# Violation of the equivalence principle induced by oscillating rest mass and
transition frequency, and its detection in atom interferometers

Jordan Gué, Aurélien Hees, Peter Wolf  SYRTE, Observatoire de Paris,
Université PSL, CNRS, Sorbonne Université, LNE, 61 avenue de l’Observatoire
75014 Paris, France

###### Abstract

We present a theoretical investigation of the expected experimental signals
produced by freely falling atoms with time oscillating mass and transition
frequency. These oscillations could be produced in a variety of models, in
particular, models of scalar dark matter (DM) non universally coupled to the
standard matter (SM) such as axion-like particles (ALP) and dilatons.
Performing complete and rigorous calculations, we show that, on one hand, two
different atomic species would accelerate at a different rate, and on the
other hand, they would produce a non-zero differential phase shift in atom
interferometers (AI). The former would produce observable signals in
equivalence principle tests like the recent MICROSCOPE mission, and we provide
a corresponding sensitivity estimate, showing that MICROSCOPE can reach beyond
the best existing searches in the ALP case. We also compare the expected
sensitivity of two future AI experiments, namely the AION-10 gradiometer and
an isotope differential AI considered for MAGIS-100, that we will refer to as
SPID. We show that the SPID setup would be more sensitive to these dark matter
fields compared to the gradiometer one, assuming equivalent experimental
parameters.

##  I Introduction

General Relativity (GR) is, alongside the Standard Model (SM) of particle
physics, one of the building blocks of fundamental physics Weinberg (1972).
One of the main assumptions of this theory is the Einstein Equivalence
Principle (EEP), which includes the weak equivalence principle (WEP), local
position invariance (LPI) and local Lorentz invariance (LLI). The WEP, also
referred as the universality of free fall (UFF), states that gravitational and
inertial masses are equivalent, implying that all bodies fall at the same rate
in the same gravitational field. While being experimentally tested at
extraordinary levels of precision by on ground measurements (e.g. Wagner  _et
al._ (2012)) or space tests Touboul  _et al._ (2022), many theories beyond GR
and the SM predict its violation Damour and Polyakov (1994); Fayet (2018,
2019). More generally, the EEP postulates that all types of energy (i.e all SM
fields) are universally coupled to gravity, such that the underlying gravity
theory must be metric Will (2014). Since EEP is not based on a fundamental
symmetry of the Universe, it is also expected to be broken at some scale, for
example by the space-time variation of fundamental constants Uzan (2011).

Time oscillating fundamental constants arise in many theories Uzan (2011);
Damour and Donoghue (2010, 2010), in particular models of massive scalar
fields non-universally coupled to SM. Many of these new fields are good dark
matter (DM) candidates, as they are oscillating at their Compton frequency
$\omega$, thus behaving as cold DM at late cosmological times, when
$\omega>H$, the Hubble constant. In particular, these oscillations can lead to
spacetime variation of the rest mass and transition frequency of atoms, with
magnitudes depending on various parameters including mass and charge numbers,
making such oscillations not universal. Oscillating constants have been widely
studied in the case of a scalar dilatonic field non universally coupled to SM
fields, e.g in Damour and Donoghue (2010). Recently, Kim and Perez (2022)
showed that the coupling between an axion-like particle (ALP), another
promising ULDM candidate Marsh (2016), and the gluon field induces a variation
of the pion mass, leading to a variation of the rest mass and transition
frequency of atoms as well. Therefore, we will be interested in the search for
these two ULDM candidates through various couplings, on one hand using
classical tests of UFF, and on the other hand through atom interferometry.
Atom interferometers (AI) provide quantum tests of the EEP where one can
measure the acceleration and/or transition frequency of atoms freely falling
in the Earth gravitational field, by using the wave behavior of matter. In
such experiments, light pulses are used to split, reflect and recombine the
atomic wavepackets, replacing the beam-splitters and mirrors of usual optical
interferometers. The phase shift between the two wavepackets after
recombination relies on the momentum exchange between atom and light, the time
between the various light pulses, the transition frequencies, and the
acceleration of the wavepackets.

The first ambition of this paper is to derive carefully the classical
differential acceleration between two tests masses and the phase shift in
several AI setups, taking into account all oscillating effects. For the
former, we show the signal is predominantly dependent on the atomic rest mass
oscillation amplitude, while for the latter, it depends on the AI
configuration setup. The first one under consideration is the two-photon
transition configuration, whose signal is a combination of both rest mass and
transition frequency oscillations, and whose dominant effect depends on the
setup. One of these contributions, coming from the oscillation of the rest
mass has already been derived in Geraci and Derevianko (2016) but we show that
a non negligible contribution dependent on the velocity of the atoms in the
galactic DM rest frame must be added, which would improve the constraints
presented in Geraci and Derevianko (2016) by 4 orders of magnitude. The second
case regards single photon transition interferometers, in particular
gradiometers, as proposed in Graham  _et al._ (2013) and then considered for
the search of ultralight dark matter (ULDM) fields in the AION (Atom
Interferometer Observatory and Network) experiment Badurina  _et al._ (2020,
2022). In that setup, two interferometric sequences are performed at different
elevation, using the same atomic species in both of them. This makes possible
the use of the same laser beam for both interactions, which strongly
suppresses the effect of laser phase noise. In addition, a very large momentum
transfer (LMT) from laser beams to atomic wavepackets is possible, making such
experiments very sensitive to transition frequency oscillations, as explicitly
shown in Arvanitaki  _et al._ (2018). However, the signal is limited by the
reduced time the atomic wavepackets spend in their excited state. The second
single photon transitions setup under consideration is based on the same LMT
sequence as in gradiometers, but would consist on the differential phase shift
measurement between two atomic isotopes which perform the AI sequence at the
same elevation. We name this variation SPID for Single Photon transition
Isotope differential AI. This kind of setup has already been proposed for the
search of ULDM in MAGIS-100 Abe  _et al._ (2021), which is a facility
operating both gradiometers and the SPID variation. Finally, for the two ULDM
candidates studied, ALP and dilatons, we show that already existing
experiments testing UFF or involving AI measurements can put constraints on
their couplings with SM fields. Using the experimental parameters of AION-10,
Badurina  _et al._ (2020, 2022), we demonstrate that the SPID variation would
be more sensitive to the couplings of these ULDM fields with SM fields
compared to the gradiometer version of AION-10. Whilst the SPID variation has
been previously proposed by the MAGIS-100 collaboration Abe  _et al._ (2021),
no detailed calculations and modelling have been provided, and the sensitivity
to couplings between the dilaton field and SM fields was not calculated, which
we do in this paper.

##  II Observational signatures induced by oscillations in atom rest mass and
transition frequency

In this section, we consider a very generic phenomenological model where both
the rest mass of an atom $A$ and its internal energy oscillate. We derive the
observable signatures of such oscillations in some systems of interest and we
will then apply those results in two different theoretical models in Sec. III.

We start by introducing a phenomenological parametrization of the oscillation
of both the atom rest mass $m_{A}$ and its transition frequency $\omega_{A}$
as

$\displaystyle m_{A}(t)$ $\displaystyle=m^{0}_{A}\left(1+Q^{A}_{M}\cos(\omega
t+\phi_{0})\right)\,,$ (1a) $\displaystyle\omega_{A}(t)$
$\displaystyle=\omega^{0}_{A}\left(1+Q^{A}_{\omega}\cos(\omega
t+\phi_{0})\right)\,,$ (1b)

where $\omega$ is the oscillation frequency, $\phi_{0}$ a phase,
$m^{0}_{A},\omega^{0}_{A}$ are respectively the unperturbed rest mass and
transition frequency of $A$ and $Q^{A}_{M},Q^{A}_{\omega}$ are respectively
the mass and frequency charge of the atom $A$. These characterizes the
relative amplitude of oscillation, supposed much smaller than unity.

The motion of particles can be described by the point mass action, i.e

$\displaystyle
S_{\mathrm{mat}}=-\sum_{A}\int_{A}d\tau(m_{A}(t)c^{2}+E_{\mathrm{int}}(t))\,,$
(2)

where the first term represents the rest mass and the second term is the
internal energy contribution ($E_{\mathrm{int}}=0$ when the atom is in its
ground state and $E_{\mathrm{int}}=\hbar\omega_{A}$ when the atom is in the
excited state). $d\tau$ is the proper time interval defined as
$c^{2}d\tau^{2}=-g_{\alpha\beta}dx^{\alpha}dx^{\beta}$, where $g_{\mu\nu}$ is
the space-time metric.

For an atom of rest mass $m_{A}$ and in an internal state characterized by an
energy $E_{\mathrm{int}}$, the Lagrangian derived from Eq.(2) becomes

$L=-\left(m_{A}(t)c^{2}+E_{\mathrm{int}}(t)\right)\Big{(}1-\frac{v_{A}^{2}(t)}{2c^{2}}\Big{)}\,,$
(3)

to first order in $(v_{A}/c)^{2}$ (where $v_{A}$ is the coordinate velocity of
the atom) and considering a flat space-time, i.e
$g_{\mu\nu}=\eta_{\mu\nu}=\mathrm{diag}(-1,1,1,1)$, the Minkowskian metric.

In the following, we show how oscillating rest mass and transition frequency
characterized by Eq. (1) produce observational signatures in different
experiments.

###  II.1 Classical trajectories of test masses

In the classical derivation, as long as the atom is not in its energetic
ground state, both oscillations Eq. (1) contribute to a violation of the
equivalence principle through Eq. (3). The Euler-Lagrange equation applied to
the Lagrangian from Eq. (3) leads to the conservation of the momentum of body
$A$, i.e. $\left(m_{A}(t)+\frac{E_{\mathrm{int}}(t)}{c^{2}}\right)\times
v_{A}(t)=\mathrm{cst}$. The oscillation of the mass and internal energy of the
atom will therefore induce a perturbation to the acceleration which, to first
order in both the $Q$ factors and in $E_{\mathrm{int}}/m_{A}^{0}c^{2}$, is

$\displaystyle\left[\vec{a}_{A}\right]_{\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{UFF}\mkern-1.5mu}\mkern
1.5mu}\approx\omega\vec{v}_{A}\left(Q^{A}_{M}+\frac{\hbar\omega^{0}_{A}}{m^{0}_{A}c^{2}}Q^{A}_{\omega}\right)\sin(\omega
t+\phi_{0})\,,$ (4)

where $\vec{a}_{A}=d\vec{v}_{A}/dt$ is the acceleration. In general,
$m^{0}_{A}c^{2}/\hbar\omega^{0}_{A}>10^{10}$, while $Q^{A}_{M}/Q^{A}_{\omega}$
is of the order of $10^{3}$ at maximum (see Table. 2) implying that the first
term of Eq. (4) is dominant. It follows a differential acceleration between
two macroscopic bodies of different composition $A$ and $B$ with the same
initial velocities given by

$\displaystyle\Delta\vec{a}$
$\displaystyle=\vec{a}_{A}-\vec{a}_{B}\approx\omega
v_{0}\left(Q^{A}_{M}-Q^{B}_{M}\right)\sin(\omega t+\phi_{0})\hat{e}_{v}\,,$
(5)

where $\vec{v}_{A}=\vec{v}_{B}\equiv\vec{v}_{0}=v_{0}\hat{e}_{v}$ is the
unperturbed velocity, i.e. the velocity at zeroth order in $Q_{M}$.

If the two test masses have different charges $Q_{M}$, Eq. (5) implies a non
zero differential acceleration between them, corresponding to a UFF violation.
The signature induced by the differential acceleration from Eq. (5) can be
searched for using UFF measurements using macroscopic test masses such as
torsion balances on Earth Wagner  _et al._ (2012), the MICROSCOPE experiment
Touboul  _et al._ (2017) or Lunar Laser Ranging WILLIAMS  _et al._
(2009)111Note that Lunar Laser Ranging measures a combination of both the weak
and strong equivalence principle..

###  II.2 Phase observable in atom interferometry

As derived in the previous section, the time-dependent mass and internal
frequency Eq. (1) produces a differential acceleration between two atoms.
Standard UFF tests search for such a differential acceleration of macroscopic
test masses. Atom interferometry (AI) constitutes the quantum equivalent of
such classical experiments. Therefore, as we shall see in this section, AI
experiments are also sensitive to oscillations from Eq. (1).

In this section, we will derive the observable signatures produced by
oscillating atomic rest mass and transition energy on various AI
configurations. In the most generic case, AI consists of generating
interference between atomic wavepackets that are split and then recombined
using atom-laser interactions. We classify the various AI schemes into two
classes: (i) AI schemes using two photon transitions and (ii) AI schemes using
single photon transitions. In the former, the transition from a stable
momentum state to another stable momentum state is done using at least two
photons, while in the latter, only one photon is needed for such a transition.

We will present the main results, i.e. the impact of the rest mass and atomic
transition frequency oscillations on the measurable phase of an AI. The
detailed calculations are performed following the methodology described in
Storey and Cohen-Tannoudji (1994) and are presented in details in Appendix A
while in the main text, we will focus on the discussion of the results.

####  II.2.1 Two-photons transitions

We start by studying two different two-photons transition interferometers.

The first interferometric scheme considered is known as two-photon transition
Raman interferometry. Its sequence is depicted in Fig. 1. In this setup, two-
level free falling atoms $A$ enter the interferometer with an initial momentum
$\hbar k$ and in their internal energetic ground state, noted $|g\rangle$
(i.e. their initial state is defined as $|g,\hbar k\rangle$). After entering
the interferometer, they interact with a pair of laser waves $L_{1}$ and
$L_{2}$ with respective frequencies
$\omega_{\mathrm{L_{1}}},\omega_{\mathrm{L_{2}}}$, whose energy difference is
resonant with the transition of the $|e\rangle\rightarrow|g\rangle$ where
$|e\rangle$ is the excited state of the two-levels atom, i.e
$\omega^{0}_{A}=\omega_{\mathrm{L_{1}}}-\omega_{\mathrm{L_{2}}}$. This means
that the atom first absorbs a photon from $L_{1}$ and stimulately emits a
photon in $L_{2}$. This process splits the atoms into two spatially distant
wavepackets, which means the state of each atom becomes the superposition of
the two internal states: (i) the ground state that remains unchanged $|g,\hbar
k\rangle$ and (ii) the excited state that has changed its momentum because of
its interaction with the two lasers $|e,\hbar(k+k_{\mathrm{eff}})\rangle$,
where
$\hbar\vec{k}_{\mathrm{eff}}=\hbar(\vec{k}_{\mathrm{L_{1}}}+\vec{k}_{\mathrm{L_{2}}})$
denotes the effective momentum transfer, considering both lasers.

This whole experimental manoeuvre is called a $\pi/2$ laser pulse since the
transition amplitude from ground state to excited state corresponds to a
probability $1/2$, i.e the state of the atoms is now a half-half superposition
of the two different states.

The atom freely propagates inside the interferometer and at time $t=T$, both
wavepackets undergo a $\pi$ pulse, which will invert the state of all atoms.
In other words, depending on the internal state of the atom prior to the $\pi$
pulse, two state transitions happen :
$|e,\hbar(k+k_{\mathrm{eff}})\rangle\rightarrow|g,\hbar k\rangle$ and
$|g,\hbar k\rangle\rightarrow|e,\hbar(k+k_{\mathrm{eff}})\rangle$.

A final laser-atom interaction happens at time $t=2T$, where a second $\pi/2$
pulse divides the two incoming wavepackets into four different ones: two of
them are in the state $|g,\hbar k\rangle$ and the remaining two are in the
state $|e,\hbar(k+k_{\mathrm{eff}})\rangle$. The study of interference pattern
between the wavepackets in the same state allows one to measure a phase shift
difference.

The second two-photon transfer AI scheme considered is the two-photons
transition Bragg-type interferometer. This scheme is similar to the Raman
interferometry presented above except that the atoms remain in the same energy
state during all the inteferometric path, i.e, the laser pulses only change
the external state of the atom (their momentum).

For both interferometers described above, the effective wavevector
$k_{\mathrm{eff}}$ depends on the setup of the experiment. If
counterpropagating lasers are used, the atom absorbs a photon in one direction
as a result of the interaction with the first laser, and emits another photon
in the opposite direction during the interaction with the second laser,
implying $k_{\mathrm{eff}}=k_{\mathrm{L_{1}}}+k_{\mathrm{L_{2}}}$. In general,
this effective wave vector is multiple orders of magnitude larger than the
transition frequency of the atom, i.e $k_{\mathrm{eff}}\gg\omega^{0}_{A}/c$.
On the opposite, if co-propagating laser waves are used in the experimental
setup, the absorption and emission directions are the same, implying
$k_{\mathrm{eff}}=k_{L_{1}}-k_{L_{2}}$. In that case,
$k_{\mathrm{eff}}=\omega^{0}_{A}/c$.

As already discussed in Geraci and Derevianko (2016), the velocity kick
experienced by the atom during its interaction with the laser beams is
perturbed by the effective mass of the atom at the time of the kick. In
addition to that effect, the beams being locked to a given frequency
reference, their own frequency oscillates as
$\omega_{L}(t)=\omega^{0}_{L}(1+Q^{L}_{\omega}\cos(\omega t+\phi_{0}))$,
neglecting the additional phase coming from the propagation of photons to the
atom (in the following sections, the travelling distance of the photon to
reach the freely falling atoms will be of the order of 10 m maximum, which
would induce a significant phase for oscillation frequencies $\omega\gtrapprox
10^{6}$ rad/s, way above the DM frequencies of interest for this paper, see
Sec. V.). Therefore, if the atom interacts with the laser beam at time $t$,
still considering $Q^{A}_{M},Q^{L}_{\omega}\ll 1$,

$\displaystyle v^{A}_{\mathrm{kick}}(t)=\frac{\hbar
k_{\mathrm{eff}}(t)}{m_{A}(t)}$ $\displaystyle\approx\frac{\hbar
k_{\mathrm{eff}}}{m^{0}_{A}}\Big{(}1+(Q^{L}_{\omega}-Q^{A}_{M})\cos(\omega
t+\phi_{0})\Big{)}\,$ $\displaystyle\equiv v^{A}_{\mathrm{kick,\ 0}}+\delta
v^{A}_{\mathrm{kick}}(t)\,,$ (6)

where $v^{A}_{\mathrm{kick,\ 0}}=\hbar k_{\mathrm{eff}}/m^{0}_{A}$ is the
nominal kick velocity, i.e without perturbation, and $\delta
v^{A}_{\mathrm{kick}}(t)$ is the perturbed contribution to the total velocity
kick imparted to the atom.

