CALT-TH/2022-011

# Atom Interferometer Tests of Dark Matter

Yufeng Du  Walter Burke Institute for Theoretical Physics, California
Institute of Technology, Pasadena, CA 91125, USA  Clara Murgui  Walter Burke
Institute for Theoretical Physics, California Institute of Technology,
Pasadena, CA 91125, USA  Kris Pardo  Jet Propulsion Laboratory, California
Institute of Technology, Pasadena, CA 91109, USA  Yikun Wang  Walter Burke
Institute for Theoretical Physics, California Institute of Technology,
Pasadena, CA 91125, USA  Kathryn M. Zurek  Walter Burke Institute for
Theoretical Physics, California Institute of Technology, Pasadena, CA 91125,
USA

###### Abstract

Direct detection experiments for dark matter are increasingly ruling out large
parameter spaces. However, light dark matter models with particle masses $<$
GeV are still largely unconstrained. Here we examine a proposal to use atom
interferometers to detect a light dark matter subcomponent at sub-GeV masses.
We describe the decoherence and phase shifts caused by dark matter scattering
off of one “arm” of an atom interferometer using a generalized dark matter
direct detection framework. This allows us to consider multiple channels:
nuclear recoils, hidden photon processes, and axion interactions. We apply
this framework to several proposed atom interferometer experiments. Because
atom interferometers are sensitive to extremely low momentum deposition and
their coherent atoms give them a boost in sensitivity, these experiments will
be highly competitive and complementary to other direct detection methods. In
particular, atom interferometers are uniquely able to probe a dark matter sub-
component with $m_{\chi}\lesssim 10~{}\rm{keV}$. We find that, for a mediator
mass $m_{\phi}=10^{-10}m_{\chi}$, future atom interferometers could close a
gap in the existing constraints on nuclear recoils down to
$\bar{\sigma}_{n}\sim 10^{-50}~{}\rm{cm}^{2}$ for $m_{\chi}\sim
10^{-5}-10^{-1}~{}\rm{MeV}$ dark matter masses.

##  I Introduction

Although we have seen dark matter (DM) through its astrophysical and
cosmological effects [1, 2, 3, 4, 5, 6, 7, 8], we have yet to directly detect
DM. The traditional direct detection methods, which rely on measuring nuclear
recoils, have succeeded in placing ever stronger limits on Weakly Interacting
Massive Particles (WIMPs), but have not led to any detections [9, 10, 11, 12,
13]. In the last several years, motivated by theories of dark sectors [14, 15,
16, 17, 18, 19, 20, 21, 22, 23], many new types of direct detection channels
and experiments have been proposed to probe lighter DM masses. Electronic
transitions in a variety of materials [24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], molecular excitations [45, 46,
47], and phonon excitations [48, 49, 50, 51, 52] have been identified as
promising new directions.

Recently, there has been an interest in using quantum sensors, and especially
those in space, to detect dark matter [e.g., 53, 54]. Atom interferometers, in
particular, offer an interesting and complementary pathway to dark matter
detection compared to conventional searches. These experiments use falling
atoms to measure differential forces along two paths, and they can be used as
accelerometers [e.g., 55, 56]. Recent suggestions for DM detection with atom
interferometers have focused on ultralight DM with couplings to either the
photon or the electron. These couplings would cause a temporal oscillation in
fundamental constants that would induce phase shifts in an atom interferometer
[57, 58, 59]. Notably, these tests would only be applicable to specific types
of atom interferometers – those that rely on atomic transitions and shared
lasers.

Another proposal considers the decoherence caused by DM scattering in atom
interferometers [60, 61]. This would be applicable to all types of atom
interferometers. Previous work has shown that future atom interferometers
would provide promising constraints on DM that interacts with the Standard
Model (SM) through nuclear recoils [61]. Atom interferometers have two useful
features that enable these constraints: the coherence of the atoms allows for
an $N^{2}$ enhancement in the scattering rate, and they do not have a minimum
energy threshold. This means that atom interferometers can probe to extremely
low momentum depositions, and thus low dark matter masses.

In this paper, we consider both the decoherence and phase signatures in more
depth, and expand the atom interferometer detection mechanism to further
scattering channels. Fig. 1 gives a schematic description of the detection
strategy. We standardize and generalize the formalism of Ref. [61] using the
methods of Ref. [62]. We then consider nuclear recoils, hidden photon
processes, and coherent axion scattering. Our main results are summarized in
Fig. 2, which shows that atom interferometers can be uniquely powerful for
probing nuclear recoil of DM with masses $m_{\chi}\lesssim 1~{}\rm{MeV}$.
Although the constraints through the other channels are not competitive, we
include rough analytic estimates.

The paper is organized as follows. In Section II we discuss the overall
method, explain our general formalism, and describe the atom interferometer
experiments we consider in this paper. We then apply our formalism in Section
III to nuclear recoils (III.1), and dark photon processes and coherent axion
scattering (III.2). We discuss how these atom interferometer constraints
compare to other relevant constraints in Section IV, and we conclude in
Section V.

Throughout this paper, we assume natural units, $\hbar=c=1$ and consider a DM
particle, $\chi$, that is a subcomponent of the total DM in the Universe:
$\Omega_{\chi}=0.05~{}\Omega_{\rm{DM}}$. Light DM with $m_{\chi}\lesssim
1~{}\rm{MeV}$ interacting with nucleons at a large enough rate to be
detectable in an atom interferometer will typically come into thermal
equilibrium at some point in the history of the Universe, giving rise to a
variety of constraints, discussed in detail in Ref. [63]; considering a dark
matter subcomponent evades these bounds. The code used for our calculations
and figures is publicly available at:
https://github.com/kpardo/atom_interferometer_dm.

##  II Direct Detection with Atom Interferometers

In this section, we review the effects of DM scattering on atom
interferometers. We develop the general formula that describes the scattering
rate measurable by an atom interferometer, and then discuss the specific atom
interferometer experiments we consider in the paper.

In essence, DM scattering will cause decoherence and phase shifts in an atom
interferometer (see Fig. 1). Consider a Mach-Zender atom interferometer [for a
general review of atom interferometers, see Ref. 56]. Analogously to the
conventional Mach-Zender interferometer, here the de-Broglie wave nature of
the atoms plays the role of light, while laser pulses act as a beamsplitters
and mirrors. The atoms start in a coherent wavepacket, then are split by a
laser pulse into two different wavepackets that travel along two different
paths. Two more laser pulses reverse the motion of these wavepackets and
recombine them. The state of the recombined wavepacket is then read, with the
relative populations depending on the initial prepared state, the timing of
the laser pulses, and any differential forces felt along the different paths.

The main observables from an atom interferometer are the visibility, $V$, and
phase, $\phi$. The visibility is a measure of the decoherence of the system,
and the phase measures the path differences between the two arms. Previously,
atom interferometer measurements would require calibration of the fringe over
several measurements with varying laser phase to give a proper measurement of
either the phase or visibility [see, e.g., 55]. However, modern atom
interferometers are capable of measuring a spatial fringe, and thus $V$, with
each measurement [some ways of doing this are shown in 64, 65]. While $\phi$
cannot be measured from a single spatial fringe without prior knowledge, the
relative phase between two different measurements or between two atom
interferometers on the same laser can be measured [e.g., 66, 67].

Figure 1: Left: Cartoon of dark matter detection with atom interferometry.
Three laser pulses are used to separate, redirect, and recombine the atom
wavepackets. Atom interferometers are a promising avenue for detecting DM
because they are sensitive to low momentum transfers and the coherence of the
atom clouds allows for an enhancement in the DM scattering rates. Right, Top:
Cartoon of expected fringe measurement (e.g., probability of the state
$|\psi\rangle$ at the end of the atom interferometer sequence as a function of
final laser phase) for the main observables from DM scattering. The black line
gives the expected fringe if there are no DM or other effects. The blue
(orange) points give simulated data points that show the effect of decoherence
(phase shifts) on the fringe. Right, Bottom: Observable effects as a function
of the DM momentum transfer ($q$) and the spatial distance between the
wavepacket paths of the atoms ($\Delta x$). The blue line shows the
decoherence induced by DM scattering (see Eqn. 8), and the orange line shows
the phase shift (see Eqn. 9). Both are maximally affected by DM at $q\sim
1/\Delta x$, which is where we expect the DM to “resolve” the clouds.

