# The centaur programmer - How Kasparov’s Advanced Chess spans over to the
software development of the future

Pedro Alves  Lusófona UniversityCampo Grande, 376LisbonPortugal1700-097
<EMAIL_ADDRESS>and  Bruno Pereira Cipriano  Lusófona UniversityCampo
Grande, 376LisbonPortugal1700-097<EMAIL_ADDRESS>
###### Abstract.

We introduce the idea of Centaur Programmer, based on the premise that a
collaborative approach between humans and AI will be more effective than AI
alone, as demonstrated in centaur chess tournaments where mixed teams of
humans and AI beat sole computers.

The paper introduces several collaboration models for programming alongside an
AI, including the guidance model, the sketch model, and the inverted control
model, and suggests that universities should prepare future programmers for a
more efficient and productive programming environment augmented with AI.

We hope to contribute to the important discussion about the diverse ways
whereby humans and AI can work together in programming in the next decade, how
universities should handle these changes and some legal implications
surrounding this topic.

Position Paper, Artificial Intelligence, Programming

##  1\. Introduction

The victory of a computer (Deep Blue) against Gary Kasparov, the world chess
champion, in 1997 was a milestone in AI history and led many to predict the
substitution of humans by machines in a wide range of activities (Hassabis,
2017). The digital revolution had been dismissing repetitive, highly
standardized tasks with little need for intellectual skills, but now it seemed
that even highly complex tasks could be performed by machines, at low cost and
high efficiency.

One such task is programming itself. While many workers were already concerned
about maintaining their jobs, this was not on the minds of the millions of
programmers fueling the digital revolution. After all, someone has to create
the systems that support this revolution. Recently, LLM-based tools (e.g.,
GPT-3) have shown great effectiveness in programming tasks, being able to
solve exercises at the level of an average student in a Computer Engineering
course (Finnie-Ansley et al., 2022). New versions promise to improve this
performance (OpenAI, 2023). Is this the Deep Blue vs. Gary Kasparov moment of
programming? Will even programmers themselves be replaced by AI?

We cannot predict the future, but we can learn from the past. Returning to the
world of chess, after Kasparov’s defeat, there were several experiments with
competitions involving humans and AI, which were called ”Centaur Chess”111Also
known as ”Advanced Chess” in analogy to the mythological figure of the Centaur
that was half human, half horse. From these experiments, one, in particular,
brought interesting findings. In 2005, there was a chess tournament in which
all kinds of entities could compete: computers, chess champions, and mixed
teams composed of humans and computers. It was already expected that computers
would beat chess champions, but what was not expected was that mixed teams
would beat computers. In other words, the combination of human + AI seemed to
be more effective than AI alone. The idea of the Centaur as a model for future
work was starting to take shape (Case, 2018).

In this paper, we argue that something similar will happen in programming. A
team formed by programmers (humans) and AI will be more effective than AI
alone, hence the term Centaur Programmer.

We propose several models under which this collaboration may occur, show how
universities may adapt to this new reality, draw attention to legal and
ethical implications and finish with some conclusions about what the future
may hold for this topic.

##  2\. Collaboration Models

The collaboration between human and AI has already been successfully proven in
other areas. Maurice Conti demonstrated that, with the help of AI, he was able
to design very efficient equipment that would never have seen the light of day
if the teams were purely made up of humans or AI (Heaven, 2018). The process
was simple: humans defined objectives and constraints related to a certain
problem, and AI evolved possible solutions with constant validation from
humans.

This collaboration model, which we can call the ”guidance model,” is one of
the most interesting possibilities also for the programming area. The
programmer starts the process, indicating to the AI what the objectives and
assumptions are, and the AI responds with solutions. With the help of the
programmer, these solutions evolve until they reach the desired result. The
final solution may or may not be better than a solution purely imagined by the
programmer, but it will probably be achieved much more quickly. For now, we
are mainly talking about efficiency gains rather than new solutions. But AI
technologies are evolving rapidly, and, in the near future, we may actually
have better-designed programs with fewer bugs and more efficient if created in
a collaborative model.

Other collaboration models may be explored in software development. In the
”sketch model,” the programmer outlines the program’s structure and the AI
fills in the gaps. While tools like GitHub Copilot (based on the GPT
technology) currently seem to fit this model (Peng et al., 2023), we believe
they have a limited role, acting on a function-by-function basis rather than
holistically (i.e., ”I will create this class because it will be useful in N
other parts of the program”).

In the ”inverted control model,” the communication direction is reversed: the
AI asks the programmer what they intend to do until it objectively understands
the goal and constraints before implementation.

All of these models require a rethinking of the software development cycle.
Programmers are already accustomed to ”boosting” their efficiency through a
”query model” where they simply ask questions to the computer (e.g., Stack
Overflow). The initial reaction to tools like ChatGPT has been to transpose
this model - when information is lacking, the programmer asks ChatGPT in a
similar manner to the query they would make on Google or Stack Overflow. But
is this the most efficient model? In this model, the programmer still does
most of the typing, wasting the potential gains that a true programmer-centaur
can have.

##  3\. The role of universities

If these new models are to profoundly transform the software development
industry, that transformation must begin in universities, where future
programmers are being educated. Universities should start exploring and
integrating these new collaboration models into their software development
curricula to prepare the future workforce for a more efficient and productive
programming environment. It is not enough to prepare future programmers, they
must prepare future centaur programmers.

We believe that this transformation will be gradual and experimental. In the
first phase, we will have the model of the ”virtual tutor”, where the student
uses AI to help them learn. This tutor can explain to the student what a
certain piece of code does or what a particular error means. It can help them
verify if a certain solution is correct or propose improvements to the code
developed by the student. It can even convert a program to different
programming languages. In any case, we are talking about a learning-centered
process - the student wants to know more and AI helps them in this process,
complementing the role of the teacher. After all, the idea of having a
personal tutor dedicated to each student would have seemed utopian until
recently due to the costs it would entail. Not anymore with AI.

But the ”virtual tutor” model, while important, has nothing to do with the
idea of the centaur-programmer, as the objectives are different. The centaur-
programmers aim to augment their ability through collaboration with AI. This
can translate into greater efficiency (producing solutions more quickly than
they would without AI) and/or greater quality (producing solutions of better
quality than they would alone). Returning to the initial concept of ”centaur
chess”, the goal was to win the game, not so much to learn to play chess.

Therefore, a second phase will emerge in which students actually learn to
augment their abilities with the help of AI. The analogy of the scientific
calculator has been widely used and seems appropriate. The calculator alone is
of little use because it does not know which calculations to perform. And the
human alone is inefficient because it takes too long to perform certain
calculations. A human equipped with a calculator is, in fact, a centaur that
uses a collaboration model accepted in the academic world.

Is this analogy valid? A calculator simply reproduces the same mechanical
process as a student performing mathematical operations (albeit much more
quickly). However, tools like ChatGPT can create new content from a set of
interactions. Students can abuse these tools to solve complete programming
exercises without actually knowing how to program. This has even led several
universities to simply ban/prohibit the use of these tools (Jimenez, 2023).
But if the industry is massively adopting the use of these tools due to the
potential increase in productivity that results from it, shouldn’t
universities follow the same path?

Returning to the analogy, there is a key point that remains valid. Both the
calculator and ChatGPT (or other forms of AI) are dependent on a human to
produce something useful. In fact, the level of usefulness that can be
extracted from these tools depends directly on the ability of the human using
them. An advanced mathematics student can certainly use a scientific
calculator more effectively and usefully than a primary school student - here
we use the term ”useful” to refer to the ability to solve a real problem
rather than just solve academic exercises. The same goes for experienced
programmers who will use their analytical and creative ability to create
programs that solve real problems. The pieces that make up these programs may
be developed by AI, but knowing which pieces to build and how to connect them
will continue to be the exclusive responsibility of the human. If it was
already clear that universities should invest more in developing these
capabilities, it now becomes even more evident.

##  4\. Legal and ethical implications

There has been much discussion about the ethical and legal implications of
using AI as a tool, particularly within the subset of AI based on LLMs, as it
is trained on large quantities of information generated by humans. On the one
hand, the unauthorized use of content raises legal questions. On the other
hand, it introduces biases that replicate patterns of behavior in society and
may lead to racism, xenophobia, or other types of discrimination.

Here, too, the figure of the centaur programmer is important as it assumes
prior (human) validation of the content produced by AI. This validation
derives from an assumption that seems essential to us: the content produced by
a collaboration between humans and AI should ultimately be the sole
responsibility of humans. For example, humans may prefer to use LLMs trained
on a dataset previously curated by them to ensure that: (1) there are no legal
problems with the unauthorized use of content; (2) there is no incorrect or
biased information feeding the training engine.

In this context, Universities also play a critical role in developing and
openly discussing the ethical guidelines for the use of AI technologies. It
makes sense for universities to host and train their own LLMs, based on data
produced by the university itself or previously curated by it, making sure it
adheres to the previously defined guidelines.

##  5\. Conclusion

Centaur-programmers are coming. They will be humans able to augment their
programming abilities by collaborating with AI. They will accomplish the
software requirements much faster and with better quality than their sole
human counterparts. They will be trained by their university to explore
different collaboration models with AI, such as the guidance and the sketch
model. They will also have a strong critical spirit and highly ethical values
that will act as a safeguard to the content produced by AI. Because, in the
end, the code may have been created by AI but its the human programmer who
will take responsibility for it.

## References

  * (1)
  * Case (2018) Nicky Case. 2018\.  How To Become A Centaur.  _Journal of Design and Science_ (Jan. 2018).  https://doi.org/10.21428/61b2215c
  * Finnie-Ansley et al. (2022) James Finnie-Ansley, Paul Denny, Brett A. Becker, Andrew Luxton-Reilly, and James Prather. 2022\.  The Robots Are Coming: Exploring the Implications of OpenAI Codex on Introductory Programming. In _Australasian Computing Education Conference_. ACM, Virtual Event Australia, 10–19.  https://doi.org/10.1145/3511861.3511863
  * Hassabis (2017) Demis Hassabis. 2017\.  Artificial Intelligence: Chess match of the century.  _Nature_ 544, 7651 (April 2017), 413–414.  https://doi.org/10.1038/544413a Number: 7651 Publisher: Nature Publishing Group.
  * Heaven (2018) Douglas Heaven. 2018\.  The designer changing the way aircraft are built.  https://www.bbc.com/future/article/20181129-the-ai-transforming-the-way-aircraft-are-built
  * Jimenez (2023) Kayla Jimenez. 2023\.  ’This shouldn’t be a surprise’ The education community shares mixed reactions to ChatGPT.  https://www.usatoday.com/story/news/education/2023/01/30/chatgpt-going-banned-teachers-sound-alarm-new-ai-tech/11069593002/
  * OpenAI (2023) OpenAI. 2023.  GPT-4 Technical Report.  https://doi.org/10.48550/arXiv.2303.08774 arXiv:2303.08774 [cs].
  * Peng et al. (2023) Sida Peng, Eirini Kalliamvakou, Peter Cihon, and Mert Demirer. 2023.  The Impact of AI on Developer Productivity: Evidence from GitHub Copilot.  http://arxiv.org/abs/2302.06590 arXiv:2302.06590 [cs].