The calculation of the observable phase shift at the output of the
interferometer follows closely the ones presented in Storey and Cohen-
Tannoudji (1994) and is detailed in Appendix A. This method relies on Feynman
path integrals and can be used for Lagrangians which are at most quadratic in
the position and velocity Storey and Cohen-Tannoudji (1994) and considering
atomic plane waves at initial time $t=t_{0}$. As shown in Eq. (3), in our
case, the Lagrangian is quadratic in the velocity and there is no dependence
on the position, hence this framework can be safely used. Note that in the
case of oscillations induced by coupling with a massive quantum field, all
calculations must be made in its own rest frame, to make sure the dependence
in the position of the particle is not more complicated, see Sec. III.

As derived in e.g. Wolf and Tourrenc (1999); Wolf and Borde (2004), there are
three independent contributions to the total phase shift in an AI: (i) the
separation phase noted $\Phi_{\mathrm{u}}$ which corresponds to a spatial
incoincidence between the two output wavepackets, (ii) the laser phase
$\Phi_{\mathrm{\ell}}$ which gathers the additional phase factors of the
laser, due to displacement of the light-matter interaction vertices and (iii)
the propagation phase shift denoted $\Phi_{\mathrm{s}}$ which is essentially
the phase accumulated by the atom wavepacket over the full interferometric
path. All of these contributions must be calculated accurately to predict the
phase shift at the output of the interferometer.

Figure 1: Two-photon Raman transition interferometry sequence considered in
the present paper. The atomic wavepackets start in the ground state
$|g\rangle$ and the laser pulses change the internal energy state. The Bragg-
type equivalent has the same space-time diagram, but the atom wavepackets do
not change their internal energy state. In black are shown the atomic paths
without any perturbation, i.e straight lines. In green is shown the perturbed
motion of the atoms induced by Eq. (1), with exaggerated amplitude of
oscillation.

In Raman-type interferometers, the oscillation of the internal energy impacts
mainly the propagation phase of atoms when they are in their excited internal
states, i.e on paths 2 and 3 in Fig. 1. As it can be seen in Eq. (4), it also
modifies the equations of motion of the atom, implying a small contribution on
the laser and separation phases, but these will be suppressed by a factor
$\hbar\omega^{0}_{A}/m^{0}_{A}c^{2}$. In Bragg-type interferometers, the
oscillation of the internal energy does not contribute as the energy state of
the atoms is unchanged. The rest mass contribution arises on all the three
different contributions to the phase shift, and it can be shown that the total
phase shift of an atom A at the end of the interferometric sequence shown in
Fig. 1 is (see Appendix. A for the detailed calculation)

$\displaystyle\Delta\Phi^{\mathrm{Bragg}}_{\mathrm{A}}$
$\displaystyle=\frac{4}{\omega}\left(k_{\mathrm{eff}}v_{0}(Q^{A}_{M}-Q^{M}_{M})\hat{e}_{v}\cdot\hat{e}_{\mathrm{kick}}+\omega^{0}_{A}Q^{L}_{\omega}+\frac{\hbar
k^{2}_{\mathrm{eff}}}{2m^{0}_{A}}Q^{A}_{M}\right)\sin^{2}\left(\frac{\omega
T}{2}\right)\sin(\omega T+\phi_{0})-\,$ $\displaystyle\frac{2\hbar
k^{2}_{\mathrm{eff}}T}{m^{0}_{A}}Q^{L}_{\omega}\sin\left(\frac{\omega
T}{2}\right)\sin\left(\frac{\omega T}{2}+\phi_{0}\right)\,,$ (7a)
$\displaystyle\Delta\Phi^{\mathrm{Raman}}_{\mathrm{A}}$
$\displaystyle=\frac{4}{\omega}\left(k_{\mathrm{eff}}v_{0}(Q^{A}_{M}-Q^{M}_{M})\hat{e}_{v}\cdot\hat{e}_{\mathrm{kick}}-\omega^{0}_{A}(Q^{A}_{\omega}-Q^{L}_{\omega})+\frac{\hbar
k^{2}_{\mathrm{eff}}}{2m^{0}_{A}}Q^{A}_{M}\right)\sin^{2}\left(\frac{\omega
T}{2}\right)\sin(\omega T+\phi_{0})-\,$ $\displaystyle\frac{2\hbar
k^{2}_{\mathrm{eff}}T}{m^{0}_{A}}Q^{L}_{\omega}\sin\left(\frac{\omega
T}{2}\right)\sin\left(\frac{\omega T}{2}+\phi_{0}\right)\,,$ (7b)

at lowest order in $v_{0}/c$ and where $Q^{M}_{M}$ is the mass charge of the
reference mirror of the AI and $\hat{e}_{\mathrm{kick}}$ is the direction of
the kick imparted to atoms. In practice, the mirror is used as a reference for
the phase of the laser.

Note that in double diffraction interferometers, i.e when two pairs of laser
beams transfer opposite momentum to the atom Giese  _et al._ (2013), such that
the spatial separation between the two coherent wavepackets is twice as large,
the total phase shift is the same with the change $k_{\mathrm{eff}}\rightarrow
2k_{\mathrm{eff}}$, as expected.

If the mass charges $Q_{M}$ are composition-dependent, the first term in Eq.
(7), proportional to $v_{0}$, in the following denoted as the mass term, is a
signature of the violation of the Einstein EP. If the charge is universal i.e
$Q^{M}_{M}=Q^{A}_{M}$, this term vanishes as expected. However, both terms
quadratic in the effective wavevector and respectively proportional to the
mass charge of the atom A and the frequency charge of the laser still remain,
implying that these are not related to a violation of the Einstein EP. In
fact, they can be understood as a non-local measurement, whose macroscopic
counterpart would be to compare the free fall of two test masses with
different velocities. In the following, we will neglect these term, since we
will consider a practical situation where $v_{0}\sim 10^{5}\>\mathrm{m/s}\gg
10^{-2}\>\mathrm{m/s}\sim\hbar k_{\mathrm{eff}}/m^{0}_{A}$. The final term
appearing in the Raman phase shift is related to the oscillation of the atomic
and laser frequencies and will be denoted as the frequency term. Depending on
the setup, the $Q^{L}_{\omega}$ and $Q^{A}_{\omega}$ charges can be the same,
cancelling this term completely. In the Bragg case, the frequency term remains
and the dominant effect will depend on the setup.

Two-photon transition AI are commonly operating using hyperfine transitions of
alkaline-Earth atoms (e.g the hyperfine transition of Rb, Cs, K atoms Ahlers
_et al._ (2022); Gauguet  _et al._ (2008); Gillot  _et al._ (2016)). For both
co-propagating and counter-propagating configurations, the laser beams are
locked onto the optical transition of an atomic ensemble of the same species
as the atoms in free fall inside the interferometer, and the frequency
difference $\omega_{\mathrm{L_{1}}}-\omega_{\mathrm{L_{2}}}=\omega^{0}_{A}$ is
provided by a radio-frequency source (in the GHz range). Then $Q^{L}_{\omega}$
is the charge of that source, which, depending on the experimental
configuration, may or may not be the same as $Q^{A}_{\omega}$ the charge of
the hyperfine transition of the atoms in the AI. As discussed previously, in
the case of co-propagating lasers, the effective wavevector corresponds to the
frequency transition of the atom, i.e $k_{\mathrm{eff}}=\omega^{0}_{A}/c$.
Therefore, the mass term of Eq. (7b) is suppressed by a factor $v_{0}/c$
compared to the frequency term. For counter-propagating laser beams, the
effective wavevector $k_{\mathrm{eff}}\gg\omega^{0}_{A}/c$, implying that the
mass term of Eq. (7b) is much bigger than in the co-propagating case. In that
case, both mass and frequency terms are relevant and need to be taken into
account. The mass term has already been derived in Geraci and Derevianko
(2016), in the case of oscillating mass coming from a coupling between matter
and a classical oscillating dark matter field. However, the calculation in
Geraci and Derevianko (2016) is performed in the lab frame, where $v_{0}\sim
10$ m/s while we argue that it should be performed in the galactocentric frame
(i.e. in the scalar field rest frame) where $v_{0}\sim 10^{5}$ m/s. Indeed, in
the lab frame, the Lagrangian becomes dependent on the atom position due to a
$\cos\left(\omega t-kx\right)$, meaning that the method derived in Storey and
Cohen-Tannoudji (1994) cannot be used. Changing the initial atom velocity
$v_{0}$ to the DM galactic velocity would improve the expected sensitivity to
the charge $Q^{A}_{M}$ by several orders of magnitude (see next sections for a
practical example). In addition, our calculations take into account the
reference mirror contribution, which was not the case in Geraci and Derevianko
(2016). This is an important contribution because in the case of universal
rest mass oscillations, the mass term of Eq. (7) cancels.

Dual atom interferometers using two atomic species $A$ and $B$ with different
mass and frequency charges will measure the difference of the interferometric
phases, whose amplitude is given by

$\displaystyle\left|\Delta\Phi^{\mathrm{Bragg}}_{\mathrm{AB}}\right|$
$\displaystyle=\frac{4}{\omega}\left|\left[k^{A}_{\mathrm{eff}}(Q^{A}_{M}-Q^{M}_{M})-k^{B}_{\mathrm{eff}}(Q^{B}_{M}-Q^{M}_{M})\right]v_{0}\hat{e}_{v}\cdot\hat{e}_{\mathrm{kick}}+\omega^{0}_{A}Q^{L,A}_{\omega}-\omega^{0}_{B}Q^{L,B}_{\omega}\right|\sin^{2}\left(\frac{\omega
T}{2}\right)\,,$ (8a)
$\displaystyle\left|\Delta\Phi^{\mathrm{Raman}}_{\mathrm{AB}}\right|$
$\displaystyle=\frac{4}{\omega}\left|\left[k^{A}_{\mathrm{eff}}(Q^{A}_{M}-Q^{M}_{M})-k^{B}_{\mathrm{eff}}(Q^{B}_{M}-Q^{M}_{M})\right]v_{0}\hat{e}_{v}\cdot\hat{e}_{\mathrm{kick}}-\left(\omega^{0}_{A}(Q^{A}_{\omega}-Q^{L,A}_{\omega})-\omega^{0}_{B}(Q^{B}_{\omega}-Q^{L,B}_{\omega})\right)\right|\sin^{2}\left(\frac{\omega
T}{2}\right)\,,$ (8b)

where we neglected the subdominant terms $\propto k^{2}_{\mathrm{eff}}$ and
where $Q^{L,A}_{\omega},Q^{L,B}_{\omega}$ are respectively the frequency
charge of the beams used for the transition of the atomic species A and B. The
mass term of Eq. (8) is the quantum equivalent of the classical calculation
Eq. (5), while the second term of Eq. (8b), proportional to the frequency
charges $Q_{\omega}$ have no classical counterpart. Assuming mass and
frequency charges of same order of magnitude, the frequency terms of Eq. (8b)
will dominate in co-propagating laser Raman interferometers, while for
counter-propagating laser Raman interferometers, both terms contribute to the
phase shift.

####  II.2.2 Single photon transition

We will now focus on interferometric setup that involves single photon
transitions. The first setup considered is known as a gradiometer and has
already been studied in Graham  _et al._ (2013); Badurina  _et al._ (2022). In
addition, we will also study an isotope differential interferometric scheme
that leads to an improved sensitivity to mass and frequency oscillations.

##### Gradiometers

Figure 2: Space-time diagram of a single photon transition, as proposed in
Graham  _et al._ (2013) with $n=2$ (using the convention of Arvanitaki  _et
al._ (2018), see text.) Two lasers are used, one located at $x=0$, with
emission (in red) towards the second laser located at $x=L$ (with blue
emission). The frequencies of the lasers are selected carefully to interact
only with the wavepacket of interest (see text). The black dots indicate the
location of matter-wave interactions.

The first single photon transition AI scheme considered consists in
gradiometers, i.e a setup where two atom interferometers are stacked at
different altitudes, for the study of e.g gravity gradients. We are interested
in the setup initially proposed by Graham  _et al._ (2013) and then studied in
e.g Arvanitaki  _et al._ (2018); Badurina  _et al._ (2022). Practically, we
consider two ensembles of atoms $A$ (one for each interferometer) located
respectively at $x_{1}$ and $x_{2}$, all initially in the state
$|g,\hbar\vec{k}\rangle$ and two lasers, one at coordinate $x=0$ with
effective wavevector $\vec{k}_{1}$ and the other at coordinate $x=L$ with
effective wavevector $\vec{k}_{2}$. In those single photon interactions
configurations, $k_{1}\approx k_{2}=k=\omega^{0}_{A}/c$.

At an initial time $t_{0}$, the first laser sends a beam which interacts with
both atom ensembles and which corresponds to a $\pi/2$ pulse. Then, the second
laser beam sends a $\pi$ Doppler shifted laser pulse in order to interact only
with the excited state wavepackets, whose motion induces a change in
transition frequency, and to convert entirely this wavepacket to the ground
state. After this sequence, all wavepackets are in the ground state, however
one of them has gained momentum $2\hbar k$. Following the convention of
Arvanitaki  _et al._ (2018); Badurina  _et al._ (2022), at the end of the
sequence, the fast wavepackets have received a Large Momentum Transfer (LMT)
photon kick of order 2222Note that in the original proposal by Graham  _et
al._ (2013), the LMT has a slightly different definition.. If another pair of
$\pi$ pulses (with the first one from the bottom laser and the second one from
the top laser in the same way as before) is sent to the faster half of the
atom, one makes a large momentum transfer (LMT) beam splitter of order 4. More
generally, if $m$ pairs of $\pi$ pulses are sent, the order of the LMT beam
splitter is $2(m+1)$ and the faster wavepacket has gained total momentum
$2(m+1)\hbar k$. In other words, the order $n$ of the LMT is defined as
$n=2(m+1)$.

Later at time $t=T$, a sequence of state inversion similar to the one used in
two-photon transitions interferometers is performed, but this time, with three
different $\pi$ pulses, the first one coming from the bottom laser, the second
one from the top laser and the last one from the bottom laser again. However,
in order to slow down the faster wavepacket, $m$ pairs of $\pi$ pulses are
added before this sequence, such that it loses $2(m+1)\hbar k$ momentum.
Symmetrically, $m$ other pairs of LMT pulses are added after the state
inversion to accelerate the other wavepacket, such that it gains $2(m+1)\hbar
k$ momentum.

Finally at time $t=2T$, a sequence of pulses opposite to the one sent at the
initial $t=0$ is sent to the wavepackets, i.e $m$ LMT pairs of $\pi$ pulses
are sent to the wavepackets before the final $\pi-\pi/2$ pulses used for
recombination. This whole sequence is depicted in Fig. 2 with $n=2$ (i.e
$m=0$) and only one interferometer.

In gradiometers, two such interferometers are stacked at different altitudes,
separated by a distance $\Delta r$, and the same laser beam is used for the
laser-atom interaction in both interferometers, which is depicted in Fig. 3.

Figure 3: Space-time diagram of a gradiometer, where two interferometers as
shown in Fig. 2 are stacked at different altitudes. In this figure,
trajectories of the atom in its ground state are represented by dashed lines,
while full lines represent paths where the atom is in its excited state.

The main advantage of this setup is that laser phase noise is entirely
cancelled when measuring the differential phase shift between the two
interferometers.

Assuming $\omega L/c\ll 1$ and $nL/c\ll T$ where the LMT kick is of order n,
and $L$ is the baseline separation between the two lasers, which we assume for
simplicity to be the distance between the two interferometers, the
differential phase shift can be computed following the exact same methodology
as in Appendix A and reads

$\displaystyle|\Delta\Phi^{\mathrm{Grad}}_{\mathrm{A}}|\approx\frac{4n\omega^{0}_{A}\Delta
rQ^{A}_{\omega}}{c}\sin^{2}\left(\frac{\omega T}{2}\right)\,,$ (9)

We recover the phase shift amplitude already derived in Graham  _et al._
(2013); Badurina  _et al._ (2022).

##### Single photon transition isotope differential AI (SPID)

In this section, we focus on another interferometric scheme, which is a
variation of the interferometric sequence presented in Sec. II.2.2. We will
show that this experimental setup is more sensitive to oscillations in atom
rest mass and transition frequency compared to regular gradiometers. This type
of setup has already been proposed, without any detail, for ULDM detection in
MAGIS-100 Abe  _et al._ (2021). The goal of this section is to show the
expected signals of such a setup, in order to compare it directly with
gradiometers, as the one expected to be used in AION-10 Badurina  _et al._
(2022).

Contrary to usual gradiometers which only use one single species of atom, we
consider two different atom isotopes, each of them undergoing individually the
interferometric path described earlier in Sec. II.2.2. The setup is presented
in Fig. 4. The two interferometers overlap at the same elevation, so the
experiment will test the universality of free fall between the two isotopes.
This setup employs single photon transition (meaning we will consider only
optical transitions) and measures the differential acceleration between two
isotopes, so we will refer to this setup as Single Photon transition Isotope
Differential (SPID).

Typically, an optical transition has a frequency of the order of $\sim$1014 Hz
while the typical frequency shift between two isotopes is of order $\sim$109
Hz (i.e. typically 5-6 orders of magnitude smaller than the nominal transition
frequency, see Takano  _et al._ (2017) for Sr). Then, a unique laser source
can be used in this setup and separated in two different beams: one which is
directly used in the laser-atom interactions inside the interferometer of the
first isotope; and the other one whose frequency is shifted, e.g. using an
electro-optic modulator (EOM), in order to interact with the second isotope.

Figure 4: Simplified setup for the SPID experiment. For the sake of
simplicity, we assume two different isotopes with respective transition
frequencies $f_{1}=f_{\mathrm{tr}}=10^{14}$ Hz,
$f_{2}=f_{\mathrm{tr}}+f_{\mathrm{EOM}}=(1+10^{-5})\times 10^{14}$ Hz. The
laser is locked on the transition frequency of isotope 1 and split in three:
the first output is used for the isotope 1 AI (in red), the second output
enters an Acousto-Optic Modulator (AOM) to shift its frequency to account for
the Doppler shift $f_{\mathrm{Dop}}$ of isotope 1 (in blue). The third output
enters an EOM to shift its frequency by $f_{\mathrm{EOM}}$ to account for the
isotope shift in order to be used with the second isotope. Then, it is again
split in two : one output which corresponds to the transition frequency of
isotope 2 (in dashed red) and the other one enters a second AOM to account for
the Doppler shift of isotope 2 (in dashed blue).