We will assume that both the phase and visibility can be reliably measured,
and we will consider the case where the only relevant force causing
decoherence comes from scattering with DM particles, which can be considered
to be a thermal bath. The first laser pulse splits the population of the atoms
and prepares the system $|\Psi\rangle$ in a quantum superposition of two
states: the “blue” and the “purple” wavepackets, following the same color
coding as in Fig. 1. In particular, any DM scattering with either the “purple”
or “blue” wavepacket will cause decoherence in the final state. The density of
states in the {purple, blue} basis is given by:

$\rho=\frac{1}{2}\left(\begin{matrix}1&\gamma\\\
\gamma^{*}&1\end{matrix}\right)\;,$ (1)

where $\gamma$ is the decoherence factor, which contains both the decoherence
and phase effects. Specifically, this factor can be decomposed into a
dimensionless decoherence, $s$ (real, positive), and a phase, $\phi$ (real),
via: $\gamma=\exp(-s+i\phi)$.

When the wavepackets recombine, the evolved state $|\tilde{\Psi}\rangle$ is
measured in the basis of the initial state $|\Psi\rangle$ with a probability
of

$P(|\tilde{\Psi}\rangle,|\Psi\rangle)=\text{Tr}\\{\rho\,|\Psi\rangle\langle\Psi|\\}=\frac{1}{2}(1+\mathrm{Re}(\gamma))=\frac{1}{2}(1+e^{-s}\cos\phi)\;.$
(2)

The visibility of an experiment, $V=e^{-s}$, is given by measuring the
amplitude of the fringe. As explained above, the absolute phase cannot be
measured – instead a differential phase must be measured by comparing two
measurements at different times or measurements of different atom
interferometers on the same laser. DM scattering will affect both of these
observables.

The decoherence factor will be directly related to the number of DM scattering
events. Let $R$ give the rate of dark matter scattering per target mass that
leads to decoherence or phase shifts in the atom interferometer. Then, over a
measurement time, $t_{\rm{exp}}$, the decoherence factor accumulated will be:

$\gamma=\exp\left[-m_{T}\int_{0}^{t_{\rm{exp}}}R~{}dt\right]\;,$ (3)

where $m_{T}$ gives the target mass – the mass of the atom cloud in the
interferometer.

Now, we turn our attention to calculating the rate per target mass, $R$.
Consider a non-relativistic DM particle with mass $m_{\chi}$ that has some
spin-independent scattering interaction with SM nucleons, mediated by a field
of mass $m_{\phi}$. For reasons that will be fully discussed later, we stress
that the observables we focus on in this paper are only effective for spin-
independent interactions. The process occurs as shown in the Feynman diagram
below, with the corresponding reference cross-section $\bar{\sigma}$,

$\begin{gathered}\includegraphics[width=73.7146pt]{RefXSec.pdf}\end{gathered}:\quad\bar{\sigma}=\frac{y_{\chi}^{2}\,y_{n}^{2}}{4\pi}\frac{\mu^{2}}{(m_{\chi}^{2}v_{0}^{2}+m_{\phi}^{2})^{2}},$
(4)

where $y_{i}$ are the dimensionless coupling factors,
$\mu=m_{\chi}m_{N}/(m_{\chi}+m_{N})$ is the reduced mass of the DM and each
nucleon, and $v_{0}=230~{}\rm{km/s}$ is the local average velocity of the DM.
When necessary, we set $y_{\chi}=\sqrt{4\pi}$ in this paper.

As we will see, atom interferometers have the best constraining power at low
$m_{\chi}$. We will be most interested in the limit $m_{\chi}\ll m_{N}$. Thus,
we replace the reduced mass with $\mu\approx m_{\chi}$ throughout. In
addition, we will assume, without loss of generality, that the DM $\chi$ is a
real scalar. Other scenarios can be adapted to the reference cross section
from Eqn. 4 by redefining the coupling $y_{\chi}$.

We would like to find the rate of DM scattering events that lead to
decoherence or phase shifts in an atom interferometer. The differential rate
per target mass is given by [see, e.g., 62]:

$\frac{dR}{d\omega}=\frac{1}{\rho_{T}}\frac{\rho_{\chi}}{m_{\chi}}\int
d^{3}\mathbf{v}~{}f(\mathbf{v})\frac{d\Gamma}{d\omega}\;,$ (5)

where $\rho_{T}$ is the density of the target material, $\omega$ is the energy
deposition, $\rho_{\chi}$ is the DM density, and $f(\mathbf{v})$ is the DM
velocity distribution function. The differential interaction rate,
$d\Gamma/d\omega$, can be written using the formalism of Ref [62] as:

$\frac{d\Gamma}{d\omega}=\frac{\pi\bar{\sigma}}{\mu^{2}}\int\frac{d^{3}{\bf
q}}{(2\pi)^{3}}\mathcal{F}_{\rm{med}}^{2}(\mathbf{q})S(\mathbf{q},\omega)\delta(\omega-\omega_{\mathbf{q}})\;,$
(6)

where $\mathbf{q}$ is the momentum deposition, $\bar{\sigma}$ is a reference
cross section, $\mathcal{F}_{\rm{med}}(\mathbf{q})$ is the mediator form
factor, and $S(\mathbf{q},\omega_{\mathbf{q}})$ is the dynamic structure
factor. It is this last factor that encodes atom interferometer-specific
physics.

The dynamic structure factor encodes the response of a detector to a given
momentum deposition [see, e.g., Eqn. 14 of 62]. It includes terms that
conserve energy in the target and account for a momentum transfer dependence
in the target material. In addition, the dynamic structure factor should
encode the restrictions of an atom interferometer as a DM detector. Namely,
only scattering events that produce momentum transfer along the atom
interferometer separation direction, $\mathbf{\Delta x}$, produce decoherence
and phase shifts. The probability that a scattering event will contribute to
the decoherence factor is [68, 69, 60, 61]:

$p_{\rm{decoh}}=1-\exp\left[i\mathbf{q}\cdot\mathbf{\Delta x}\right]\;.$ (7)

To get a better idea of this effect, let us consider the real and imaginary
parts separately. If we take only the real component of $p_{\rm{decoh}}$ and
integrate over the angle between $\mathbf{q}$ and $\mathbf{\Delta x}$, we
find:

$\int_{-1}^{1}\mathrm{Re}(p_{\rm{decoh}})~{}d\cos\theta_{\mathbf{q\Delta
x}}=1-\mathrm{sinc}(q\Delta x)\;,$ (8)

where $q\equiv\|\mathbf{q}\|$ and $\Delta x\equiv\|\mathbf{\Delta x}\|$. This
is the “decoherence effect” plotted in Fig. 1. Note that the decoherence is
maximal when the DM “resolves” the clouds, $q\sim 1/\Delta x$, and quickly
vanishes for $q<1/\Delta x$.

Integrating over the imaginary part in this same way would give us zero. Some
anisotropy in the DM flux is needed for its interaction to induce a relative
phase between the two clouds. Taking into account that anisotropy, and
anticipating the full derivation of the rate of DM interactions, we
parameterize the effect responsible for generating a non-zero shift in the
relative phase as:

$\begin{split}\int_{-1}^{1}\mathrm{Im}(p_{\rm{decoh}})\,e^{-\displaystyle\tilde{v}_{e}q\Delta
x\cos\theta_{\bf q\Delta x}}~{}d\cos\theta_{\mathbf{q\Delta
x}}&=2\,\frac{\tilde{v}_{e}\sin(q\Delta x)\cosh(\tilde{v}_{e}q\Delta
x)-\cos(q\Delta x)\sinh(\tilde{v}_{e}q\Delta x)}{q\Delta
x(\tilde{v}_{e}^{2}+1)}\;,\end{split}$ (9)

where $\tilde{v}_{e}=v_{e}/(v_{0}^{2}m_{\chi}\Delta x)$ is a scaled version of
the Earth’s velocity, and we have assumed that ${\bf\Delta x}$ is aligned with
${\bf v_{e}}$. Like the decoherence effect, the phase effect is maximal at
$q\sim 1/\Delta x$. However, it oscillates rapidly about $0$ for $q>1/\Delta
x$.