# Refined Ultra-Light Scalar Dark Matter Searches with Compact Atom
Gradiometers

Leonardo Badurina<EMAIL_ADDRESS>Theoretical Particle Physics
and Cosmology Group, Department of Physics, King’s College London, Strand,
London, WC2R 2LS, UK  Diego Blas  Grup de Física Teòrica, Departament de
Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain,  Institut
de Fisica d’Altes Energies (IFAE), The Barcelona Institute of Science and
Technology,
Campus UAB, 08193 Bellaterra (Barcelona), Spain  Theoretical Particle Physics
and Cosmology Group, Department of Physics, King’s College London, Strand,
London, WC2R 2LS, UK  Christopher McCabe  Theoretical Particle Physics and
Cosmology Group, Department of Physics, King’s College London, Strand, London,
WC2R 2LS, UK

###### Abstract

Atom interferometry is a powerful experimental technique that can be employed
to search for the oscillation of atomic transition energies induced by ultra-
light scalar dark matter (ULDM). Previous studies have focused on the
sensitivity to ULDM of km-length atom gradiometers, where atom interferometers
are located at the ends of very-long baselines. In this work, we generalise
the treatment of the time-dependent signal induced by a linearly-coupled
scalar ULDM candidate for vertical atom gradiometers of any length and find
correction factors that especially impact the ULDM signal in short-baseline
gradiometer configurations. Using these results, we refine the sensitivity
estimates for AION-10, a compact $10\text{\,}\mathrm{m}$ gradiometer that will
be operated in Oxford, and discuss optimal experimental parameters that
enhance the sensitivity to linearly-coupled scalar ULDM. After comparing the
sensitivity reach of devices operating in broadband and resonant modes, we
show that well-designed compact atom gradiometers are able to explore regions
of dark matter parameter space that are not yet constrained.

††preprint:

##  I Introduction

Despite overwhelming astrophysical and cosmological evidence for its existence
through gravitationally induced effects, dark matter (DM) has yet to be
understood at a fundamental level [1]. Indeed, the fundamental constituents of
DM and their interactions are completely unknown, and the DM mass could range
from particles as light as 10-22 eV to asteroid-mass primordial black holes
[2, 3]. To unravel the constituents of DM, direct searches for DM particles
via non-gravitational interactions have been among the highest priorities in
particle physics, astrophysics and cosmology [4].

Because of null results in direct searches for weak mass-scale particles in
colliders and direct detection searches (see, e.g., the constraints from the
xenon-based experiments [5, 6]), increasing experimental and theoretical
interest has shifted to ultra-light bosons with a sub-eV mass. Owing to their
large occupation number in the Milky Way, such ultra-light dark matter (ULDM)
candidates are expected to behave as coherent oscillating waves, and thus
would lead to new phenomenological signatures [7]. Prominent among these
ultra-light bosons are many well-motivated DM candidates [8, 9], such as
dilatons, moduli and the relaxion [10], as well as the pseudoscalar QCD axion
and axion-like-particles (ALPs) [11], and vector bosons [12, 13].

Multiple phenomenological ‘portals’ have been proposed to test the wide range
of well-motivated theoretical models over as wide a range of masses as
possible. In this paper we focus on the interactions between ultra-light
scalar DM with electrons or photons, which can lead to oscillations in
fundamental constants (see e.g. [14, 15, 16]). These DM-induced oscillations
can in turn alter atomic transition energies, which can be searched for with
atom interferometer experiments [17, 18].

An atom interferometer can be succinctly described as an experiment that
compares the phase between coherent spatially delocalised quantum
superpositions of atom clouds [19]. Two spatially separated atom
interferometers that are referenced by the same laser pulse via single-photon
atomic transitions can be operated as a gradiometer [20]. The distinctive
signal of a coherent oscillating DM field can be tested with a gradiometer by
finding the difference between the phases measured by two atom interferometers
[18]. A key advantage of the gradiometer experimental set-up 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 interferometers’ laser noise cancels in a differential
measurement [20].

A number of single-photon atom gradiometers have been proposed or are
currently under construction, including AION-10 [21], MAGIS-100 [22, 23], MIGA
[24], ELGAR [25], ZAIGA [26], and AEDGE [27]. Many of these experiments are
envisaged as a test-bed for technologies that could later be employed in much
larger km-scale interferometers, which is considered to be the scale required
in order to operate as a mid-band gravitational wave detector [28, 29, 30,
31]. For example, within the AION project, AION-10 is a
$10\text{\,}\mathrm{m}$ instrument that will be operated in Oxford and there
is a roadmap in place to later develop $100\text{\,}\mathrm{m}$ and km
versions [21].

While the DM-induced phase difference has been calculated for km-scale
gradiometers, a careful treatment has not been provided for more compact
designs, such as AION-10. Remarkably this compact set-up could be sensitive to
a currently unconstrained region of DM models, making this analysis even more
relevant. The purpose of this paper is therefore to refine the previous
calculations in [21], and to provide updated projections of the AION-10
sensitivity to linear ultra-light scalar DM–electron and photon interactions.

This paper is structured as follows. In section II we define the ULDM model
and the parameters that can be constrained experimentally. In section III, we
present the refined DM-phase calculation for short-baseline gradiometers. In
section IV, we discuss how the sensitivity of a compact gradiometer depends on
tuneable experimental parameters. Then in section V, we utilise the short-
baseline results to provide updated sensitivity estimates for the AION-10
experiment. Finally, we present our conclusions in section VI. Two appendices
provide further details and derivations of calculations to support the results
in section III.

##  II  Ultra-light Scalar Dark Matter

Ultra-light dark matter (ULDM) can be modeled as a temporally and spatially
oscillating non-relativistic classical field, which follows from its high
occupation number and its small mean velocity and velocity dispersion
(characteristic of DM in the Milky Way) [9].

The velocity $\mathbf{v}$ of the virialized DM has a typical magnitude of
$10^{-3}$ in natural units and follows some astrophysical velocity
distribution $f(\mathbf{v})$. In this work, the precise form of the velocity
distribution does not need to be specified; we only use the fact that the
spread of DM velocities $\sigma_{v}\approx
160~{}\mathrm{km}\,\mathrm{s}^{-1}\approx 5\times 10^{-4}$ (in natural units)
[32] and that the field is non-relativistic. Temporally, the field will be
coherent over a coherence time $\tau_{c}=2\pi/m_{\phi}\sigma_{v}^{2}$, while
spatially it will also be coherent over a coherence length
$\lambda_{c}=2\pi/m_{\phi}\sigma_{v}$ [33].

Within a coherence length and time of the field, the light scalar $\phi$ is
modelled by the classical non-relativistic solution of the equations of motion
for a Lagrangian describing a free massive scalar particle

$\phi(t,\mathbf{x})\simeq\frac{\sqrt{2\rho_{\mathrm{DM}}}}{m_{\phi}}\cos{\left(\omega_{\phi}t-\textbf{k}_{\phi}\cdot\textbf{x}+\theta\right)}\,,$
(1)

where
$\omega_{\phi}=m_{\phi}(1+|\textbf{v}|^{2}/2)+\mathcal{O}(m_{\phi}^{2}|\textbf{v}|^{3})$,
$\textbf{k}_{\phi}=m_{\phi}\textbf{v}$, and $\theta$ is an unknown phase that
is assumed to be random.111We use natural units with $\hbar=c=1$. We see that
the field is characterized by its temporal angular frequency, which is set by
the dark matter mass $m_{\phi}$ up to small kinetic energy corrections. We
have assumed that the field saturates the entire local DM density
$\rho_{\mathrm{DM}}$ so that the amplitude is equal to
$\sqrt{2\rho_{\mathrm{DM}}}/m_{\phi}$. In our numerical results, we use the
canonical value
$\rho_{\mathrm{DM}}=$0.3\text{\,}\mathrm{GeV}\,\mathrm{cm}^{-3}$$ [34].

In this paper, we will only consider the leading linear-couplings of $\phi$
with the Standard Model (SM) fields. We will consider the effective Lagrangian
at an energy scale of $\sim 1$ GeV describing an ultra-light scalar DM
particle coupled linearly to SM fields [35]. Explicitly, the Lagrangian that
we consider is

$\mathcal{L}\supset\mathcal{L}_{\mathrm{SM}}+\mathcal{L}_{\phi}\,,$ (2)

in which the pertinent SM Lagrangian terms for electrons and photons are

$\mathcal{L}_{\mathrm{SM}}\supset-\frac{1}{4e^{2}}F_{\mu\nu}F^{\mu\nu}-m_{e}\overline{\psi}_{e}\psi_{e}\,,$
(3)

where $e$ is the electric charge and $m_{e}$ is the electron mass,
$F_{\mu\nu}$ is the Maxwell field-strength tensor, and the linear interactions
between the ultra-light scalar DM and the photon and electron fields are

$\displaystyle\mathcal{L}_{\phi}$
$\displaystyle\supset\phi(t,\mathbf{x})\sqrt{4\pi
G_{N}}\left[\frac{d_{e}}{4e^{2}}F_{\mu\nu}F^{\mu\nu}-d_{m_{e}}m_{e}\overline{\psi}_{e}\psi_{e}\right].$
(4)

Following the convention in Refs. [36, 37], we have parameterised the $d_{e}$
and $d_{m_{e}}$ couplings relative to Newton’s gravitational constant $G_{N}$.

The addition of these interactions to the SM Lagrangian results in a time-
varying electron mass and fine structure constant [37, 15],

$\displaystyle m_{e}(t,\mathbf{x})$
$\displaystyle=m_{e}\left[1+d_{m_{e}}\sqrt{4\pi
G_{N}}\phi(t,\mathbf{x})\right]$ (5) $\displaystyle\alpha(t,\mathbf{x})$
$\displaystyle\approx\alpha\left[1+d_{e}\sqrt{4\pi
G_{N}}\phi(t,\mathbf{x})\right]\,.$ (6)

It is this time-dependence that will give rise to the observable scalar DM
signal in an atom gradiometer.

##  III ULDM Signal in Atom Gradiometers

The properties of atoms largely depend on the electron mass $m_{e}$ and fine
structure constant $\alpha$. For example, for optical electronic transitions,
the transition frequency $\omega_{A}$ scales as $\omega_{A}\propto
m_{e}\alpha^{2+\xi_{A}}$ to a high degree [14], where $\xi_{A}$ is a
calculable parameter governing the electronic transition. As a result of the
DM-induced oscillations in $m_{e}$ and $\alpha$, $\omega_{A}$ will oscillate
as

$\omega_{A}(t,\mathbf{x})\simeq\omega_{A}+\Delta\omega_{A}(t,\mathbf{x})\,,$
(7)

where

$\Delta\omega_{A}(t,\mathbf{x})=\omega_{A}\sqrt{4\pi
G_{N}}\left[d_{m_{e}}+(2+\xi_{A})d_{e}\right]\phi(t,\mathbf{x})\,.$ (8)

For our later calculations, we will focus on the AION-10 setup, which will
operate on the $\mathrm{5s^{2}\,{}^{1}S_{0}\leftrightarrow 5s5p\,^{3}P_{0}}$
clock-transition in ${}^{87}\mathrm{Sr}$. For this transition, we have
$\omega_{A}=2.697\times 10^{15}~{}\mathrm{rad}/\mathrm{s}$ (corresponding to a
wavelength $\lambda_{A}\approx 698$ nm) [38] and $\xi_{A}\approx 0.06$ [39].

The total phase difference $\Phi$ between two paths of an atom interferometer
can be written as the sum of three contributions (see e.g., [40]),

$\Phi=\Phi_{\mathrm{propagation}}+\Phi_{\mathrm{laser}}+\Phi_{\mathrm{separation}}\,.$
(9)

The propagation phase, $\Phi_{\mathrm{propagation}}$, arises from the free-
fall evolution of the atom’s external (e.g. linear momentum) and internal
(e.g. energy levels) degrees of freedom between light pulses; the laser phase,
$\Phi_{\mathrm{laser}}$, originates from atom-laser interactions used to
manipulate the atom’s wavefunction at each beamsplitter and mirror sequence;
and the separation phase, $\Phi_{\mathrm{separation}}$, accounts for the wave
packet separation of the two ‘arms’ of the interferometer at the final
beamsplitter pulse.

ULDM-induced oscillations of fundamental constants would be observable in the
evolution of the internal degrees of freedom of atoms and so contribute to the
propagation phase. Since the electronic transition energy would receive a
spacetime-dependent correction $\Delta\omega_{A}(t,\mathbf{x})$, an ULDM-
induced signal will be accumulated in the propagation phase of the excited
state ($\mathrm{5s5p\,^{3}P_{0}}$) relative to that of the ground state
($\mathrm{5s^{2}\,{}^{1}S_{0}}$) (see Appendix A for further details).
Explicitly, for each path segment in which the atom is in the excited state,
the ULDM-induced phase contribution is [18]

$\Phi_{s_{1}}^{s_{2}}\equiv\int_{s_{1}}^{s_{2}}{\Delta\omega_{A}(t,\mathbf{x})\,ds}\,,$
(10)

where the integral is taken along the geodesic of the atom between spacetime
points $s_{1}$ and $s_{2}$. These points are found by intersecting the atom’s
and laser’s geodesics.