The calculation of the differential phase shift for such a setup can be
computed following the exact same methodology as in Appendix A. If we assume
the optical transition frequencies to be close, i.e
$\omega^{0}_{A}\approx\omega^{0}_{B}\equiv\omega_{0}$ and the initial velocity
to be much larger than the velocity kick333We recall that we are working in
the DM rest-frame (galactic frame) so $v_{0}\sim 10^{-3}c$. $v_{0}\gg\hbar
k_{\mathrm{eff}}/m^{0}$, the differential phase shift between the two
interferometers reads

$\displaystyle\Delta\Phi^{\mathrm{SPID}}_{\mathrm{AB}}\approx-\frac{4nv_{0}\omega_{0}}{\omega
c}\left((Q^{A}_{M}-Q^{B}_{M})\hat{e}_{v}\cdot\hat{e}_{\mathrm{kick}}\sin^{2}\left(\frac{\omega
T}{2}\right)\sin(\omega T+\phi_{0})-\frac{\omega
L}{2v_{0}}(Q^{A}_{\omega}-Q^{B}_{\omega})\sin\left(\frac{\omega
T}{2}\right)\sin\left(\frac{\omega T}{2}+\phi_{0}\right)\right)\,$ (10a)
$\displaystyle\rightarrow|\Delta\Phi^{\mathrm{SPID}}_{\mathrm{AB}}|\approx\frac{4n\omega_{0}v_{0}}{\omega
c}\left|(Q^{A}_{M}-Q^{B}_{M})\hat{e}_{v}\cdot\hat{e}_{\mathrm{kick}}\right|\sin^{2}\left(\frac{\omega
T}{2}\right)$ (10b)

where we kept only lowest order terms in $\omega L/c$. In the second line, we
have neglected the second term since the frequency charge of both isotopes are
the same in the theoretical models considered in the next section. In
addition, we did not take into account the effect of an oscillating EOM
frequency through $Q^{\mathrm{EOM}}_{\omega}\neq 0$. Nonetheless, its effect
is suppressed by a factor $\omega L/v_{0}\times\Delta\omega^{0}/\omega^{0}\leq
10^{-7}$ compared to the leading term, where $\Delta\omega^{0}$ is the isotope
shift. In such a case (and similarly for $L=0$), we recover the Bragg
interferometers phase shift derived in Eq. (7a) with
$n\omega_{0}/c=k_{\mathrm{eff}}$, as expected.

The advantage of this setup compared to usual gradiometers can be immediately
visualized by comparing Eq. (10b) and Eq. (9): the SPID setup does not suffer
from a small factor $\omega\Delta r/v_{0}$. More precisely, the ratio of
amplitude of signals between this variation and usual gradiometers is roughly
$v_{0}/(\omega\Delta
r)\times(Q^{A}_{M}-Q^{B}_{M})/Q^{A}_{\omega}\sim\left(1\mathrm{\>rad.s^{-1}}/\omega\right)$
for the following values: $\Delta r\sim 5$ m, $v_{0}\sim 10^{-3}c$,
$\hat{e}_{v}\cdot\hat{e}_{\mathrm{kick}}\sim\mathcal{O}(1)$ and mass and
frequency charges presented in Table. 2). This implies that, at low angular
frequency ($\omega<1$ rad/s), the signal of the single photon transition
isotope differential AI will be larger than the one in a gradiometer. The
simple reason for this difference in signal amplitudes is that the gradiometer
leading order phase shift in Eq. (9) is proportional to the frequency charge
$Q_{\omega}$, or in other words, it is proportional to the time in which the
wavepackets are in their excited state. As can be noticed from Fig. 2, this
happens for a limited time, of the order of $nL/c$. Conversely, the signal
amplitude of the SPID variation is proportional to mass charges $Q_{M}$ whose
effect is imprinted in the phase shift, whatever the internal state, i.e for a
time $\sim T$.

In addition, as we shall see in Sec. III, in some of the theoretical models
considered, the frequency charge of optical transition is zero,
$Q_{\omega}=0$, making the gradiometer insensitive to such models, at leading
order, while for all models of interest, the difference of mass charges of
isotopes $Q^{A}_{M}-Q^{B}_{M}\neq 0$ (see Table. 2).

In the next sections, we show how these theoretical results Eqs. (5), (8),(9)
and (10b) could constrain dark matter models which would produce such time
oscillations, namely axions and dilatons.

##  III Mass and frequency charges for some DM candidates

In the previous section, we have considered a particular phenomenological time
oscillation of atomic rest mass and transition frequencies, see Eq. (1). We
have derived the observable signatures produced by such an oscillations on
various types of observables. In this section, we will focus on two
particularly well studied and motivated DM candidates: the pseudo-scalar field
axion and the scalar field dilaton. We will show that these DM candidates
induce oscillations of atomic rest mass and transition frequency, and derive
the expression of the mass and frequency charges appearing in Eq. (1) from the
fundamental parameters of the theory.

###  III.1 Axion

The QCD axion was originally proposed to solve the strong CP problem Peccei
and Quinn (1977) by promoting the $\theta$-vacuum of QCD to a dynamical field
Marsh (2016). A new U(1) symmetry is introduced and spontaneously broken at a
given energy scale $f_{a}$. The axion $a$ is the pseudo Goldstone boson of
mass $m_{a}$ of this broken symmetry. It is defined as

$\displaystyle\theta=\sqrt{\hbar c}\frac{a}{f_{a}}\,.$ (11)

The QCD axion mass is inversely proportional to $f_{a}$. A particle similar to
the QCD axion, with the exception that it does not solve the strong CP
problem, i.e whose mass is not proportional to $1/f_{a}$ is called an Axion
Like Particle (ALP).

A scalar ALP field follows the usual Klein-Gordon equation in an expanding
universe, and when $m_{a}c^{2}/\hbar\gg H$, the field oscillates at its
Compton frequency $\omega_{a}$ and behaves as Cold Dark Matter Marsh (2016).

Let us consider an ALP in the galactic centered reference frame

$a=a_{0}\cos(\omega_{a}t+\phi_{a})\,.$ (12)

with $a_{0},\omega_{a},\phi_{a}$ respectively the amplitude, frequency and
phase of the field. In this equation, the field $a$ has units of
$\sqrt{\mathrm{J/m}}$.

Similarly to Kim and Perez (2022), we consider a model where the ALP and
gluons couple as follows

$\mathcal{L}_{\mathrm{int}}=\frac{\sqrt{\hbar
c}g^{2}_{s}}{32\pi^{2}}\frac{a}{f_{a}}\tilde{G}^{\mu\nu}G_{\mu\nu}\,,$ (13)

in SI units and where $G_{\mu\nu}$ is the gluon strength tensor field,
$\tilde{G}^{\mu\nu}$ its dual and $g_{s}$ is the dimensionless strong
coupling.

If the ALP is identified as DM, its amplitude of oscillation $a_{0}$ is fixed
by the local DM density $\rho_{\mathrm{DM}}$ i.e.444The scalar field $a$
energy density is
$\rho=\langle\frac{\dot{a}^{2}}{2c^{2}}+\frac{\omega^{2}_{a}a^{2}}{2c^{2}}\rangle=\frac{\omega^{2}_{a}a^{2}_{0}}{2c^{2}}$
where $\langle\rangle$ represents the average over several field oscillations.

$a_{0}=\frac{\sqrt{2\rho_{\mathrm{DM}}}c}{\omega_{a}}\,.$ (14)

####  III.1.1 Axionic mass charge

In Kim and Perez (2022), it is shown that the interaction Lagrangian from Eq.
(13) induces a dependency of the mass of pions to the axion field, which
implies a dependency of the mass of nucleons and atomic binding energy on the
axion field. As a consequence, the mass of any atom will also depend on the
axion field and its coupling strength with gluons. In this sense, we can
define the dimensionless axionic mass charge of the atom A as

$\displaystyle[Q^{A}_{M}]_{a}=\frac{\partial\ln
m_{A}}{\partial\left(\theta^{2}\right)}\,,$ (15)

such that the mass of the atom A has a small time dependency

$\displaystyle dm_{A}(t)$
$\displaystyle=m^{0}_{A}[Q^{A}_{M}]_{a}d\theta^{2}(t)\,,$ (16a) with
$m^{0}_{A}$ the nominal mass of the atom. We can use Eqs. (11), (12) and (14)
to derive the full expression of the mass of the atom A, i.e $\displaystyle
m_{A}(t)=m^{0}_{A}(1+[Q^{A}_{M}]_{a}\theta^{2}(t))\,$ (16b)
$\displaystyle\equiv m^{0}_{A}\left(1+\frac{\hbar
c^{3}\rho_{\mathrm{DM}}[Q^{A}_{M}]_{a}}{f^{2}_{a}\omega^{2}_{a}}\cos(2\omega_{a}t+2\phi_{a})\right)\,,$
(16c)

at first order in axionic mass charge and where we made a reparametrization of
the nominal mass of the atom at the last line, i.e $m^{0}_{A}\rightarrow
m^{0}_{A}\left(1+\hbar
c^{3}\rho_{\mathrm{DM}}[Q^{A}_{M}]_{a}/f^{2}_{a}\omega^{2}_{a}\right)$. The
time dependency of atomic masses induced by an ALP can therefore be
parametrized using the parametrization used in Sec. II (see Eq. (1a)) by using
the changes of variables $\omega\rightarrow 2\omega_{a}$ and
$Q^{A}_{M}\rightarrow\hbar
c^{3}\rho_{\mathrm{DM}}[Q^{A}_{M}]_{a}/f^{2}_{a}\omega^{2}_{a}$.

Eq. (15) describes how much the mass of an atom depends on the axion field. In
the following, we derive the expression of the axionic charge of an atom. To
do so, we will combine the work of Kim and Perez (2022) who derived how the
pion mass depends on the axion field with the explicit calculation of the
dependency of the nucleons rest mass and the binding energy on the pion mass
from Damour and Donoghue (2010).

The rest mass of an atom, with charge number Z and neutron number N, can be
parameterized as

$\displaystyle m_{\mathrm{atom}}$
$\displaystyle=m_{\mathrm{const.}}+E_{\mathrm{bind}}\,$ (17a)
$\displaystyle=Z(m_{p}+m_{e})+Nm_{n}+E_{\mathrm{bind}}\,,$ (17b)

where $m_{p},m_{e},m_{n}$ are respectively the rest masses of the proton, the
electron and the neutron, and where $m_{\mathrm{const.}},E_{\mathrm{bind}}$
represent respectively the rest mass of the particle constituents of the atom
(proton, neutron, electron) and the nuclear binding energy.

In Kim and Perez (2022), it is shown the pion mass is $\theta$ dependent and
influence the nucleon mass $m_{\mathrm{N}}$ through

$\displaystyle\frac{\partial\ln m_{N}}{\partial\ln m_{\pi}^{2}}$
$\displaystyle\approx 0.06\,,$ (18a) $\displaystyle\frac{\partial\ln
m_{\pi}^{2}}{\partial\left(\theta^{2}\right)}$
$\displaystyle=-\frac{m_{u}m_{d}}{2(m_{u}+m_{d})^{2}}=-0.109\,$ (18b)
$\displaystyle\Rightarrow\frac{\partial\ln
m_{N}}{\partial\left(\theta^{2}\right)}$ $\displaystyle\approx-0.065\,,$ (18c)

such that, for an atom made of (N+Z) nucleons, the contribution of the
nucleons rest mass to the atom rest mass to the axionic charge is given by

$[Q^{\mathrm{atom}}_{M}]_{a}\Big{|}_{\mathrm{const.}}=\frac{\partial\ln
m_{N}}{\partial\left(\theta^{2}\right)}=Q_{N}\approx-0.065\,,$ (19)

meaning that it is independent of the number of nucleons inside the atom, or
in other words of the atomic species. If we consider only the impact of the
axion field on the rest mass of nucleons inside the atoms, the macroscopic
observable (5) would be independent of the atomic species. In other word, at
this level of approximation (i.e considering only the rest mass of nucleons),
following Eq. (4), all atom will undergo an additional acceleration,
proportional to the universal axionic charge, implying e.g. a null
differential acceleration Eq. (5). However, following Eq. (7), this nucleon
rest mass contribution would induce a non-zero phase shift in AI, but not
relative to a violation of the UFF, as explained earlier and in agreement with
the macroscopic measurement.

Let us now focus on the contribution of the binding energy of the nuclei to
the axionic mass charge and show that it is composition dependent.

As computed in Damour and Donoghue (2010), the binding energy of the nuclei
depends to first order on the mass of the pions and therefore, following Eq.
(18), on the ALP. We will now use the results from Section IV of Damour and
Donoghue (2010) to infer the analytical expression of the dependency of the
binding energy to the ALP and to the mass number $A$ and the charge number
$Z$.

Three different interactions contribute to the binding energy Damour and
Donoghue (2010): the central force $E_{\mathrm{central}}$ coming from the
isospin symmetric central nuclear force, the asymmetry energy
$E_{\mathrm{asym}}$, i.e the residual energy from the asymmetry between
neutrons and protons inside the nucleus and the Coulomb force
$E_{\mathrm{Coulomb}}$ depending on how tightly the nucleons are packed
together, leading to

$\displaystyle E_{\mathrm{bind}}$
$\displaystyle=E_{\mathrm{central}}+E_{\mathrm{asym}}\frac{(A-2Z)^{2}}{A}+\,$
$\displaystyle E_{\mathrm{Coulomb}}\frac{Z(Z-1)}{A^{1/3}}\,,$ (20)

In appendix B, we review how each of these contribution depends on the mass of
the pions and therefore, using Eq. (18) on the axion field, to derive the non-
universal axionic mass charge of an atom due to the binding energy, which
reads

$\displaystyle[Q^{\mathrm{atom}}_{M}]_{a}$
$\displaystyle\approx-0.070+10^{-3}\times\left(\frac{3.98}{A^{1/3}}+\right.\,$
$\displaystyle\left.2.22\frac{(A-2Z)^{2}}{A^{2}}+0.015\frac{Z(Z-1)}{A^{4/3}}\right)\,.$
(21)

####  III.1.2 Axionic frequency charge

In this section, we derive the axionic frequency charge (see Eq. 1b) for
various transitions frequencies. We will consider two different types of
atomic transitions, which do not depend on the same physical parameters: (i)
the hyperfine transitions and (ii) the optical ones.

Similarly to the previous section, we define the axionic frequency charge of
the atom as

$\displaystyle[Q^{\mathrm{atom}}_{\omega}]_{a}$
$\displaystyle\equiv\frac{\partial\ln\omega_{\mathrm{atom}}}{\partial(\theta)^{2}}\,,$
(22)

and derive the full expression of the frequency of the atom A as

$\displaystyle\omega_{\mathrm{atom}}(t)\equiv\omega^{0}_{\mathrm{atom}}\left(1+\frac{\hbar
c^{3}\rho_{\mathrm{DM}}[Q^{\mathrm{atom}}_{\omega}]_{a}}{f^{2}_{a}\omega^{2}_{a}}\cos(2\omega_{a}t+2\phi_{a})\right)\,.$
(23)

Again, one can notice that this time dependency can be parametrized using Eq.
(1b) by the changes of variables $\omega\rightarrow 2\omega_{a}$ and
$Q^{A}_{\omega}\rightarrow\hbar
c^{3}\rho_{\mathrm{DM}}[Q^{\mathrm{atom}}_{\omega}]_{a}/f^{2}_{a}\omega^{2}_{a}$.

We will first consider hyperfine atomic transitions
($\omega^{H}_{\mathrm{atom}}$) which are impacted by the axion-gluon coupling
from Eq. (13) as Kim and Perez (2022)

$\displaystyle\frac{\partial\ln\omega^{H}_{\mathrm{atom}}}{\partial(\theta)^{2}}$
$\displaystyle=\left(\frac{\partial\ln g}{\partial\ln
m^{2}_{\pi}}-\frac{\partial\ln m_{p}}{\partial\ln
m^{2}_{\pi}}\right)\frac{\partial\ln m^{2}_{\pi}}{\partial(\theta)^{2}}\,,$
(24)

where $g$ is the nucleon g-factor whose dependence on pion mass is given by
Kim and Perez (2022)

$\displaystyle\frac{\partial\ln g}{\partial\ln m^{2}_{\pi}}$
$\displaystyle=K_{n}\frac{\partial\ln g_{n}}{\partial\ln
m^{2}_{\pi}}+K_{p}\frac{\partial\ln g_{p}}{\partial\ln
m^{2}_{\pi}}-0.17K_{b}\,,$ (25a) with $\displaystyle\frac{\partial\ln
g_{n}}{\partial\ln m^{2}_{\pi}}$ $\displaystyle\approx-0.25\,,$ (25b)
$\displaystyle\frac{\partial\ln g_{p}}{\partial\ln m^{2}_{\pi}}$
$\displaystyle\approx-0.17\,,$ (25c)

where $g_{n}$ ($g_{p}$) respectively the neutron (proton) gyromagnetic factors
and $K_{n}$, $K_{p}$ and $K_{b}$, coefficients computed from chiral
perturbation theory (nuclear shell model). These coefficients are not
measurable, but they are related to observable parameters, namely, the
sensitivity coefficient of the nuclear magnetic moment to light quarks masses
$\kappa_{q}$, to strange quarks masses $\kappa_{s}$ and to light quarks masses
over QCD energy scale $\kappa$ through Flambaum and Tedesco (2006)

$\displaystyle\kappa_{q}$ $\displaystyle=-0.118K_{n}-0.087K_{p}\,,$ (26a)
$\displaystyle\kappa_{s}$ $\displaystyle=0.0013K_{n}-0.013K_{p}\,,$ (26b)
$\displaystyle\kappa$ $\displaystyle=-0.12K_{n}-0.10K_{p}-0.11K_{b}\,.$ (26c)

Using Eq. (18), (24), (25) and (26), we define the dimensionless axionic
frequency charge of the atom as

$\displaystyle[Q^{\mathrm{atom}}_{\omega}]_{a}$
$\displaystyle\approx\left(-16.8\kappa-5.69\kappa_{q}+25.1\kappa_{s}+0.65\right)\times
10^{-2}\,.$ (27)

In particular, Eq. (27) is relevant for 87Rb hyperfine transition, whose
associated $\kappa$ parameters are
$\kappa=-0.016,\kappa_{q}=-0.046,\kappa_{s}=-0.010$ Flambaum and Tedesco
(2006).

