# From RATs to riches: mitigating anthropogenic and synanthropic
noise in atom interferometer searches for ultra-light dark matter

John Carlton<EMAIL_ADDRESS>Christopher McCabe  Theoretical Particle
Physics and Cosmology Group, Department of Physics, King’s College London, UK

###### Abstract

Atom interferometers offer promising new avenues for detecting ultra-light
dark matter (ULDM). The exceptional sensitivity of atom interferometers to
fluctuations in the local gravitational potential exposes them to sources of
noise from human (anthropogenic) and animal (synanthropic) activity, which may
obscure signals from ULDM. We characterise potential anthropogenic and
synanthropic noise sources and examine their influence on a year-long
measurement campaign by AION-10, an upcoming atom interferometer experiment
that will be located at the University of Oxford. We propose a data cleaning
framework that identifies and then masks anthropogenic and synanthropic noise.
With this framework, we demonstrate that even in noisy conditions, the
sensitivity to ULDM can be restored to within between 10% and 40% of an atom
shot noise-limited experiment, depending on the specific composition of the
anthropogenic and synanthropic noise. This work provides an important step
towards creating robust noise reduction analysis strategies in the pursuit of
ULDM detection with atom interferometers.

††preprint:

##  I Introduction

Figure 1: Side-on (left panel) and top-down (right panel) schematic of the
Beecroft Building stairwell hosting the AION-10 tower, which is highlighted in
blue. Highlighted in red are people moving on the stairs and in the foyer, a
nearby passenger lift, traffic on the road outside, and nearby rodents, all of
which could induce a time-dependent phase shift through their gravitational
interaction with the atom clouds in the atom interferometers.

Developments in quantum sensing technologies are enabling experiments of
unprecedented precision with applications in fundamental physics [1].
Prominent among these novel sensors are atom interferometers, which have
already provided precision measurements of fundamental constants [2, 3, 4],
tested foundational aspects of quantum mechanics [5, 6, 7] and general
relativity [8, 9, 10, 11], constrained models of dark energy [12, 13, 14, 15]
and ‘fifth’ forces [16, 17], and have been proposed as sensors to search for
dark matter (DM) [18, 19, 20, 21, 22, 23, 24] and gravitational waves [25, 26,
27, 28, 29, 30, 31].

Light-pulse atom interferometers exploit the wave nature of atoms to split
clouds of cold atoms into a superposition of two quantum states that propagate
along different paths. Laser pulses manipulate the atoms before they
coherently recombine and are imaged to measure the phase and contrast acquired
by the different arms. For a more in-depth discussion of the physical
principles, see, for example, Refs. [32, 33, 34]. The phase measured by atom
interferometers is sensitive to changes in timings, atomic structure, and
local accelerations making them excellent detectors for DM and gravitational
waves. Several terrestrial atom interferometer experiments, including AION
[35], ELGAR [36], MAGIS-100 [37], MIGA [38], and ZAIGA [39], intend to probe
unexplored parameter space associated with these phenomena. More ambitious
projects, such as AEDGE [40] and STE-QUEST [41], aim to take this technology
to space for even more sensitive measurements [42].

The exquisite sensitivity of atom interferometers to changes in the local
gravitational potential is well known [43, 44]. For instance, lead bricks
placed next to atom interferometers have been shown to induce a phase shift on
the order of a radian in the interference fringes [45, 46], atom
interferometer experiments have been used to measure Newton’s gravitational
constant $G_{\mathrm{N}}$ [47, 48], and in the emerging field of quantum
cartography, atom interferometers have been used to detect underground
cavities and pipes due to the changes in local gravity gradients [49].

While this sensitivity to the gravitational potential has a myriad of
advantages, it could also lead to unwanted signals when searching for DM or
gravitational wave signatures. This is akin to the challenge faced by laser
interferometer experiments searching for gravitational waves, where signals
from nearby massive objects have been modelled to quantify their contribution
to the background noise, see e.g., Refs. [50, 51]. However, unlike laser
interferometers, atom interferometers are most sensitive to signals in the so-
called ‘mid-band frequency range’ between 0.1 and 10 Hz. Previous studies have
investigated the impact of ground and atmospheric density fluctuations on atom
interferometers in the mid-band [52, 53, 54, 55, 23]. However, the impact of
anthropogenic (human sourced) and synanthropic (animal sourced) noise in the
mid-band has yet to be explored.

This paper investigates the impact of anthropogenic and synanthropic noise on
the upcoming AION-10 atom interferometer experiment [35]. AION-10 will have a
baseline of approximately $10\text{\,}\mathrm{m}$ and operate in a gradiometer
configuration, which means that two identical atom interferometers are run
simultaneously, launching from the bottom and middle of the baseline. Both
atom interferometers will operate using the same laser source to suppress
laser phase noise and vibration [45, 56, 57]. Single-photon interactions will
excite or de-excite the atoms on the
$5\mathrm{s}^{2}\,{}^{1}\mathrm{S}_{0}\leftrightarrow
5\mathrm{s}5\mathrm{p}\,^{3}\mathrm{P}_{0}$ optical clock transition
($698\text{\,}\mathrm{nm}$ line) in strontium-87 [58, 59]. The photon
interaction splits the atom into a superposition of momentum states, which
creates the two arms of the atom interferometer. Large momentum transfer (LMT)
techniques apply laser-pulses throughout the sequence to improve the
sensitivity by increasing the separation of the interferometer arms [60, 61,
57, 62, 63].

Figure 1 shows a schematic of the stairwell of the Beecroft Building at the
University of Oxford’s Department of Physics where AION-10 will be hosted. As
this is an active university building, many sources of anthropogenic and
synanthropic noise could potentially surround the tower. This includes
traceable sources, such as people passing by the tower on the stairs or in the
foyer of the building, a passenger lift moving vertically nearby, or vehicles
passing on the road outside. In addition, there may be sources that are more
difficult to trace, such as from random animal transients (RATs), exemplified
in Fig. 1 by small rodents moving near the base of the tower. All of these
moving masses will cause changes in the local gravitational potential and
could lead to unwanted backgrounds.

Searches for ultra-light dark matter (ULDM) with AION-10 will necessitate
months or even years of data taking. Previous ULDM sensitivity projections
with atom interferometers have neglected the influence of anthropogenic or
synanthropic noise. Hence, the primary objective of this work is to
characterise potential anthropogenic and synanthropic noise sources in the
context of AION-10.

The rest of the paper is structured as follows: section II gives a brief
review of periodic ULDM signals in atom interferometers and describes the
analysis formalism in the frequency-domain for an experiment that has only
atom shot noise as a background. Section III characterises the phase
contribution from anthropogenic and synanthropic noise sources surrounding
AION-10, and describes our simulation of a year-long measurement campaign by
the AION-10 experiment. In section IV, we outline a data-cleaning and
mitigation strategy, and present the associated frequency analysis in the
presence of anthropogenic and synanthropic noise sources. Finally, in section
V, we provide a summary of our findings and discuss future directions.