The spacetime on Earth can be effectively modelled using the Schwarzschild
metric

$ds^{2}=\left(1+2\Phi_{\mathrm{E}}(r)\right)dt^{2}-\frac{1}{1+2\Phi_{\mathrm{E}}(r)}dr^{2}-r^{2}d\Omega^{2}\,,$
(11)

where $\Phi_{\mathrm{E}}(r)$ is the Earth’s gravitational field at radial
position $r$. Using the metric Eq. (11), a light pulse that leaves one of the
laser sources at position $r_{0}$ and time $t_{0}$ will arrive at the atom’s
at position $r$ at coordinate time $t(r,r_{0},t_{0})$. By solving the geodesic
equations to third order in the affine parameter $\lambda$ of the laser’s null
curve, inverting the laser’s radial geodesic $r(\lambda)$ to determine
$\lambda(r)$ to $\mathcal{O}(r^{3})$, and substituting $\lambda(r)$ self-
consistently into $t(\lambda)$, we obtain

$t(r,r_{0},t_{0})=t_{0}+\frac{r-r_{0}}{1+2\Phi_{\mathrm{E}}(r_{0})}+\frac{(r-r_{0})^{2}\partial_{r}\Phi_{\mathrm{E}}(r_{0})}{(1+2\Phi_{\mathrm{E}}(r_{0}))^{2}}\,.$
(12)

On the Earth’s surface $\Phi_{\mathrm{E}}(r_{\mathrm{Earth}})\sim 10^{-8}$,
and for all AION proposals, from AION-10 to AION-km, $r-r_{0}\lesssim 10^{-5}$
s (in natural units). Thus, in subsequent calculations we will make use of
the approximation $t(r,r_{0},t_{0})\approx t_{0}+r-r_{0}$.

Furthermore, we will consider interferometric sequences with excited state
paths of spatial length $|\Delta\textbf{r}_{e}|$ much smaller than $\Delta
t_{e}$, which is the duration of these same paths; in this case, we may
approximate the atom as being stationary when in the excited state, such that
the line element $ds$ can be expressed as $\sqrt{g_{00}(r)}dt$. For the metric
in Eq. (11), $g_{00}(r)=1+2\Phi_{\mathrm{E}}(r)\approx 1$, such that we may
further use the approximation $ds\approx dt$. Given that $|\textbf{k}_{\phi}|$
is suppressed by a factor of 10-3 relative to the angular frequency $\omega$,
we may neglect the spatial dependence of $\phi$, and thus of
$\Delta\omega_{A}$, and rewrite Eq. (10) as

$\displaystyle\Phi_{t_{1}}^{t_{2}}$
$\displaystyle\equiv\int_{t_{1}}^{t_{2}}{\Delta\omega_{A}(t)dt}$ (13)
$\displaystyle=\overline{\Delta\omega_{A}}\int_{t_{1}}^{t_{2}}\cos(\omega_{\phi}t+\theta)dt\;,$
(14)

where we have defined

$\overline{\Delta\omega_{A}}=\omega_{A}\sqrt{4\pi
G_{N}}\left[d_{m_{e}}+(2+\xi_{A})d_{e}\right]\frac{\sqrt{2\rho_{\mathrm{DM}}}}{m_{\phi}}\,,$
(15)

as the magnitude of $\Delta\omega_{A}(t,\mathbf{x})$.

Figure 1: Schematic spacetime diagram of the atom gradiometer concept with
$n=2$ large momentum transfer (LMT) kicks. The experiment consists of a pair
of cold-atom interferometers performing single-photon transitions between a
ground state (blue) and an excited state (red). Height, or radial position, is
shown on the vertical axis, and the time axis is horizontal. The laser pulses
(wavy lines) traveling across the baseline from opposite sides are used to
divide, redirect, and recombine the atomic states, yielding interference
patterns that are sensitive to the modulation of the atomic transition
frequency caused by coupling to DM. Beamsplitter ($\pi/2$) pulses, which begin
and end the sequence, are shown in black, while $\pi$-pulses are shown in
light grey. Atom-light interactions with $\pi/2$-pulses are shown as black
squares, while atom-light interactions with $\pi$-pulses are shown with grey
circles. The gradiometer separation $\Delta r$ is also shown.

Using Eq. (13) and summing over all paths in which the atom is in the excited
state, we can find the total ULDM-induced phase difference for a single-photon
atom interferometer labelled by $p$. Following a sequence as depicted in Fig.
1 (which is for $n=2$), the phase of the $p^{\rm{th}}$ interferometer is

$\begin{split}\Phi_{p}=\sum_{m=1}^{n/2}&\left[\Phi_{T-(n-2m+1)L+r_{p,l}}^{T-(n-2m-1)L-r_{p,l}}+\Phi_{2T-(n-2m)L-r_{p,f}}^{2T-(n-2m)L+r_{p,f}}\right.\\\
&\left.-\Phi_{T+(2m-1)L-r_{p,u}}^{T+(2m-1)L+r_{p,u}}-\Phi_{(2m-2)L+r_{p,i}}^{2mL-
r_{p,i}}\right]\;,\end{split}$ (16)

where $n$ is the number of large momentum transfer kicks (LMT) [41], $L$ is
the baseline separation between the laser and the mirror, and $T$ is the
interrogation time, which is approximately half the duration of a single
interferometric sequence that lasts for a total time $2T+L$. With our
convention, there are $4n-1$ pulses in total, of which two are $\pi/2$-pulses
and the remaining $4n-3$ are $\pi$-pulses. These are distributed such that at
the beginning and the end of the sequence, there are $(n-1)$ $\pi$-pulses,
while around time $T$ in the middle of the sequence there are $(2n-1)$
$\pi$-pulses (the ‘mirror sequence’) with $n-1$ $\pi$-pulses before time $T$
and $n$ $\pi$-pulses after. With our labelling convention, the $\pi/2$-pulses
are emitted at times $0$ and $2T$. The position vectors label the path taken
by the excited state of the $p$th interferometer: $r_{p,i}$, $r_{p,l}$,
$r_{p,u}$ and $r_{p,f}$ refer to the initial, lower, upper and final path of
the $p^{\mathrm{th}}$ interferometer as viewed in the laboratory frame,
respectively. Finally, the sum is over positive integers, as the number of LMT
kicks, $n$, is even to ensure that the atom is in the ground state along the
longest paths.

The ULDM signal for the atom gradiometer corresponds to the difference between
the total phase differences of two atom interferometers. Using Eq. (16), the
atom gradiometer signal is defined as

$\Phi_{\rm{s}}=\Phi_{1}-\Phi_{2}\;.$ (17)

Since $\omega_{\phi}L\ll 1$ and in the limit that the launch velocity is the
same in both interferometers, we show in Appendix B that Eq. (17) can be
reduced to the relatively compact expression:

$\begin{split}\Phi_{\mathrm{s}}=\frac{\Delta
r}{L}\Big{\\{}\Big{[}&\Phi^{T+L}_{T-(n-1)L}-\Phi^{nL}_{0}\Big{]}\\\
\qquad&-\Big{[}\Phi^{2T+L}_{2T-(n-1)L}-\Phi^{T+nL}_{T}\Big{]}\Big{\\}}\;,\end{split}$
(18)

where we defined

$\Delta r=r_{2,i}-r_{1,i}=r_{2,u}-r_{1,u}=r_{2,l}-r_{1,l}=r_{2,f}-r_{1,f}$
(19)

as the gradiometer separation. Having the same launch velocity in both
interferometers ensures that all of the equality signs hold in Eq. (19). We
can also obtain a compact expression for the signal amplitude (again, see
Appendix B for details):

$\begin{split}\overline{\Phi}_{\rm{s}}=8\frac{\overline{\Delta\omega_{A}}}{m_{\phi}}\frac{\Delta
r}{L}&\Bigg{|}\sin\left[\frac{m_{\phi}nL}{2}\right]\sin\left[\frac{m_{\phi}T}{2}\right]\\\
&\quad\times\sin\left[\frac{m_{\phi}(T-(n-1)L)}{2}\right]\Bigg{|}\;.\end{split}$
(20)

In this expression, we have neglected the sub-leading kinetic corrections to
the DM angular frequency, so that $\omega_{\phi}=m_{\phi}$.

Setting $\Delta r=L$ in Eq. (18) and Eq. (20) we recover the result presented
in Ref. [18], in which the authors considered experimental configurations in
which two interferometers are separated by a distance that is comparable to
the length of the baseline (i.e. $\Delta r\approx L$). While this
approximation is valid for long-baseline setups, the $\Delta r/L$ factor
presented in this work becomes an important correction to the ULDM signal in
compact atom gradiometers. For example, in the long-baseline setup considered
in Ref. [18], $\Delta
r/L=$980\text{\,}\mathrm{m}$/$1000\text{\,}\mathrm{m}$=0.98$, whereas for a
compact atom gradiometer such as AION-10 $\Delta
r/L=$5\text{\,}\mathrm{m}$/$10\text{\,}\mathrm{m}$=0.5$.

For a compact gradiometer where $nL\ll T$ and $m_{\phi}nL\ll 1$, it is a good
approximation to simplify Eq. (20) further to $\overline{\Phi}_{\rm{s}}\approx
4\,\overline{\Delta\omega_{A}}\,n\Delta r\,\sin^{2}\left[m_{\phi}T/2\right]$.
This approximate form has the advantage that it is relatively straightforward
to determine the ULDM DM mass that maximises $\overline{\Phi}_{\rm{s}}$.
Although it naively seems that this should occur at $m_{\phi}=\pi/T$, this is
not correct as there is an additional mass term implicit in
$\overline{\Delta\omega_{A}}$ (recall from Eq. (15) that
$\overline{\Delta\omega_{A}}\propto m_{\phi}^{-1})$. When taking this
additional mass-term into account, we find that $\overline{\Phi}_{\rm{s}}$ is
maximised for $m_{\phi}\approx 2.33/T$.

##  IV  Sensitivity Dependence on Tuneable Experimental Parameters

In the previous section, we obtained in Eq. (20) a compact expression for the
signal amplitude from a single interferometric sequence that lasts for a time
$2T+L$. In a realistic experiment, this sequence will be repeated many times
giving rise to $N$ measurements over the course of a measurement campaign set
by the integration time $T_{\mathrm{int}}$. Furthermore, the measured signal
will consist of a contribution from background noise sources in addition to
any ULDM-induced phase. In this section, we provide a general discussion of
how the sensitivity to ULDM parameters scales with tuneable experimental
parameters in a compact atom gradiometer. We then apply the results to the
AION-10 setup.

To extract information from a discrete series of measurements collected at
different points in time, the estimator of the power spectral density (PSD) is
commonly used (see e.g. [42]). The PSD is useful because it contains
information on the frequency spread and amplitude of the analysed data. We
define $\Phi_{m}$ as the phase measured from a gradiometer sequence that
begins at time $m\Delta t$, where $\Delta t$ is the temporal separation
between successive measurements (i.e. $1/\Delta t$ is the sampling rate), and
we assume that measurements are made continuously for the total integration
time, i.e., $T_{\mathrm{int}}=N\Delta t$. By first taking the discrete Fourier
transform $\tilde{\Phi}_{k}$ of the measured signal, defined as

$\tilde{\Phi}_{k}=\sum_{m=0}^{N-1}\Phi_{m}\exp{\left(-\frac{2\pi
imk}{N}\right)}\;,$ (21)

where $k$ is an integer taking values from $0$ to $N-1$, we can define the PSD
as

$S_{k}=\frac{(\Delta
t)^{2}}{T_{\rm{int}}}\left|\tilde{\Phi}_{k}\right|^{2}\;.$ (22)

We will use the signal-to-noise ratio (SNR) as an estimator of the signal
strength. We define the angular frequency $\omega=2\pi k/T_{\rm{int}}$, such
that the SNR at the frequency $\omega_{0}$ takes the form

$\mathrm{SNR}=\frac{S_{\mathrm{s}}(\omega_{0})}{\sigma_{S_{\mathrm{n}}}(\omega_{0})}\;,$
(23)

where $S_{\mathrm{s}}(\omega)$ is the PSD of the ULDM signal and
$\sigma_{S_{\mathrm{n}}}(\omega)$ is the standard deviation of the noise PSD.