Let us now consider optical transitions whose frequency does depend neither on
the nucleon g-factor nor on the proton mass at lowest order. This implies that
the axionic frequency charge in that case is 0. Higher order contributions
would lead to non-zero axionic frequency charge, in particular if one
considers its dependence on the fine structure constant $\alpha$ which arises
at loop level Beadle  _et al._ (2023), and is therefore highly suppressed.

###  III.2 Dilaton scalar field

44footnotetext: The pairing energy is subdominant for all atoms compared to
the other interactions Damour and Donoghue (2010), therefore we neglect it.

Another example of scalar dark matter candidate is the dilaton Arvanitaki  _et
al._ (2015); Stadnik and Flambaum (2015); *stadnik:2015yu; *Flambaum16. In the
most general case, this field is non universally coupled to matter, which
makes some “constants” of Nature to evolve with the field Damour and Polyakov
(1994); Damour  _et al._ (2002); Damour and Donoghue (2010); Arvanitaki  _et
al._ (2015); Stadnik and Flambaum (2015); *stadnik:2015yu; *Flambaum16; Hees
_et al._ (2018).

In this paper, we will only consider linear coupling555Quadratic coupling has
also been widely studied, e.g in Stadnik and Flambaum (2015, 2016); Hees  _et
al._ (2018). between the dimensionless dilatonic field $\phi$ and several SM
fields, namely the matter fermions, the gluon and the photon fields through
the following Lagrangian Damour and Donoghue (2010)

$\displaystyle\mathcal{L}_{\mathrm{int}}$
$\displaystyle=\phi\left[\frac{d_{e}}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu}-\frac{d_{g}\beta_{3}}{2g_{s}}G_{\mu\nu}G^{\mu\nu}-\right.\,$
$\displaystyle\left.\sum_{i=e,u,d}(d_{m_{i}}+\gamma_{m_{i}}d_{g})m_{i}\bar{\Psi}_{i}\Psi_{i}\right]\,,$
(28)

where $\mu_{0}$ is the magnetic permeability, $F_{\mu\nu}$ the electromagnetic
strength tensor, $\beta_{3}$ the running coupling function of the QCD gauge
coupling and $\Psi_{i}$ the different fermionic fields. All the $d_{i}$ are
dimensionless couplings between the dilaton and the SM fields which will,
amongst others, modify the mass of atoms through the variation of several
“constants” of Nature, namely the fine structure constant $\alpha$, the
electron mass $m_{e}$, the quark masses and the QCD energy scale
$\Lambda_{\mathrm{QCD}}$ Damour and Donoghue (2010). More generally, a
“constant” $X$ is modified by Eq. (III.2) as Damour and Donoghue (2010)

$\displaystyle X\rightarrow X\left(1+d_{X}\phi\right)\,.$ (29)

At cosmological scales, the dilaton behaves as a classical oscillating field

$\displaystyle\phi=\phi_{0}\cos(\omega_{\phi}t+\Phi)\,,$ (30)

whose amplitude $\phi_{0}$ is directly related to the DM energy density, i.e
Arvanitaki  _et al._ (2015); Stadnik and Flambaum (2015); Hees  _et al._
(2018)

$\displaystyle\phi_{0}=\frac{\sqrt{8\pi
G\rho_{\mathrm{DM}}}}{\omega_{\phi}c}\,,$ (31)

where $G$ is the Newton’s gravitational constant.

####  III.2.1 Dilatonic mass charges

The dependency of the constants $X$ on the scalar field $\phi$, see Eq. (29),
induces variations of the rest mass of an atom which is characterized by
Damour and Donoghue (2010)

$\displaystyle[Q^{\mathrm{atom}}_{M}]_{\phi}$ $\displaystyle=\frac{\partial\ln
m_{\mathrm{atom}}}{\partial\phi}\,.$ (32)

In Damour and Donoghue (2010), it has been shown that the
$[Q^{\mathrm{atom}}_{M}]_{\phi}$ coupling function depends on the dilatonic
charges $Q_{M,X}$ and on the dilaton/matter coupling coefficients $d_{X}$
as666Note that following Damour and Donoghue (2010), we neglect the
contribution of the QED trace anomaly that would add another small
contribution Nitti and Piazza (2012).

$\displaystyle[Q^{\mathrm{atom}}_{M}]_{\phi}$
$\displaystyle=\left(Q^{\mathrm{atom}}_{M,m_{e}}(d_{m_{e}}-d_{g})+Q^{\mathrm{atom}}_{M,e}d_{e}+\right.\,$
$\displaystyle\left.Q^{\mathrm{atom}}_{M,\hat{m}}(d_{\hat{m}}-d_{g})+Q^{\mathrm{atom}}_{M,\delta
m}(d_{\delta m}-d_{g})\right)\,,$ (33)

with the expressions of the dilatonic charges given by

$\displaystyle Q_{M,\hat{m}}$
$\displaystyle=0.093-\frac{0.036}{A^{1/3}}-0.02\frac{(A-2Z)^{2}}{A^{2}}-\,$
(34a) $\displaystyle 1.4\times 10^{-4}\frac{Z(Z-1)}{A^{4/3}}\,,$
$\displaystyle Q_{M,\delta m}$ $\displaystyle=0.0017\frac{A-2Z}{A}\,,$ (34b)
$\displaystyle Q_{M,m_{e}}$ $\displaystyle=5.5\times 10^{-4}\frac{Z}{A}\,,$
(34c) $\displaystyle Q_{M,e}$
$\displaystyle=\left(-1.4+8.2\frac{Z}{A}+7.7\frac{Z(Z-1)}{A^{4/3}}\right)\times
10^{-4}\,,$ (34d)

where A and Z represent respectively the mass and atomic numbers of the atom,
$\hat{m}$ is the mean of the up and down quark masses and $\delta m$ their
difference.

####  III.2.2 Dilatonic frequency charges

In a similar way, the dilatonic frequency charge is defined as

$\displaystyle[Q^{\mathrm{atom}}_{\omega}]_{\phi}$
$\displaystyle=\frac{\partial\ln\omega_{\mathrm{atom}}}{\partial\phi}\,.$ (35)

Similarly to the mass charge, the frequency charge depends on dilatonic
charges $Q_{\omega,X}$ and on the dilaton/matter coupling coefficients $d_{X}$
(see e.g. Hees  _et al._ (2018))

$\displaystyle[Q^{\mathrm{atom}}_{\omega}]_{\phi}$
$\displaystyle=\left(Q^{\mathrm{atom}}_{\omega,m_{e}}\left(d_{m_{e}}-d_{g}\right)+Q^{\mathrm{atom}}_{\omega,e}d_{e}+\right.\,$
$\displaystyle\left.Q^{\mathrm{atom}}_{\omega,\hat{m}}\left(d_{\hat{m}}-d_{g}\right)+Q^{\mathrm{atom}}_{\omega,\delta
m}\left(d_{\delta m}-d_{g}\right)\right)\,.$ (36)

In case of hyperfine transitions, we have
$f^{hyp}_{\mathrm{atom}}\propto\alpha^{k_{\alpha}}(m_{e}/m_{p})(m_{q}/\Lambda_{\mathrm{QCD}})^{k_{q}}$
Flambaum and Tedesco (2006), where $k_{\alpha},k_{q}$ represent respectively
the sensitivity coefficients of the hyperfine transition to the fine structure
constant and to the ratio of the light quark masses $m_{q}$ to the QCD mass
scale $\Lambda_{\mathrm{QCD}}$ ratio. Subsequently, the corresponding
dilatonic frequency charges are

$\displaystyle Q^{\mathrm{hyp}}_{\omega,m_{e}}$ $\displaystyle=1\,,$ (37a)
$\displaystyle Q^{\mathrm{hyp}}_{\omega,e}$ $\displaystyle=k_{\alpha}\,,$
(37b) $\displaystyle Q^{\mathrm{hyp}}_{\omega,\hat{m}}$
$\displaystyle=-0.037+k_{q}\,,$ (37c) $\displaystyle
Q^{\mathrm{hyp}}_{\omega,\delta m}$ $\displaystyle=0.0017+k_{q}\,,$ (37d)

where we used the dependency of the proton mass to the light quark masses
$\partial\ln m_{p}/\partial\ln\hat{m}=0.037$, $\partial\ln
m_{p}/\partial\ln\delta m=-0.0017$ Damour and Donoghue (2010).

The frequencies of optical transition dependence depend mainly on the electron
mass and on the fine structure constant $f^{opt}_{\mathrm{atom}}\propto
m_{e}\alpha^{2+\epsilon_{\mathrm{atom}}}$ Arvanitaki  _et al._ (2015) (where
$\epsilon_{\mathrm{atom}}$ accounts for the relativistic correction of the
dependence of $f_{\mathrm{atom}}$ to $\alpha$). Therefore, the frequency
dilatonic charges write

$\displaystyle Q^{\mathrm{opt}}_{\omega,m_{e}}$ $\displaystyle=1\,,$ (38a)
$\displaystyle Q^{\mathrm{opt}}_{\omega,e}$
$\displaystyle=2+\epsilon_{\mathrm{atom}}\,,$ (38b) $\displaystyle
Q^{\mathrm{opt}}_{\omega,\hat{m}}$ $\displaystyle=0\,,$ (38c) $\displaystyle
Q^{\mathrm{opt}}_{\omega,\delta m}$ $\displaystyle=0\,.$ (38d)

In conclusion, similarly to Sec. III.1, the oscillating mass and transition
frequency of the atom A in the dilatonic framework can be written as

$\displaystyle m_{\mathrm{atom}}(t)$ $\displaystyle\equiv
m^{0}_{\mathrm{atom}}\left(1+\frac{\sqrt{8\pi
G\rho_{\mathrm{DM}}}[Q^{\mathrm{atom}}_{M}]_{\phi}}{\omega_{\phi}c}\cos(\omega_{\phi}t+\Phi)\right)\,,$
(39a) $\displaystyle\omega_{\mathrm{atom}}(t)$
$\displaystyle\equiv\omega^{0}_{\mathrm{atom}}\left(1+\frac{\sqrt{8\pi
G\rho_{\mathrm{DM}}}[Q^{\mathrm{atom}}_{\omega}]_{\phi}}{\omega_{\phi}c}\cos(\omega_{\phi}t+\Phi)\right)\,,$
(39b)

where we have used Eqs. (30-32) and (35) and where both mass and frequency
charges are respectively given by Eqs. (III.2.1) and (III.2.2). These time
dependencies can easily be mapped into the parmeterization from Eq. (1) used
in Sec. II by the simple changes of variables $\omega\rightarrow\omega_{\phi}$
and $Q^{A}_{M}$ (resp. $Q^{A}_{\omega}$) $\rightarrow\left(\sqrt{8\pi
G\rho_{\mathrm{DM}}}/\omega_{\phi}c\right)[Q^{A}_{M}]_{\phi}$ (resp.
$[Q^{A}_{\omega}]_{\phi}$).

##  IV Description of existing and future experiments

The previous sections were dedicated to a theoretical derivation of the
signatures induced by some models of ULDM on several types of experiments. In
this section, we will present several existing or future experiments, based
either on the classical measurement of differential acceleration between
macroscopic bodies or on atom interferometry. The main goal of this section is
to present an overview of the experiments, their main characteristic and also
provide an estimate of their noise level and integration time. These will be
needed in the future section to estimate the sensitivity of the experiments to
the various axionic and dilatonic couplings.

The experiments considered in this paper are the following:

  1. 1.

the MICROSCOPE space experiment Touboul  _et al._ (2017, 2022).

  2. 2.

the AI experiment operating at Stanford Asenbaum  _et al._ (2020).

  3. 3.

the AION-10 experiment Badurina  _et al._ (2020).

  4. 4.

the SPID variation presented in Sec. II.2.2

The first two are existing experiments with available data, while AION-10 is a
proposed experimental scheme with futuristic experimental parameters. We will
derive the noise level of the SPID variation, in particular by taking into
account the effect of the EOM, which was not studied for the expected
sensitivity of MAGIS-100 Abe  _et al._ (2021). Since we aim at comparing this
variation with the gradiometer one of AION-10, we will also assume the same
experimental parameters as AION-10.

####  IV.0.1 MICROSCOPE

MICROSCOPE is a space mission that was launched to test the UFF in space
Touboul  _et al._ (2017). The payload contains (amongst other) two concentric
test masses, one made of Platinum and the other one made of Titanium. By
closely monitoring the relative displacement between the two test masses along
the symmetry axis, one can measure the differential acceleration between them
and estimate the Eötvos $\eta$ parameter. The final result of MICROSCOPE is
Touboul  _et al._ (2022)

$\eta=(-1.5\pm 2.7)\times 10^{-15}\,,$ (40a) at a 1$\sigma$ confidence-level.

The full MICROSCOPE dataset used to constrain the UFF is made of 17 sessions
Rodrigues  _et al._ (2022). The noise PSD of the differential acceleration of
the session 404 is given by le Bars  _et al._ (2019)

$\displaystyle S_{a}(f)=2.2\times 10^{-24}f^{-1}+2.3\times
10^{-17}f^{4}\>\mathrm{\left(m/s^{2}\right)^{2}/Hz}\,,$ (40b)

for frequencies between $10^{-5}$ Hz and 0.3 Hz. The $f^{-1}$ slope noise
comes from thermal effects of the gold wire connecting the test masses to the
cage, while the high frequency noise in $f^{4}$ is the second derivative of
the position measurement white noise Touboul  _et al._ (2017).

The UFF signal is modulated by spinning the satellite at a frequency chosen to
minimize the noise, $f_{\mathrm{EP}}\sim 3$ mHz. Note that $f_{\mathrm{EP}}$
is in reality a linear combination of the spin frequency $f_{\mathrm{spin}}$
and orbital frequency $f_{\mathrm{orb}}$, with $f_{\mathrm{spin}}\gg
f_{\mathrm{orb}}\sim 1.5\times 10^{-4}$ Hz Touboul  _et al._ (2017) for
session 404. Three different spin frequencies exist depending on the session
Bergé (2023), such that one should take both contributions into account for
the determination of $f_{\mathrm{EP}}$ for other sessions.

The bucket frequency of this acceleration noise corresponds approximately to
$f_{\mathrm{bucket}}\sim 30$ mHz le Bars  _et al._ (2019); Touboul  _et al._
(2017).

To take into account the contribution of the full experiment, i.e the 17
different sessions, we make the hypothesis that all of them have the same
level of acceleration noise Eq. (40b) while we consider the orbital period to
be constant $T_{\mathrm{single\>orbit}}\approx 5946$ s Rodrigues  _et al._
(2022). This hypothesis is sufficient for the rough sensitivity analysis we do
here, but will have to be revisited when doing a complete MICROSCOPE data
analysis in search for DM candidates.

Using the total number of orbits of the experiment, i.e
$N_{\mathrm{orbits}}=1362$ Rodrigues  _et al._ (2022), we find that the total
integration time is $T_{\mathrm{int}}=T_{\mathrm{single\>orbit}}\times
N_{\mathrm{orbits}}\equiv 8.1\times 10^{6}$ s.

As an experiment measuring the differential acceleration between two test
masses, the corresponding theoretical signal is given by Eq. (5).

####  IV.0.2 Atom interferometry in the Stanford Tower

The most stringent constrain on the Eötvös parameter $\eta$ obtained from atom
interferometry experiment is achieved using a Bragg atom interferometry
experiment in the Stanford Tower. In this experiment, the relative
acceleration of freely falling clouds of two isotopes of Rubidium (85Rb and
87Rb) is measured Asenbaum  _et al._ (2020). The differential interferometric
phase measurement between these two inteferometers leads to a constraint on
$\eta$ given by Asenbaum  _et al._ (2020)

$\displaystyle\eta=(1.6\pm 5.2)\times 10^{-12}\,.$ (41a) Both atoms are
launched upwards inside the Stanford Tower for an interferometric sequence of
total duration $2T=1.91$ s, corresponding to the total time for the atoms to
fall back Asenbaum  _et al._ (2020).

Double diffraction interferometry is performed using two lasers both resonant
with the $|5^{2}S_{1/2}\rangle\rightarrow|5^{2}P_{3/2}\rangle$ 87Rb
transition, i.e. their wavevectors are $k_{1}\approx k_{2}\approx
2\pi/\lambda$, with $\lambda=780$ nm. Three different momentum splittings have
been used: $\\{4\hbar k,8\hbar k,12\hbar k\\}$. All are in agreement with no
EP violation Asenbaum  _et al._ (2020). In the following, we will consider the
highest momentum transfer since this will enhance the signal searched for,
which corresponds to using an effective wavevector
$k_{\mathrm{eff}}=24\pi/\lambda$ in Eq. (7a).