##  II Ultra-light dark matter signals

For masses below $\sim 1\,\mathrm{eV}$, a bosonic ULDM field within our galaxy
can be modelled as a set of classical waves [64]. The coherent oscillation of
these ULDM waves can give rise to a variety of time-dependent signals that can
be probed with atom interferometers [35, 37, 22]. This includes the
possibility of time-dependent accelerations between atoms in theories of
vector and potentially tensor candidates [18, 65], the time-dependent
precession of nuclear spins in the case of pseudoscalars [66], or time-
dependent oscillations of fundamental ‘constants’ in the case of scalar ULDM
candidates [19, 20, 21]. These signals are in general characterised by a
frequency set by the ULDM mass, an amplitude dependent on the local ULDM
density and the ULDM mass, and a coherence time that depends on the ULDM mass
and the ULDM virial velocity in our galaxy.

In this work, we take as our benchmark model a scalar ULDM candidate,
$\varphi(t,\mathbf{x})$, but our analysis in sections III and IV can in
principle be straightforwardly extended to other ULDM candidates that induce a
phase with a regular periodic pattern. We denote the scalar ULDM mass by
$m_{\varphi}$, and we assume that its characteristic speed is
$|\bm{v}|\sim\mathcal{O}(10^{-3})$.111Here, we follow the convention in the
dark matter literature and use natural units with $\hbar=c=1$. However, at
other points, we will include explicit factors of $\hbar$. We also assume the
scalar ULDM accounts for all of the local dark matter density, so we set
$\rho_{\varphi}=0.3~{}\mathrm{GeV}/\mathrm{cm}^{3}$ [67]. Following Refs. [68,
69], we assume linear interactions between $\varphi(t,\mathbf{x})$ and the
Standard Model fields,

$\mathcal{L}\supset\varphi(t,\bm{x})\sqrt{4\pi
G_{\mathrm{N}}}\left[\frac{d_{e}}{4e^{2}}F_{\mu\nu}F^{\mu\nu}-d_{m_{e}}m_{e}\bar{\psi}_{e}\psi_{e}\right],$
(1)

where $e$ is the electric charge ($\alpha=e^{2}/(4\pi)$ is the fine-structure
constant), $F^{\mu\nu}$ is the Maxwell field strength tensor, $m_{e}$ is the
electron mass, $\psi_{e}$ is the electron spinor field, and $d_{e}$ and
$d_{m_{e}}$ are coupling constants parameterising the ULDM interaction with
photons and electrons, respectively.222Quadratic interactions are also viable
(see e.g., Refs. [70, 71, 72]) but will not be considered in this work.

The linear interactions introduce time-dependent corrections to the
electromagnetic fine-structure constant and the electron mass [73]:

$\displaystyle\alpha(t,\bm{x})$
$\displaystyle\approx\alpha\left[1+d_{e}\sqrt{4\pi
G_{\mathrm{N}}}\,\varphi(t,\bm{x})\right],$ (2) $\displaystyle
m_{e}(t,\bm{x})$ $\displaystyle=m_{e}\left[1+d_{m_{e}}\sqrt{4\pi
G_{\mathrm{N}}}\,\varphi(t,\bm{x})\right].$ (3)

Upon modelling the scalar ULDM field as a classical oscillating field,

$\varphi(t,\bm{x})\approx\frac{\sqrt{2\rho_{\varphi}}}{m_{\varphi}}\cos(\omega_{\varphi}t-\bm{k_{\varphi}}\cdot\bm{x}+\theta),$
(4)

where $\omega_{\varphi}\approx m_{\varphi}(1+|\bm{v}|^{2}/2)$,
$\bm{k}_{\varphi}=m_{\varphi}\bm{v}$ and $\theta$ is a random phase, we see
that to leading order in $|\bm{v}|$, the angular frequency of the time-
dependent correction to $\alpha$ and $m_{e}$ is set by $m_{\varphi}$. Since
$\bm{k}_{\varphi}\cdot\Delta\bm{x}\ll 1$ for all values of $m_{\varphi}$
accessible to terrestrial atom interferometers, where $\Delta\bm{x}$ is the
distance scale over which the atoms propagate, we can safely ignore the
$\bm{k_{\varphi}}\cdot\bm{x}$ term and focus solely on the time-dependence of
the ULDM field.

Strictly speaking, the amplitude for the oscillating field in eq. (4) only
holds in the regime where the total integration time is longer than the
coherence time [74]. Otherwise, the amplitude has a stochastic nature that can
be modelled with a Rayleigh parameter (see e.g., Refs. [75, 76, 77]). However,
the inclusion of a Rayleigh parameter does not impact the mitigation
strategies discussed in this work so for simplicity, we use eq. (4)
throughout.

Time-dependent variations in the electron mass or fine structure constant lead
to time-dependent transition energies in atoms. For optical electronic
transitions of the type that will be used in AION-10, the time-dependent
variation induced by scalar ULDM is

$\displaystyle\omega_{A}(t)$
$\displaystyle\simeq\omega_{A}+\overline{\Delta{\omega_{A}}}\cos(\omega_{\varphi}t+\theta)\,,$
(5) $\displaystyle\overline{\Delta\omega_{A}}$
$\displaystyle=\omega_{A}\sqrt{4\pi
G_{\mathrm{N}}}\left[d_{m_{e}}+(2+\xi_{A})d_{e}\right]\frac{\sqrt{2\rho_{\varphi}}}{m_{\varphi}}\,,$
(6)

where $\omega_{A}$ is the transition frequency in the absence of
$\varphi(t,\bf{x})$, and $\xi_{A}\simeq 0.06$ for the strontium-87 clock
transition [78].

Figure 2: Left: Phase difference between the upper and lower atom
interferometers from an oscillating ULDM field (upper panel) with
$m_{\varphi}\simeq 4\times 10^{-16}$ eV
($f_{\varphi}=$0.1\text{\,}\mathrm{Hz}$$) and $d_{m_{e}}=1$, and from atom
shot noise (ASN, lower panel) assuming a phase noise
$\delta\phi=10^{-4}~{}$\text{\,}\mathrm{rad}\mathrm{/}\sqrt{\mathrm{Hz}}$$.
The points represent 50 simulated measurements, assuming $\Delta
t=$2\text{\,}\mathrm{s}$$. The solid grey line in the upper panel highlights
the periodic nature of the differential phase induced by ULDM, compared to the
stochastic phase induced by ASN in the lower panel. Right: Square-root of the
PSD, $\sqrt{S_{k}}$, for the ULDM phase signal (pink spike), and the ASN
(purple curve) assuming $T_{\mathrm{int}}=1/3$ yr. The parameters for the ULDM
signal and ASN are the same as in the left panels. The yellow line shows the
root mean square for ASN, which is constant in frequency space, as expected
for white noise.