The decoherence factor probability is independent of the overall probability
that an incoming scattering event produces a given final state. In other
words, the dynamic structure factor is:

$S(\mathbf{q},\omega)\equiv\frac{p_{\rm{decoh}}}{V}\sum_{f}|\langle
f|\mathcal{F}_{T}(\mathbf{q})|i\rangle|^{2}~{}2\pi\delta(E_{f}-E_{i}-\omega_{\mathbf{q}})\;,$
(10)

where $\mathcal{F}_{T}$ is the target form factor and the delta function
enforces energy conservation.

The last factor in determining the general rate equation is the DM phase-space
distribution. We will follow Ref. [62] and take:

$f(\mathbf{v})=\frac{1}{N_{0}}\exp\left(-\frac{(\mathbf{v}+\mathbf{v}_{e})^{2}}{v_{0}^{2}}\right)\Theta(v_{\rm{esc}}-\|\mathbf{v}+\mathbf{v}_{e}\|)\;,$
(11)

where $\|\mathbf{v}_{e}\|=240~{}\rm{km/s}$ is the Earth’s
velocity111Technically, $\mathbf{v}_{e}$ should be replaced by the velocity of
each experiment, for the space-based experiments. However, many of these
mission concepts do not have finalized orbits. They are likely to be solar
orbits with semi-major axis $\sim 1~{}\rm{AU}$. To first order in orbital
eccentricity, this would give them similar velocities to the Earth. The
experiment on the International Space Station (BECCAL) is one exception to
this. However, it’s orbital velocity around the Earth is within $\lesssim
10\%$ of the Earth’s velocity we assume here. Thus we assume Earth’s velocity
for all of our experiments., $\Theta(x)$ is the Heaviside function,
$v_{\rm{esc}}=600~{}\rm{km/s}$ is the escape velocity of the Galaxy, and
$N_{0}$ is the normalization factor:

$N_{0}=\pi^{3/2}v_{0}^{3}\left[\mathrm{erf}\left(\frac{v_{\rm{esc}}}{v_{0}}\right)-\frac{2}{\sqrt{\pi}}\frac{v_{\rm{esc}}}{v_{0}}\exp\left(-\frac{v_{\rm{esc}}^{2}}{v_{0}^{2}}\right)\right]\;.$
(12)

We can simplify the integration by remembering that there is a relationship
between the velocity and the momentum transfer. Regardless of the target, the
energy deposition from a given DM momentum transfer and velocity is:

$\omega_{\mathbf{q}}=qv\cos\theta_{\mathbf{qv}}-\frac{q^{2}}{2m_{\chi}}\;,$
(13)

where $\theta_{\mathbf{vq}}$ is the angle between $\mathbf{v}$ and
$\mathbf{q}$, and $v=\|\mathbf{v}\|$.

Thus, the energy deposition delta function can be traded for one in velocity.
In general, we can define [62]:

$\displaystyle\begin{split}g(\mathbf{q},\omega)&\equiv\int
d^{3}\mathbf{v}~{}f(\mathbf{v})2\pi\delta(\omega-\omega_{\mathbf{q}})\\\
&=\frac{2\pi^{2}v_{0}^{2}}{N_{0}q}\left[\exp\Big{(}-v_{-}^{2}(\mathbf{q},\omega)/v_{0}^{2}\Big{)}-\exp(-v_{\rm
esc}^{2}/v_{0}^{2})\right]\;,\end{split}$ (14)

where:

$v_{-}(\mathbf{q},\omega)=\mathrm{min}\left[\frac{1}{q}\left|\mathbf{q}\cdot\mathbf{v}_{e}+\omega+\frac{q^{2}}{2m_{\chi}}\right|,v_{\rm{esc}}\right]\;.$
(15)

The energy deposition in this equation is set by the conservation of energy
equation in the dynamic structure factor, which is different for the various
channels we discuss in this paper.

The rate equation for DM scattering that is measurable via either decoherence
or phase shifts in an atom interferometer is then:

$R=\frac{1}{\rho_{T}V}\frac{\rho_{\chi}}{m_{\chi}}\frac{\pi\bar{\sigma}}{m_{\chi}^{2}}\int\frac{d^{3}\mathbf{q}}{(2\pi)^{3}}\mathcal{F}_{\rm{med}}^{2}({\bf
q})\left(1-\exp(i\mathbf{q}\cdot\mathbf{\Delta x})\right)\sum_{f}|\langle
f|\mathcal{F}_{T}(\mathbf{q})|i\rangle|^{2}g(\mathbf{q},E_{f}-E_{i})\;.$ (16)

To get the total decoherence factor, which includes both the decoherence and
phase shift effects, we integrate over the measurement time and multiply by
the total target mass. If we ignore any daily modulation, then there is no
explicit time-dependence in the rate. This then gives a total decoherence of:

$s-i\phi=\frac{\rho_{\chi}}{m_{\chi}}\frac{\pi\bar{\sigma}}{m_{\chi}^{2}}t_{\rm{exp}}\int\frac{d^{3}\mathbf{q}}{(2\pi)^{3}}\mathcal{F}_{\rm{med}}^{2}({\bf
q})\left(1-\exp(i\mathbf{q}\cdot\mathbf{\Delta x})\right)\sum_{f}|\langle
f|\mathcal{F}_{T}(\mathbf{q})|i\rangle|^{2}g(\mathbf{q},E_{f}-E_{i})\;,$ (17)

where $t_{\rm{exp}}$ is the measurement time, and the volume factor in the
rate equation cancels with the volume factor in the dynamic structure factor.

We stress here that, unlike other direct detection experiments, there is no
minimum energy threshold for these atom interferometer experiments. Unlike
traditional direct detection setups, atom interferometers do not directly
measure the energy deposition – only the visibility and phase. This allows
atom interferometers to probe much lighter DM masses than traditional direct
detection setups. In practice, the decoherence observable loses sensitivity to
light DM once the mass is below scales $m_{\chi}v_{0}\lesssim 1/\Delta x$. On
the other hand, the phase has no minimum. The main limitations for this method
comes from other backgrounds.

There are many possible decoherence and phase shift mechanisms for atom
interferometers [e.g., see Ref. 70]. To definitively make a DM detection, an
atom interferometer would need to see a signal that varies with the expected
DM flux [61]. For a terrestrial experiment, this would give the so-called
“daily modulation” caused by the Earth’s rotation with respect to the incoming
DM flux. For a space-based experiment, like most of the ones we consider here,
the exact modulation will depend on the orbit. Low-Earth orbits will lead to
modulation signals with timescales on the order of the orbit (e.g., for an
experiment on the International Space Station, the timescale is $\sim{90}$
minutes). Other orbits would need to be studied in depth. We do not consider
this effect in-depth – we assume that if the decoherence and phase effects can
be measured in a single shot, then their variance with time will also be
measurable (but see Appendix A for a short description of how to include the
daily modulation). Thus, we study here a necessary, but not sufficient, metric
for detecting DM with atom interferometers.

We say that an atom interferometer has sensitivity to a given DM model when
the estimated signal is larger than the expected noise (i.e., we set a signal-
to-noise threshold of 1). Thus, the overall sensitivity to a given cross
section is given by solving for the cross section in the equation:

$\frac{(X-X_{\rm{bkg}})^{2}}{\sigma_{X}^{2}}=1\;,$ (18)

where $X$ denotes either the decoherence, $s$, or the phase, $\phi$.
$X_{\rm{bkg}}$ is the average value of either observable without any DM
scattering effects, while $\sigma_{X}$ is the noise for each observable. We
note that we do not study backgrounds here such as cosmic rays and solar
photons. Ref. [71] claimed, via an order of magnitude estimate, that the
decoherence between atom clouds caused by cosmic rays are not a significant
background. Solar photons, in the forward scattering limit, could be a
background source for phase shifts due to their lower energies (which can
mimic dark matter kinematics) and high fluxes. However, their interaction rate
will be suppressed because the atom clouds are neutral, with interactions
occurring through the polarizability, which is generally suppressed. We leave
detailed study for future work, but note in the interim that our reach curves
can be simply re-scaled with the background event rate.