The resolution per shot on the differential acceleration is $\sigma_{\Delta
a}=1.4\times 10^{-11}g$ Asenbaum  _et al._ (2020), with $g=9.81$ m/$s^{2}$ the
Earth gravitational acceleration on ground. We assume white noise (see below)
with a corresponding acceleration noise PSD

$\displaystyle S_{\Delta a}=\frac{2\sigma^{2}_{\Delta
a}}{f_{s}}\approx\left(7.7\times 10^{-11}\ g\right)^{2}\ \mathrm{Hz}^{-1}\,,$
(41b)

with $f_{s}$ the sampling frequency of the experiment, defined as
$f_{s}=1/T_{\mathrm{cycle}}$ where $T_{\mathrm{cycle}}=15$ s is the duration
of one interferometric sequence, including atom preparation, launch and free
fall Asenbaum  _et al._ (2020). The raw measurement in this experiment is
actually a differential phase shift $\Delta\Phi$ which is related to the
differential acceleration between the atoms $\Delta a$ through
$\Delta\Phi=k_{\mathrm{eff}}T^{2}\Delta a$ Storey and Cohen-Tannoudji (1994).
Therefore, we can infer the ASD of the phase shift using

$\displaystyle\sqrt{S_{\Phi}}=k_{\mathrm{eff}}T^{2}\sqrt{S_{\Delta a}}\,.$
(41c)

The final uncertainty of this differential acceleration measurement is mostly
limited by electromagnetic effects, coming from the Bragg lasers and non
homogeneous magnetic field Asenbaum  _et al._ (2020). Knowing the final
uncertainty of the differential measurement (41a) we derive the total
“effective” experiment time $T_{\mathrm{int}}$ under our white noise
assumption i.e. assuming that individual experimental cycles are uncorrelated.
Then, the number of cycles $N$ can be derived as

$\displaystyle N=\frac{1}{g^{2}}\left(\frac{\sigma_{\Delta
a}}{\sigma_{\eta}}\right)^{2}\approx 77\,,$ (41d)

and

$\displaystyle T_{\mathrm{int}}=N\times T_{\mathrm{cycle}}\approx
1148\mathrm{\>s}\,.$ (41e)

####  IV.0.3 AION-10

The Atom Interferometer Observatory and Network (AION) is an experimental
program to search for ULDM and gravitational waves in the $10^{-1}-10$ Hz
range using atom interferometry Badurina  _et al._ (2020). AION-10 is a 10
meter-long single-photon atom gradiometer instrument which will use
${}^{87}\mathrm{Sr}$ atoms and that will be built in Oxford Badurina  _et al._
(2022). Contrary to the other experiments previously introduced, AION-10 will
operate in a distant future, assuming a much better control on noise than
current experiments. Following Badurina  _et al._ (2022), we will use the
following experimental parameters ultimately envisaged for AION-10:

$\displaystyle T_{\mathrm{int}}$ $\displaystyle=10^{8}\mathrm{\>s}\,,$ (42a)
$\displaystyle T$ $\displaystyle=0.74\mathrm{\>s}\,,$ (42b) $\displaystyle
S_{\Phi}(f)$ $\displaystyle=10^{-8}\mathrm{\>rad^{2}/Hz}\,,$ (42c)
$\displaystyle n$ $\displaystyle=1000\,,$ (42d)
$\displaystyle\omega_{\mathrm{Sr}}$ $\displaystyle=2.697\times
10^{15}\mathrm{\>rad/s}\,,$ (42e) $\displaystyle\Delta r$
$\displaystyle=4.86\mathrm{\>m}\,,$ (42f) $\displaystyle L$
$\displaystyle=10\mathrm{\>m}\,,$ (42g)

respectively the interrogation time, the free evolution time, the gradiometer
phase noise PSD, the number of LMT kicks, the optical transition frequency
used, the gradiometer separation, and the total size.

####  IV.0.4 SPID

The SPID experiment described in Sec. II.2.2 consists of two colocated
interferometers with an interferometric sequence similar to the one used in
gradiometers, as AION-10 but using two different isotopes.

We consider this setup variation for two different reasons: (i) this setup is
considered as one running mode of MAGIS-100 Abe  _et al._ (2021), and we want
to derive the expected sensitivity of such a large scale experiment to axion
and dilaton signals, and (ii) we would like to compare the gradiometer and the
SPID setups for the same experimental parameters to assess which one is the
best to search for axion and dilaton DM.

For this reason, we consider first the experimental parameters from one
running mode of MAGIS 100. They are listed in Abe  _et al._ (2021), see their
Fig. 3. Their noise levels are much lower than current experiments, similar to
AION-10. MAGIS-100 will use two isotopes of Strontium (87Sr and 88Sr) in a
100-meter baseline. The transition under consideration in Abe  _et al._ (2021)
is $|5^{1}S_{0}\rangle\rightarrow|5^{3}P_{1}\rangle$ with frequency
$\omega_{\mathrm{Sr}}=2.73\times 10^{15}$ rad/s. In terms of acceleration
noise level and order of LMT, we consider the upgraded parameters, i.e

$\displaystyle\sqrt{S_{a}}(f)$ $\displaystyle=6\times
10^{-17}\mathrm{g/}\sqrt{\mathrm{Hz}}\,,$ (43a) $\displaystyle n$
$\displaystyle=1000\,.$ (43b) In addition, for a $L=100$ m high tower, the
free-fall time of atoms is given by $2T=\sqrt{8L/g}$, which implies $T\sim
4.5$ s. Following Eq. (41c), the phase noise PSD is $\displaystyle
S_{\Phi}(f)$
$\displaystyle=\left(\frac{n\omega_{\mathrm{Sr}}T^{2}}{c}\right)^{2}S_{a}(f)\sim
10^{-8}\>\mathrm{rad^{2}/Hz}\,,$ (43c)

i.e a similar phase noise PSD as AION-10. Finally, the full integration time
of the experiment corresponds to 1 year of observation,
$T_{\mathrm{int}}\approx 3.16\times 10^{7}$ s.

The second reason why we consider the SPID setup is to directly compare the
expected sensitivity on axion and dilaton couplings of two experiments with
the exact same experimental parameters but operating a gradiometer on one hand
and the SPID variation on the other hand. For this matter, we will compare the
sensitivity of the current version of AION-10, i.e a gradiometer, with a SPID
setup using the same noise levels. In this case, in addition to the phase
noise, we also include the EOM noise, which is used to shift the frequency of
the laser to account for the isotope shift ($\sim 1$ GHz) as discussed in Sec.
II.2.2 and depicted in Fig. 4. An alternative scheme would be to phase-lock
two lasers with a $\sim 1$ GHz frequency offset. In either case, this setup
generates an additional frequency fluctuation $\sigma_{f}(t)$ in the second
beam, which comes from the EOM internal noise and from the phase noise of the
GHz reference frequency. Whilst at high Fourier frequencies this additional
noise stays well below the interferometer phase noise announced in Badurina
_et al._ (2022) for AION-10, it adds a component, which dominates at low
frequency, at $S^{\mathrm{GHz}}_{\Phi}(f)\approx
10^{-13}\left(f/\mathrm{Hz}\right)^{-2}\,\mathrm{rad^{2}/Hz}$ Barke (2015);
Hartnett  _et al._ (2012); Xie  _et al._ (2017). Since this noise is
uncorrelated with the gradiometer phase noise the total phase noise of the
experiment is then
$S^{\mathrm{SPID}}_{\Phi}(f)=S^{\mathrm{GHz}}_{\Phi}(f)+S^{\mathrm{AION}}_{\Phi}(f)$.
For the remaining experimental parameters, such as the total time of
experiment, the flight time of atoms and the number of LMT kicks, we will
assume the same values as AION-10, see Eq. (42). We will also consider the use
of four different pairs of isotopes of alkaline-Earth neutral atoms commonly
used as optical clocks Ludlow  _et al._ (2015), and we will assess the impact
of the choice of the atoms on the ULDM sensitivity: (i) two bosonic isotopes
of Strontium ${}^{86}\mathrm{Sr}-^{88}\mathrm{Sr}$ based on the
$5s^{2}\>{}^{1}S_{0}\rightarrow 5s5p\>^{3}P_{1}$ transition whose frequency is
$\omega_{\mathrm{Sr}}=2.73\times 10^{15}$ rad/s Le Targat  _et al._ (2006);
Ludlow  _et al._ (2015); (ii) two bosonic isotopes of Calcium
${}^{40}\mathrm{Ca}-^{44}\mathrm{Ca}$ using the $4s4s\>^{1}S_{0}\rightarrow
4s4p\>^{3}P_{1}$ transition with frequency $\omega_{\mathrm{Ca}}=2.87\times
10^{15}$ rad/s Sterr  _et al._ (2004) ; (iii) one mixture of fermionic and
bosonic isotope of Ytterbium ${}^{171}\mathrm{Yb}-^{176}\mathrm{Yb}$ using the
$6s^{2}\>{}^{1}S_{0}\rightarrow 6s6p\>^{3}P_{1}$ transition of frequency
$\omega_{\mathrm{Yb}}=3.39\times 10^{15}$ rad/s Atkinson  _et al._ (2019);
Jones  _et al._ (2023) and (iv) two bosonic isotopes of Mercury,
${}^{196}\mathrm{Hg}-^{202}\mathrm{Hg}$ using the
$6s^{2}\>{}^{1}S_{0}\rightarrow 6s6p\>^{3}P_{1}$ transition of frequency
$\omega_{\mathrm{Hg}}=7.45\times 10^{15}$ rad/s Witkowski  _et al._ (2019).
All these parameters are summarized in Tab. 1.

Isotopes | Transition | Frequency (rad/s)
---|---|---
86Sr, 88Sr | $5s^{2}\>{}^{1}S_{0}\rightarrow 5s5p\>^{3}P_{1}$ | $2.73\times 10^{15}$
40Ca, 44Ca | $4s4s\>^{1}S_{0}\rightarrow 4s4p\>^{3}P_{1}$ | $2.87\times 10^{15}$
171Yb, 176Yb | $6s^{2}\>{}^{1}S_{0}\rightarrow 6s6p\>^{3}P_{1}$ | $3.39\times 10^{15}$
196Hg, 202Hg | $6s^{2}\>{}^{1}S_{0}\rightarrow 6s6p\>^{3}P_{1}$ | $7.45\times 10^{15}$
Table 1: Various isotope pairs used in the SPID setup, for direct comparison
with AION-10

##  V  Sensitivity of various experiments to axion and dilaton fields

In this section, we will derive the sensitivity of the experiments listed in
the previous section on the various axion and dilaton couplings discussed in
Sec. III. These estimates are obtained by comparing the theoretical signal
presented in Sec. II to the published experimental noise presented in Sec. IV
(or predictions regarding AION-10 and MAGIS-100). The expected sensitivity of
MAGIS-100 to another DM candidate, namely to a U(1) $B-L$ field has already
been demonstrated in Abe  _et al._ (2021), but we show that the expected
sensitivity of SPID to axion and dilaton couplings is also very competitive.
We insist that the results obtained in this section needs to be interpreted as
sensitivity analysis or expected experimental reach, while actual constraints
would require careful statistical searches for the signal, with a full
analysis of statistical and systematic uncertainties.

###  V.1 Experimental features impacting the signal

In this section, we will review the various experimental parameters that
impact the theoretical signal presented in Sec. II.

First, the observable signatures induced by an axion or dilaton field depend
on the properties of the atoms used in the experiment. More precisely, they
depend directly on the axionic or dilatonic charges that have been introduced
in Sec. III. In Tab. 2, we present all non-universal axionic and dilatonic
charges of all atomic species used in the set of experiments presented in Sec.
IV. The axionic charges are derived from Eqs. (21) and (27) for hyperfine
atomic transitions. The dilatonic charges are derived from Eqs. (34), (37) and
(38). Note that, in the experiments considered in Sec. IV, 85Rb is only used
in a two-photons transition Bragg-type interferometer which is independent of
the frequency charge at leading order, see Eq. (7a) and comments after.
Similarly, 195Pt and 48Ti are only used in a classical test of the UFF which
is also independent of the frequency charge (at leading order, see Eq. (4).
For these reasons, the frequency charges of these atoms are not provided in
Tab. 2.

Second, as can be noticed from Eqs. (5), (7) and (10b), some observables
depend on the initial velocity of the bodies. Note that the method used to
perform the calculations in Sec. II.2 is only valid for Lagrangians that are
at most quadratic in the position and velocity Storey and Cohen-Tannoudji
(1994). In the case of massive axion and dilaton, this is the case only in the
galactocentric frame, which we assume to be the scalar field rest frame. In
any other frame (in particular in a lab-centered reference frame), the field
will behave as $\cos\left(\omega t-\vec{k}\cdot\vec{x}\right)$ leading to a
Lagrangian not at most quadratic in the position. In the galactocentric frame,
the atoms and laser have an initial velocity corresponding to the velocity of
the Sun in the DM halo, i.e

$\displaystyle\vec{v}_{0}=v_{\mathrm{DM}}\hat{e}_{\mathrm{DM}}\approx
10^{-3}c\>\hat{e}_{\mathrm{DM}}\,.$ (44)

The Sun velocity in the galactic halo points towards $\alpha$ Cygni, the
biggest star of the Cygnus constellation Miuchi  _et al._ (2020), which
corresponds to a right ascension
$\alpha_{\mathrm{DM}}=310.36\degree\mathrm{W}$ and declination
$\delta_{\mathrm{DM}}=45.28\degree\mathrm{N}$ van Leeuwen (2007).

The observable signatures depend also on the orientation of the experimental
setup with respect to this initial velocity. Indeed, for classical tests of
the universality of free fall between two macroscopic bodies, one measures
$\Delta\vec{a}\propto\vec{v}_{0}$ (see Eq.(5)) projected onto the sensitive
axis of the instrument. Therefore, the signal depends on the projection of the
velocity of the bodies in the galactocentric frame with the sensitive axis of
the experiment. Similarly, all the AI experiments depend on the scalar product
between the initial galactocentric velocity of the atomic clouds with the
direction of the velocity kick undergone in the interferometric schemes, i.e.
on $\hat{e}_{v}\cdot\hat{e}_{\mathrm{kick}}$, see Eqs. (7), (8) and (10b).

All the AI-based experiments operate at constant location, loc, on Earth with
longitude $\lambda_{\mathrm{loc}}$ and latitude $\phi_{\mathrm{loc}}$. We
assume the velocity kick to be directed vertically, i.e. in the Earth’s
reference frame
$\hat{e}_{\mathrm{kick}}=\left(\cos(\lambda_{\mathrm{loc}})\cos(\phi_{\mathrm{loc}}),\sin(\lambda_{\mathrm{loc}})\cos(\phi_{\mathrm{loc}}),\sin(\phi_{\mathrm{loc}})\right)$.
As first approximation, we consider the declination as equivalent to the
terrestrial latitude, i.e $\delta_{\mathrm{DM}}\approx\phi_{\mathrm{DM}}$,
such that the dot product $\hat{e}_{v}\cdot\hat{e}_{\mathrm{kick}}$ is simply
given by
$\cos(\phi_{\mathrm{loc}})\cos(\phi_{\mathrm{DM}})\cos(\lambda_{\mathrm{loc}}-\lambda_{\mathrm{DM}}(t))+\sin(\phi_{\mathrm{loc}})\sin(\phi_{\mathrm{DM}})$.
$\lambda_{\mathrm{DM}}(t)$ is the longitude of $\alpha$ Cygni at the time $t$
of the experiment. Indeed, while $\alpha_{\mathrm{DM}}$ is fixed, the former
follows the Earth rotation with frequency $\omega_{E}\sim 7\times 10^{-5}$ Hz
and is therefore time dependent

$\displaystyle\lambda_{\mathrm{DM}}(t)$ $\displaystyle=\omega_{E}t+\varphi\,,$
(45a) where the phase $\varphi$ corresponds to the longitude of $\alpha$ Cygni
at the origin of time reference considered. For short experiments with an
experimental time much smaller than a day (i.e. relevant for Stanford’s
experiment, see Sec IV.0.2), the dot product is roughly constant and depends
therefore on the exact time of the day when the experiment was conducted. In
order to infer a sensitivity estimate, we will only consider the mean value of
the dot product which is given by
$\sin(\phi_{\mathrm{loc}})\sin(\phi_{\mathrm{DM}})$, i.e
$\displaystyle\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{kick}}\Big{|}_{\mathrm{Stanford}}$
$\displaystyle\approx 0.43$ (45b) For longer time experiments, like the SPID
variation, the dot product evolves with time, such that the signal evolves as
$\displaystyle s(t)$
$\displaystyle\propto\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{kick}}\cos(\omega_{\mathrm{DM}}t+\phi_{0})\,$
(45c)
$\displaystyle=[\cos(\phi_{\mathrm{loc}})\cos(\phi_{\mathrm{DM}})\cos(\omega_{E}t+\varphi)+\,$
$\displaystyle\sin(\phi_{\mathrm{loc}})\sin(\phi_{\mathrm{DM}})]\cos(\omega_{\mathrm{DM}}t+\phi_{0}),$
(45d) i.e the Earth rotation modulates the signal at frequency
$f_{e}=\omega_{E}/2\pi$. We will make two approximations for our estimates.
For all the DM frequencies of interest $\omega_{\mathrm{DM}}$, the Earth
rotation is a slowly varying function, because
$\omega_{E}<\omega_{\mathrm{DM}}$. Therefore, we will assume that the signal
will manifest itself by a single peak at frequency
$f_{\mathrm{DM}}=\omega_{\mathrm{DM}}/2\pi$ in Fourier space. In addition, we
will be interested only by the maximum value of the dot product, which is only
a function of the different latitudes.
$\displaystyle\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{kick}}=\mathrm{Max}\left(|\cos(\phi_{\mathrm{loc}}\pm\phi_{\mathrm{DM}})|\right)\,.$
(45e) For the various locations under consideration in this paper, we have
$\displaystyle\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{kick}}\Big{|}_{\mathrm{Oxford}}$
$\displaystyle\approx 0.99\,,$ (45f)
$\displaystyle\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{kick}}\Big{|}_{\mathrm{Fermilab}}$
$\displaystyle\approx 1.00\,,$ (45g) where we assume the SPID variation
operates at the same location as AION-10, i.e Oxford Badurina  _et al._ (2020)
for consistent comparison.