Although several background components will contribute to the noise PSD
$S_{\mathrm{n}}(\omega)$ (see for example the detector systematics discussion
in Ref. [22]), the design goal of an experiment’s detection system is that the
dominant phase noise is from atom shot noise. For a single interferometer, the
noise variance is $\sigma^{2}=(C^{2}N_{a})^{-1}$ for phase differences close
to $\pi/2$, where $N_{a}$ is the number of atoms in the cloud and $C\leq 1$ is
the interferometer contrast [43, 44]. The interferometer contrast $C$ is
important because it characterises the amplitude of the oscillation of the
number of atoms in the ground/excited state, from which the interferometer
phase is inferred (see e.g. [45]). By the Wiener-Khinchin theorem, this
implies that the noise PSD for a single interferometer is $\Delta
t/C^{2}N_{a}$, while for an atom gradiometer employing identical
interferometers, $S_{\mathrm{n}}=2\Delta t/C^{2}N_{a}$. Because atom shot
noise is white noise, the standard deviation of the gradiometer noise PSD is

$\sigma_{S_{\mathrm{n}}}=S_{\mathrm{n}}=\frac{2\Delta t}{C^{2}N_{a}}\;,$ (24)

which is frequency-independent by definition. Hence, at $\mathrm{SNR}=1$ the
peak of the ULDM-induced PSD is within one standard deviation of the mean of
the noise PSD.

Next, we consider the PSD for the ULDM signal, $S_{\mathrm{s}}(\omega)$. From
the PSD defining equation, Eq. (22), we can see that the maximum of
$S_{\mathrm{s}}(\omega)$ will be proportional to
$\overline{\Phi}_{\rm{s}}^{2}$. The frequency spread of
$S_{\mathrm{s}}(\omega)$ is related to the reciprocal of the coherence time,
$2\pi/\tau_{c}\sim 10^{-7}\,(m_{\phi}/10^{-15}~{}\mathrm{eV})~{}\mathrm{Hz}$,
while the experiment’s frequency resolution is given by
$2\pi/T_{\mathrm{int}}$.

In the regime $T_{\mathrm{int}}<\tau_{c}$, the ULDM signal is dominated by a
single frequency, which implies that the PSD is approximately
$S_{\mathrm{s}}(\omega_{0})\approx
T_{\mathrm{int}}|\overline{\Phi}_{\rm{s}}|^{2}$ at a frequency $\omega_{0}$
largely set by $m_{\phi}$, and zero elsewhere. Hence, the experiment’s
sensitivity to the coupling constants increases with integration time and is
given by

$d_{\phi}\simeq\frac{\sqrt{\mathrm{SNR}}}{\overline{\Phi}_{\rm{R}}}\times\sqrt{\frac{S_{n}}{T_{\mathrm{int}}}}\;,$
(25)

where in order to isolate the dependence on the couplings $d_{m_{e}}$ and
$d_{e}$, we have defined

$\displaystyle\overline{\Phi}_{\rm{s}}$
$\displaystyle=d_{\phi}\overline{\Phi}_{\rm{R}}$ (26) $\displaystyle d_{\phi}$
$\displaystyle=d_{m_{e}}+(2+\xi_{A})d_{e}\;,$ (27)

where $\overline{\Phi}_{\rm{R}}$ is the remaining part of the signal amplitude
with the couplings factored out.

In the limit $T_{\mathrm{int}}>\tau_{c}$, it might naively be expected that
the experiment’s sensitivity to $d_{\phi}$ follows a similar argument.
However, this is not the case since the PSD of the entire signal will no
longer correspond to a spike in frequency space, but will have a finite width
and profile dictated by the DM velocity distribution. In this case, a
likelihood profile analysis could be used to extract the experiment’s
sensitivity to $d_{\phi}$ (see e.g. [46]). Alternatively, Bartlett’s method
[42, 47] can be applied to find individual PSDs from data streams of duration
$\tau_{\rm{Bart}}\lesssim\tau_{c}$, which are then averaged. An advantage of
Bartlett’s method over a likelihood-based analysis is that the detailed form
of the DM speed distribution does not need to be specified. Bartlett’s method
reduces the frequency resolution of the signal PSD to a spike, while also
reducing the standard deviation of the noise PSD by a factor
$\sqrt{T_{\mathrm{int}}/\tau_{\rm{Bart}}}$. In the limit
$\tau_{\rm{Bart}}=\tau_{c}$, the experiment’s sensitivity to the couplings
strengths is therefore given by

$d_{\phi}\simeq\frac{\sqrt{\mathrm{SNR}}}{\overline{\Phi}_{\rm{R}}}\times\sqrt{\frac{S_{n}}{\sqrt{\tau_{c}T_{\mathrm{int}}}}}\;.$
(28)

For terrestrial atom gradiometers, Newtonian gravity gradient noise (GGN) is
expected to exceed atom shot noise at frequencies less than approximately
${10}^{-1}\text{\,}\mathrm{H}\mathrm{z}$, which corresponds to a mass
$m_{\phi}\approx 4\times 10^{-16}$ eV. If the GGN noise cannot be mitigated,
this will impose a lower limit on the frequencies that can be probed [18]. In
our treatment, we take a conservative approach and only show projections above
${10}^{-1}\text{\,}\mathrm{H}\mathrm{z}$. For compact atom gradiometers that
operate on the time scale of years (i.e.
$T_{\mathrm{int}}\gtrsim\mathrm{few}\times 10^{7}~{}\mathrm{s}$), the total
integration time will in general exceed the coherence time of an ULDM signal
for all ULDM masses of interest. Hence, the scaling of the experimental
parameters for a linear scalar ULDM-electron or photon interaction is
described by Eq. (28).

Pulling all parts of this discussion together, we find that the experiment’s
maximum sensitivity to a linear scalar ULDM-electron or photon interaction (in
the limit $T_{\mathrm{int}}>\tau_{c}$) scales with experimental parameters in
the following way:

$d^{\mathrm{best}}_{\phi}\sim\Bigg{(}\frac{1}{T}\Bigg{)}^{5/4}\frac{1}{C\,n\Delta
r}\,\,\Bigg{(}{\frac{\Delta
t}{N_{a}}}\Bigg{)}^{1/2}\Bigg{(}\frac{1}{T_{\mathrm{int}}}\Bigg{)}^{1/4}\;.$
(29)

The additional $m_{\phi}$ dependence in $\tau_{c}$ has two effects: the
maximum sensitivity occurs for ULDM masses $m_{\phi}\approx 2.04/T$, which is
slightly below $m_{\phi}\approx 2.33/T$ (where $\overline{\Phi}_{s}$ is
maximised); and the scaling is $T^{-5/4}$ rather than $T^{-1}$. Eq. (29)
reveals a hierarchy of importance amongst the tuneable experimental
parameters: the sensitivity scales as $T^{-5/4}$, so is most sensitive to
changes in this parameter; it varies linearly with the inverse of $C$, $n$ and
$\Delta r$; with the square root of $\Delta t$ and $N_{a}$; and with the
quartic root of $T_{\rm{int}}$, indicating the least sensitivity to this
parameter.

Figure 2: Projected sensitivities for $\mathrm{SNR}=1$ to scalar DM
interactions with electrons after an integration time of
$T_{\mathrm{int}}=10^{8}~{}\\!\rm{s}$ within the
$$0.1\text{\,}\mathrm{H}\mathrm{z}$\leq
f_{\phi}\leq$4\text{\,}\mathrm{H}\mathrm{z}$$ signal window for a
$10\text{\,}\mathrm{m}$ atom gradiometer with $n=1000$,
$S_{n}=10^{-8}~{}\\!\mathrm{Hz}^{-1}$, and: fixed gradiometer separation
$\Delta r=$4.86\text{\,}\mathrm{m}$$ for different interrogation times $T$
(left panel); optimal gradiometer separation $\Delta r$ for different
interrogation times $T$ (right panel). As $T$ increases, the sensitivity
curves move to higher values of $m_{\phi}$ and $d_{m_{e}}$. The loss in
sensitivity can be mitigated somewhat by increasing $\Delta r$. The orange
regions show parameter space that has already been excluded through searches
for violations of the equivalence principle via terrestrial torsion balance
experiments [48] and MICROSCOPE [49].

The scaling in Eq. (29) shows that the interrogation time $T$ is important as
it not only sets the ULDM mass at which the experiment has the maximum
sensitivity, but it also affects the experiment’s maximum sensitivity reach in
$d_{\phi}$. We demonstrate this explicitly in the left panel of Fig. 2, where
we have plotted the electron coupling when $\mathrm{SNR}=1$ for different
values of the the interrogation time from $0.74\text{\,}\mathrm{s}$ to
$0.2\text{\,}\mathrm{s}$ while keeping the gradiometer separation ($\Delta r$)
and all other parameters fixed. The left panel shows that as $T$ increases,
the sensitivity curves both move to higher ULDM masses and higher values of
$d_{m_{e}}$ as predicted by Eq. (29). Hence, shorter interrogation times imply
a loss in sensitivity. In these plots, we have assumed
$T_{\rm{int}}=10^{8}~{}\\!\mathrm{s}$ to employ Bartlett’s method, and set
$S_{n}=10^{-8}~{}\\!\mathrm{Hz}^{-1}$ and $n=1000$. As we will discuss in more
detail in the following section, these values are indicative of the parameters
ultimately envisaged for an atom gradiometer such as AION-10.

In the right panel, we show that this loss in sensitivity can be mitigated
somewhat by increasing the gradiometer separation. However, there is a
practical limit on how large the gradiometer separation can be, which is
provided by the baseline of the experiment. In the right panel, we have chosen
$\Delta r$ to be as large as possible while ensuring that the atoms remain
inside a $10\text{\,}\mathrm{m}$ atom gradiometer at all times.

Finally, we see that there may be some benefit in running an experiment with
different values of $T$. The oscillatory nature of the signal amplitude
$\overline{\Phi}_{\rm{s}}$ means that for fixed $T$, there will be points in
parameter space where the amplitude is significantly reduced (e.g., at
$m_{\phi}\approx 6\times 10^{-15}~{}\\!\mathrm{eV}$ when
$T=$0.74\text{\,}\mathrm{s}$$). By running with a different $T$ value, for
instance, $T=$0.3\text{\,}\mathrm{s}$$, the gaps in sensitivity can be closed.
To maximise the experiment’s sensitivity reach, it may even be desirable to
employ one baseline to run two atom interferometers with alternate launch
conditions, allowing different regions of the signal frequency to be probed
simultaneously.

##  V Optimising for AION-10

Figure 3:  Projected sensitivities for $\mathrm{SNR}=1$ of a
$10\text{\,}\mathrm{m}$ atom gradiometer to scalar ULDM interactions with
electrons after an integration time of
$T_{\mathrm{int}}=10^{8}~{}\\!\mathrm{s}$ and assuming
$S_{n}=10^{-8}~{}\\!\mathrm{Hz}^{-1}$. In purple, ‘AION-10 (this work)’, we
plot the refined sensitivity curve of AION-10. The bold lines show the power-
averaged envelope, while the fainter curves show the full expected
sensitivity. Left panel: the experimental parameters are $n=1000$, $\Delta
r=$4.86\text{\,}\mathrm{m}$$, $r_{1,i}=$0\text{\,}\mathrm{m}$$ and
$v_{i}=$4.04\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$$ resulting in
$T=$0.74\text{\,}\mathrm{s}$$, which is the maximum value of $T$ in a
$10\text{\,}\mathrm{m}$ gradiometer. Right panel: the parameters are $n=1000$,
$\Delta r=$7.8\text{\,}\mathrm{m}$$, $r_{1,i}=$0\text{\,}\mathrm{m}$$,
$v_{i}=$1.5\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$$ and
$T=$0.3\text{\,}\mathrm{s}$$, which are chosen to maximise the sensitivity at
$2\pi f_{\phi}=$10\text{\,}\mathrm{H}\mathrm{z}$$. The orange regions show
parameter space that has already been excluded through searches for violations
of the equivalence principle via terrestrial torsion balance experiments [48]
and MICROSCOPE [49]. In green, ‘AION-10 (JCAP’20)’, we plot the ‘goal’
projected sensitivity of AION-10 as presented in Ref. [21]. The insets in each
plot depict the spacetime diagram of the interferometric sequence for the
aforementioned set of experimental parameters.