First, we consider the decoherence sensitivity. The background for the
experiment, $s_{\rm{bkg}}$, is given as a function of the fringe visibility,
$V$, through: $s_{\rm{bkg}}=\ln(V^{-1})$. If there is no decoherence, then
$V=1$; however, most interferometers have at least some decoherence from
environmental effects. A conservative fringe visibility factor is $V=0.5$
[61], which we will use for all of our experiments. We assume that, per
measurement, the visibility noise is $\sigma_{V}/V=0.1$, which corresponds to
$\sigma_{s}=0.1$ per measurement. We also assume that $\sigma_{s}$ will scale
with the number of measurements, $N_{\rm{meas}}$, as: $\sigma_{s}\propto
N_{\rm{meas}}^{-1/2}$. As we discuss further in the next subsection, $N_{\rm
meas}$ is the number of drop times accumulated over the run-time (assumed to
be 1 year) of the experiment.

Now let us consider the phase sensitivity. We ignore any effects from gravity
here and set the average $\phi_{\rm{bkg}}=0$. The standard deviation of the
phase is then the minimum measurable phase shift, which is set by the number
of nucleons, typical contrast, and other backgrounds. While some experiments
directly list their minimum phase shifts, some do not. For these other
experiments, the minimum measurable acceleration can be used to calculate the
minimum phase via:

$\sigma_{\phi}=\frac{1}{4}m_{A}\Delta x\,t_{\text{exp}}a_{\text{min}},$ (19)

where $m_{A}$ is the mass of one atom within the atom interferometer, and the
momentum injected by the laser beam has been written as a function of the
separation of the wave packets and the measurement time. The factor of $1/4$
accounts for our use of the full measurement time, $t_{\rm{exp}}$, rather than
the time between pulses ($\frac{1}{2}t_{\rm{exp}}$), which is typically used
in these estimates. We list the minimum measurable phase shifts for each
mission in the last column of Table 1.

We would also like to note: there are upper bounds on the sensitivities as
well: if there is too much decoherence from DM scattering in an experiment,
then the experiment would never measure a fringe! This is then degenerate with
an experiment simply not working properly (i.e., not properly producing a
coherent state or having too many decoherence-producing backgrounds). In
principle, the space-based experiments are more subject to this problem than
terrestrial ones - if DM scattering cross sections were high enough to always
produce decoherence, then they would most likely never reach the terrestrial
detectors due to atmospheric shielding. Thus, a null test for DM in this case
would be to compare identical setups in Earth and space. We ignore these
issues in our sensitivity curves since this would be easily solved via the
above test. In addition, several atom interferometers have been shown to
properly work on the surface of the Earth, and one has been shown to work in
Low-Earth Orbit [72]. Although it may be difficult to distinguish DM from
other backgrounds, no signal of decoherence unambiguously constrains DM
interactions.

To summarize: DM scattering with momentum transfer along the separation axis
of an atom interferometer causes decoherence and phase shifts in an atom
interferometer. The rate of these interactions is given by Eqn. 16. The exact
form of the rate will depend on the particulars of the DM model and scattering
type. The measured decoherence and phase shift effects are then given by Eqn.
17. The overall sensitivity of the experiment is given by comparing either the
decoherence or the phase shift to any background signal and the expected noise
via Eqn. 18. The resulting limits are shown in Figs. 2, 3, and 4, which we
will discuss in detail in Sections III.1 & III.2.

Figure 2: DM detection prospects for atom interferometers in the nucleon cross
section vs $m_{\chi}$ plane assuming a light mediator, $m_{\phi}\propto
m_{\chi}$. The solid, colored lines show the limits from decoherence, and the
dashed, colored lines show the limits from phase shifts. The different colors
signify different experiments, where we assume 1 year of continuous
measurements for each experiment. The gray lines give the superfluid helium
prospects for the nuclear recoil (solid) and 2-phonon (dashed) cases [49], and
the Al2O3 prospects for phonon excitations (dotted) [44]. The superfluid
helium forecasts assume a massless mediator. We assume an energy threshold of
$\omega_{\rm{min}}=1~{}\rm{meV}$ for Al2O3. The solid, gray region indicates
5th force constraints from Ref. [73] and the stellar emission constraints (RG
and HB stars [74], and SN1987a [75]). Although the atom interferometer
experiments can probe to much lighter $m_{\chi}$ than shown here, we only plot
to $10~{}\rm{eV}$. Lighter DM would have velocities greater than the escape
speed of the galaxy, if we assume that the DM decouples before the QCD
temperature. For a longer discussion of these other constraints, see Section
IV. We assume that the DM particle $\chi$ is a DM subcomponent and set
$y_{\chi}=\sqrt{4\pi}$ for the 5th force and stellar emission constraints.
Figure 3: DM detection prospects for atom interferometers in the nucleon cross
section vs $m_{\chi}$ plane for fixed $m_{\phi}=1~{}\rm{eV}$. The solid,
colored lines show the limits from decoherence, and the dashed, colored lines
show the limits from phase shifts. The different colors signify different
experiments, where we assume 1 year of continuous measurements for each
experiment. The gray lines give the superfluid helium prospects for the
nuclear recoil (solid) and 2-phonon (dashed) cases [49], and the Al2O3
prospects for phonon excitations (dotted) [44]. The superfluid helium
forecasts assume a massless mediator. We assume an energy threshold of
$\omega_{\rm{min}}=1~{}\rm{meV}$ for Al2O3. The solid, gray region indicates
5th force constraints from Ref. [73]. The change in the shape of the region is
set by the transition from heavy to light mediator (as $m_{\chi}$ increases
relative to the fixed $m_{\phi}=1~{}\rm{eV}$). Figure 4: Forecasted DM-nucleon
cross section limits from atom interferometers for the heavy mediator case.
The solid, colored lines show the limits from decoherence, and the dashed,
colored lines show the limits from phase shifts. The different colors signify
different experiments, where we assume 1 year of continuous measurements for
each experiment. The gray lines give the superfluid helium prospects for the
nuclear recoil (solid) and 2-phonon (dashed) [49]. The dotted line gives the
Al2O3 prospects from phonon excitations [44]. We note that much of this
parameter space is excluded by various collider, astrophysical, and
cosmological constraints, dependent on the mediator mass and hence not shown
here [see Fig. 6 of Ref. 49, and related discussions therein]. For this plot,
we also assume that the $\chi$ makes up all of the DM. Although the dark
matter model space probed by atom interferometers for massive mediators is of
somewhat limited interest due to existing collider, astrophysical, and
cosmological constraints, we include this plot because it enables an easy
calculation of the constraints for the other processes we consider in Section
III.2, where such constraints do not necessarily apply.

###  II.1 Atom Interferometer Experiments

In this section, we describe the various atom interferometer experiments that
we consider in this paper. For the most part, these are future, space-based
experiment concepts that span a wide range in the parameters that matter for
our DM signals ($\Delta x$, $N_{\rm{nucleon}}$, $r_{\rm{cloud}}$, and
$t_{\rm{exp}}$). Many of the other experiments that we do not discuss here
(e.g., AEDGE [76], AION [77], STE-QUEST [78]) have parameters, where
published, within the ranges we consider here and also feature multiple
interferometers which are beyond the scope of the present work. We note that
our public code can be used to calculate the limits for any other experiment,
given the necessary parameters.

Table 1 gives the important parameters for all of the experiments we consider
and we discuss each of them in turn below. In analogy to other direct
detection experiments, which typically give limits assuming 1 kg-yr of
exposure, we assume an overall mission time of 1 year for all experiments. To
calculate the number of measurements that each experiment will take, $N_{\rm
meas}$, we assume no downtime between measurements and simply divide the
assumed 1 year mission time by the experimental measurement time.