The time-dependent nature of the electronic transition induces a phase in an
atom interferometer. For the scalar ULDM search in AION-10, we assume the
‘broadband’ LMT sequence discussed in Refs. [20, 21], which consists of an
initial $\pi/2$-pulse, $(4n-3)$ $\pi$-pulses from alternating directions and a
final $\pi/2$-pulse, where $n$ is the number of LMT kicks.333A $\pi/2$ $[\pi]$
pulse is a pulse of resonant radiation that has a duration of $\pi/(2\Omega)$
[$\pi/\Omega$], where $\Omega$ is the Rabi frequency. For the gradiometer
configuration, the gradiometer phase, i.e. the phase difference between the
upper and lower atom interferometers, induced by scalar ULDM for this sequence
is

$\begin{split}\Delta\phi_{\ell}&=8\,\frac{\overline{\Delta\omega_{A}}}{\omega_{\varphi}}\,\frac{\Delta
r}{L}\,\sin\left[\frac{\omega_{\varphi}nL}{2}\right]\\\
&\quad\times\sin\left[\frac{\omega_{\varphi}\left(T-(n-1)L\right)}{2}\right]\sin\left[\frac{\omega_{\varphi}T}{2}\right]\\\
&\quad\quad\times\cos\left[\omega_{\varphi}\left(\frac{2T+L}{2}+\ell\Delta
t\right)+\theta\right]\;,\end{split}$ (7)

where $\Delta r$ is the spatial distance between the two atom interferometers,
$L$ is the total baseline, $2T$ is the time between the initial and final
$\pi/2$ pulses ($T$ is the ‘interrogation time’), $\Delta t$ is the temporal
separation between successive measurements and $\ell$ is an integer that
labels the measurements [21, 23]. For $N$ measurements, $\ell=0$ corresponds
to the first measurement and $\ell=N-1$ to the last. The amplitude of
$\Delta\phi_{\ell}$ is largest when
$m_{\varphi}\simeq\left(1.5\,\mathrm{s}/T\right)\times 10^{15}\,\mathrm{eV}$,
or in terms of frequency, when
$f_{\varphi}=m_{\varphi}/(2\pi)\simeq\left(0.4\,\mathrm{s}/T\right)\,\mathrm{Hz}$.
Given $T\sim\mathcal{O}(1)$ s, AION-10 is most sensitive to scalar ULDM with
$m_{\varphi}\sim 10^{-16}$ – $10^{-14}$ eV, or equivalently, the frequency
range $f_{\varphi}\sim 0.1$ – $10$ Hz [35].

###  II.1 Frequency-domain analysis with atom shot noise

The pink points in the upper left panel of Fig. 2 show the simulated periodic
ULDM phase difference for an ULDM field with properties $d_{m_{e}}=1$,
$d_{e}=0$, $m_{\varphi}\simeq 4\times 10^{-16}$ eV
($f_{\varphi}=$0.1\text{\,}\mathrm{Hz}$$), and $\theta=\pi/2$. We use the
experimental parameters corresponding to the AION-10 (goal) scenario [35]: a
baseline $L=$10\text{\,}\mathrm{m}$$, a separation $\Delta
r=$5\text{\,}\mathrm{m}$$, and $n=1000$. Assuming that the atom clouds are
launched with a speed of
$v_{\mathrm{l}}=$3.86\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$, we find
$T=$0.73\text{\,}\mathrm{s}$$, and we assume $\Delta
t=$2\text{\,}\mathrm{s}$$.

The goal of AION-10 is to reach a phase noise $\delta\phi$ that is dominated
by the standard quantum limit, set by atom shot noise (ASN). The ASN scales as
$(N_{\mathrm{atom}})^{-1/2}$, where $N_{\mathrm{atom}}$ is the number of atoms
per shot. The purple points in the lower left panel of Fig. 2 show simulated
phase differences assuming only ASN with the AION-10 (goal) phase noise of
$\delta\phi=10^{-4}~{}$\text{\,}\mathrm{rad}\mathrm{/}\sqrt{\mathrm{Hz}}$$
[35].

Comparing the upper and lower panels on the left of Fig. 2 highlights the
challenge of searching for a ULDM signal in the time domain with ASN-limited
measurements. The magnitude of the ULDM phase difference is substantially
smaller than from ASN, even for $d_{m_{e}}=1$, which is orders of magnitude
larger than current constraints on this parameter [79].

Instead, the frequency domain provides a better way to search for the periodic
ULDM signal amid the ASN. For an uniformly sampling experiment, the classical
periodogram [80] is a good estimator of the power spectral density (PSD),
which provides information on the spectral content of the phase time-series.
For a total integration time $T_{\mathrm{int}}=N\Delta t$, the classical
periodogram is

$S_{k}=\frac{\Delta
t}{N}\left|\sum_{\ell=0}^{N-1}\Delta\phi_{\ell}\exp\left[-\frac{2\pi i\ell
k}{N}\right]\right|^{2},$ (8)

where $k=0,\dots,(N-1)$ labels the discrete frequencies,
$f_{k}=k/T_{\mathrm{int}}$, with a frequency resolution $\Delta
f=(T_{\mathrm{int}})^{-1}$. The maximum non-aliased frequency is given by the
Nyquist frequency $f_{\mathrm{Ny}}=(2\Delta t)^{-1}$. ULDM signals at
frequencies higher than $f_{\mathrm{Ny}}$ are still measurable through
aliasing [81]. Following common usage, we will use the terms periodogram and
PSD interchangeably to refer to $S_{k}$.

The right panel of Fig. 2 shows the square root of the PSD for ULDM (pink
‘spike’) and for ASN (purple curve). The pink spike occurs at $m_{\varphi}$,
the frequency of the phase difference given in eq. (7), and the spike’s
amplitude scales with $d_{m_{e}}$ or $d_{e}$, and with the integration time
[21]. The stochastic nature of the ASN is reflected in the purple curve, which
varies stochastically with frequency. Since the frequency resolution $\Delta
f\sim 10^{-7}\,~{}\mathrm{Hz}$, when plotted on this scale it appears as a
band, so we have also plotted in yellow the root mean square for an ensemble
of ASN-only experimental realisations. As expected for white noise, this is
constant in frequency space.

Previous ULDM sensitivity projections for atom interferometers have utilised
this frequency-domain framework with the assumption that experimental noise is
dominated by ASN (e.g., Refs. [20, 35, 37, 22]).444These projections utilise
Bartlett’s method when the integration time is greater than the ULDM coherence
time, which reduces the ULDM PSD to a spike (single frequency bin). We will
also follow this approach. While the phase response from rotational effects,
magnetic fields, black-body radiation, and from laser phase noise have
previously been considered and been shown to be sub-dominant to ASN [37], the
impact of time-dependent anthropogenic and synanthropic noise on this
formalism has yet to be investigated. We will address this in remainder of
this paper.

##  III Anthropogenic and
synanthropic signals

In this section, we describe how we calculate the phase response from
anthropogenic and synanthropic sources. For each noise source, we evaluate the
gradiometer phase; that is, the difference in the phases measured by each atom
interferometer:

$\Delta\phi_{\ell}=\phi_{\ell}^{(u)}-\phi_{\ell}^{(l)},$ (9)

where the indices $u,l$ indicate the upper and lower atom interferometer,
respectively.

We begin by describing the semi-classical formalism used to calculate the
phase for each atom interferometer, where for a quadratic Hamiltonian,
Ehrenfest’s theorem reduces to Hamilton’s equations, so the atoms follow their
classical trajectories [82]. The classical trajectory in the broadband LMT
sequence can be modelled as a Mach-Zehnder sequence in which each pulse
transfers a momentum $\pm n\hbar k$, where $k$ is the wave-vector of the
laser, to either the upper or lower arm of the atom interferometer. We assume
the atoms launch radially upwards from a spherically symmetric Earth (which
defines the $z$-direction) and solve the Euler-Lagrange equations in a
gravitational field with acceleration $g$ and a quadratic term accounting for
the Earth’s gravity gradient $\gamma_{zz}$. We ignore effects from Coriolis
forces as we assume they are corrected with a rotation compensation system,
(see, e.g., Ref. [37]). We then treat the influence of the gravitational
potential of people, objects, or RATs nearby, as perturbations on the
classical trajectories of the atoms.