For MICROSCOPE, the axis of measurement is alongside the test masses
cylinders’ longitudinal symmetry axis Touboul  _et al._ (2017). The orbital
motion of the satellite around Earth is sun-synchronous, which means that the
orientation of the orbital plane evolves with time with an annual period. As
it was mentioned in Sec. IV, the measurements are distributed on 17 different
sessions, each of them lasting $T_{\mathrm{int}}/17\sim 5$ days on average.
For such durations, we can assume the orbital plane to be fixed during each
session. In addition, the satellite spins around an axis that is orthogonal to
the axis of measurement with angular frequency $\omega_{\mathrm{spin}}$. In
total, this means the dot product can be written as

$\displaystyle\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{meas.}}(t)\Big{|}_{\mathrm{\mu
SCOPE}}$ $\displaystyle=A(t)\cos(\omega_{\mathrm{spin}}t+\psi)\,,$ (45h)

where $A(t)$ depends on the orientation of the orbital plane, i.e is fixed for
one session but changes from one session to another, and with $\psi$ an
irrelevant phase. Using MICROSCOPE’s publicly available data, we estimated
numerically the coefficient $A(t)$ for every session and we find that its
value oscillates between 0.71 and 1. For our estimates, we will consider its
mean value, i.e $|\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{meas.}}|\approx
0.85$. In conclusion, the signal presented in Eq. (5) is modulated by the
factor

$\displaystyle\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{meas.}}(t)\Big{|}_{\mathrm{\mu
SCOPE}}$ $\displaystyle\approx 0.85\cos(\omega_{\mathrm{spin}}t+\psi)\,.$
(45i)

Thus, the observable signal in the MICROSCOPE experiment is oscillating at the
combination of the DM frequency and the spin frequency, i.e. has harmonics at
the two frequencies $\omega_{\mathrm{DM}}\pm\omega_{\mathrm{spin}}$.
Therefore, for DM frequencies in Eq. (5) such that
$\omega_{\mathrm{DM}}\ll\omega_{\mathrm{spin}}$, we will consider that the
signal in MICROSCOPE oscillates at $\omega_{\mathrm{spin}}$, whereas if
$\omega_{\mathrm{DM}}\gg\omega_{\mathrm{spin}}$, we will make our estimate
with the signal oscillating at $\omega_{\mathrm{DM}}$. As explained in Sec.
IV, $\omega_{\mathrm{spin}}$ depends on the measurement session since three
different spinning frequencies have been used during the full mission. They
differ by a factor 5 roughly Bergé (2023). For our rough sensitivity
estimates, we use the data from session 404 as a basis, therefore we will
assume a constant $\omega_{\mathrm{spin}}\sim 18.4$ mrad/s.

|  | Axionic charges |  | Dilatonic charges |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---
Experiment | Species | $Q_{M}$ |  $Q_{\omega}$ | $Q_{M,e}$ | $Q_{M,m_{e}}$ | $Q_{M,\hat{m}}$ | $Q_{M,\delta m}$ | $Q_{\omega,e}$ | $Q_{\omega,\hat{m}}$ | $Q_{\omega,\delta m}$
|  | [$\times 10^{-3}$] |  [$\times 10^{-3}$] | [$\times 10^{-3}$] | [$\times 10^{-3}$] | [$\times 10^{-3}$] | [$\times 10^{-3}$] |  | [$\times 10^{-3}$] | [$\times 10^{-3}$]
MICROSCOPE |  195Pt Bergé  _et al._ (2018) | -69.065 |  $-$ | 4.278 | 0.220 | 85.25 | 0.340 | $-$ | $-$ | $-$
|  48Ti Bergé  _et al._ (2018) | -68.770 |  $-$ | 2.282 | 0.253 | 82.58 | 0.138 | $-$ | $-$ | $-$
Stanford |  87Rb | -68.920 |  9.30 Flambaum and Tedesco (2006) | 2.869 | 0.234 | 83.95 | 0.254 | 2.34 Guéna  _et al._ (2012) | 18.0 Guéna  _et al._ (2012) | 20.7 Guéna  _et al._ (2012)
|  85Rb | -68.924 |  $-$ | 2.961 | 0.239 | 83.98 | 0.220 | $-$ | $-$ | $-$
AION-10/SPID |  40Ca | -68.715 |  0 | 2.409 | 0.275 | 82.08 | 0 | 2.02 Angstmann  _et al._ (2004) | 0 | 0
|  44Ca | -68.738 |  0 | 2.116 | 0.250 | 82.29 | 0.155 | 2.02 | 0 | 0
|  86Sr | -68.933 |  0 | 3.074 | 0.243 | 84.06 | 0.198 | 2.06 | 0 | 0
|  87Sr | -68.932 |  0 | 3.027 | 0.240 | 84.05 | 0.215 | 2.06 Bloom  _et al._ (2014) | 0 | 0
|  88Sr | -68.930 |  0 | 2.980 | 0.238 | 84.03 | 0.232 | 2.06 | 0 | 0
|  171Yb | -69.054 |  0 | 4.114 | 0.225 | 85.14 | 0.308 | 2.31 Hinkley  _et al._ (2013) | 0 | 0
|  176Yb | -69.043 |  0 | 3.957 | 0.219 | 85.05 | 0.348 | 2.31 | 0 | 0
|  196Hg | -69.077 |  0 | 4.469 | 0.224 | 85.35 | 0.312 | 2.81 Angstmann  _et al._ (2004) | 0 | 0
|  202Hg | -69.066 |  0 | 4.291 | 0.218 | 85.25 | 0.353 | 2.81 | 0 | 0
All AI | SiO2 | -68.442 |  $-$ | 1.607 | 0.275 | 79.62 | 0.003 | $-$ | $-$ | $-$
Table 2: Axionic/dilatonic charges for species of atoms used in experiments
considered (see text) (SiO2 represents the reference mirror, composed of 47%
Si and 53% O). For atoms involved in Raman AI, the transitions are hyperfine
for 87Rb and optical for the rest. Axionic charges are derived from Eqs. (21)
and (27), while dilatonic charges are derived from Eqs. (34),(37) and (38).
The value of $[Q_{\omega,m_{e}}]_{\phi}$ is universal for all atomic
transitions, see Eqs. (37) and (38), so it is not provided. For isotopes, we
assume the same $Q_{\omega,e}$ value (see text).

In addition, the DM has a velocity distribution in the galaxy, which implies
that the scalar field oscillation will have a coherence time $\tau(\omega)$
given by Derevianko (2018)

$\displaystyle\tau(\omega)=\frac{2\pi
c^{2}}{\sigma^{2}_{v}\omega}\approx\frac{2\pi\times 4\times
10^{6}}{\omega}\,,$ (46)

where $\sigma_{v}\approx 5\times 10^{-4}c$ is the DM galactic velocity
dispersion Evans  _et al._ (2019). As we shall see in the next paragraphs, the
coherence time will affect the sensitivity of the experiment.

We will now present the general principle used to derive the sensitivity
curves. The sensitivity of a given experiment to a general DM-SM coupling
$\chi$ at a given DM Compton frequency $\omega_{\mathrm{DM}}$ depends on: (i)
the time of integration of the experiment, (ii) possibly on the coherence time
of the DM field, (iii) the PSD of the noise at $\omega_{\mathrm{DM}}$ and (iv)
on the expected amplitude of the theoretical signal produced by the DM
candidate. In the following, we will assume a constant threshold SNR value of
1 for all experiments under consideration. Let us write very generically the
signal searched for $s(t)$ as

$\displaystyle
s(t)=\chi\left[X_{s}\right]\cos\left(\omega_{\mathrm{DM}}t+\varphi\right)\,,$
(47)

where we factorize the coupling from the rest of the signal. If the time of
integration is much shorter than the coherence time, i.e
$T_{\mathrm{int}}\ll\tau(\omega_{\mathrm{DM}})$, this equation is fully valid
and the experiment sensitivity on the coupling is

$\displaystyle\chi$
$\displaystyle=\frac{\sqrt{\mathrm{SNR}}}{[X_{s}]}\sqrt{\frac{S_{s}}{T_{\mathrm{int}}}},\,.$
(48a) Note that in this regime, where
$T_{\mathrm{int}}\ll\tau(\omega_{\mathrm{DM}})$, a correction factor to the
sensitivity arises due to the stochastic nature of the amplitude of the field
Centers  _et al._ (2021). In our case where we consider a 68% detection
threshold (i.e. SNR=1), this correction factor induces a loss in signal of
$\sim 1.5$.

On the other hand, when the integration time is much longer than the coherence
time of the field, i.e $T_{\mathrm{int}}\gg\tau(\omega_{\mathrm{DM}})$, this
means that the signal searched for and parametrized by Eq. (47) is no longer
coherent, i.e. it should in principle be modeled as a sum of several
stochastic harmonics, see Foster  _et al._ (2018). Another method to analyse
the data is to cut the dataset in fragments with duration smaller than
$\tau(\omega_{\mathrm{DM}})$ and search for a coherent signal in each of these
blocks of data. In such a case, the experimental sensitivity to the coupling
is reduced and becomes Budker  _et al._ (2014)

$\displaystyle\chi$
$\displaystyle=\frac{\sqrt{\mathrm{SNR}}}{[X_{s}]}\sqrt{\frac{S_{s}}{\sqrt{T_{\mathrm{int}}\tau(\omega_{\mathrm{DM}})}}}\,.$
(48b)

Finally, we will assume the local DM energy density to be
$\rho_{\mathrm{DM}}=0.4$ GeV/cm3 McMillan (2011).

###  V.2 Sensitivity of the experiments to the axion-gluon coupling

First of all, let us remind that all signals considered in this paper are
quadratic in the coupling $1/f_{a}$.

Several laboratory experiments Schulthess  _et al._ (2022); Roussy  _et al._
(2021); Zhang  _et al._ (2022); Abel  _et al._ (2017) already constrain the
$f_{a}$ coupling of the axion, which are all shown by the red-brown full lines
in Fig. 5.

Figure 5: Current lab constraints on $1/f_{a}$ axion coupling from O’Hare
(2020) (the constraint from Zhang  _et al._ (2022) has been rescaled for
consistent value of local DM energy density), including new sensitivity
estimate resulting from this work, i.e from Stanford and MAGIS-100 atom
interferometry experiments and EP classical test from MICROSCOPE. The expected
sensitivity of the AION-10-SPID-like experiment is shown in four different
colors, each using four different pairs of isotopes, denoted ”AION-10-SPID”
(see text).

####  V.2.1  MICROSCOPE

Using Eqs. (5) and (16c), we can express the amplitude of the differential
acceleration between two test-masses A and B as

$\displaystyle|\Delta\vec{a}(t)|$ $\displaystyle=\frac{2\hbar
c^{3}\rho_{\mathrm{DM}}v_{\mathrm{DM}}}{f^{2}_{a}\omega_{a}}\left|\left([Q^{A}_{M}]_{a}-[Q^{B}_{M}]_{a}\right)\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{meas.}}(t)\Big{|}_{\mathrm{\mu
SCOPE}}\right|\equiv\frac{1}{f^{2}_{a}}[X_{a}]\,,$ (49)

considering Eq. (45i).

In Fig. 5, the expected sensitivity of MICROSCOPE is shown by the orange full
line. As a reminder, all the experimental parameters we used to obtain this
curve are explicited in Sec. IV and V.1. This curve presents two breaking
point frequencies. The first one $f\sim 1$ mHz corresponds to half of
$f_{\mathrm{spin}}\sim 3$ mHz. As we discussed in the previous section, we
consider two different frequency regimes, depending on whether the signal
frequency is higher or lower than $f_{\mathrm{spin}}$. In the axion case, the
signal Eq. (49) oscillates at twice the axion field frequency, therefore the
breaking point is $f_{\mathrm{spin}}/2$. The second breaking point arising at
$f\sim 15$ mHz corresponds to half of the bucket frequency of the acceleration
noise PSD, for the same reason as above. Note that there is no breaking point
frequency associated with the coherence time of the field because it would
arise at a frequency larger than the bandwidth of the noise PSD, at around 500
mHz. As it can be seen from this curve, a complete re-analysis of MICROSCOPE’s
data would enable to constrain a new region of the parameter space, over
approximately two orders of magnitude in mass, compared to existing laboratory
experiments.

####  V.2.2  Atom interferometry

For the atom interferometry experiments, i.e Stanford Tower, AION-10,
MAGIS-100, and the SPID variation, we use Eqs. (8), (9) and (10b) to express
respectively the amplitude of the phase shift between atomic species A and B
in differential two-photon transition AI (Stanford experiment), gradiometers
(AION-10) and SPID with both AION-10 and MAGIS-100 experimental parameters. As
shown in Section. III.1.2, the axionic frequency charge in optical transitions
is 0, which explains cancellation of the frequency term in the phase
difference of the SPID variation, Eq. (10a) for the ALP model. In addition,
the Stanford experiment is a Bragg-transition differential AI involving 2
isotopes, we will therefore assume
$k^{A}_{\mathrm{eff}}=k^{B}_{\mathrm{eff}}\equiv
k^{\mathrm{AB}}_{\mathrm{eff}}$ and with the same laser source
$Q^{L,A}_{\omega}=Q^{L,B}_{\omega}$. Then, the phase shifts read

$\displaystyle\Delta\Phi^{\mathrm{Stanford}}_{\mathrm{AB}}$
$\displaystyle\approx\frac{2\hbar
c^{3}\rho_{\mathrm{DM}}v_{\mathrm{DM}}k^{\mathrm{AB}}_{\mathrm{eff}}}{\omega^{3}_{a}f^{2}_{a}}\Big{|}\left([Q^{A}_{M}]_{a}-[Q^{B}_{M}]_{a}\right)\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{kick}}\Big{|}_{\mathrm{Stanford}}\Big{|}\sin^{2}(\omega_{a}T)\equiv\frac{1}{f^{2}_{a}}[X_{\Phi}]_{\mathrm{Stanford}}\,,$
(50a) $\displaystyle\Delta\Phi^{\mathrm{AION-10}}_{\mathrm{A}}$
$\displaystyle\approx\frac{4n\omega^{0}_{A}\Delta r\hbar
c^{2}\rho_{\mathrm{DM}}[Q^{A}_{\omega}]_{a}}{\omega^{2}_{a}f^{2}_{a}}\sin^{2}(\omega_{a}T)\equiv\frac{1}{f^{2}_{a}}[X_{\Phi}]_{\mathrm{AION-10}}\,,$
(50b) $\displaystyle\Delta\Phi^{\mathrm{MAGIS-100}}_{\mathrm{AB}}$
$\displaystyle\approx\frac{2n\omega_{0}\hbar
c^{2}\rho_{\mathrm{DM}}v_{\mathrm{DM}}}{\omega^{3}_{a}f^{2}_{a}}\left|\left([Q^{A}_{M}]_{a}-[Q^{B}_{M}]_{a}\right)\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{kick}}\Big{|}_{\mathrm{Fermilab}}\right|\sin^{2}(\omega_{a}T)\equiv\frac{1}{f^{2}_{a}}[X_{\Phi}]_{\mathrm{MAGIS-100}}\,$
(50c) $\displaystyle\Delta\Phi^{\mathrm{AION-10-SPID}}_{\mathrm{AB}}$
$\displaystyle\approx\frac{2n\omega_{0}\hbar
c^{2}\rho_{\mathrm{DM}}v_{\mathrm{DM}}}{\omega^{3}_{a}f^{2}_{a}}\left|\left([Q^{A}_{M}]_{a}-[Q^{B}_{M}]_{a}\right)\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{kick}}\Big{|}_{\mathrm{Oxford}}\right|\sin^{2}(\omega_{a}T)\equiv\frac{1}{f^{2}_{a}}[X_{\Phi}]_{\mathrm{AION-10-SPID}}\,,$
(50d)

at first order in the axionic charges and where we again factorized the axion-
gluon coupling from the rest.

In Fig. 5, we present the sensitivity of the Stanford Tower Asenbaum  _et al._
(2020), MAGIS-100 experiment Abe  _et al._ (2021) and the SPID AI setup with
AION-10 experimental parameters, denoted “AION-10-SPID”.

One can notice that MICROSCOPE is approximately two to three orders of
magnitude more sensitive than Stanford. This is consistent considering that
the signal is quadratic in $1/f_{a}$, that MICROSCOPE constrains the Eötvös
parameter $\eta$ three orders of magnitude better than the Stanford experiment
and that the difference in axionic mass charges of species used in the
experiments is 2 orders of magnitude larger for MICROSCOPE.

Contrary to the gradiometer setup of AION-10 which is insensitive to the
axion-gluon coupling (at leading order, the sensitivity is proportional to the
axionic frequency charge of the optical transition of Sr, which is 0), the
SPID variation or MAGIS-100 would have the largest sensitivity to this
coupling compared to existing laboratory experiments. Such an experiments
would improve the current lab constraint Schulthess  _et al._ (2022) by 2
orders of magnitude over a mass range of 4 orders of magnitude
($10^{-17}-10^{-13}$ eV). In this mass range, the MAGIS-100 experiment, which
uses the same setup has also a very interesting sensitivity on the coupling,
which is comparable to the one of AION-10 in the SPID setup. For lower
frequencies, the difference in sensitivity between both setups is due to the
absence of the EOM noise in the MAGIS-100 proposal, which, as explained
earlier, is limiting at these frequencies.

###  V.3 Sensitivity of the experiments to the dilaton couplings

The calculations are similar to the ones from the previous section with one
difference: the signal induced by a dilaton DM candidate is linear to the
dilaton coupling coefficient $d_{i}$, while it was quadratic in the coupling
for the axion.

####  V.3.1  MICROSCOPE

Using Eqs. (5) and (39a), we can express the amplitude of the acceleration
difference between two test-masses A and B as

$\displaystyle|\Delta\vec{a}(t)|$ $\displaystyle=\frac{\sqrt{8\pi
G\rho_{\mathrm{DM}}}v_{\mathrm{DM}}}{c}\left|\left([Q^{A}_{M}]_{\phi}-[Q^{B}_{M}]_{\phi}\right)\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{meas.}}(t)\Big{|}_{\mathrm{\mu
SCOPE}}\right|\equiv\sum_{i=e,m_{e},\hat{m},\delta m}d^{*}_{i}[X^{i}_{a}]\,,$
(51)

where $d^{*}_{i}=d_{i}-d_{g}$ for $i=m_{e},\hat{m},\delta m$ or
$d^{*}_{i}=d_{e}$ (following Eq. (III.2.1)), are the couplings we factorized
from the rest of the signal and where each function $X^{i}_{a}$ has units of
acceleration and contains the partial dilatonic mass charge $Q_{M,i}$ defined
in Eq. (III.2.1).

As it has already been discussed in Hees  _et al._ (2018), the sensitivity of
MICROSCOPE from the oscillatory behavior of the field, descrbed by (51) is not
competitive compared to the Yukawa-type fifth-force generated by the Earth on
the two test masses of MICROSCOPE (shown in Fig. 6).