The Bose-Einstein Condensate and Cold Atom Laboratory (BECCAL) is a proposed
upgrade [79] to the Cold Atom Lab (CAL), which is currently running on the
International Space Station (ISS) [80, 72]. This upgrade would improve the
current atom interferometer capabilities of CAL, including a higher number of
atoms in the condensates. BECCAL would study a few different atoms; however
for simplicity we focus on its 87Rb capabilities in this paper. We assume that
BECCAL would produce condensates with $10^{6}$ atoms, a cloud size of
$150~{}\mu\rm{m}$, and a free-fall (measurement) time of 2.6 s. We take the
separation between the atom interferometer arms to be $\Delta x=3~{}\rm{mm}$.
The sensitivity of the experiment is limited by the rotation of the ISS as it
orbits the Earth. We assume an acceleration sensitivity set by the noise in
the rotation sensor used to counter-rotate the experimental setup, which sets
$a_{\rm{min}}=3\times 10^{-12}~{}\rm{m/s}^{2}$.

Macroscopic Quantum Resonators (MAQRO) is a proposed space mission to perform
high-mass atom interferometry [81, 82]. The mission would use SiO2 molecules
in an atom cloud with $10^{10}$ nucleons and a radius of $120~{}\rm{nm}$. The
baseline separation would be $100~{}\rm{nm}$ and the free-fall measurement
time would be $100~{}\rm{s}$. We assume an acceleration sensitivity of
$10^{-13}~{}\rm{m/s}^{2}$.

The Gravity Probe and Dark energy Detection mission (GDM) is a NASA Innovative
Advanced Concepts (NIAC) Phase II mission
concept222https://www.nasa.gov/directorates/spacetech/niac/2020_Phase_I_Phase_II/Gravity_Observation_and_Dark_energy_Detection_Explorer/.
Although its primary goal would be to measure deviations from the
gravitational inverse-square law, its instruments would be well-suited for the
direct detection scheme we consider here. It would consist of a constellation
of four spacecraft, each with six atom interferometers. In this paper, we
consider the limits from just one of these interferometers. However, the large
number of interferometers, each with separation axes in different directions
would be useful for disentangling any possible DM signature from other
backgrounds. We assume an atom interferometer baseline of $\Delta
x=25~{}\rm{m}$, an interrogation time of $20~{}\rm{s}$, a cloud size of
$r_{\rm{cloud}}=1~{}\rm{mm}$, and $10^{8}$ 87Rb atoms. We assume an
acceleration sensitivity of $2.2\times 10^{-15}~{}\rm{m/s}^{2}$.

Finally, we consider a proposed terrestrial experiment, described in Ref.
[83], which we refer to as the “Pino” experiment in this paper. This table-top
experiment concept would use an all-magnetic scheme to perform a double slit
experiment with a macroscopic sphere. We take the example experimental
parameters listed in Appendix A of Ref. [83] as our parameters. Namely, we
assume a sphere of Niobium with radius 1 micron and $2\times 10^{13}$
nucleons. We assume a separation of $290~{}\rm{nm}$ and a free-fall time of
$0.483~{}\rm{s}$. The goal for this experimental concept is to measure the
decoherence from gravity. Thus, we let the self-gravity of the sphere define
the minimum acceleration: $a_{\rm{min}}=GM/r_{\rm{cloud}}^{2}=2.5\times
10^{-12}~{}\rm{m/s}^{2}$.

Mission | Target | $r_{\rm{cloud}}$ | $N_{\rm{nucleon}}$ | $\Delta x$ | $t_{\rm{exp}}$ | $a_{\rm{min}}$ | $\sigma_{\phi}$
---|---|---|---|---|---|---|---
|  | [m] |  | [m] | [s] | [m/s2] | [rad]
BECCAL |  87Rb | $1.5\times 10^{-4}$ | $8.7\times 10^{7}$ | $3\times 10^{-3}$ | 2.6 | $3\times 10^{-12}$ | $8.1\times 10^{-6}$
MAQRO | SiO2 | $1.2\times 10^{-7}$ | $10^{10}$ | $10^{-7}$ | $100$ | $1\times 10^{-13}$ | $2.4\times 10^{-10}$
GDM |  87Rb | $10^{-3}$ | $8.7\times 10^{9}$ | $25$ | $20$ | $2.2\times 10^{-15}$ | $3.8\times 10^{-4}$
Pino | Nb | $10^{-6}$ | $2.2\times 10^{13}$ | $2.9\times 10^{-7}$ | 0.483 | $2.5\times 10^{-12}$ | $1.3\times 10^{-10}$
Table 1: Mission concept parameters assumed in this paper.

##  III Dark Matter Interaction Rates in Atom Interferometers

Various interaction processes between the DM and atoms can cause the
decoherence and phase shift effects. As will be discussed in detail below,
because atoms in an atom interferometer are in a Bose-Einstein condensate
state, the DM-atom interaction rate will receive a $N^{2}$ enhancement in the
coherent scattering limit, where $N$ is the number of nucleons or electrons in
the target. Thus, to exploit this $N^{2}$ enhancement, we primarily seek
processes that do not break the coherence of states within one wavepacket. In
particular, we consider DM scattering via nuclear recoils (III.1), and
interactions involving dark photons and coherent axion scattering (III.2).
Only the nuclear recoils produce interesting constraints; however, we give
estimates of the constraints for the other cases.

###  III.1 Nuclear Recoil

In this section, we derive the atom interferometer observables expected from
DM scatering via nuclear recoils. We must use the physics of nuclear recoils
to set two of the terms in Eqn. 17: the target form factor, and the energy
deposition via the conservation of energy. We start with these definitions and
then walk through the simplifications needed to arrive at the final
decoherence equation, Eqn. 27. We then consider the phase observable.

We will be agnostic about the mediator in this section and take its form
factor to be:

$\mathcal{F}_{\rm{med}}(q)=\frac{(m_{\chi}v_{0})^{2}+m_{\phi}^{2}}{q^{2}+m_{\phi}^{2}}\;,$
(20)

where $v_{0}$ is the DM velocity dispersion. This reduces to the light
mediator form factor
($\mathcal{F}_{\rm{med}}(q)\rightarrow(m_{\chi}v_{0}/q)^{2}$) for
$m_{\phi}\rightarrow 0$, and to the heavy mediator form factor
($\mathcal{F}_{\rm{med}}(q)\rightarrow 1$) for $m_{\phi}\gg q$.

An important factor in setting these limits is whether the scattering is in
the coherent limit (i.e., whether the DM scatters off of the cloud as a whole
or only 1 nucleus). This is set by the momentum transfer, $q$, of the dark
matter and the relevant cloud radii. The various limits here will be important
for both finding the target form factor and the minimum velocity, $v_{-}$.

Consider an atom interferometer that has $N_{A}$ identical atoms, each with
atomic mass number $A$. The structure factor contains the response of these
$N=AN_{A}$ nucleons to a DM scattering event. It will depend on the momentum
transfer, $q$, versus the size of the atom cloud, $r_{C}$, the size of each
atom, $r_{A}$, and the size of each nucleon, $r_{N}$. In particular, there are
four regimes to consider:

  1. 1.

$q\ll 1/r_{C}$: Here, the DM does not “see” any of the separate atoms.
Instead, it sees the entire atomic cloud as one object. This means that the
rate will receive a Born enhancement: $R\propto N^{2}$.

  2. 2.

$1/r_{C}<q<1/r_{A}$: In this case, the DM sees the separate atoms, but does
not see individual nucleons. Thus, there is a Born enhancement for each of the
nucleons within an atom, but not for all of the atoms: $R\propto A^{2}N_{A}$.

  3. 3.

$1/r_{A}<q<1/r_{N}$: Here, the DM sees each separate nucleon. There is no Born
enhancement of any kind: $R\propto N$.

  4. 4.

$q\gg 1/r_{N}$: We do not consider any extra interactions in this case, and we
allow the rate to go to zero whenever the momenta is large enough to probe
within individual nucleons.