The Lagrangian characterising the atoms in the presence of the Earth’s
gravitational field and a perturbing gravitational potential $V(z,t)$ is
therefore

$L=m_{A}\left(\frac{1}{2}\dot{z}^{2}-gz+\frac{1}{2}\gamma_{zz}z^{2}-V(z,t)\right),$
(10)

where $m_{A}$ is the mass of the atom. In general, three contributions to the
phase arise: the propagation phase accrued between laser pulses; the laser
phase imprinted on the atoms by the laser pulses; and a separation phase when
the arms of the interferometer do not overlap at the end of the sequence [83].
It is straightforward to calculate the contributions to the phase in the
absence of $V(z,t)$ (see, e.g., Refs. [84, 85, 86]). The additional phase from
the perturbing gravitational potential is found from

$\begin{split}\delta\phi_{\ell}^{(u,l)}=\frac{m_{A}}{\hbar}\int_{\ell\Delta
t}^{\ell\Delta t+2T}\mathrm{d}t&\left[V\left(z^{(u,l)}_{2},t\right)\right.\\\
&\quad-\left.V\left(z^{(u,l)}_{1},t\right)\right]\,\,,\end{split}$ (11)

where the indices $1,2$ denote the classical paths travelled by the atoms in
each arm of the interferometer in the absence of $V(z,t)$.

We parameterise the gravitational potential as

$V(z,t)=-\frac{G_{\mathrm{N}}M}{\sqrt{D(t)^{2}+\left({z(t)}-h(t)\right)^{2}}}\,,$
(12)

where $M$ is the mass of the person, object, or RAT, $D(t)$ and $h(t)$ are the
horizontal and vertical distances from the atoms, and $z(t)$ is the vertical
height of the atoms during the sequence. All vertical distances are defined
relative to the atom launch position in the lower atom interferometer.

The semi-classical formalism that we have used is exact for a quadratic
potential. Potentials that are not quadratic, which includes the gravitational
potential, will give non-classical corrections. The size of the correction can
be estimated by considering the leading correction to
$\partial_{t}\langle\hat{p}\rangle$, which is
$\tfrac{1}{2}(\partial_{r}^{3}V(r))\Delta x^{2}$, where
$\langle\hat{p}\rangle$ is the momentum operator expectation value and $\Delta
x^{2}$ is the variance of atom wave-packet size [82, 87, 88, 86]. For typical
AION-10 parameters, we find the leading non-classical correction is $\sim
10^{-16}g\left(M/80\,\mathrm{kg}\right)\left(2\,\mathrm{m}/r\right)^{4}\left(\Delta
x^{2}/1\,\mathrm{mm}^{2}\right)$. Given that 10 m atom interferometers operate
at a sensitivity $\sim 10^{-12}g$ (see, e.g., Ref. [89]), the semi-classical
formalism is sufficiently accurate for the gravitational potentials that we
consider.

###  III.1 Characterising background-induced phases

Figure 3: Upper part of each panel shows the simulated time-dependent phases
induced by the gravitational potential of anthropogenic and synanthropic noise
sources. From top-to-bottom, we consider: a passenger lift (elevator), a
person moving on stairs and in the foyer, a car, and five random animal
transients (RATs). The time-dependence of the distance parameters are plotted
in the lower part of each panel. The position of the two atom interferometer
launch points are shown as dashed lines at $h=$0\text{\,}\mathrm{m}$$ and
$h=$5\text{\,}\mathrm{m}$$. The largest differential phase is induced by the
person on the stairs, due to their proximity to the AION-10 tower.

Having defined the semi-classical formalism, we are now in a position to
determine the phase response from anthropogenic and synanthropic sources that
could impact AION-10. Figure 3 shows the simulated phase response from a
single incidence of five possible noise sources: a passenger lift; a person
climbing stairs next to the tower; a person walking across the foyer next to
the top of the tower; a car travelling past on a nearby road; and a small
animal moving close to the base of the tower.555The gravity gradient gives a
constant contribution $\Delta\phi_{\ell}=nk\gamma_{zz}\delta rT^{2}$, which we
do not include when plotting $\Delta\phi_{\ell}$, so as to clarify the impact
of the noise sources. Laser-pulse sequences can be engineered to minimise the
gravity gradient [90, 91], but we do not investigate these schemes in this
work. The upper part of each panel shows the time-dependence of the
gradiometer phase, while the lower part shows the most relevant time-dependent
distance parameter that enters eq. (12).

The first two examples consider scenarios where the dominant effect occurs
from vertical movement parallel to the tower. The upper panel models a
passenger lift that starts from rest at the base of the AION-10 tower, moves
up past the top of tower at a constant speed of
$v=$2.5\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$, and then stops. The
behaviour of the phase mirrors this by initially recording a constant value,
changing as the lift moves, before finally plateauing to a constant value. We
model the lift with a constant mass and assume that there is no counterweight
mechanism. The second panel models the response from a person walking at
constant speed $v=$1\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ to the
base of the tower, then up the stairs next to the tower at constant speed
$v=$0.5\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$, before finally walking
away. The phase is initially small, changes significantly as the person climbs
the stairs, and tends to zero after the person walks away. Were the lift to
move in the opposite direction, or a person to walk down the stairs, the phase
curve would be reversed.

Next, we consider two scenarios where the dominant effect occurs predominantly
from horizontal movement relative to the tower: a person walking at constant
speed $v=$1\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ along the foyer
adjacent to the stairwell that hosts AION-10; and a car travelling at constant
speed $v=$10\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ on a straight
road. The shape of the phase response is similar in both cases, although there
is a relative sign change and the shape is narrower for the car as it moves
faster.

Finally, the simulated phase response from five small animals (RATs) moving
quasi-randomly under the base of the AION-10 tower is shown in the bottom
panel of Fig. 3. The phase response of the small animal induces
$\mathcal{O}(50)\,\mu\mathrm{rad}$ ‘ripples’ in the time series, which are
significantly smaller than the phase from the other sources. However, the
total induced response is similar in magnitude to atom shot noise
($\mathcal{O}(250)\,\mu\mathrm{rad}$ cf. Fig. 2). As such, we include RATs in
our analysis to represent a class of noise sources that give a contribution
somewhat similar to atom shot noise, and which may be difficult to mitigate
against in analysis strategies.

It is relatively straightforward to gain intuition for the results in Fig. 3
by considering a limiting case of the phase response calculation where
analytic results can be obtained. To that end, for this discussion we will
ignore the effect of the Earth’s gravity gradient on the classical trajectory
of the atoms since it results in more complicated expressions with little gain
in intuition. First, by considering eqs. (11) and (12), it is trivial to
observe that for an object of constant mass $M$, $V(z,t)\propto M$ so
$\delta\phi_{\ell}^{(u,l)}\propto M$ and hence $\Delta\phi_{\ell}\propto M$,
i.e. the phase response is simply linear in $M$.