####  V.3.2 Atom Interferometry

Regarding atom interferometry experiments, we use (9) and (10b) and we express
respectively the amplitude of the phase shift between atomic species A and B
in differential two-photon transition AI, gradiometers and the SPID setup,
with respectively AION-10 and MAGIS-100 experimental parameters. As it can be
noticed from Eqs.(38), most of the dilatonic frequency charges are universal.
Additionally, the calculation of the relativistic correction of the dependence
to $\alpha$ of the optical frequency encoded in the parameter $\epsilon$ (see
Eq. (38) and explicited in Dzuba  _et al._ (1999)) shows that $\epsilon$ does
not depend on the number of neutrons. This means that isotopes with the same
optical frequency transition share the same parameter $\epsilon$, implying
that in the same way as in the axion case, the frequency term of the phase
shift for the SPID variation in Eq. (10a) can be neglected. Then, the
different phase shifts read

$\displaystyle\Delta\Phi^{\mathrm{Stanford}}_{\mathrm{AB}}$
$\displaystyle\approx\frac{4\sqrt{8\pi
G\rho_{\mathrm{DM}}}v_{\mathrm{DM}}k^{\mathrm{AB}}_{\mathrm{eff}}}{\omega^{2}_{\phi}c}\Big{|}\left([Q^{A}_{M}]_{\phi}-[Q^{B}_{M}]_{\phi}\right)\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{kick}}\Big{|}_{\mathrm{Stanford}}\Big{|}\sin^{2}\left(\frac{\omega_{\phi}T}{2}\right)\equiv
d^{*}_{i}[X^{i}_{\Phi}]_{\mathrm{Stanford}}\,,$ (52a)
$\displaystyle\Delta\Phi^{\mathrm{AION-10}}_{\mathrm{A}}$
$\displaystyle\approx\frac{4n\omega^{0}_{A}\Delta r\sqrt{8\pi
G\rho_{\mathrm{DM}}}[Q^{A}_{\omega}]_{\phi}}{\omega_{\phi}c}\sin^{2}\left(\frac{\omega_{\phi}T}{2}\right)\equiv
d^{*}_{i}[X^{i}_{\Phi}]_{\mathrm{AION-10}}\,,$ (52b)
$\displaystyle\Delta\Phi^{\mathrm{MAGIS-100}}_{\mathrm{AB}}$
$\displaystyle\approx\frac{4n\omega_{0}\sqrt{8\pi
G\rho_{\mathrm{DM}}}v_{\mathrm{DM}}}{\omega^{2}_{\phi}c^{2}}\left|\left([Q^{A}_{M}]_{\phi}-[Q^{B}_{M}]_{\phi}\right)\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{kick}}\Big{|}_{\mathrm{Fermilab}}\right|\sin^{2}\left(\frac{\omega_{\phi}T}{2}\right)\equiv
d^{*}_{i}[X^{i}_{\Phi}]_{\mathrm{MAGIS-100}}\,,$ (52c)
$\displaystyle\Delta\Phi^{\mathrm{AION-10-SPID}}_{\mathrm{AB}}$
$\displaystyle\approx\frac{4n\omega_{0}\sqrt{8\pi
G\rho_{\mathrm{DM}}}v_{\mathrm{DM}}}{\omega^{2}_{\phi}c^{2}}\left|\left([Q^{A}_{M}]_{\phi}-[Q^{B}_{M}]_{\phi}\right)\hat{e}_{\mathrm{DM}}\cdot\hat{e}_{\mathrm{kick}}\Big{|}_{\mathrm{Oxford}}\right|\sin^{2}\left(\frac{\omega_{\phi}T}{2}\right)\equiv
d^{*}_{i}[X^{i}_{\Phi}]_{\mathrm{AION-10-SPID}}\,,$ (52d)

where we sum over the various coupling constants, i.e
$i=\\{e,m_{e},\hat{m},\delta m\\}$ and where we again factorized the couplings
from the rest of the signal containing partial dilatonic mass and/or frequency
charges defined in Eqs. (III.2.1),(III.2.2) depending on the experiment.

Figure 6: Current constraints on all the dilatonic couplings of interest in
this paper : $d_{e}$ (top left), $d_{m_{e}}-d_{g}$ (top right),
$d_{\hat{m}}-d_{g}$ (bottom left), $d_{\delta m}-d_{g}$ (bottom right) from
Touboul  _et al._ (2022); Wagner  _et al._ (2012); Hees  _et al._ (2016);
Kennedy  _et al._ (2020), with 95% confidence level. The expected
sensitivities of AION-10 are shown in dark blue dashed line while the expected
sensitivity of AION-10 using the SPID variation, noted ”AION-10-SPID” are
respectively shown in green, light blue, purple and magenta dashed lines,
depending on the isotope pair used. Finally, the expected sensitivity of
MAGIS-100 is shown in dark green. The light orange background represents
already constrained regions of the parameter space (see text).

On the dilatonic side, laboratory constraints such as Torsion balances Wagner
_et al._ (2012), MICROSCOPE777As discussed earlier, the best MICROSCOPE
sensitivity comes from the static term of the field, i.e when considering a
fifth force generated by Earth on the two test masses, while we only focused
on the oscillatory term in this paper. Touboul  _et al._ (2022) or hyperfine
and optical clocks Hees  _et al._ (2016); Kennedy  _et al._ (2020) are
currently the most stringent constraints on the different dilatonic couplings.

In Fig. 6, we do not present the sensitivity of the Stanford tower experiment,
as it is not competitive compared to the best existing constraints. As it was
discussed in Badurina  _et al._ (2022), AION-10 would improve the current
constraint on the $d_{m_{e}}-d_{g}$ coupling, over a mass range approximately
between $7\times 10^{-16}$ eV and $4\times 10^{-15}$ eV.

With the same experimental parameters as AION-10, operating the SPID setup
would improve the current constraints on various dilatonic couplings. By
respectively using Ca and Hg isotopes pairs, the constraint on $d_{e}$ would
be improved by a factor $\sim 2$ and 3 respectively, over a mass range
covering more than two orders of magnitude (approximately from $8\times
10^{-18}$ eV to $10^{-15}$ eV), compared to MICROSCOPE, the current best
constraint in this mass range. Regarding the $d_{m_{e}}-d_{g}$ coupling, the
use of Ca and Hg isotopes respectively would improve the best constraints, by
a factor 2.5 and 1.5 respectively over the same mass range. Finally, all pairs
of isotopes presented would improve the current best constraint on $d_{\delta
m}-d_{g}$ by one order of magnitude, depending on the isotope pair used, over
a mass range covering 4 orders of magnitude (from $5\times 10^{-19}$ eV to
$3\times 10^{-15}$ eV).

One can notice also that operating both gradiometers and SPID at AION-10 would
give complementary sensitivities for the search of the $d_{m_{e}}-d_{g}$
coupling of the dilaton. Indeed, while the gradiometer would improve the
current constrain in the $\sim 7\times 10^{-16}-3\times 10^{-15}$ eV/$c^{2}$
mass range, the SPID setup’s sensitivity at lower mass is better, as described
in the last paragraph. In addition, SPID would be more sensitive than AION-10
to the $d_{e},d_{\hat{m}}-d_{g}$ and $d_{\delta m}-d_{g}$ couplings in the
full range of masses of interest (i.e lower than $10^{-14}$ eV/$c^{2}$).

Regarding the sensitivity of MAGIS-100 operating the SPID setup, one can
notice that it would overall give the best constrain at low masses. In the
same way as for the axion-gluon coupling, the reason is because the MAGIS-100
community has not considered the use of an EOM to account for the isotope
shift and this addition would lower the sensitivity at low masses, by the same
$m^{-1}_{\mathrm{DM}}$ slope as the ”AION-10-SPID” curves. Despite this
difference, one can notice that MAGIS-100 would reach better sensitivity on
most couplings, compared to the AION-10-SPID variation. The reason is that for
low masses, such that $\omega_{\phi}T\ll 1$, the signal is quadratic in the
free fall time T, and MAGIS-100 using a much longer baseline than AION-10
(100m vs 10m), the free fall time is roughly 6 times longer for MAGIS-100,
resulting in increased signal. Note that compared to the sensitivity curves on
dilaton couplings presented in Abe  _et al._ (2021), but using the gradiometer
setup of MAGIS-100, the curve presented in this paper would constrain a larger
DM mass range. Indeed, while the former has a peak sensitivity around a mass
$10^{-15}$ eV and quickly loses sensitivity for lower masses, the SPID setup
of MAGIS-100 would have a constant sensitivity of the same order of magnitude
for at least one order of magnitude of mass ($10^{-17}-10^{-16}$ eV), if we
assume the loss of sensitivity due to the EOM.

##  VI Conclusion and outlook

In this paper, we derive the complete form of the expected signals in
experiments involving differential acceleration between two test masses and in
various AI setups in the general framework of atoms with time oscillating mass
and transition frequency. We show that if the mass and frequency charges,
which characterize respectively the amplitude of oscillation of the rest mass
and transition frequency of the atom, are not universal among atoms, non zero
signals in such experiments can be expected. If observed, such signals would
constitute a violation of the equivalence principle. We study two particular
models that lead to time varying mass and transition frequencies: axion like
particles (ALP) and dilatons.

We demonstrated a new way of constraining ALP dark matter coupling to gluons.
In particular, we showed that the MICROSCOPE experiment has sufficient
sensitivity to constrain this coupling at an unprecedented level, compared to
already existing laboratory experiments. This, in our opinion, warrants a
corresponding detailed analysis of MICROSCOPE data to search for a potential
ALP signal.

We also show the sensitivity to dilaton and ALP couplings of various AI
setups. In particular, we derive the sensitivity of MAGIS-100-like setups Abe
_et al._ (2021), denoted as SPID in this paper, to dilaton and axion
couplings. Compared to the MAGIS-100 proposal, we argue that additional noise
coming from the generation of the laser frequency shift (e.g. using an EOM) to
account for the isotope shift is necessary. This implies an additional noise
source which is limiting at low DM mass, i.e for $m_{\mathrm{DM}}$
c${}^{2}<10^{-17}$ eV. Nonetheless, using AION-10 experimental parameters, we
show that the SPID variation would be able to constrain both ALP coupling to
gluons and dilaton couplings to photons, electrons and quarks
$d_{e},d_{m_{e}}-d_{g},d_{\delta m}-d_{g}$ to remarkable levels compared to
existing laboratory experiments and future experiments, including AION-10.
Indeed, while AION-10 would not be sensitive to the axion-gluon at first
order, the SPID experiment would improve the best laboratory constraints by
roughly 2 orders of magnitude over 4 orders of magnitude mass range. Regarding
the dilatonic couplings, the SPID experiment would largely surpass the
expected sensitivity of AION-10 on $d_{e},d_{\hat{m}}-d_{g},d_{\delta
m}-d_{g}$ couplings over the full mass range, while being more sensitive than
AION-10 on the $d_{m_{e}}-d_{g}$ coupling at masses $m_{\phi}\leq 10^{-15}$
eV/$c^{2}$. We also show that the MAGIS-100 sensitivity to all of these
couplings is very competitive for DM masses in the $10^{-19}-10^{-16}$ range
due to its larger baseline. It should be noted that the possible addition of
the EOM would decrease its sensitivity for $m_{\mathrm{DM}}$
c${}^{2}<10^{-17}$ eV.

Despite being extremely sensitive to various DM candidates-SM fields
couplings, the SPID setup would also test the universality of free fall, with
unprecedented level. Assuming MAGIS-100 experimental set of parameters
described in Section. IV.0.4, the free fall acceleration on ground $g=9.81$
m/s2, a total cycle time $T_{\mathrm{cycle}}=20$ s, to account for the atom
preparation, free fall and measurement, the corresponding constraint on the
Eötvös parameter would reach

$\displaystyle\eta=\frac{1}{gk_{\mathrm{eff}}T^{2}}\sqrt{\frac{S^{\mathrm{MAGIS-100}}_{\phi}}{2T_{\mathrm{cycle}}}}\approx
9\times 10^{-18}\,,$ (53)

which would improve the current bound from MICROSCOPE Touboul  _et al._ (2022)
by more than 2 orders of magnitude. However, this estimate is certainly over-
optimistic as it does not take into account additional DC effects on the
isotopes, such as gravity gradients, temperature gradients (blackbody
radiation), wave-front aberrations, magnetic field gradients, etc…

###### Acknowledgements.

The authors acknowledge Leonardo Badurina, Diego Blas and Robin Corgier for
helpful discussions. This work was supported by the Programme National GRAM of
CNRS/INSU with INP and IN2P3 cofunded by CNES.

##  Appendix A Full calculation of the two photon transition AI phase shift

We consider an atom A whose nominal rest mass and transition frequency
$m^{0}_{A},\omega^{0}_{A}$ are perturbed such that they oscillate in phase as
in Eq. (1), i.e $m^{0}_{A}\left(1+Q^{A}_{M}\cos(\omega
t+\phi_{0})\right),\omega^{0}_{A}\left(1+Q^{A}_{\omega}\cos(\omega
t+\phi_{0})\right)$, where the charges $Q^{A}_{M},Q^{A}_{\omega}$ represent
the respective amplitude of oscillation of the rest mass and transition
frequency. We show how this perturbation produces a phase at the end of a two-
photon transition atom interferometer, namely we derive the results found in
Eq.(7). Without loss of generality, we will assume a Raman interferometer,
such that the transition frequency of the atom is relevant. Note that the
Bragg interferometer phase would be easily recovered by setting
$Q^{A}_{\omega}=0$. We also make all calculations at first order in the
perturbations $Q_{M},Q_{\omega}$.

We describe the quantum state of the different atom wavepackets by a
wavefunction $\Psi$, which we break down into two different wavefunctions
$\Psi_{\mathrm{I}}$ and $\Psi_{\mathrm{II}}$, where I,II respectively the
up/down paths in Fig. 1, depending on the classical path the atom followed.
Considering free particles at the input of the interferometer, the atomic
plane waves have the form Storey and Cohen-Tannoudji (1994)

$\displaystyle\Psi_{\mathrm{init}}(t_{0},\vec{x}_{0})=\Psi_{0}e^{i\Phi(t_{0},\vec{x}_{0})}\,,$
(54)

with the amplitude of the wavefunction $\Psi_{0}$,
$\Phi(t_{0},\vec{x}_{0})=\vec{k}\cdot\vec{x}_{0}-\omega t_{0}-\varphi$ with
$\omega,\vec{k},\varphi$ respectively its angular frequency, wavevector and
constant phase. At the output of the interferometer, at a time $T_{f}\geq 2T$,
considering all the different phase contributions listed previously, the
atomic plane wave along the trajectory j read

$\displaystyle\Psi_{\mathrm{j}}(T_{f},x_{f})=\Psi_{\mathrm{init}}(T_{d},x_{f})\times
e^{i\Phi_{\mathrm{j}}}\,,$ (55)

where $\Phi_{\mathrm{j}}=\Phi_{\mathrm{sj}}+\Phi_{\mathrm{\ell
j}}+\Phi_{\mathrm{uj}}$ represents the trajectory dependent phase factor.
Then, at some detection time $T_{\mathrm{d}}\geq T_{\mathrm{f}}$, a detector
measures by fluorescence the number of atoms on each quantum state, which is
essentially the measurement of the probability that the two wavepackets are in
the same quantum state, i.e

$\displaystyle\int\Big{|}\Psi_{\mathrm{I}}(T_{\mathrm{d}},\vec{x}_{\mathrm{d}})+\Psi_{\mathrm{II}}(T_{\mathrm{d}},\vec{x}_{\mathrm{d}})\Big{|}^{2}dS\,,$
(56)

where the integral is taken over the detector area S. Plugging Eq. (55) into
Eq. (56) and neglecting loss of contrast due to decoherence, we find that the
measurement result is proportional to (1 + $\cos\Delta\Phi$) where
$\Delta\Phi=\Phi_{\mathrm{I}}-\Phi_{\mathrm{II}}$ is the phase difference
between the two wave functions at $T_{\mathrm{d}}$. For simplicity, we
consider $T_{\mathrm{d}}\equiv 2T$.

We first derive the perturbed equations of motion of the atom following by the
perturbation to the acceleration Eq. (4), to get the motion followed by the
different wavepackets along the trajectories presented in Fig. 1. They read

$\displaystyle\vec{v}_{A}(t,t_{0})\approx\vec{v}_{0}\left(1-Q^{A}_{M}\left(\cos(\omega
t+\phi_{0})-\cos(\omega t_{0}+\phi_{0})\right)\right)\,,$ (57a)
$\displaystyle\vec{x}_{A}(t,t_{0})\approx\vec{x}_{0}+\vec{v}_{0}(t-t_{0})-Q^{A}_{M}\frac{\vec{v}_{0}}{\omega}\left(\sin(\omega
t+\phi_{0})-\sin(\omega t_{0}+\phi_{0})-\omega(t-t_{0})\cos(\omega
t_{0}+\phi_{0})\right)\,,$ (57b)

at first order in the perturbation $Q^{A}_{M}$, where we discarded the
frequency term from the acceleration, as it is negligible, and with
$\vec{x}_{0}=\vec{x}(t_{0}),\vec{v}_{0}=\vec{v}(t_{0})$ respectively the
initial position and velocity of the atom when entering the interferometer.