To maximise a potential ULDM-induced signal in a single-photon atom
gradiometer with frequency $\gtrsim${10}^{-1}\text{\,}\mathrm{H}\mathrm{z}$$,
a careful selection of experimental parameters is required. This may at first
seem trivial given that Eq. (29) suggests that the experiment’s sensitivity is
maximised at $m_{\phi}\approx 2.04/T$ for high sampling rates, large atom
cloud populations, high LMT, high contrast, large separations between atom
interferometers, and long interrogation and integration times. However, the
experimental sensitivity also indirectly depends on the initial launch
velocity ($v_{i}$) and the position from which the atom clouds are initially
launched ($r_{1,i}$ and $r_{2,i}$, following the notation used in Fig. 1). In
our analysis, we require that the paths of different interferometers do not
overlap as this completely avoids the possibility of collisional losses. This
requirement, together with the constraint that atomic states must be confined
within the baseline of the experimental apparatus, implies that the
interrogation time $T$ becomes a function of the other experimental parameters
governing the interferometric sequence. Although the AION-10 design is not yet
final, we will use experimental parameters that are indicative of the
ambitious goals envisaged in the final phase of AION-10 to describe how the
search for optimal AION-10 experimental parameters can be carried out.

For a baseline $L=$10\text{\,}\mathrm{m}$$ and assuming that $n=1000$, we
carried out the search for optimal parameters by scanning over values of
$v_{i}$, $r_{1,i}$ and $r_{2,i}$. We assume that both clouds are launched with
the same velocity and that the atoms are subject to a constant gravitational
acceleration $g\simeq 9.81~{}\\!\mathrm{m}/\mathrm{s}^{2}$. In calculating
atom trajectories, we have also assumed that all of the momentum transfer from
the $n$ LMT kicks is transmitted instantaneously while in reality this would
occur over a timescale $\mathcal{O}(500~{}\\!\mathrm{\mu s})$ [41]. However,
this gives a negligible correction to the trajectories over timescales $\sim
T$.

There is a configuration that maximises $T$ where analytic results can be
obtained. In this configuration, the lower interferometer is positioned at
$r_{i,1}=0$ and the atoms start and end at $r_{1,f}=0$, which implies that the
sequence is symmetric around $T$. The upper interferometer will start at
$r_{2,i}=\Delta r$, and its upper arm is made to reach $r_{2,u}=L$ after a
time $T$. Finally, the upper arm of the lower interferometer is made to reach
the same position as the lower arm of the top interferometer at time $T$,
i.e., $r_{1,u}=r_{2,l}$. This configuration occurs if
$2v_{i}=\sqrt{8gL+9v_{n}^{2}}-4v_{n}$, $gT=v_{i}+v_{n}/2$ and $\Delta
r=v_{n}T$, where $v_{n}=n\hbar k_{A}/m_{A}$ is the speed imparted to the
${}^{87}\mathrm{Sr}$ atom after $n$ LMT kicks, which depends on the angular
wavenumber $k_{A}$ of the clock transition and the ${}^{87}\mathrm{Sr}$ atom
mass $m_{A}$. The sensitivity projection when $\mathrm{SNR}=1$ for the maximum
$T$ configuration is shown by the purple line in the left panel of Fig. 3. For
$L=$10\text{\,}\mathrm{m}$$ and $n=1000$, we find that
$T=$0.74\text{\,}\mathrm{s}$$,
$v_{i}=$4.04\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$$ and $\Delta
r=$4.86\text{\,}\mathrm{m}$$. The inset shows the spacetime diagram of the
interferometric sequence and confirms that the trajectories remain within the
baseline at all times during the sequence. We have also checked numerically
that this configuration results in the maximum signal amplitude
$\Phi_{\rm{s}}$.

In calculating the purple line in the left panel of Fig. 3, we have assumed
$T_{\rm{int}}=10^{8}~{}\\!\mathrm{s}$ to employ Bartlett’s method with a
gradiometer’s noise power spectral density
$S_{n}=10^{-8}~{}\\!\mathrm{Hz}^{-1}$. The plots and troughs of the
sensitivity curves that were apparent in Fig. 2 are again found in Fig. 3 and
are shown by the lighter purple lines. To facilitate the comparison with other
results in the literature, the darker, straighter lines show the power-
averaged envelope using the approximation $|\sin x|=\min\\{x,1/\sqrt{2}\\}$ in
Eq. (20), which smooths out the peaks and troughs arising from the fast
oscillation of the trigonometric functions. However, it should be understood
that this approximation is a visual aid and that limits display the peak and
trough structure.

The orange shaded regions in Fig. 3 show parameter space that has already been
excluded through searches from terrestrial torsion balance experiments [48]
and MICROSCOPE [49] so we see that with these parameter values, there is the
possibility of probing unconstrained regions of parameter space. We plot
results for the DM mass range
$${10}^{-15}\text{\,}\mathrm{e}\mathrm{V}$\lesssim
m_{\phi}\lesssim${10}^{-13}\text{\,}\mathrm{e}\mathrm{V}$$ but they could in
principle be extended to even larger masses. While the lower bound is set by
the frequency at which gravity gradient noise dominates, the upper bound
corresponds to the highest frequency at which the DM wave remains coherent
over successive measurements, such that $\tau_{c}\gtrsim\Delta t$ [18, 50].
For a fiducial value $\Delta t\simeq$10\text{\,}\mathrm{s}$$, this corresponds
to $m_{\phi}\lesssim${10}^{-9}\text{\,}\mathrm{e}\mathrm{V}$$.

The green lines in Fig. 3 show the estimates based on a previous set of
calculations in Ref. [21]. Unlike in this paper, the calculations in Ref. [21]
did not include the $\Delta r/L$ correction in Eq. (20) and used a larger
value of $T$ that was not consistent with the requirement that the atom
trajectories remain within the baseline at all times. The result is that the
refined calculations shown by the purple line is at larger values of the
electron coupling.

While the results in the left panel of Fig. 2 show the sensitivity for the
maximum $T$ configuration, it is possible to arrange for different sequences.
In particular, a different launch scheme could be employed to focus on
different regions of parameter space. In the right panel of Fig. 2, for
example, we show a configuration that maximises the sensitivity at a signal
angular frequency of $10\text{\,}\mathrm{H}\mathrm{z}$. This means that we
have found the experimental parameters that result in the lowest electron
coupling at this frequency (corresponding to a mass $m_{\phi}\simeq 7\times
10^{-15}~{}\\!\mathrm{eV}$). This configuration was found by scanning over
values of $v_{i}$, $r_{i,1}$ and $r_{i,2}$, resulting in
$T=$0.3\text{\,}\mathrm{s}$$, $\Delta r=$7.8\text{\,}\mathrm{m}$$,
$r_{1,i}=$0\text{\,}\mathrm{m}$$ and
$v_{i}=$1.5\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$$. The spacetime diagram of
the interferometric sequence is shown in the right-panel inset, and shows that
in this case, the trajectories remain inside the baseline at all times but the
sequence ends at a different position compared to the launch position.
Comparing the left and right panels of Fig. 2 we see that the power-averaged
envelope has shifted to larger frequencies so that the refined calculation in
the right panel is closer to the older result from Ref. [21] over a larger
region of parameter space. However, this has come at the cost of reduced
sensitivity to signal frequencies in the range from 0.1 Hz to 1 Hz.

Figure 4: Projected sensitivities for $\mathrm{SNR}=1$ to scalar ULDM
interactions with photons after an integration time of
$T_{\mathrm{int}}=${10}^{8}\text{\,}\mathrm{s}$$. In purple, ‘AION-10 (this
work)’, we plot the refined sensitivity projection for AION-10 for the maximum
$T$ parameters $n=1000$, $\Delta r=$4.86\text{\,}\mathrm{m}$$,
$r_{1,i}=$0\text{\,}\mathrm{m}$$ and
$v_{i}=$4.04\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$$ resulting in
$T=$0.74\text{\,}\mathrm{s}$$. In green, ‘AION-10 (JCAP’20)’, we plot the
AION-10 ‘goal’ projected sensitivity from Ref. [21]. The bold lines show the
power-averaged sensitivity, while the fainter curves show the full expected
sensitivity. The orange regions show parameter space that has already been
excluded through searches for violations of the equivalence principle via
terrestrial torsion balance experiments [48] and MICROSCOPE [49]. The inset
depicts the spacetime diagram of the interferometric sequence.

As shown in the insets in Fig. 3, the atom trajectories display differences
depending on the angular frequency at which the sensitivity is optimised. At
larger $T$ (or equivalently, when optimising at lower frequencies), the area
enclosed by the atom paths is maximised. This will favour schemes in which the
interferometers are separated by $\sim L/2$ and the atoms in each travel a
maximum displacement of $\sim L/2$. On the contrary, at smaller $T$
(optimising at higher frequency), the separation between the interferometers
is maximised. This will favour schemes in which the interferometers are
separated by $\sim L$. In this case, operating a further atom interferometer
at $\sim L/2$ could be envisaged, as this would provide a further independent
measurement of the DM field and may lead to better control of the GGN.

For completeness, in Fig. 4 we also present the projected gradiometer’s
optimal sensitivity to scalar DM coupled to photons for the maximum $T$
configuration, which optimises the sensitivity at lower values of $m_{\phi}$.
The experimental parameters are the same as those used in the left panel of
Fig. 3. As in Fig. 3, we again find that the refined calculation in this work
is at higher values of the coupling compared to the calculation in Ref. [21].
For the photon coupling, the constraints from atom interferometery are less
stringent than the space-based constraints from MICROSCOPE but have the
potential to match the sensitivity from terrestrial torsion balance
experiments.

###  V.1 Resonant mode with multiple spacetime diamonds

All of the results so far have used the sequence shown in Fig. 1, which
consists of a single closed spacetime ’diamond’. We will refer to this
sequence as the ‘broadband mode’ owing to the sensitivity over a wide range of
frequencies. However, an alternative sequence that enhances the sensitivity at
certain frequencies could also be employed [29]. This ‘resonant mode’ scheme
consists of $Q$ closed spacetime diamonds each lasting for a time $\sim 2T$.
Two examples of the closed spacetime diagrams for $Q=2$ and $Q=20$ are shown
in the insets in Fig. 5. The resonant mode employs $n_{\rm{tot}}=2Q(2n-1)+1$
laser pulses in total, which consists of two $\pi/2$-pulses that define the
start and end of the interferometric sequence and are emitted at times $0$ and
$2QT$, and $\left[2Q(2n-1)-1\right]$ $\pi$-pulses. The number of LMT-kicks
during a single diamond in the sequence is again denoted by $n$. The signal
amplitude induced by ULDM for resonant mode searches takes the form

$\begin{split}\overline{\Phi}^{\,Q}_{\rm{s}}&=8\frac{\overline{\Delta\omega_{A}}}{m_{\phi}}\frac{\Delta
r}{L}\Bigg{|}\sin\left[\frac{m_{\phi}nL}{2}\right]\sin\left[\frac{m_{\phi}T}{2}\right]\\\
&\qquad\times\sin\left[\frac{m_{\phi}(T-(n-1)L)}{2}\right]\frac{\sin\left[Qm_{\phi}T\right]}{\sin\left[m_{\phi}T\right]}\Bigg{|}.\end{split}$
(30)

The derivation of this expression is given in Appendix C. The proposed
advantage of the resonant mode is that there is an enhancement by a factor $Q$
at $m_{\phi}=\pi/T$ over a mass range $\sim\pi/(QT)$ relative to the broadband
search using the same set of experimental parameters. The additional $Q$
enhancement and the resonant behaviour arises from the
$\sin\left[Qm_{\phi}T\right]/\sin\left[m_{\phi}T\right]$ term in Eq. (30),
which was not present in Eq. (20).