With all of this in mind, we take the structure factor to be:

$\sum_{f}|\langle
f|\mathcal{F}_{T}(\mathbf{q})|i\rangle|^{2}=\sum_{i,j=1,...,N_{A}}\langle
e^{-i\Delta\mathbf{y}_{ij}\cdot\mathbf{q}}\rangle_{\rm{target}}=N[1+AF_{A}^{2}(qr_{A})+A(N_{A}-1)F^{2}(qr_{C})]\;,$
(21)

where $F_{A}(x)$ is the atomic form factor, $F(x)$ is the cloud form factor,
and $\Delta y_{ij}$ are the interparticle spacings. We take the atomic form
factor to be the Helm form factor [84]:

$F_{A}(q)=\frac{3j_{1}(qr_{A})}{qr_{A}}e^{-q^{2}s_{p}^{2}/2}\;,$ (22)

where $j_{1}(x)$ is a spherical Bessel function, $r_{A}\approx A^{1/3}\times
1.2~{}\rm{fm}$ is the typical size of an atom, and $s_{p}\approx
0.9~{}\rm{fm}$ is the skin depth.

The cloud form factor is typically taken as [60, 85]:

$F(x)=\frac{3j_{1}(x)}{x}\;.$ (23)

Note that for low momentum transfer, $F^{2}(q\,r_{C})\rightarrow 1$. For high
momentum transfer, $F^{2}(q\,r_{C})\rightarrow 1/(q\,r_{C})^{4}$.

Now, let us consider the conservation of energy. For a given momentum transfer
$q$, the change in energy of a target nucleus is:
$E_{f}-E_{i}=q^{2}/(2m_{i})$, where $i$ refers to either the full atomic cloud
or one atom, as discussed above. However, note that for the dark matter masses
that we consider in this paper $m_{\chi}\ll m_{i}$ regardless of the target
$i$. Thus, we can approximate $E_{f}-E_{i}=\omega\simeq 0$. With the energy
deposition given by Eqn. 13, the minimum velocity is then given by:

$v_{-}({\bf
q})\approx\mathrm{min}\left[\frac{1}{q}\left|\mathbf{q}\cdot\mathbf{v}_{e}+\frac{q^{2}}{2m_{\chi}}\right|,v_{\rm{esc}}\right]\;.$
(24)

Note that $v_{-}$ depends on both the magnitude and direction of $\mathbf{q}$.
Although the direction will not have a large impact on the decoherence rate
calculation, it will be important for the phase calculation.

####  III.1.1 Decoherence

The decoherence is given by the real part of Eqn. 17. As was shown by Ref.
[61], the daily modulation only affects the decoherence rate by a factor $2$.
We also find that it has little effect on the final limits (see Appendix A).
Thus, for ease of understanding the underlying physics, we set
$\mathbf{v}_{e}=0$ for the decoherence calculations.

Putting all of the above together gives a decoherence rate of:

$\displaystyle\begin{split}R_{\rm{decoh}}=\frac{1}{\rho_{T}V}\frac{\rho_{\chi}}{m_{\chi}}\frac{\pi\bar{\sigma}v_{0}^{2}(q_{0}^{2}+m_{\phi}^{2})^{2}}{2N_{0}m_{\chi}^{2}m_{\phi}^{4}}\int
dq\frac{q}{\left((q/m_{\phi})^{2}+1\right)^{2}}N[1+AF_{A}^{2}(qr_{A})+A(N_{A}-1)F^{2}(qr_{C})]\\\
\times\left[\exp\left(-\frac{v_{-}^{2}(q)}{v_{0}^{2}}\right)-\exp\left(-\frac{v_{\rm
esc}^{2}}{v_{0}^{2}}\right)\right]\int_{-1}^{1}d\cos\theta\left(1-\cos(\mathbf{q}\cdot\mathbf{\Delta
x})\right)\;,\end{split}$ (25)

where we define $\theta$ as the angle between ${\bf q}$ and ${\bf\Delta x}$.
The cosine integral can be done analytically, yielding:

$\displaystyle\begin{split}R_{\rm{decoh}}=\frac{1}{\rho_{T}V}\frac{\rho_{\chi}}{m_{\chi}}\frac{\pi\bar{\sigma}v_{0}^{2}(q_{0}^{2}+m_{\phi}^{2})^{2}}{N_{0}m_{\chi}^{2}m_{\phi}^{4}}\int
dq\frac{q}{\left((q/m_{\phi})^{2}+1\right)^{2}}N[1+AF_{A}^{2}(qr_{A})+A(N_{A}-1)F^{2}(qr_{C})]\\\
\times\left[\exp\left(-\frac{v_{-}^{2}(q)}{v_{0}^{2}}\right)-\exp\left(-\frac{v_{\rm
esc}^{2}}{v_{0}^{2}}\right)\right]\left(1-\frac{\sin(q\Delta x)}{q\Delta
x}\right).\end{split}$ (26)

The overall decoherence is given by plugging this into Eqn. 17:

$\displaystyle\begin{split}s=\frac{\rho_{\chi}}{m_{\chi}}\frac{\pi\bar{\sigma}v_{0}^{2}(q_{0}^{2}+m_{\phi}^{2})^{2}}{N_{0}m_{\chi}^{2}m_{\phi}^{4}}t_{\rm{exp}}\int
dq\frac{q}{\left((q/m_{\phi})^{2}+1\right)^{2}}N[1+AF_{A}^{2}(qr_{A})+A(N_{A}-1)F^{2}(qr_{C})]\\\
\times\left[\exp\left(-\frac{v_{-}^{2}(q)}{v_{0}^{2}}\right)-\exp\left(-\frac{v_{\rm
esc}^{2}}{v_{0}^{2}}\right)\right]\left(1-\frac{\sin(q\Delta x)}{q\Delta
x}\right).\end{split}$ (27)

To build intuition, we now consider various approximations of this integral.
For each of these, we will also consider two scenarios for the mediator mass:
$m_{\phi}\propto m_{\chi}$ and fixed $m_{\phi}$. These correspond to Figures 2
& 3, respectively.

  1. 1.

$m_{\chi}\rightarrow 0$ – For either mediator case, this limit implies that
both $q\,r_{C}\ll 1$ and $q\,\Delta x\ll 1$. The first of these allows the
last term in the target form factor to dominate, which makes the rate go as
$N^{2}$. The second of these will set the last term in Eqn 26 to go as
$(q\Delta x)^{2}$ (to second order in the Taylor expansion). In addition,
$v_{-}\rightarrow 0$. Taking the light mediator limit of the form factor, we
find that the limit on the cross section scales as:

$\lim_{m_{\chi}\to 0}\bar{\sigma}_{n}\propto\frac{1}{N^{2}m_{\chi}(\Delta
x)^{2}},$ (28)

where we only write the factors that depend on the DM mass and the experiment
explicitly. This behavior applies to both $m_{\phi}\propto m_{\chi}$ and
$m_{\phi}=\text{constant}$. In this limit of $m_{\chi}\to 0$, these two
mediator scenarios correspond to the massless mediator and heavy mediator
cases, respectively; however, both have the same form factor limit:
$\lim_{m_{\chi}\to 0}{\cal F}_{\text{med}}\sim 1$.

Note that the figures do not show this scaling for the GDM experiment. Because
of its large $\Delta x$, there is a range of low $m_{\chi}$ where $q\Delta
x\gtrsim 1$, and thus the DM will still cause decoherence. In this regime, the
bounds from GDM then scale as $\bar{\sigma}\propto m_{\chi}/N^{2}$, as shown
in the figures. At low enough $m_{\chi}$, below the scales shown in the
figures, GDM would then follow the above scaling.

  2. 2.

$m_{\chi}\to\infty$ – For $m_{\phi}\propto m_{\chi}$, the rate benefits from
the momentum transfer growing proportionally to the dark matter mass. In this
limit, $q\,r_{C}\gg 1$ and $q\,\Delta x\gg 1$. The first of these means that
the first term in the target form factor dominates, so $R_{\rm{decoh}}\propto
N$. The second of these will set the probability of causing decoherence to
$1$. Taking the light mediator limit of the form factor we find:

$\lim_{m_{\chi}\to\infty}\bar{\sigma}_{n}\propto\frac{m_{\chi}}{N}\quad\rm{for~{}}m_{\phi}\propto
m_{\chi}\;.$ (29)

Thus the cross section limit at high masses is only dependent on the number of
nuclei in the experiments and is largely insensitive to the rest of the
experimental setup. Note that we only see this limit for GDM and BECCAL in
some panels of Fig. 2 – we do not reach high enough masses to see this
behavior for the other experiments. Instead, the higher masses in our plots in
Fig. 2 are explained by limit 3.

