halo Model (SHM) [46,47] given by where σ v is the velocity dispersion which is set, at the solar position, by the value of the local standard of rest v 0 = √ 2σ v ≈ 238 km/s, and v obs ≈ 252 km/s is the average speed of the Earth relative to the halo rest frame [48]. Although the DM speed distribution is characterised by a cut-off at the escape velocity, v esc ∼ 800 km/s in the Earth's frame, and may feature additional, anisotropic components (e.g., [49][50][51][52]), the simple form in Eq. (3.5) with no cut-off and no anisotropies is sufficient for our discussion.
The atom interferometer's integration time sets the frequency resolution, ∆f = 1/T int, which in turn sets the ULDM speed ∆v that can be resolved, since ∆f = m φ v∆v/2π, where m φ is the mass of the scalar field. The former relation follows from the properties of the discrete Fourier transform of the data, while the latter follows from the ULDM's kinetic energy definition. In an experiment capable of resolving the ULDM speed resolution to an accuracy of ∆v, the scalar ULDM field can be written as where ρ DM is the local dark matter density to which we assign the value 0.3 GeV/cm 3, ω a m φ (1 + v 2 a /2) is the angular frequency of the ULDM wave for a speed v a, and the sum is over experimentally-resolvable speeds which are indexed by the variable a [53]. The variable θ a ∈ [0, 2π) is a random phase, which we assume is uniformly distributed, while α a is Rayleigh distributed with α 2 a = 2; its probability density function is given by (3.8) Importantly for the subsequent discussion, the random phase θ a and the Rayleigh variable α a are independent [53]. In Eq. (3.6) we neglected the spatial variation of the ULDM wave. We justify this by noting that on average the wave vector is suppressed by a factor of v ∼ 10 −3 relative to the angular frequency. Furthermore, the lengths of the baselines that we consider are significantly smaller than the ULDM's de Broglie wavelength, such that any spatiallydependent contribution to the phase of the DM wave is highly subdominant. Indeed, the frequency range of interest corresponds to ULDM masses between 10 −17 eV and 10 −10 eV, which would be associated with de Broglie wavelengths of length 10 11 km and 10 4 km, respectively.
The phase shift induced by linearly-coupled scalar ULDM for the gradiometer AG-(i, j) where the two AIs are separated by a distance ∆z (i,j) = z i − z j can be parametrised as Here, φ a,m ⊃ ω a m∆t + θ a is the phase of the DM wave at the end of the sequence. The amplitude A a is dependent on the experimental variables n, T, and L characterising the interferometric sequence, and on phenomenological parameters, the most important of which is the ULDM-Standard Model coupling strength d φ. In the limit nL T and ω a nL 1, which are always satisfied in planned interferometers [14,15], (3.10) The complete expressions for A a and ω a are given in Appendix A.
Since α a and θ a are independent, over long timescales the signal is characterised by a vanishing expectation value, Φ and a covariance given by Both of these properties follow from the statistical properties of the amplitude and random phase. A complete derivation of Eqs. (3.11) and (3.12) is provided in Appendix A.1.

Gravity gradient noise from seismic effects
Time-dependent perturbations to the gravitational potential around the free-falling atom clouds cause a phase shift to build up between two paths of an atom interferometer. These perturbations in the gravitational potential originate from terrestrial density fluctuations close to each atom interferometer, as illustrated in Fig. 2, and give rise to a noise source commonly referred to as gravity gradient noise (GGN). For the experimental configuration and frequency range considered in this paper, the dominant source of GGN is expected to consist of ground density perturbations induced by ambient horizontally-propagating seismic waves that are confined near the Earth's surface by horizontal geological strata [24, Figure 2. Schematic representation of the seismic field for a fundamental Rayleigh mode with angular frequency ω a travelling horizontally in the directionk. The Rayleigh wave will induce underground density variations δρ at positions labelled by r. The density variations induce perturbations in the gravitational potential, which serve as a noise source for