Figure 4: Simulated differential phase induced by a ULDM signal, atom shot
noise, and anthropogenic and synanthropic noise sources. The left and right
panels show a simulated time series during the daytime and nighttime,
respectively. The lower panels focus on the first 20 minutes of the time
series. The phase is dominated by the anthropogenic noise sources.

Next, consider a stationary object of mass $M$ in the regime where $D,h\gg
z(t)$. Expanding the potential, the first non-zero contribution to
$\Delta\phi_{\ell}$ arises from
$V(z,t)=-\tfrac{1}{2}G_{\mathrm{N}}Mz^{2}(D^{2}-2h^{2})/(D^{2}+h^{2})^{5/2}$,
and gives a phase

$\Delta\phi_{\ell}\simeq-nkT^{2}\,G_{\mathrm{N}}M\Delta
r\,\frac{D^{2}-2h^{2}}{(D^{2}+h^{2})^{5/2}}\;.$ (13)

In the limit $D\gg h$, we see that the gradiometer phase is negative and
scales with an inverse cube law with distance from the atom interferometers.
This power-law scaling with distance relative to the linear scaling with mass
is the reason that a single person walking on the stairs next to the tower
induces a much larger phase than the more massive passenger lift that is
further away. We also see from eq. (13) that we get a sign change when
$D\lesssim h$, which is the behaviour we see in Fig. 3 for the person (foyer)
relative to the car: both have the same value of $h$ but the minimum $D$ value
is $5\text{\,}\mathrm{m}$ for the person (foyer) and $20\text{\,}\mathrm{m}$
for the car. For the relatively small values of $D$ and $h$ used in Fig. 3,
next-order corrections to $\Delta\phi_{\ell}$ modify the relation such that
the sign changes at $D\sim h$ rather than $D=\sqrt{2}h$.

###  III.2 Simulated daytime and nighttime time series

Over an extended period of data taking, many anthropogenic and synanthropic
sources will contribute to the total phase readout. Figure 4 shows a
simulation of extended data-taking periods: the left panels show the time
series from a busy period during the day, while the right panels show the time
series from a quiet period at night when the occupation levels in the building
are significantly lower. The upper panels show a five hour time series, while
the lower panels zoom in on the first 20 minutes.

We make the following assumptions for the daytime simulation. Firstly, we
assume the lift moves randomly between seven floors within the building at a
rate of $20$ movements per hour between 9-10am and 12.30-2pm, five movements
per hour between 10-12.30pm, and we assume the mass varies randomly between
$1000$ and $2000\text{\,}\mathrm{k}\mathrm{g}$ to account for a different
number of people during each movement; Secondly, we assume that five people
per hour use the stairs next to the tower and that each person has a mass
ranging from $50$ to $100\text{\,}\mathrm{k}\mathrm{g}$. Thirdly, we model
$100$ people per hour walking across the foyer or on floors above the tower
between 9-10am and 12.30-2pm, and 25 people per hour between 10-12.30pm.
Finally, we assume $60$ vehicles pass per hour, with a mass randomly selected
between $1000$ and $5000\text{\,}\mathrm{k}\mathrm{g}$.

For the nighttime simulation, our assumptions are similar but the rates are
reduced. Firstly, the lift moves once per hour at a random time throughout the
night. Secondly, owing to the large induced phase from the people on the
stairs, we assume that the stairs next to the tower are closed at night.
Thirdly, we model up to ten people in the foyer and up to ten cars passing
throughout the night. In both the daytime and nighttime simulations, we
include the contribution from atom shot noise, RATs and ULDM, but their
contribution is not visible on the scale in Fig. 4.

In all panels in Fig. 4, we see that the time series is characterised by large
transients on short timescales, $\mathcal{O}(1)\,\mathrm{min}$, that often
resemble spikes or oscillations, together with longer periods (minutes or even
hours) where the time series is relatively flat. The spiked features can be
understood with reference to Fig. 3: the change in one plateau to another
results from a movement of the lift; the transient down and up behaviour
follows from a person on the stairs; and the Gaussian shapes are people on the
foyer or vehicles passing by. Unsurprisingly, the daytime simulated time
series has significantly more variation compared to the nighttime simulation.

##  IV Analysis strategy and
sensitivity projections

Removing noise from time series data is a problem encountered in many fields
(see, e.g., Refs. [92, 93, 94, 95, 96, 97, 98]). While a menagerie of
techniques are known, we utilise an approach that is aligned with our intended
outcome: to achieve sensitivity to ULDM that approaches an atom shot noise-
limited experiment. Our discussion in this section has two goals. As we are
working with simulated data, which contains a representative rather than fully
comprehensive set of possible noise sources, our first goal is to provide a
high-level discussion of the data cleaning strategy. Our expectation is that
the methods will be extended upon application to real data but the principles
will remain. Our second goal is to address the potential impact of
anthropogenic and synanthropic noise sources on ULDM sensitivity projections.

To proceed, we exploit information about the ULDM signal and noise sources.
Firstly, we have shown in section III that anthropogenic or synanthropic noise
sources induce time-dependent contributions to the differential phase on
$\mathcal{O}(1)\,\mathrm{min}$ timescales, separated by longer periods of
stability.

Secondly, any noise-reduction procedure must ensure that distortions to the
periodic structure in the ULDM signal are minimised. This is because the
amplitude of the ULDM signal in the time-domain is much smaller than atom shot
noise (cf. Fig. 2), so our analysis relies on recovering the periodic signal
through a frequency domain analysis.

Thirdly, the sensitivity to the coupling constants parameterising the ULDM
interactions with photons or electrons scales with the total integration time
as $(T_{\rm{int}})^{-1/4}$ when the integration time is greater than the ULDM
coherence time, or $(T_{\rm{int}})^{-1/2}$ in the opposite regime [21]. This
means that removing parts of the time series and thereby reducing
$T_{\rm{int}}$ results in a relatively small loss in sensitivity. In addition,
an even more drastic choice of restricting to nighttime running (i.e., to 8
hours instead of 24 hours), when the time series is significantly cleaner,
reduces sensitivity to most of the mid-band frequency range by only $30\%$,
rather than a factor of three.666We have not considered the impact of
anthropogenically induced seismic noise but minimising this source also
motivates running at night.

All of these considerations motivate nighttime running combined with the
application of masks to remove the anthropogenic- or synanthropic-induced
transients from the time series. An immediate implication of using masks is
that we have to analyse a time series with gaps. Equivalently, this can be
viewed as a non-uniformly sampled time series. Analysis of ‘gapped’ or non-
uniformly sampled time series is common in astronomy, since ground-based
observations occur at night and clouds or weather can prevent observations for
short periods. Therefore, rather than using the classical periodogram, we
instead use an estimator of the PSD that is commonly used in the astrophysics
community and which is designed for non-uniformly sample data: the Lomb-
Scargle periodogram [99, 100].

In the rest of this section, we describe our data cleaning strategy, review
properties of the Lomb-Scargle periodogram, and examine the PSD derived from
time series encompassing both anthropogenic and synanthropic noise sources,
both with and without applying the data cleaning techniques. Finally, we
present the impact of anthropogenic and synanthropic noise sources on ULDM
sensitivity projections after the data cleaning strategies have been employed.