Now let us consider the $m_{\phi}=\text{constant}$ case. We will set
$m_{\phi}=1\text{ eV}\gtrsim 1/r_{C}$, as we do in Fig. 3. Unlike in the
previous case, for a fixed mediator mass the rate benefits from low momentum
transfer, $q<r_{C}^{-1}$. Then, we expect the following scaling for the masses
$m_{\chi}>m_{\phi}/v_{0}\sim 1~{}\rm{keV}$:

$\bar{\sigma}\propto\frac{m_{\phi}^{4}r_{C}^{2}}{N^{2}m_{\chi}}\quad\rm{for~{}}m_{\phi}=\text{constant}\;.$
(30)

Naively it might be surprising that the limits on the reference cross section
improve linearly with $m_{\chi}$, as Fig. 3 displays. However, because we
define $\bar{\sigma}\propto(y_{\chi}y_{\phi})^{2}/m_{\chi}^{2}$ for
$m_{\chi}\gg m_{\phi}$ (see Eq. 4), the constraints on the couplings
$(y_{\chi}y_{\phi})^{2}$ weaken linearly with $m_{\chi}$ in this limit. This
relationship between the couplings and the reference cross section also
explains why the 5th force constraints become more aggressive at higher
$m_{\chi}$ for fixed $m_{\phi}$ (see Section IV and Ref. [63] for more
discussion).

For the fixed mediator case where $m_{\chi}<m_{\phi}/v_{0}\sim 1~{}\rm{keV}$,
the cross-section is expected to scale as:

$\bar{\sigma}\propto\frac{m_{\chi}^{3}r_{C}^{2}}{N^{2}}\quad\rm{for~{}}m_{\phi}=\text{constant}\;.$
(31)

  3. 3.

There is a third limit for $m_{\chi}\propto m_{\phi}$ that lies in between the
first two limits: when $q<r_{C}^{-1}$ and $q\sim(\Delta x)^{-1}$. Since
$q<r_{C}^{-1}$, the rate still has an $N^{2}$ enhancement; however, since
$q\sim(\Delta x)^{-1}$, the probability of producing decoherence is also high.
In this intermediate regime, which is between the two “knees”, the two
mediator cases behave differently.

For $m_{\phi}\propto m_{\chi}$, $q<r_{C}^{-1}$ boosts the $m_{\chi}^{-3}$
dependence of the rate, which dominates with respect to the $m_{\chi}^{-1}$
discussed in the second limit. Thus, we expect:

$\bar{\sigma}\propto\frac{r_{C}^{2}}{N^{2}R_{\phi\chi}^{4}}m_{\chi}^{3}.$ (32)

where $R_{\phi\chi}=m_{\phi}/m_{\chi}$ is the constant ratio assumed between
the mediator and the dark matter masses.

The sweet spot for sensitivity of the atom interferometers occurs in the
threshold between limit 1 and 3, corresponding to a change in the sign of the
slope in the curves in Fig. 2. These two limits overlap at the $m_{\chi}$ with
best sensitivity, where the rate is boosted by both a large probability of
decoherence and the $N^{2}$ Born enhancement. Equating both limits we expect
the best constraint for a given experiment to happen at:

$m_{\chi}^{\text{knee}}\sim\frac{1}{R_{\phi\chi}\sqrt{\Delta x\,r_{C}}}.$ (33)

####  III.1.2 Phase

As mentioned above, the phase effect relies on anistropy of the DM flux. Thus,
we must include the Earth’s velocity, $v_{e}$, when calculating the phase
effect. This gives a phase rate of:

$\displaystyle\begin{split}R_{\rm{phase}}=\frac{1}{\rho_{T}V}\frac{\rho_{\chi}}{m_{\chi}}\frac{\pi\bar{\sigma}v_{0}^{2}(q_{0}^{2}+m_{\phi}^{2})^{2}}{2N_{0}m_{\chi}^{2}m_{\phi}^{4}}\int
dq\frac{q}{\left((q/m_{\phi})^{2}+1\right)^{2}}N[1+AF_{A}^{2}(qr_{A})+A(N_{A}-1)F^{2}(qr_{C})]\\\
\times\int_{-1}^{1}d\cos\theta\left(-\sin(\mathbf{q}\cdot\mathbf{\Delta
x})\right)\left[\exp\left(-\frac{v_{-}^{2}(q,\theta)}{v_{0}^{2}}\right)-\exp\left(-\frac{v_{\rm
esc}^{2}}{v_{0}^{2}}\right)\right].\end{split}$ (34)

Since the presence of the $v_{e}$ term in Eqn. 24 promotes the integrand to an
odd function, the dark matter flux anistropy is critical for obtaining a non-
zero rate.

We now analyze the behavior of the rate in the limit of different DM masses.
As in the decoherence case, we discuss limits in the context of two mediator
scenarios, shown in Figures 2 & 3: where $m_{\phi}$ is some constant ratio of
$m_{\chi}$, and where $m_{\phi}$ is fixed to some value.

  1. 1.

$m_{\chi}\rightarrow 0$ – As before, in this limit $q\,r_{C}\ll 1$ and
$q\,\Delta x\ll 1$. The first of these makes the rate go as $N^{2}$, and the
second of these allows us to expand $\sin(\mathbf{q}\cdot\mathbf{\Delta
x})\simeq q\,\Delta x\cos\theta$ (to first order in the Taylor expansion) in
Eqn. 34. Hence, the cross section constraints in this limit scale as:

$\lim_{m_{\chi}\to 0}\bar{\sigma}\propto\frac{1}{N^{2}\Delta x}.$ (35)

This behaviour applies independently of the $m_{\phi}$ scenario. This is
because, as before, $\lim_{m_{\chi}\to 0}{\cal
F}_{\text{med}}\sim\text{constant in }m_{\chi}$ in both cases. Note that the
sensitivity of the phase measurement in this limit is expected to be constant.
This constant shift in the phase induced by the low DM momentum transfer can
be understood as the classical effect of having a wave with large wavelength
acting on two clouds separated by a physical distance of $\Delta x$. We only
see this constant behavior at the lower $m_{\phi}$ ratios in Fig. 2 because
the $m_{\phi}/m_{\chi}$ ratio sets the intercept with the next limit case (see
below).

  2. 2.

$m_{\chi}\to\infty$ – $R_{\text{phase}}\to 0$ when $q\,\Delta x\gg 1$ because
of the oscillatory behavior of the sinusoidal function (see right panel of
Fig. 1 for illustration). Thus, we are interested in the limit where
$m_{\chi}\to\infty$, but $v\ll v_{0}$. This implies that $q\,\Delta x\lesssim
1$ will bound the growth of $q$, which will then be decoupled from $m_{\chi}$.
Furthermore, because we are now in the $q\,\Delta x\lesssim 1$ regime, we may
expect the $N^{2}$ term in the target form factor will dominate (since for
most of the missions $r_{C}\sim\Delta x$). Hence, we expect the cross section
constraint to scale as:

$\lim_{m_{\chi}\to\infty}\bar{\sigma}\propto\frac{(\Delta
x)^{3}\,m_{\chi}^{4}}{N^{2}r_{\phi\chi}^{4}}\quad\text{ for }m_{\phi}\propto
m_{\chi}\;,$ (36)

where an extra $1/(m_{\chi}\Delta x)$ in the rate comes from the phase
integral, as Eqn. 9 explicitly shows333Notice that $\tilde{v}_{e}$ factorizes
out in the right-hand side of Eqn. 9 due to the expansion of the hyperbolic
sine and it scales inversely proportional to $m_{\chi}\Delta x$.. The
$\bar{\sigma}_{n}\propto m_{\chi}^{4}$ scaling determines the slope of the
curves in the logarithmic plots of Fig. 2. The intercept is fixed by $\Delta
x$, $N$, and the ratio $r_{\phi\chi}=m_{\phi}/m_{\chi}$.