Figure 5: Comparison of the projected sensitivities of a
$10\text{\,}\mathrm{m}$ atom gradiometer to scalar DM interactions with
electrons when operated in broadband and resonant modes for
$T_{\mathrm{int}}=${10}^{8}\text{\,}\mathrm{s}$$. The green curve shows the
broadband mode sensitivity, previously shown as the ‘AION-10 (this work)’
curve in the right panel of Fig. 3. The purple curves shows the resonant mode
sensitivity for $Q=2$, $T=$0.3\text{\,}\mathrm{s}$$, $n=500$, $\Delta
r=$8.0\text{\,}\mathrm{m}$$, $r_{1,i}=$0.0\text{\,}\mathrm{m}$$ and
$v_{i}=$4.25\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$$. The blue curve shows
the resonant mode sensitivity for $Q=20$, $T=$0.03\text{\,}\mathrm{s}$$,
$n=50$, $\Delta r=$8.0\text{\,}\mathrm{m}$$,
$r_{1,i}=$0.0\text{\,}\mathrm{m}$$ and
$v_{i}=$5.7\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$$. The insets depict the
spacetime diagrams for each resonant mode sequence. The orange regions show
the excluded parameter space from torsion balance experiments [48] and
MICROSCOPE [49].

In Fig. 5 we compare the projected sensitivity from broadband and resonant
mode searches for two different values of $Q$. In this comparison, we assume
that the maximum total number of laser pulses is $n_{\mathrm{max}}=4000$ and,
for each value of $Q$, we choose the largest even value of $n$ that satisfies
the inequality $2Q(2n-1)+1\leq n_{\mathrm{max}}$. This implies that as $Q$
increases, $n$ must decrease but the maximum signal amplitude will remain
similar since $\overline{\Phi}^{\,Q}_{\rm{s}}\propto nQ$. The purple line in
Fig. 5 shows the sensitivity curve for a resonant mode configuration operating
with $Q=2$, $T=$0.3\text{\,}\mathrm{s}$$, $n=500$, $\Delta
r=$8.0\text{\,}\mathrm{m}$$, $r_{1,i}=$0.0\text{\,}\mathrm{m}$$ and
$v_{i}=$4.25\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$$; in blue, we show the
sensitivity curve for a resonant mode configuration operating with $Q=20$,
$T=$0.03\text{\,}\mathrm{s}$$, $n=50$, $\Delta r=$8.0\text{\,}\mathrm{m}$$,
$r_{1,i}=$0.0\text{\,}\mathrm{m}$$ and
$v_{i}=$5.7\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$$. The corresponding
spacetime diagrams are show in the inset. As in our previous calculations, we
have assumed an integration time of ${10}^{8}\text{\,}\mathrm{s}$ and a noise
power spectral density of ${10}^{-8}\text{\,}\mathrm{H}\mathrm{z}^{-1}$.

In green, we show the broadband mode sensitivity that was shown previously (as
the purple line) in the right panel of Fig. 3. Comparing the broadband and
resonant search modes, we find that neither resonant mode configuration
substantially increases the sensitivity relative to the broadband mode. For
$Q=2$, the maximum sensitivity reach is approximately equal to the one
available to broadband searches at integer multiples of $m_{\phi}\sim\pi/T$,
but is smaller for other frequencies. One possible advantage of the $Q=2$
sequence is that it begins and ends at the same location, which was not the
case for the broadband sequence (cf. inset in Fig. 3 right). This would be an
advantage for an experimental configuration where the detection and readout
optics are located at the same location (e.g. at the sidearm base). Higher $Q$
values result in a smaller value of $T$, which is why the best sensitivity for
$Q=20$ is shifted to higher values of $m_{\phi}$. The $Q=20$ curve also more
clearly displays the resonant structure as the sensitivity away from
$m_{\phi}=\pi/T$ is substantially weaker. While the high-$Q$ mode allows for
the possibility of probing a specific frequency, for compact gradiometers, the
curve’s minimum will in general lie above the exclusion limits from torsion
balance experiments and MICROSCOPE. Given current exclusion limits, we
therefore find that no significant improvement can be found when operating a
compact atom gradiometer in resonant mode instead of broadband mode.

##  VI Conclusions

Single-photon atom gradiometry is a powerful experimental technique that can
be used to search for oscillations of fundamental constants caused by ultra-
light dark matter (ULDM). While the existing formalism for calculating the
gradiometer sensitivity to ULDM is applicable to
$\mathcal{O}(\mathrm{km})$-baseline atom gradiometers in which two
interferometers are separated by a distance that is comparable to the length
of the baseline, the calculations presented in this work provide a careful
treatment that is also applicable to gradiometers that have a more compact
design. In particular, we have been careful to include the contribution to the
ULDM-induced phase from all segments where the atom is in the excited state,
thus finding an additional $\Delta r/L$ correction-term in Eq. (20). In
addition, we have chosen physically realistic experimental parameters that
ensure the atoms remain confined with the compact baseline at all times.

Using these results, we have refined the sensitivity estimates for AION-10, a
compact $10\text{\,}\mathrm{m}$ gradiometer that will be operated in Oxford.
Using the ambitious experimental parameters envisaged for the final phase of
AION-10, we find that AION-10 has the potential to probe currently
unconstrained values of the electron-ULDM coupling for ULDM masses around
$10^{-15}~{}\\!\mathrm{eV}$ (cf. Figs. 2 and 3). We have also provided a
discussion of the how the sensitivity scales with tuneable experimental
parameters. This allows for a relatively straightforward remapping of our
projected sensitivities to other parameter choices so that if, for instance,
AION-10 ultimately achieves $n=100$ LMT kicks instead of $n=1000$, or if the
number of strontium atoms per cloud is $N_{a}=10^{6}$ instead of
$N_{a}=10^{8}$, Eq. (29) implies that the projections to the electron coupling
will be weaker by a factor of 10 relative to the projections shown in Figs. 2
or 3.

We also provided a comparison of the sensitivity that can be achieved with an
interferometric sequence that involves multiple closed diamonds in spacetime,
the so-called resonant mode. As Fig. 5 showed, utilising a multiple diamond
configuration did not ultimately result in an improvement in sensitivity over
the broadband sequence.

Although our focus has been on the signal induced by ultra-light scalar dark
matter, we end by commenting on the potential impact of our study for
gravitational wave searches with atom gradiometers. The $\Delta r/L$
correction will also arise in a careful calculation of the signal induced by a
gravitational wave travelling through an atom gradiometer. This correction can
be particularly relevant in long-baseline, single-photon, multi-gradiometer
configurations (i.e. when more than two atom interferometers are located
within a single baseline and are referenced by the same laser), which have
been envisaged to improve GGN characterisation. A combined analysis of the
signal-phases recorded by different pairs of interferometers would have to
take into account the different gradiometer separations to ensure accurate
predictions.

## Acknowledgements

We are grateful to members of the AION Collaboration for many fruitful
discussions and we thank Ankit Beniwal, Elliot Bentine, Thomas Edwards, John
Ellis, Tiffany Harte and Thomas Hird for comments on the manuscript. LB, DB
and CM acknowledge support from the Science and Technology Facilities Council
(STFC) Grant ST/T00679X/1. In addition, LB is a recipient of an STFC quota
studentship and CM is supported by the STFC Grant ST/N004663/1. DB
acknowledges support from the Fundación Jesus Serra and the Instituto de
Astrofísica de Canarias under the Visiting Researcher Programme 2021 agreed
between both institutions. IFAE is partially funded by the CERCA program of
the Generalitat de Catalunya.

##  Appendix A Scalar ULDM phase calculation

As a result of the linear interactions introduced in Eq. (4), all electronic
energy levels in an atom receive an oscillating ULDM-induced correction
proportional to the DM field. Therefore, the energies of the ground and
excited states in the two level system considered in this work, namely
$\mathrm{5s^{2}\,{}^{1}S_{0}}$ and $\mathrm{5s5p\,^{3}P_{0}}$ in
${}^{87}\mathrm{Sr}$ respectively, are functions of space and time and
oscillate at a frequency largely set by the DM mass. Neglecting the spatial
variation of the field, we define the ULDM-induced time variation of the
$\mathrm{5s^{2}\,{}^{1}S_{0}}$ and $\mathrm{5s5p\,^{3}P_{0}}$ state energies
as $\Delta\omega_{g}(t)$ and $\Delta\omega_{e}(t)$, respectively, and they
satisfy the relation
$\Delta\omega_{e}(t)=\Delta\omega_{g}(t)+\Delta\omega_{A}(t)$, where
$\Delta\omega_{A}(t)$ is the oscillating transition energy correction defined
in Eq. (8).

The ULDM-induced oscillations of fundamental constants would be observable in
the evolution of the internal degrees of freedom of the atom between atom-
light interaction points. Neglecting the spatial evolution of the DM field,
the ULDM-induced phase contribution between times $t_{1}$ and $t_{2}$ along
any path is

$\Phi_{t_{1},(g,e)}^{t_{2}}\equiv\int_{t_{1}}^{t_{2}}{\Delta\omega_{(g,e)}(t)dt}\,,$
(31)

where the subscripts $g$ and $e$ label the ground and excited state energies,
respectively.

In the limit that the separation between the upper and lower arms of the
interferometer at the end of the sequence is small, to leading order the
interferometer sequence can be modelled as a closed loop (see Fig. 1).
Combining Eq. (31) with the definitions of $\Delta\omega_{g,e}$, and computing
the difference between the sum of all ULDM-induced phase contributions along
each interferometer arm, the total ULDM-induced phase difference of the
$p^{\mathrm{th}}$ interferometer relative to the atom’s ground state takes the
form

$\Phi_{p,\mathrm{tot}}\approx\Phi_{p}+\oint_{C}\Delta\omega_{g}(t)\,dt=\Phi_{p}\;.$
(32)

In the limit that the separation phase is negligible, the integral is a
closed-loop integral over $\Delta\omega_{g}(t)$, which is identically zero.
Here, $\Phi_{p}$ is the quantity defined in Eq. (16), which is the total ULDM-
induced phase difference of the $p^{\mathrm{th}}$ single-photon atom
interferometer accumulated in the propagation phase of the excited state
($\mathrm{5s5p\,^{3}P_{0}}$) relative to the ground state
($\mathrm{5s^{2}\,{}^{1}S_{0}}$).

Thus, in the limit that the separation phase is negligible, the total ULDM-
induced phase difference of a lone single-photon atom interferometer only
depends on the phase accumulated by the excited state relative to the ground
state of a two-level system.

##  Appendix B Signal calculation for compact atom gradiometers

In the range of frequencies probed by atom gradiometers,
$10^{-1}\,\mathrm{Hz}\lesssim\omega_{\phi}\lesssim 10^{5}\,\mathrm{Hz}$, the
spatial dependence of the ULDM field is highly subdominant and can be safely
neglected, i.e. $\mathbf{k}_{\phi}\cdot\mathbf{r}\ll 1$. Therefore, the
expression for the gradiometer signal, Eq. (17), contains pairwise
contributions that are identical up to time integration limits, where the
difference depends on the position of the atomic wave-packets during atom-
laser interactions. Explicitly, using Eq. (16) and grouping terms, Eq. (17)
can be expressed as

$\displaystyle\Phi_{\mathrm{s}}=$
$\displaystyle\sum_{m=1}^{n/2}\left[\Phi_{T+(2m-n-1)L+r_{1,l}}^{T+(2m-n+1)L-r_{1,l}}-\Phi_{T+(2m-n-1)L+r_{2,l}}^{T+(2m-n+1)L-r_{2,l}}\right]+\sum_{m=1}^{n/2}\left[\Phi_{2T-(n-2m)L-r_{1,f}}^{2T-(n-2m)L+r_{1,f}}-\Phi_{2T-(n-2m)L-r_{2,f}}^{2T-(n-2m)L+r_{2,f}}\right]$
(33)
$\displaystyle-\sum_{m=1}^{n/2}\left[\Phi_{T+(2m-1)L-r_{1,u}}^{T+(2m-1)L+r_{1,u}}-\Phi_{T+(2m-1)L-r_{2,u}}^{T+(2m-1)L+r_{2,u}}\right]-\sum_{m=1}^{n/2}\left[\Phi_{(2m-2)L+r_{1,i}}^{2mL-
r_{1,i}}-\Phi_{(2m-2)L+r_{2,i}}^{2mL-r_{2,i}}\right]\;.$

Each of the four terms in square brackets can be manipulated into a simpler
form by performing the integrals in the ULDM-induced phase contribution as
defined in Eq. (13), using trigonometric identities, and summing over the LMT
kicks. We will show the manipulations explicitly only for the final square
bracket term as it is straightforward to adapt the argument for the other
terms.