Figure 5: Our data cleaning approach to eliminate transient noise in a
simulated time series. The upper left panel shows the raw time series
containing transients in the phase induced by anthropogenic noise and
(invisible on this scale) changes induced by synanthropic noise. Transients
are identified by calculating $\sigma_{\Delta\phi}$, the rolling standard
deviation (upper right). Events that surpass the transient detection
threshold, indicated by the magenta line, are masked, as indicated in the
lower panels: diagonal lined hatching represents masking applied to
anthropogenic noise while the dotted hatching represents masked synanthropic
noise. The bottom left panel shows the result of detrending the masked time
series. The cleaned time series is now distributed around a common baseline
and is ready for analysis with the Lomb-Scargle periodogram.

###  IV.1 Cleaning the time series

Our data cleaning approach is summarised in Fig. 5. We start with the raw time
series shown in the upper left panel of Fig. 5, which has been created to have
multiple transients spread throughout the time period. The first stage is to
identify and remove transients from the time series. Several approaches have
been developed to identify transients, including moving average techniques,
median filtering, change point detection and machine learning algorithms (see,
e.g., Ref. [101]). Moving averages are generally more effective at detecting
short-term, localised transients of the type that appear in our raw time
series [102]. Therefore, we adopt a moving average of the standard deviation
algorithm, which for simplicity, we refer to as the rolling standard
deviation.

We define the rolling standard deviation for the $\ell^{\mathrm{th}}$ phase in
the series as

$\sigma_{\Delta\phi_{\ell}}=\sqrt{\frac{1}{N-1}\sum_{i=\ell-(N-1)/2}^{\ell+(N-1)/2}\left(\Delta\phi_{i}-\overline{\Delta\phi}_{\ell}\right)^{2}\,},$
(14)

where $N$ is an odd integer that defines the window size and
$\overline{\Delta\phi_{\ell}}$ is the rolling mean, defined as

$\overline{\Delta\phi_{\ell}}=\frac{1}{N}\sum_{i=\ell-(N-1)/2}^{\ell+(N-1)/2}\Delta\phi_{i}\;.$
(15)

The choice of the window size is important. While a larger $N$ can provide
more stability in the rolling standard deviation, a value that is too large
may miss localised transients. We found that a step size of $N=101$ strikes a
good balance between these two considerations.

The upper right panel of Fig. 5 shows the rolling standard deviation for the
raw time series shown in the upper left panel. Transients are identified when
they cross a transient threshold, shown by the solid pink line. Our threshold
is set as the mean-plus-five-standard-deviations of an ASN-limited experiment.
Transients caused by anthropogenic noise are orders of magnitude above the
threshold so are easily identified. In contrast, transients caused by
synanthropic noise appear just above threshold. Events that surpass the
rolling standard deviation are masked. For the time series in Fig. 5, the
masked regions are represented by hatched patterns in the lower panels, where
diagonal hatching represents anthropogenic noise and dotted hatching
represents synanthropic noise.

The final stage involves detrending the masked time series to eliminate any
long-term drifts and establish a common baseline. We achieve this by fitting a
linear regression model to each region of the time series between the masks.
Subtracting the linear fit from each region effectively detrends the time
series. This detrending approach works well with our simulated data. However,
it may require modifications when applied to real data, where quadratic trends
or other longer-term drift effects could be more pronounced and require
special consideration. The cleaned time series looks similar to the ASN-only
time series shown in Fig. 2, but with gaps at the regions where the masks have
been applied.

###  IV.2 Lomb-Scargle periodogram

We employ the Lomb-Scargle periodogram to analyse the spectral features of the
cleaned time series. The Lomb-Scargle periodogram is usually defined as

$\begin{split}S^{\mathrm{LS}}_{k}&=\frac{1}{2}\frac{\left(\sum_{\ell}\Delta\phi_{\ell}\cos{\left(2\pi
f_{k}\left[t_{\ell}-\tau\right]\right)}\right)^{2}}{\sum_{\ell}\cos^{2}(2\pi
f_{k}[t_{\ell}-\tau])}\\\
&\qquad\quad+\frac{1}{2}\frac{\left(\sum_{\ell}\Delta\phi_{\ell}\sin(2\pi
f_{k}[t_{\ell}-\tau])\right)^{2}}{\sum_{\ell}\sin^{2}(2\pi
f_{k}[t_{\ell}-\tau])}\,,\end{split}$ (16)

where $\ell$, as before, is an integer running over the values from $0$ to
$N-1$, $t_{\ell}$ denotes the time of the $\ell^{\mathrm{th}}$ measurement,
$f_{k}$ denotes discrete frequencies, and $\tau$ is given for each frequency
$f_{k}$ by

$\tau=\frac{1}{4\pi f_{k}}\tan^{-1}\left(\frac{\sum_{\ell}\sin(4\pi
f_{k}t_{\ell})}{\sum_{\ell}\cos(4\pi f_{k}t_{\ell})}\right)\;.$ (17)

See Ref. [103] for a more comprehensive discussion.

As defined here, $S^{\mathrm{LS}}_{k}$ has units of $\mathrm{rad}^{2}$, which
is different to our definition of the classical periodogram, $S_{k}$, which
has units $\mathrm{rad}^{2}/\mathrm{Hz}$. The Lomb-Scargle periodogram allows
for differing values of $\Delta t$ throughout the time series, so dropping the
$\Delta t$ prefactor that was included in eq. (8) avoids ambiguity. In our
simulation, however, we assume a structured observation regime in which we fix
$\Delta t=$2\text{\,}\mathrm{s}$$ for all observations. Therefore, to make a
closer connection with the earlier results, we show
$\mathrm{PSD}_{\rm{LS}}\equiv\Delta t\,S^{\mathrm{LS}}_{k}$ in our figures.

In practical terms, we calculate $S^{\mathrm{LS}}_{k}$ using the astropy
package [104], with the ‘psd’ normalisation and ‘fast’ method, which utilises
the fast Fourier transform. This formalism fits a sinusoid to the data at each
frequency with a higher power in the periodogram being associated with a
better fit. It uses the floating-mean model to fit for the mean of data and
uses only one Fourier term in the fit. We set
‘$\text{samples\\_per\\_peak}=1$’ such that $f_{k}=k/T_{\rm{int}}$, which
implies that our frequency grid and Nyquist frequency are the same for both
the Lomb-Scargle and classical periodograms.

###  IV.3 Frequency-domain analysis with anthropogenic and synanthropic noise

Figure 6:  Upper panel: the square root of the Lomb-Scargle PSD for a time
series that includes contributions from a ULDM field, ASN, and anthropogenic
noise associated with nighttime running. Lower panel: the PSD is extended to
include the contribution from synanthropic noise in addition to the
aforementioned components. The pink line shows the PSD of the raw time series
without any cleaning strategies applied, assuming nighttime running for 365
days. The purple line shows the PSD of the time series after the
implementation of our cleaning strategy. The yellow (green) lines show the
root mean square with (without) cleaning. The cleaned PSDs closely resemble
the ASN-only case indicating that the data cleaning method has successfully
identified and removed anthropogenic and synanthropic transients.