We start by the calculation of the position and velocity of the atom at the
end of the trajectory portion 1 in Fig. 1. From Eq.(57), the atom does not
undergo any kick velocities from the laser pulses, hence after a time T, its
equations of motion read

$\displaystyle\vec{x}^{(1)}_{A}(T)$
$\displaystyle=\vec{v}_{0}\left(T-\frac{Q^{A}_{M}}{\omega}(\sin(\omega
T+\phi_{0})-\sin(\phi_{0})-\omega T\cos(\phi_{0}))\right)\,$ (58a)
$\displaystyle\vec{v}^{(1)}_{A}(T)$
$\displaystyle=\vec{v}_{0}\left(1-Q^{A}_{M}\left(\cos(\omega
T+\phi_{0})-\cos(\phi_{0})\right)\right)\,.$ (58b)

On the other hand, the wavepacket on portion 2 in Fig. 1 has undergone a kick
velocity with amplitude $v_{\mathrm{kick,0}}+\delta v_{\mathrm{kick}}(t=0)$,
following Eq. (6), with

$\displaystyle\delta\vec{v}_{\mathrm{kick}}(t=0)=\vec{v}_{\mathrm{kick,0}}\left(Q^{L}_{\omega}-Q^{A}_{M}\right)\cos(\phi_{0})\,,$
(59)

with $\vec{v}_{\mathrm{kick,0}}$ along an arbitrary direction $\hat{x}$, hence

$\displaystyle\vec{x}^{(2)}_{A}(T)=(\vec{v}_{0}+\vec{v}_{\mathrm{kick,0}})T-Q^{A}_{M}\frac{\sin(\omega
T+\phi_{0})-\sin(\phi_{0})}{\omega}(\vec{v}_{0}+\vec{v}_{\mathrm{kick,0}})+T(Q^{A}_{M}\vec{v}_{0}-Q^{L}_{\omega}\vec{v}_{\mathrm{kick,0}})\cos(\phi_{0})\,,$
(60a)
$\displaystyle\vec{v}^{(2)}_{A}(T)=\vec{v}_{0}+\vec{v}_{\mathrm{kick,0}}-Q^{A}_{M}\cos(\omega
T+\phi_{0})(\vec{v}_{0}+\vec{v}_{\mathrm{kick,0}})+\left(Q^{A}_{M}\vec{v}_{0}+Q^{L}_{\omega}\vec{v}_{\mathrm{kick,0}}\right)\cos(\phi_{0})\,,$
(60b)

At time T, both wavepackets undergo a kick velocity with opposite direction,
such that their momenta states are exchanged. Following Eq. (6), this means
that at the end of portion 1, the atom undergoes a kick velocity of amplitude
$v_{\mathrm{kick,0}}+\delta v_{\mathrm{kick}}(t=T)$, with

$\displaystyle\delta\vec{v}_{\mathrm{kick}}(t=T)=\vec{v}_{\mathrm{kick,0}}\left(Q^{L}_{\omega}-Q^{A}_{M}\right)\cos(\omega
T+\phi_{0})\,,$ (61)

with same direction $\hat{x}$, hence, at the end of the portion 3, its
coordinates read

$\displaystyle\vec{x}^{(3)}_{A}(2T)=(2\vec{v}_{0}+\vec{v}_{\mathrm{kick,0}})T+\frac{Q^{A}_{M}}{\omega}\left(\vec{v}_{\mathrm{kick,0}}\sin(\omega
T+\phi_{0})+\vec{v}_{0}\sin(\phi_{0})+2\vec{v}_{0}\omega
T\cos(\phi_{0})-(\vec{v}_{0}+\vec{v}_{\mathrm{kick,0}})\sin(2\omega
T+\phi_{0})\right)+\,$ $\displaystyle
Q^{L}_{\omega}\vec{v}_{\mathrm{kick,0}}T\cos(\omega T+\phi_{0})\,,$ (62a)
$\displaystyle\vec{v}^{(3)}_{A}(2T)=\vec{v}_{0}+\vec{v}_{\mathrm{kick,0}}-Q^{A}_{M}\left((\vec{v}_{0}+\vec{v}_{\mathrm{kick,0}})\cos(2\omega
T+\phi_{0})-\vec{v}_{0}\cos(\phi_{0})\right)+Q^{L}_{\omega}\vec{v}_{\mathrm{kick,0}}\cos(\omega
T+\phi_{0})\,.$ (62b)

The atom at the end of the portion 2 undergoes a kick velocity in the other
direction, i.e of amplitude $v_{\mathrm{kick,0}}+\delta
v_{\mathrm{kick}}(t=T)$, but with opposite direction (-$\hat{x}$), compared to
the previous laser kicks, hence

$\displaystyle\vec{x}^{(4)}_{A}(2T)=(2\vec{v}_{0}+\vec{v}_{\mathrm{kick,0}})T-\frac{Q^{A}_{M}}{\omega}\left(\vec{v}_{\mathrm{kick,0}}\sin(\omega
T+\phi_{0})+\vec{v}_{0}\sin(2\omega
T+\phi_{0})-(\vec{v}_{0}+\vec{v}_{\mathrm{kick,0}})\sin(\phi_{0})-2\vec{v}_{0}\omega
T\cos(\phi_{0})\right)-\,$ $\displaystyle
Q^{L}_{\omega}\vec{v}_{\mathrm{kick,0}}T\left(\cos(\omega
T+\phi_{0})-2\cos(\phi_{0})\right)\,,$ (63a)
$\displaystyle\vec{v}^{(4)}_{A}(2T)=\vec{v}_{0}\left(1-Q^{A}_{M}\left(\cos(2\omega
T+\phi_{0})-\cos(\phi_{0})\right)\right)-Q^{L}_{\omega}\vec{v}_{\mathrm{kick,0}}\left(\cos(\omega
T+\phi_{0})-\cos(\phi_{0})\right)\,,$ (63b)

at the end of portion 4.

###  A.1 Propagation phase contribution

The first component of phase shift is the one coming from the phase
accumulated by atoms throughout the whole interferometric paths taking into
account modified equations of motion and perturbed kicks. In the special case
of quadratic lagrangian in the position and velocity of the atom at maximum,
this phase is by the principle of least action the integral of the lagrangian
over the path from initial point $i$ to final point $f$
$(t_{i},x_{i})\rightarrow(t_{f},x_{f})$

$\displaystyle\Phi_{\mathrm{s}}=\frac{1}{\hbar}\int_{t_{i}}^{t_{f}}L(x,\dot{x})dt\,,$
(64)

where the lagrangian is defined in Eq. (3). The internal energy term of the
lagrangian being associated with the oscillation of the transition energy, it
only contributes when the atom is on the excited state, i.e on paths 2 and 3
in Fig. 1. Then, the phase accumulated by the atoms on the path I of the
interferometer is

$\displaystyle\Phi_{\mathrm{sI}}$
$\displaystyle=-\omega^{0}_{A}\int_{0}^{T}dt(1+Q^{A}_{\omega}\cos(\omega
t+\phi_{0}))-\frac{m^{0}_{A}c^{2}}{\hbar}\,$
$\displaystyle\left[\int_{0}^{T}dt\left(1+Q^{A}_{M}\cos(\omega
t+\phi_{0})\right)\left(1-\frac{|\vec{v}^{(2)}_{A}(t)|^{2}}{2c^{2}}\right)+\right.\,$
$\displaystyle\left.\int_{T}^{2T}dt\left(1+Q^{A}_{M}\cos(\omega
t+\phi_{0})\right)\left(1-\frac{|\vec{v}^{(4)}_{A}(t)|^{2}}{2c^{2}}\right)\right]\,,$
(65)

while the atom wavepacket on the path II accumulates a phase

$\displaystyle\Phi_{\mathrm{sII}}$
$\displaystyle=-\omega^{0}_{A}\int_{T}^{2T}dt(1+Q^{A}_{\omega}\cos(\omega
t+\phi_{0}))-\frac{m^{0}_{A}c^{2}}{\hbar}\,$
$\displaystyle\left[\int_{0}^{T}dt\left(1+Q^{A}_{M}\cos(\omega
t+\phi_{0})\right)\left(1-\frac{|\vec{v}^{(1)}_{A}(t)|^{2}}{2c^{2}}\right)+\right.\,$
$\displaystyle\left.\int_{T}^{2T}dt\left(1+Q^{A}_{M}\cos(\omega
t+\phi_{0})\right)\left(1-\frac{|\vec{v}^{(3)}_{A}(t)|^{2}}{2c^{2}}\right)\right]\,,$
(66)

where $v^{(1)},v^{(2)},v^{(3)},v^{(4)}$ are respectively the atom velocities
along portions 1, 2, 3 and 4 in Fig. 1, which can all be calculated explicitly
using the first part of this appendix.

Then, the propagation phase shift between the two perturbed trajectories is
simply

$\displaystyle\Phi_{\mathrm{s}}$
$\displaystyle=\Phi_{\mathrm{sI}}-\Phi_{\mathrm{sII}}=-\frac{4}{\omega}\left[k_{\mathrm{eff}}\left(v_{0}\hat{e}_{v}\cdot\hat{e}_{\mathrm{kick}}+\frac{\hbar
k_{\mathrm{eff}}}{2m^{0}_{A}}\right)Q^{A}_{M}+\omega^{0}_{A}Q^{A}_{\omega}\right]\sin^{2}\left(\frac{\omega
T}{2}\right)\sin(\omega T+\phi_{0})+\,$ $\displaystyle
4k_{\mathrm{eff}}TQ^{L}_{\omega}\left(v_{0}\hat{e}_{v}\cdot\hat{e}_{\mathrm{kick}}+\frac{\hbar
k_{\mathrm{eff}}}{2m^{0}_{A}}\right)\sin\left(\frac{\omega
T}{2}\right)\sin\left(\frac{\omega T}{2}+\phi_{0}\right)\,,$ (67a)

where we used $m^{0}_{A}v_{\mathrm{kick}}=\hbar k_{\mathrm{eff}}$ at zeroth
order in the perturbation, following Eq. (6) and where we defined
$\vec{v}_{0}=v_{0}\hat{e}_{v}$ and
$\vec{v}_{\mathrm{kick}}=v_{\mathrm{kick}}\hat{e}_{\mathrm{kick}}$.

###  A.2 Laser phase contribution

We now consider the phase from light-matter interaction between the laser and
the atoms. A referenced mirror is used in the interferometer as a reference to
measure the phase of the laser. Considering the mirror is freely falling
during the entire interferometric process, its own mass composition is
affected by the oscillating mass behavior Eq. (1), i.e
$m_{M}=m^{0}_{M}\left(1+Q^{M}_{M}\cos(\omega t+\phi_{0})\right)$, with
$m^{0}_{M},Q^{M}_{M}$ respectively its unperturbed mass and mass charge,
implying it follows the same perturbed equations of motion as the atom
Eq.(57). At each space-time points of light-matter interaction is associated a
phase, which contributes to the laser phase shift. These points are denoted
$A,B,C,D_{2}$ in Fig. 1, since without loss of generality, we assume the
measurement of the interference pattern between the two wavepackets ending up
on the internal ground state $|g\rangle$.

Keeping in mind the wavefunction expansion form
$\Phi_{L}(t,\vec{x})=\vec{k}_{L}\cdot\vec{x}-\omega_{L}t-\varphi$, with
$\omega_{L},\vec{k}_{L}$ the light angular frequency and wavevector
respectively, the total light phase felt by the atom along a path $X$ is given
by

$\displaystyle\Phi_{\ell X}$
$\displaystyle=\sum_{j=0}^{n}\Phi_{L}(t_{j}-t_{i},\vec{x}_{j}-\vec{x}_{i})\,$
(68)
$\displaystyle=\sum_{j=0}^{n}\\{\vec{k}_{\mathrm{j,eff}}\cdot[-(\vec{x}_{\mathrm{j,M}}(t_{j})-\vec{x}_{\mathrm{j,At}}(t_{j}))]-\varphi_{\mathrm{j,eff}}(t_{j})\\}\,,$
(69)

where we sum on the total number of points interaction of path X (occurring at
times $\\{0,T,2T\\}$) and where $t_{i},\vec{x}_{i}$ correspond respectively to
the time and position of laser at emission of the photon while
$t_{j},\vec{x}_{j}$ are the time and position of the atom at the time of
reception. At the second line, we made the assumption that the time of
emission and reception are the same and we assumed that at each interaction
point j, $k_{\mathrm{j,eff}},\varphi_{\mathrm{j,eff}}$ are respectively the
effective wavevector associated with the effective energy transfer from the
photons to the atom and the phase of the laser, all measured in the laboratory
frame, $x_{\mathrm{j,M}}$ is the mirror coordinate and $x_{\mathrm{j,At}}$ the
perturbed vertex of the atomic path. Both of these quantities are time
dependent in our framework since the laser frequency is locked on an atom
ensemble whose frequency oscillates through Eq. (1). In particular the phase
of the laser is defined as the integral of its frequency, i.e

$\displaystyle\varphi_{\mathrm{j,eff}}(t)$
$\displaystyle=\int^{t}_{0}dt^{\prime}\omega_{L}(t^{\prime})\equiv\int^{t}_{0}dt^{\prime}\omega^{0}_{L}\left(1+Q^{L}_{\omega}\cos(\omega
t^{\prime}+\phi_{0})\right)\,.$ (70)

As shown in Fig. 1, the laser-atom interactions happen at points A and C for
path I and points B and $D_{2}$ for path II.

Then, the laser shift summed on the vertices of the path I takes the form

$\displaystyle\Phi_{\mathrm{\ell I}}$
$\displaystyle=-\left[\vec{k}_{\mathrm{eff}}(T)\left(-(\vec{x}_{\mathrm{M}}(T)-\vec{x}_{\mathrm{C}})\right)-\varphi_{\mathrm{eff}}(T)\right]+\,$
$\displaystyle\vec{k}_{\mathrm{eff}}(0)(-(\vec{x}_{\mathrm{M}}(0)-\vec{x}_{\mathrm{A}}))-\varphi_{\mathrm{eff}}(0)\,,$
(71a) while the laser phase summed on all vertices of the path II is
$\displaystyle\Phi_{\mathrm{\ell II}}$
$\displaystyle=-\left[\vec{k}_{\mathrm{eff}}(2T)\left(-(\vec{x}_{\mathrm{M}}(2T)-\vec{x}_{\mathrm{D_{2}}})\right)-\varphi_{\mathrm{eff}}(2T)\right]\,$
$\displaystyle+\vec{k}_{\mathrm{eff}}(T)(-(\vec{x}_{\mathrm{M}}(T)-\vec{x}_{\mathrm{B}}))-\varphi_{\mathrm{eff}}(T)\,.$
(71b)

Note that the minus signs in front of both first terms in Eqs. (71a) and (71b)
come from a phase factor in the case of transition from excited to ground
state ($|e\rangle\rightarrow|g\rangle$) (see Wolf and Tourrenc (1999)).
Assuming for simplicity that the mirror has initial position $x_{M}(0)=0$ and
same velocity $v_{0}$ as atoms, the total laser phase is simply

$\displaystyle\Phi_{\mathrm{\ell}}$ $\displaystyle=\Phi_{\mathrm{\ell
I}}-\Phi_{\mathrm{\ell II}}\,$ (72a)
$\displaystyle=\frac{4}{\omega}\left(k_{\mathrm{eff}}v_{0}\left(Q^{A}_{M}-Q^{M}_{M}\right)\hat{e}_{v}\cdot\hat{e}_{\mathrm{kick}}+\frac{\hbar
k^{2}_{\mathrm{eff}}}{m^{0}_{A}}Q^{A}_{M}+Q^{L}_{\omega}\left(\omega^{0}_{A}-\frac{\hbar
k^{2}_{\mathrm{eff}}\omega T}{m^{0}_{A}}\cot\left(\frac{\omega
T}{2}\right)\right)\right)\sin^{2}\left(\frac{\omega T}{2}\right)\sin(\omega
T+\phi_{0})\,.$ (72b)

###  A.3 Separation phase contribution

Following Eq. (56) and considering that, at the end of the classical paths at
$t=T_{d}$, the wave packet of the path $i$=$\\{$I,II$\\}$ can be expressed as
Wolf and Borde (2004)

$\displaystyle\Psi_{i}=\Psi_{0}e^{i\left(\Phi_{si}+\Phi_{\ell
i}+\frac{p_{i}}{\hbar}(x-x_{i})\right)}\,,$ (73)

where $\Phi_{s},\Phi_{\ell}$ are respectively the propagation and laser phase
contributions and where $p_{i},x_{i}$ are respectively the momentum and
position of the wavepacket just after the last $\pi/2$ pulse at $t=T_{d}$.
Then, it can be shown easily that for small difference in momentum between the
two wavepackets $\Delta\vec{p}=\vec{p}_{\mathrm{I}}-\vec{p}_{\mathrm{II}}$
(which we show in the following), the separation phase shift can be expressed
as

$\displaystyle\Phi_{u}$
$\displaystyle=\Delta\vec{p}\cdot\vec{x}_{\mathrm{det.,COM}}(2T)-\frac{\Delta(\vec{p}\cdot\vec{x})}{\hbar}\,,$
(74)

where we compute the different of momenta $\vec{p}=\hbar\vec{k}$ and position
$\vec{x}$ between $D_{1}$ and $D_{2}$ in Fig. 1, the two wavepackets in the
same energy state, i.e $|g\rangle$ in our calculation.
$\vec{x}_{\mathrm{det.,COM}}$ represents the detector center of mass position
at time $t=T_{d}=2T$, i.e in our case it is simply
$\vec{x}_{\mathrm{det.,COM}}(2T)=2\vec{v}_{0}T$.

Since at the end of portion 3, the wavepacket is in the excited state, we must
take into account an additional kick at time $t=2T$ of amplitude
$v_{\mathrm{kick,0}}+\delta v_{\mathrm{kick}}(t=2T)$ and direction -$\hat{x}$
to this wavepacket, in order to put it back to the ground state, such that it
state corresponds to the point $D_{2}$ in Fig. 1. Since, we assume the
detection to be immediately after the kick, the position of the wavepacket
will not be impacted by it. The final velocity of this wavepacket read

$\displaystyle\vec{v}^{(D_{2})}_{A}(2T)$
$\displaystyle=\vec{v}_{0}\left((1-Q^{A}_{M}\left(\cos(2\omega
T+\phi_{0})-\cos(\phi_{0})\right)\right)-\,$ $\displaystyle
Q^{L}_{\omega}\vec{v}_{\mathrm{kick,0}}\left(\cos(2\omega
T+\phi_{0})-\cos(\omega T+\phi_{0})\right)\,,$ (75)

resulting in a different velocity compared to the other wavepacket
($\vec{v}^{(4)}_{A}(2T)$). Then, the separation phase is a function of both
differences of position and velocities at the end of the two paths

$\displaystyle\Delta\vec{x}(2T)=\vec{x}^{(4)}_{A}(2T)-\vec{x}^{(3)}_{A}(2T)\,$
(76a)
$\displaystyle=4\vec{v}_{\mathrm{kick}}\left(Q^{L}_{\omega}T-\frac{Q^{A}_{M}}{\omega}\sin\left(\frac{\omega
T}{2}\right)\right)\sin\left(\frac{\omega T}{2}\right)\sin(\omega
T+\phi_{0})\,$ (76b)
$\displaystyle\Delta\vec{v}(2T)=\vec{v}^{(4)}_{A}(2T)-\vec{v}^{(D_{2})}_{A}(2T)\,$
(76c)
$\displaystyle=-4\vec{v}_{\mathrm{kick}}Q^{L}_{\omega}\sin^{2}\left(\frac{\omega