We begin by performing the integral in the general expression for the ULDM-
induced phase contribution, $\Phi_{t_{1}}^{t_{2}}$, defined in Eq. (13), and
simplifying with a sum-to-product trigonometric identity:

$\displaystyle\Phi_{t_{1}}^{t_{2}}$
$\displaystyle=\overline{\Delta\omega_{A}}\int_{t_{1}}^{t_{2}}\cos(\omega_{\phi}t+\theta)dt$
(34)
$\displaystyle=2\frac{\overline{\Delta\omega_{A}}}{\omega_{\phi}}\sin\left(\frac{\omega_{\phi}(t_{2}-t_{1})}{2}\right)\cos\left(\frac{\omega_{\phi}(t_{1}+t_{2})}{2}+\theta\right)\;.$
(35)

This implies that

$\displaystyle\sum_{m=1}^{n/2}\left[\Phi_{(2m-2)L+r_{1,i}}^{2mL-
r_{1,i}}-\Phi_{(2m-2)L+r_{2,i}}^{2mL-r_{2,i}}\right]$ (36)
$\displaystyle=2\frac{\overline{\Delta\omega_{A}}}{\omega_{\phi}}\Big{[}\sin\bigl{(}\omega_{\phi}L-\omega_{\phi}r_{1,i}\bigr{)}-\sin\bigl{(}\omega_{\phi}L-\omega_{\phi}r_{2,i}\bigr{)}\Big{]}\sum_{m=1}^{n/2}\cos\bigl{(}\omega_{\phi}L(2m-1)+\theta\bigr{)}$
(37)
$\displaystyle=2\frac{\overline{\Delta\omega_{A}}}{\omega_{\phi}}\left[\frac{\sin\bigl{(}\omega_{\phi}L-\omega_{\phi}r_{1,i}\bigr{)}-\sin\bigl{(}\omega_{\phi}L-\omega_{\phi}r_{2,i}\bigr{)}}{\sin\left(\omega_{\phi}L\right)}\right]\sin\left(\frac{n\omega_{\phi}L}{2}\right)\cos\left(\frac{n\omega_{\phi}L}{2}+\theta\right)\;,$
(38)

where to reach the final line, we have used a modified form of Lagrange’s
identity:

$\sum_{m=1}^{N}\cos(mA+B)=\frac{\sin(AN/2)}{\sin(A/2)}\cos\left(\frac{A+2B+AN}{2}\right)\;.$
(39)

We recognise that Eq. (38) contains the term

$\Phi^{nL}_{0}=2\frac{\overline{\Delta\omega_{A}}}{\omega_{\phi}}\sin\left(\frac{n\omega_{\phi}L}{2}\right)\cos\left(\frac{n\omega_{\phi}L}{2}+\theta\right)\;,$
(40)

which allows us to write

$\sum_{m=1}^{n/2}\left[\Phi_{(2m-2)L+r_{1,i}}^{2mL-
r_{1,i}}-\Phi_{(2m-2)L+r_{2,i}}^{2mL-
r_{2,i}}\right]=\Phi^{nL}_{0}\left[\frac{\sin\bigl{(}\omega_{\phi}L-\omega_{\phi}r_{1,i}\bigr{)}-\sin\bigl{(}\omega_{\phi}L-\omega_{\phi}r_{2,i}\bigr{)}}{\sin\left(\omega_{\phi}L\right)}\right]\;.$
(41)

Similar manipulations can be used on the other terms to arrive at the
expression

$\displaystyle\Phi_{\mathrm{s}}$
$\displaystyle=\Phi^{T+L}_{T-(n-1)L}\left[\frac{\sin{(\omega_{\phi}L-\omega_{\phi}r_{1,l})}-\sin{(\omega_{\phi}L-\omega_{\phi}r_{2,l})}}{\sin\left(\omega_{\phi}L\right)}\right]+\Phi^{2T+L}_{2T-(n-1)L}\left[\frac{\sin\left(\omega_{\phi}r_{1,f}\right)-\sin\left(\omega_{\phi}r_{2,f}\right)}{\sin\left(\omega_{\phi}L\right)}\right]$
(42)
$\displaystyle\qquad-\Phi^{T+nL}_{T}\left[\frac{\sin\left(\omega_{\phi}r_{1,u}\right)-\sin\left(\omega_{\phi}r_{2,u}\right)}{\sin\left(\omega_{\phi}L\right)}\right]-\Phi^{nL}_{0}\left[\frac{\sin{(\omega_{\phi}L-\omega_{\phi}r_{1,i})}-\sin{(\omega_{\phi}L-\omega_{\phi}r_{2,i})}}{\sin\left(\omega_{\phi}L\right)}\right]\,.$

So far, we have made no assumption about the position vectors. However, if two
atom interferometers have the same experimental parameters, as we have been
assuming in this paper, then some additional simplifications can be made so
that $\Phi_{\mathrm{s}}$ only depends on the distance between the atom
interferometers, $\Delta r$, which was previously defined in Eq. (19), and the
mean distance $\bar{r}_{a}=(r_{1,a}+r_{2,a})/2$, where $a$ labels the location
of the atom during the sequence. In this case, we can use the sum-to-product
trigonometric identities to write

$\begin{split}\Phi_{\mathrm{s}}=\frac{2\sin\left(\omega_{\phi}\Delta
r/2\right)}{\sin\left(\omega_{\phi}L\right)}&\Big{[}\Phi^{T+L}_{T-(n-1)L}\cos\left(\omega_{\phi}L-\omega_{\phi}\bar{r}_{l}\right)+\Phi^{2T+L}_{2T-(n-1)L}\cos\left(\omega_{\phi}\bar{r}_{f}\right)\\\
&\qquad-\Phi^{T+nL}_{T}\cos\left(\omega_{\phi}\bar{r}_{u}\right)-\Phi^{nL}_{0}\cos\left(\omega_{\phi}L-\omega_{\phi}\bar{r}_{i}\right)\Big{]}\;.\end{split}$
(43)

As discussed earlier in the paper, since $\omega_{\phi}L\ll 1$ and
$r_{1,i},r_{1,u},r_{1,l},r_{1,f},r_{2,i},r_{2,u},r_{2,l},r_{2,f}<L$, the sine
and cosine terms in Eq. (43) can be expanded to give

$\Phi_{\mathrm{s}}=\frac{\Delta
r}{L}\Big{\\{}\left[\Phi^{T+L}_{T-(n-1)L}-\Phi^{nL}_{0}\right]-\left[\Phi^{2T+L}_{2T-(n-1)L}-\Phi^{T+nL}_{T}\right]\Big{\\}}+\mathcal{O}\Big{(}\left(\omega_{\phi}L\right)^{2}\Big{)}\,.$
(44)

In AION-10, where $L\approx 10$ m, the next-to-leading term in the expansion
at $\omega_{\phi}\sim 1~{}\mathrm{Hz}$ is suppressed by a factor of
$(\omega_{\phi}L)^{2}\sim 10^{-15}$; hence, to very high precision the ULDM-
induced differential phase is given by the leading-order term in $\Delta r/L$
in Eq. (44), which is the expression that is given in Eq. (18).

Finally, we can perform some further straightforward trigonometric
manipulations on Eq. (44) to arrive at a compact expression for
$\Phi_{\mathrm{s}}$ and therefore show the origin of Eq. (20). Starting with
Eq. (35) and by using cosine sum-to-product identities, we find

$\displaystyle\left[\Phi^{T+L}_{T-(n-1)L}-\Phi^{nL}_{0}\right]$
$\displaystyle=-4\frac{\overline{\Delta\omega_{A}}}{\omega_{\phi}}\sin\left[\frac{\omega_{\phi}nL}{2}\right]\sin\left[\frac{\omega_{\phi}(T-(n-1)L)}{2}\right]\sin\left[\frac{\omega_{\phi}(T+L)}{2}+\theta\right]$
(45) $\displaystyle\left[\Phi^{2T+L}_{2T-(n-1)L}-\Phi^{T+nL}_{T}\right]$
$\displaystyle=-4\frac{\overline{\Delta\omega_{A}}}{\omega_{\phi}}\sin\left[\frac{\omega_{\phi}nL}{2}\right]\sin\left[\frac{\omega_{\phi}(T-(n-1)L)}{2}\right]\sin\left[\frac{\omega_{\phi}(3T+L)}{2}+\theta\right]\;.$
(46)

Substituting these expressions into the leading order term in Eq. (44),
together with a final application of a sine sum-to-product identity, we arrive
at

$\Phi_{\mathrm{s}}=8\frac{\overline{\Delta\omega_{A}}}{\omega_{\phi}}\frac{\Delta
r}{L}\sin\left[\frac{\omega_{\phi}nL}{2}\right]\sin\left[\frac{\omega_{\phi}(T-(n-1)L)}{2}\right]\sin\left[\frac{\omega_{\phi}T}{2}\right]\cos\left[\omega_{\phi}T+\frac{\omega_{\phi}L}{2}+\theta\right]\;.$
(47)

The last term in Eq. (47) contains information about the phase of the DM field
and will not be observable over the measurement campaign. In fact, because we
assume that we are working in the regime where the coherence time of the
signal over the relevant mass range is shorter than the projected
interrogation time, all phase information will be lost. The estimator of the
signal’s power spectral density (PSD) would then be the appropriate analysis
to extract information from the time-dependent signal collected over the
course of a measurement campaign [42, 46]. In particular, this quantity
contains information on the signal’s frequency spread and is proportional to
the amplitude squared of the analysed signal at $\omega_{\phi}$ [46]. The
amplitude of the signal (i.e. Eq. (47)) then takes the form as presented in
Eq. (20), explicitly

$\displaystyle\overline{\Phi}_{\rm{s}}$
$\displaystyle=\sqrt{\frac{1}{\pi}\int_{0}^{2\pi}\Phi_{s}^{2}\,d\theta}$ (48)
$\displaystyle=8\frac{\overline{\Delta\omega_{A}}}{m_{\phi}}\frac{\Delta
r}{L}\Bigg{|}\sin\left[\frac{m_{\phi}nL}{2}\right]\sin\left[\frac{m_{\phi}T}{2}\right]\sin\left[\frac{m_{\phi}(T-(n-1)L)}{2}\right]\Bigg{|}\;,$
(49)

where we have neglected sub-leading kinetic corrections to the angular
frequency by making the approximation $\omega_{\phi}\approx m_{\phi}$.

##  Appendix C Signal calculation for the case of multiple diamonds

The ULDM-induced phase calculation presented in Appendix B applies to the
‘single-diamond’ configuration shown in Fig. 1. However, we also considered in
section V.1 the resonant mode search that employs a sequence with $Q$ copies
of the single-diamond configuration (see the left inset in Fig. 5 for an
example with $Q=2$). In this appendix, we derive the form of the ULDM-induced
phase for a sequence with $Q$-diamonds, where each diamond contains $n$ LMT
kicks. This implies that the full sequence has a total of $[2Q(2n-1)+1]$ laser
pulses and we use the labelling convention that the final $\pi/2$-pulse is
emitted after a time $2QT$.