The two panels in Fig. 6 show the square root of the Lomb-Scargle PSD for a
simulated time series with a ULDM field with parameters $m_{\varphi}\simeq
4\times 10^{-16}\,\mathrm{eV}$ ($f_{\varphi}=0.1\,\mathrm{Hz}$) and
$d_{m_{e}}=1$, ASN with phase noise
$\delta\phi=10^{-4}\,\mathrm{rad}/\sqrt{\mathrm{Hz}}$ and anthropogenic noise
(upper panel), and with both anthropogenic and synanthropic noise (lower
panel). The simulated time series assumes that data taking occurs during an
eight-hour period at nighttime each day over the course of one year, for a
total (raw) integration time of $1/3~{}\mathrm{yr}$.

The assumptions made in generating the nighttime anthropogenic noise are
discussed in section III.2. The synanthropic noise in the time series that
enters the lower panel of Fig. 6 is generated with the following assumptions:
we simulate five RATs with a mass $0.25\text{\,}\mathrm{kg}$ running on
pseudo-random paths underneath the AION-10 detector. Each RAT on average runs
on a pseudo-random path 300 times per night. The paths are randomly generated
on a $10\text{\,}\mathrm{m}$ $\times$ $10\text{\,}\mathrm{m}$ horizontal grid
that is $15\text{\,}\mathrm{m}$ below the atom launch position in the lower
atom interferometer.

In both panels, the pink curve shows the Lomb-Scargle PSD obtained from the
raw time series without any cleaning strategies applied (when plotted on this
scale, it appears as a large band). The green line shows the root mean square.
By comparing the upper and lower panels, we see that the pink and green lines
are similar, showing that, as we would expect for both scenarios, the
uncleaned PSD is dominated by anthropogenic noise. We also observe that the
mean and standard deviation of the uncleaned PSD is at least two orders of
magnitude larger than the ASN-only example shown in Fig. 2. The result is that
the ULDM spike with the stated parameters cannot be observed with the
uncleaned PSD.

We next consider the purple curves in both panels of Fig. 6, which are the
Lomb-Scargle PSDs obtained after the application of our data cleaning
approach. The yellow lines show the root mean square. In the upper panel, we
observe a cleaned PSD that closely resembles the ASN-only case: the mean and
standard deviation are both
$10^{-4}\,$\text{\,}\mathrm{rad}\mathrm{/}\sqrt{\mathrm{Hz}}$$ and are
independent of frequency for the frequency range plotted. This indicates that
the data cleaning method has successfully identified and removed anthropogenic
noise. As a result, atom shot-noise limited sensitivity is achieved, albeit
with a reduction in the integration time owing to the presence of masks. The
masks have reduced the number of points in the raw time series and therefore
the effective integration time by $16.3\%$. Furthermore, the ULDM spike is
clearly identified in the cleaned PSD at 0.1 Hz, which indicates minimal
distortion to the ULDM signal. The amplitude of the ULDM spike in the upper
panel is
$2.07\text{\times}{10}^{-2}\text{\,}\mathrm{rad}\mathrm{/}\sqrt{\mathrm{Hz}}$,
compared to
$2.27\text{\times}{10}^{-2}\text{\,}\mathrm{rad}\mathrm{/}\sqrt{\mathrm{Hz}}$
in Fig. 2. Since $S_{k}\propto T_{\mathrm{int}}$ [21], the reduction in power
in the ULDM spike follows as a result of masking part of the raw time series.

In the lower panel, where we account for synanthropic noise alongside
anthropogenic noise, the cleaned PSD exhibits a close resemblance to the ASN-
only case for frequencies above approximately $0.04\,\mathrm{Hz}$. However, at
lower frequencies, the PSD rises due to the presence of synanthropic noise.
Additionally, the inclusion of synanthropic noise leads to more of the time
series being masked, resulting in a further reduction of the ULDM spike
amplitude to
$1.67\text{\times}{10}^{-2}\text{\,}\mathrm{rad}\mathrm{/}\sqrt{\mathrm{Hz}}$,
as 45.7% of the points are masked in this instance.

###  IV.4 Impact on ULDM sensitivity projections

Figure 7: $\mathrm{SNR}=1$ sensitivity projections on the ULDM-electron
coupling from simulated data assuming an eight-hour measurement window at
nighttime over 365 days. The green line shows the sensitivity projection from
a time series with contributions from anthropogenic and synanthropic noise,
along with phase noise from ASN, without any time-series-cleaning strategies
applied. The projections after implementing our cleaning strategy are shown in
purple and pink. The purple line includes contributions from anthropogenic
noise and phase noise from ASN, while the pink line also includes synanthropic
noise.

Figure 7 shows three sensitivity projections in terms of the ULDM-electron
coupling as a function of the ULDM frequency and mass. We use the signal-to-
noise ratio (SNR) as an estimator of the signal strength and show projections
with $\mathrm{SNR}=1$ [21]. For all three projections, we assume that data is
collected during an eight-hour window at nighttime for 365 days. The green
line shows the sensitivity that would be achieved from the raw time series
containing ASN with phase noise
$\delta\phi=10^{-4}~{}\mathrm{rad}/\sqrt{\mathrm{Hz}}$ together with
anthropogenic and synanthropic noise in the absence of any cleaning
strategies. In contrast, the purple and pink lines demonstrate the sensitivity
after masking the transients and detrending the time series. The purple line
assumes the presence of anthropogenic noise, while the pink line assumes that
synanthropic noise is also present.

There are a number of features to note in Fig. 7. Firstly, the impact of the
cleaning strategy on the sensitivity projections is dramatic. Without
cleaning, the sensitivity projection on $d_{m_{e}}$ is larger by approximately
two-to-three orders of magnitude. However, with a cleaning strategy in place,
the impact of anthropogenic and synanthropic noise can be substantially
mitigated.

Secondly, the presence of synanthropic noise leads to a slight reduction in
overall sensitivity compared to the anthropogenic-only case. As discussed in
section IV.3, the additional masks applied in the case of synanthropic noise
result in a slightly smaller ULDM spike amplitude. In the regime where the
integration time is smaller than the ULDM coherence time, approximately
$0.5\text{\,}\mathrm{Hz}$ for the chosen parameters, the sensitivity scales
with $(T_{\rm{int}})^{-1/2}$. Consequently, masking about $45\%$ of the time
series instead of $16\%$ reduces the sensitivity by a factor of 1.2, which is
consistent with the behaviour observed between approximately
$0.04\text{\,}\mathrm{Hz}$ and $0.5\text{\,}\mathrm{Hz}$. In contrast, above
$0.5\text{\,}\mathrm{H}\mathrm{z}$, the integration time exceeds the ULDM
coherence time, causing the sensitivity to scale with $(T_{\rm{int}})^{-1/4}$.
As a result, the sensitivity is only reduced by a factor of 1.1, which again
corresponds with the observed behaviour in Fig. 7.