Repeating the steps in Appendix B but for a sequence that starts at time
$(2q-2)T$, it is straightforward to show that the induced phase for the
$q^{\mathrm{th}}$-diamond, which lasts for a duration of approximately $2T$,
is

$\displaystyle\Phi_{\mathrm{s}}^{q}$ $\displaystyle=\frac{\Delta
r}{L}\Big{\\{}\left[\Phi^{(2q-1)T+L}_{(2q-1)T-(n-1)L}-\Phi^{(2q-2)T+nL}_{(2q-2)T}\right]-\left[\Phi^{2qT+L}_{2qT-(n-1)L}-\Phi^{(2q-1)T+nL}_{(2q-1)T}\right]\Big{\\}}$
(50)
$\displaystyle=8\frac{\overline{\Delta\omega_{A}}}{\omega_{\phi}}\frac{\Delta
r}{L}\sin\left[\frac{\omega_{\phi}nL}{2}\right]\sin\left[\frac{\omega_{\phi}(T-(n-1)L)}{2}\right]\sin\left[\frac{\omega_{\phi}T}{2}\right]\cos\left[(2q-1)\omega_{\phi}T+\frac{\omega_{\phi}L}{2}+\theta\right]\;.$
(51)

Here, we have ignored the negligible corrections
$\mathcal{O}\left(\left(\omega_{\phi}L\right)^{2}\right)$. The full induced
phase is obtained by summing the contribution from all diamonds, which, with
the help of Eq. (39), gives

$\sum_{q=1}^{Q}\Phi_{\mathrm{s}}^{q}=8\frac{\overline{\Delta\omega_{A}}}{\omega_{\phi}}\frac{\Delta
r}{L}\sin\left[\frac{\omega_{\phi}nL}{2}\right]\sin\left[\frac{\omega_{\phi}T}{2}\right]\sin\left[\frac{\omega_{\phi}(T-(n-1)L)}{2}\right]\frac{\sin\left[Q\omega_{\phi}T\right]}{\sin\left[\omega_{\phi}T\right]}\cos\left[Q\omega_{\phi}T+\frac{\omega_{\phi}L}{2}+\theta\right]\;.$
(52)

From this expression and when the sub-leading kinetic corrections are ignored,
we obtain the resonant mode signal amplitude

$\overline{\Phi}^{\,Q}_{\rm{s}}=8\frac{\overline{\Delta\omega_{A}}}{m_{\phi}}\frac{\Delta
r}{L}\Bigg{|}\sin\left[\frac{m_{\phi}nL}{2}\right]\sin\left[\frac{m_{\phi}T}{2}\right]\sin\left[\frac{m_{\phi}(T-(n-1)L)}{2}\right]\frac{\sin\left[Qm_{\phi}T\right]}{\sin\left[m_{\phi}T\right]}\Bigg{|},$
(53)

which is the amplitude of the signal presented in Eq. (30). This result
differs from Eq. (49) by the factor
$\sin\left[Qm_{\phi}T\right]/\sin\left[m_{\phi}T\right]$.

## References

  * [1] G. Bertone, D. Hooper and J. Silk, _Particle dark matter: Evidence, candidates and constraints_ , _Phys. Rept._ 405 (2005) 279 [hep-ph/0404175].
  * [2] M. Battaglieri et al., _US Cosmic Visions: New Ideas in Dark Matter 2017: Community Report_ , in _U.S. Cosmic Visions: New Ideas in Dark Matter_ , 7, 2017, 1707.04591.
  * [3] A. M. Green and B. J. Kavanagh, _Primordial Black Holes as a dark matter candidate_ , _J. Phys. G_ 48 (2021) 4 [2007.10722].
  * [4] European Astroparticle Physics Strategy 2017-2026, https://www.appec.org/roadmap.
  * [5] XENON Collaboration, E. Aprile et al., _Dark Matter Search Results from a One Ton-Year Exposure of XENON1T_ , _Phys. Rev. Lett._ 121 (2018) 111302 [1805.12562].
  * [6] PandaX Collaboration, Y. Meng et al., _Dark Matter Search Results from the PandaX-4T Commissioning Run_ , 2107.13438.
  * [7] E. G. M. Ferreira, _Ultra-light dark matter_ , _Astron. Astrophys. Rev._ 29 (2021) 7 [2005.03254].
  * [8] J. Jaeckel and A. Ringwald, _The Low-Energy Frontier of Particle Physics_ , _Ann. Rev. Nucl. Part. Sci._ 60 (2010) 405 [1002.0329].
  * [9] L. Hui, _Wave Dark Matter_ , 2101.11735.
  * [10] P. W. Graham, D. E. Kaplan and S. Rajendran, _Cosmological Relaxation of the Electroweak Scale_ , _Phys. Rev. Lett._ 115 (2015) 221801 [1504.07551].
  * [11] D. J. E. Marsh, _Axion Cosmology_ , _Phys. Rept._ 643 (2016) 1 [1510.07633].
  * [12] A. E. Nelson and J. Scholtz, _Dark Light, Dark Matter and the Misalignment Mechanism_ , _Phys. Rev. D_ 84 (2011) 103501 [1105.2812].
  * [13] 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].
  * [14] A. Arvanitaki, J. Huang and K. Van Tilburg, _Searching for dilaton dark matter with atomic clocks_ , _Phys. Rev. D_ 91 (2015) 015015 [1405.2925].
  * [15] Y. V. Stadnik and V. V. Flambaum, _Searching for dark matter and variation of fundamental constants with laser and maser interferometry_ , _Phys. Rev. Lett._ 114 (2015) 161301 [1412.7801].
  * [16] Y. V. Stadnik and V. V. Flambaum, _Can dark matter induce cosmological evolution of the fundamental constants of Nature?_ , _Phys. Rev. Lett._ 115 (2015) 201301 [1503.08540].
  * [17] A. A. Geraci and A. Derevianko, _Sensitivity of atom interferometry to ultralight scalar field dark matter_ , _Phys. Rev. Lett._ 117 (2016) 261301 [1605.04048].
  * [18] 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].
  * [19] S. Abend, M. Gersemann, C. Schubert, D. Schlippert, E. M. Rasel, M. Zimmermann, M. A. Efremov, A. Roura, F. A. Narducci and W. P. Schleich, _Atom interferometry and its applications_ , 2001.10976.
  * [20] 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].
  * [21] L. Badurina et al., _AION: An Atom Interferometer Observatory and Network_ , _JCAP_ 05 (2020) 011 [1911.11755].
  * [22] MAGIS-100 Collaboration, J. Coleman, _Matter-wave Atomic Gradiometer InterferometricSensor (MAGIS-100) at Fermilab_ , _PoS_ ICHEP2018 (2019) 021 [1812.00482].
  * [23] M. Abe et al., _Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100)_ , 2104.02835.
  * [24] B. Canuel et al., _Exploring gravity with the MIGA large scale atom interferometer_ , _Sci. Rep._ 8 (2018) 14064 [1703.02490].
  * [25] B. Canuel et al., _ELGAR—a European Laboratory for Gravitation and Atom-interferometric Research_ , _Class. Quant. Grav._ 37 (2020) 225017 [1911.03701].
  * [26] M.-S. Zhan et al., _ZAIGA: Zhaoshan long-baseline atom interferometer gravitation antenna_ , _IJMPD_ 29 (2020) 1940005 [1903.09288].
  * [27] AEDGE Collaboration, Y. A. El-Neaj et al., _AEDGE: Atomic Experiment for Dark Matter and Gravity Exploration in Space_ , _EPJ Quant. Technol._ 7 (2020) 6 [1908.00802].
  * [28] 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].
  * [29] 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].
  * [30] P. W. Graham and S. Jung, _Localizing Gravitational Wave Sources with Single-Baseline Atom Interferometers_ , _Phys. Rev. D_ 97 (2018) 024052 [1710.03269].
  * [31] J. Ellis and V. Vaskonen, _Probes of gravitational waves with atom interferometers_ , _Phys. Rev. D_ 101 (2020) 124013 [2003.13480].
  * [32] N. W. Evans, C. A. J. O’Hare and C. McCabe, _Refinement of the standard halo model for dark matter searches in light of the Gaia Sausage_ , _Phys. Rev. D_ 99 (2019) 023012 [1810.11468].
  * [33] G. P. Centers et al., _Stochastic fluctuations of bosonic dark matter_ , 1905.13650.
  * [34] J. I. Read, _The Local Dark Matter Density_ , _J. Phys. G_ 41 (2014) 063101 [1404.1938].
  * [35] D. B. Kaplan and M. B. Wise, _Couplings of a light dilaton and violations of the equivalence principle_ , _JHEP_ 08 (2000) 037 [hep-ph/0008116].
  * [36] T. Damour and J. F. Donoghue, _Phenomenology of the Equivalence Principle with Light Scalars_ , _Class. Quant. Grav._ 27 (2010) 202001 [1007.2790].
  * [37] T. Damour and J. F. Donoghue, _Equivalence Principle Violations and Couplings of a Light Dilaton_ , _Phys. Rev. D_ 82 (2010) 084033 [1007.2792].
  * [38] H. Hachisu, G. Petit, F. Nakagawa, Y. Hanado and T. Ido, _Si-traceable measurement of an optical frequency at the low 10−16 level without a local primary standard_ , _Opt. Express_ 25 (2017) 8511.
  * [39] E. J. Angstmann, V. A. Dzuba and V. V. Flambaum, _Relativistic effects in two valence electron atoms and ions and the search for variation of the fine structure constant_ , _Phys. Rev. A_ 70 (2004) 014102 [physics/0404042].
  * [40] 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].
  * [41] J. Rudolph, T. Wilkason, M. Nantel, H. Swan, C. M. Holland, Y. Jiang, B. E. Garber, S. P. Carman and J. M. Hogan, _Large Momentum Transfer Clock Atom Interferometry on the 689 nm Intercombination Line of Strontium_ , _Phys. Rev. Lett._ 124 (2020) 083604 [1910.05459].
  * [42] J. T. VanderPlas, _Understanding the Lomb-Scargle periodogram_ , _The Astrophysical Journal Supplement Series_ 236 (2018) 16.
  * [43] J. Le Gouët, T. Mehlstäubler, J. Kim, S. Merlet, A. Clairon, A. Landragin and F. Pereira Dos Santos, _Limits to the sensitivity of a low noise compact atomic gravimeter_ , _Applied Physics B_ 92 (2008) 133–144.
  * [44] W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen and D. J. Wineland, _Quantum projection noise: Population fluctuations in two-level systems_ , _Phys. Rev. A_ 47 (1993) 3554.
  * [45] A. Roura, _Circumventing Heisenberg’s uncertainty principle in atom interferometry tests of the equivalence principle_ , _Phys. Rev. Lett._ 118 (2017) 160401 [1509.08098].
  * [46] J. W. Foster, N. L. Rodd and B. R. Safdi, _Revealing the Dark Matter Halo with Axion Direct Detection_ , _Phys. Rev. D_ 97 (2018) 123006 [1711.10489].
  * [47] D. Budker, P. W. Graham, M. Ledbetter, S. Rajendran and A. Sushkov, _Proposal for a Cosmic Axion Spin Precession Experiment (CASPEr)_ , _Phys. Rev. X_ 4 (2014) 021030 [1306.6089].
  * [48] T. A. Wagner, S. Schlamminger, J. H. Gundlach and E. G. Adelberger, _Torsion-balance tests of the weak equivalence principle_ , _Class. Quant. Grav._ 29 (2012) 184002 [1207.2442].
  * [49] J. Bergé, P. Brax, G. Métris, M. Pernot-Borràs, P. Touboul and J.-P. Uzan, _MICROSCOPE Mission: First Constraints on the Violation of the Weak Equivalence Principle by a Light Scalar Dilaton_ , _Phys. Rev. Lett._ 120 (2018) 141101 [1712.00483].
  * [50] A. Derevianko, _Detecting dark-matter waves with a network of precision-measurement tools_ , _Phys. Rev. A_ 97 (2018) 042506 [1605.09717].