Thirdly, we observe a reduction in the sensitivity projection below
approximately $0.04\,\mathrm{Hz}$ when synanthropic noise is included,
relative to the anthropogenic-only case. This is simply a manifestation of the
rise in the noise PSD below $0.04\,\mathrm{Hz}$ that we saw in the lower panel
of Fig. 6. The low-frequency rise in the noise PSD is also responsible for the
appearance of Gaussian-like bumps in the pink line at multiples of
$2f_{\mathrm{Ny}}=$0.5\text{\,}\mathrm{Hz}$$. Owing to aliasing, the noise PSD
below $f_{\mathrm{Ny}}$ is folded (or reflected) around $f_{\mathrm{Ny}}$.
Consequently, the rise at low frequencies reappears just below
$2f_{\mathrm{Ny}}$, and this pattern repeats at higher frequencies. A similar
behaviour is also observed in the uncleaned line. However, in the uncleaned
case, the rise at low frequencies is much more pronounced, leading to more
prominent bumps at multiples of $2f_{\mathrm{Ny}}$. In contrast, bumps at
$2f_{\mathrm{Ny}}$ are absent in the anthropogenic-only case (and also in
shot-noise limited projections) since the noise remains flat at low
frequencies. As a result, when folded, the noise PSD remains flat, implying
the absence of any bump features in addition to the expected loss in
sensitivity due to the trigonometric functions in eq. (7).

Figure 8:  Relative comparison of $\mathrm{SNR}=1$ sensitivity projections on
the ULDM-electron coupling relative to an ASN-limited experiment. All
scenarios assume simulated data from an eight-hour measurement window at
nighttime over 365 days. The green line shows the relative sensitivity for a
time series with contributions from anthropogenic and synanthropic noise,
along with phase noise from ASN, without any time-series-cleaning strategies
applied. The relative sensitivity after implementing our cleaning strategy is
shown in purple and pink for scenarios with anthropogenic-only and
anthropogenic and synanthropic noise, respectively.

In Fig. 8, we present a comparison of the $\mathrm{SNR}=1$ sensitivity
projections relative to an ASN-limited experiment that collects data during an
eight-hour window at nighttime for 365 days. The purple line, which represents
the anthropogenic only scenario after masking transients and detrending the
time series, shows that data cleaning has restored the sensitivity to within
10% or better of an ASN-limited experiment. The transition at approximately
$0.5\text{\,}\mathrm{Hz}$ occurs because of the different scaling of the
sensitivity with $T_{\rm{int}}$, depending on whether the integration time is
greater or smaller than the ULDM coherence time.

The inclusion of synanthropic noise, as shown by the pink line in Fig. 8,
leads to a further reduction in sensitivity since more of the time series is
masked. In this scenario, the sensitivity is within 40% of an ASN-limited
experiment over much of the parameter space. Exceptions occur at low frequency
and at multiples of $2f_{\mathrm{Ny}}$ due to the phenomenon of folding.

Finally, the sensitivity projection obtained using the uncleaned time series,
indicated by the green line in Fig. 8, is substantially weaker. This
highlights the importance of noise reduction methods to recover sensitivity to
ultra-light dark matter candidates.

##  V Summary and outlook

Forthcoming terrestrial atom interferometer experiments will offer a new probe
of dark matter interactions with normal matter. Among these experiments is
AION-10, which will be hosted in the Beecroft Building at the University of
Oxford’s Department of Physics. AION-10 will operate amongst a background of
anthropogenic (human sourced) and synanthropic (animal sourced) sources (cf.
Fig. 1). These sources could induce time-dependent phase shifts that have the
potential to hinder searches for the periodic signature produced by ultra-
light dark matter.

In this work, we have characterised potential anthropogenic and synanthropic
noise sources in the context of AION-10. Among these sources, we found that
massive bodies moving close to AION-10, such as people walking on stairs near
the AION-10 supporting tower, can lead to the most significant transients in
the phase shift time series (cf. Fig. 3). A passenger lift within the building
or vehicles passing on a nearby road would also produce significant transient
effects. While it may be straightforward to implement a detection system to
trace the movement of people or vehicles close to the building, directly
monitoring the movement of rodents or other small animals may prove to be more
challenging. To that end, we also characterised the impact of synanthropic
sources, which we referred to as random animal transients (RATs) to highlight
their potentially untraceable nature.

Searching for ultra-light dark matter with AION-10 will necessitate several
months of continuous data taking to enable detection of a dark matter signal
with a frequency-domain analysis. We therefore simulated extended runs of the
experiment for periods during the day and night (cf. Fig. 4). While nighttime
running significantly reduced the number of transient events relative to
daytime running, additional cleaning of the time series was still required to
optimise sensitivity to ultra-light dark matter.

For that reason, we presented a high-level discussion of a data cleaning
strategy to mitigate anthropogenic and synanthropic noise sources (cf. Fig.
5). Our strategy involved identifying and masking transient effects, followed
by detrending to eliminate any long-term drifts and to establish a common
baseline. While we anticipate that the specific implementation of our cleaning
strategy may require modification when applied to real data rather than
simulated data, the basic underlying principles should remain the same.

We then performed a frequency-domain analysis on the cleaned time series with
the Lomb-Scargle periodogram, which provides an estimator of the power
spectral density for non-uniformly sampled time series. The cleaned power
spectral densities closely resembled the atom shot noise-only case (cf. Fig. 2
and Fig. 6), with only a small increase at low frequency ($\lesssim
0.04\,\mathrm{Hz}$) in the scenario where both anthropogenic and synanthropic
noise was present. This indicated that the data cleaning method successfully
identified and removed anthropogenic- and synanthropic-induced transients.

Lastly, we assessed the impact of anthropogenic and synanthropic noise sources
on ULDM sensitivity projections (cf. Fig. 7). In the absence of any data
cleaning, the sensitivity deteriorates by two-to-three orders of magnitude
compared to an atom shot noise-limited experiment. However, through the
application of masking transients and detrending the time series, we
successfully restored the sensitivity to within 10% (or 40%) of an atom shot
noise-limited experiment in the scenario where anthropogenic (or anthropogenic
and synanthropic) noise is present (cf. Fig. 8).

While our findings indicate that masking segments of the time series results
in a relatively modest reduction in AION-10’s sensitivity to ultra-light dark
matter, this approach does come with certain drawbacks. Notably, AION-10 is
anticipated to operate concurrently with other atom interferometer
experiments, like MAGIS-100. The utilisation of time-series masks could
potentially complicate the process of networking these experiments
effectively.

To address these limitations, more sophisticated alternatives may be explored,
including the application of machine learning techniques. Such techniques
could potentially offer enhanced methods of cleaning the time series without
relying on masks. For instance, a repository of anthropogenic and synanthropic
noise templates could be created based on the simulations we have developed in
the paper. Through further feature engineering, these templates could be
further refined to train a neural network to accurately extract the noise
signal from the time series. This would preserve the underlying atom shot
noise and ultra-light dark matter components intact without losing integration
time. However, these additional investigations fall beyond the scope of our
current study and warrant future research endeavours.

## Acknowledgements

We are grateful to members of the AION and MAGIS-100 collaborations for many
fruitful discussions, and to Leonardo Badurina, Ankit Beniwal, John Ellis,
Jeremiah Mitchell and Tim Kovachy for comments on the manuscript. J.C.
acknowledges support from a King’s College London NMES Faculty Studentship.
C.M. is supported by the Science and Technology Facilities Council (STFC)
Grant No. ST/T00679X/1. For the purpose of open access, the authors have
applied a Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising from this submission. No experimental datasets were
generated by this research.

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