On the other hand, when $m_{\phi}$ is fixed, the cross section sensitivity
scales as:

$\lim_{m_{\chi}\to\infty}\bar{\sigma}\propto\frac{(\Delta
x)^{3}\,m_{\phi}^{4}}{N^{2}}\quad\text{ for a fixed }m_{\phi}\;,$ (37)

which is independent of $m_{\chi}$ (as long as this limit holds). If
$m_{\phi}<1/\Delta x$, the constant scaling with $m_{\chi}$ would go as
$\bar{\sigma}\propto 1/(N^{2}\Delta x)$.

One final note before we move on to other interactions: if the DM scatters off
of the electrons in the clouds instead, we would get a similar rate: this rate
would also get an $N^{2}$ enhancement since the electrons in a BEC are also
coherent. However, we do not explicitly include these results here because the
limits from astrophysical constraints are much more constraining for electron
couplings [see, e.g., 63].

###  III.2 Hidden Photon Processes and Coherent Axion Scattering

In this section, we consider other processes that could cause decoherence and
phase shifts in an atom interferometer through coherently scattering with one
of the two wavepackets. We discuss three different processes: DM scattering
mediated by a hidden photon through kinematic mixing, hidden photon scattering
through baryon and lepton number couplings, and coherent axion scattering. The
hidden photon kinematic mixing scenario has a rate suppressed by the poor
polarizability of the atom clouds in these experiments. The rates for the
other two processes can be projected onto the rate for nuclear recoil with
heavy mediators, and thus are not optimized for detection. Overall, these
processes do not prove as compelling for atom interferometers as the nuclear
recoil with light mediators. We focus on the decoherence observable only in
this section; however, the phase shifts can be calculated using the methods of
the previous section.

  * •

Hidden photon through kinetic mixing

A hidden photon, $A^{\prime}$, couples to the SM photon through a kinetic
mixing operator ${\cal
L}\supset\frac{\kappa}{2}F^{\mu\nu}F^{\prime}_{\mu\nu}$, where $\kappa$ is the
kinetic mixing parameter, $F_{\mu\nu}$ is the photon field strength, and
$F^{\prime}_{\mu\nu}$ is that for the dark photon. The SM photons polarize the
atom clouds with polarizability $\alpha$, and the effective Hamiltonian for
the interaction between the dark photon, the SM photon, and the target medium
then reads:

$H_{\rm I}=-\alpha\,\kappa\int
d^{3}\mathbf{r}\,n(\mathbf{r})\mathbf{E}(\mathbf{r})\cdot\mathbf{E}^{\prime}(\mathbf{r})\;,$
(38)

where $\mathbf{E}$ is the electric field for the SM photons,
$\mathbf{E}^{\prime}$ is the electric field for the dark photon and
$n(\mathbf{r})$ is the number density of the atom cloud.

With this interaction Hamiltonian, we can calculate the overall decoherence
for DM scattering mediated by the hidden photon in the absence of external
fields using the real part of Eqn. 17:

$s=\left(\frac{\alpha}{a_{0}^{3}}\right)^{2}\frac{\rho_{\chi}}{m_{\chi}}\frac{\pi\bar{\sigma}}{\mu^{2}}t_{\rm{exp}}\int\frac{d^{3}\mathbf{q}}{(2\pi)^{3}}\mathcal{F}_{\rm{med}}^{2}\left(1-\cos(\mathbf{q}\cdot\mathbf{\Delta
x})\right)\sum_{f}|\langle
f|\mathcal{F}_{T}(\mathbf{q})|i\rangle|^{2}g(\mathbf{q},\omega_{1})\;,$ (39)

where $a_{0}$ is the Bohr radius. Following the formalism in [49], we define
the reference cross section $\bar{\sigma}$ as:

$\begin{gathered}\includegraphics[width=73.7146pt]{RefXsec_DPhot.pdf}\end{gathered}\quad:\quad\begin{gathered}\bar{\sigma}=\kappa^{2}\frac{\mu^{2}}{\pi}\frac{1}{(m_{\chi}^{2}v_{0}^{2}+m_{A^{\prime}}^{2})^{2}}\;,\end{gathered}$
(40)

where we have assumed the DM charge under the hidden photon to be 1. We use
the mediator form factor:

$\mathcal{F}_{\rm{med}}(q)=\frac{(m_{\chi}v_{0})^{2}+m_{A^{\prime}}^{2}}{q^{2}+m_{A^{\prime}}^{2}},$
(41)

and the structure function (convoluted with the kinematic function):

$\sum_{f}|\langle
f|\mathcal{F}_{T}(\mathbf{q})|i\rangle|^{2}g(\mathbf{q},\omega_{1})=a_{0}^{6}\frac{q^{2}}{2}\int\frac{d^{3}\mathbf{k}_{1}}{(2\pi)^{3}}\omega_{1}S(\mathbf{q}-\mathbf{k}_{1})g(\mathbf{q},\omega_{1})\;,$
(42)

where the static structure factor
$S(\mathbf{q}-\mathbf{k}_{1})=\sum_{i,j=1,...,N_{A}}\langle
e^{-i\Delta\mathbf{y}_{ij}\cdot(\mathbf{q}-\mathbf{k}_{1})}\rangle_{\rm{target}}$
follows the same form as the one derived for nuclear recoil in Eqn. 21 (see
Sec. B for a detailed derivation). Notice that $\omega_{1}$ is the energy of
an on-shell photon and obeys the dispersion relation $\omega_{1}\approx
k_{1}$. We perform a numerical evaluation of the decoherence effect with
$\alpha/a_{0}^{3}\sim\mathcal{O}(100)$, and find that the limits are $\gtrsim$
20 orders of magnitude lower than the nuclear recoil limits in the light
$m_{\chi}$ regime.

  * •

Hidden photon through baryon and lepton number couplings

A hidden photon can couple coherently to the total baryon number of the atom
cloud through:

$\mathcal{L}\supset\frac{g_{B}}{3}\bar{q}\gamma_{\mu}qZ_{B}^{\mu},$ (43)

where $g_{B}$ is the gauge coupling constant.

The rate for $Z_{B}$ scattering, and thus the overall decoherence, mediated by
a nucleon, has the same dynamical structure function as nuclear recoil given
by Eqn. 21. However, the reference cross section in this case is defined as:

$\begin{gathered}\includegraphics[width=86.72267pt]{DiagramZB.pdf}\end{gathered}:\quad\bar{\sigma}=\frac{3}{16\pi
m_{N}^{2}}\left(\frac{g_{B}}{3}\right)^{4}\;,$ (44)

with a trivial mediator form factor $\mathcal{F}_{\rm{med}}(q)=1$. We note
that this cross section scaling is similar to that of nuclear recoil mediated
by a heavy mediator. This is a consequence of the vector nature of the
baryonic force on quarks; if a non-zero axial interaction is present, the
cross section would be enhanced by a factor $m_{N}^{2}/m_{Z_{B}}^{2}$. The
sensitivity curves for a heavy mediator are shown in Fig. 4, which can be
converted to limits on $g_{B}$ using the reference cross section above.

Similarly to the baryon number coupling scenario, if the hidden photon couples
to the net lepton number, the rate would also benefit from the same coherent
scattering enhancement. However, as mentioned at the end of Sec. III.1, the
electronic interactions of DM are more constrained that those involving
baryons [63].

  * •

Coherent axion scattering

Generally, couplings between axion DM and fermions are spin-dependent, which
would break the coherence within one wavepacket and suppress the scattering
rate. However, as has been recently pointed out in Ref. [86], there is a spin-
independent interaction between axions and nucleons:

$\mathcal{L}\supset\frac{a^{2}}{8f_{a}^{2}}\sum_{N=p,n}\delta
m_{N}\,\bar{N}N\;,$ (45)

where $a$ is the axion field, $f_{a}$ is the axion decay constant, and the
mass parameter $\delta m_{N}\sim\mathcal{O}(10)$ MeV characterizes the quark
mass contribution to the nucleon mass. Importantly, such an interaction is not
additionally suppressed by the axion mass $m_{a}$, as is normally expected
from the expansion of the axion-nucleon interacting term. Instead, it is
proportional to the quark masses, encoded in $\delta m_{N}$, which also
violate the shift-symmetry in the UV Lagrangian. This spin-independent
interaction, combined with the large occupation number of axion DM, at first
seems like a promising candidate for detecting axion DM with atom
interferometers.

This effective interaction produces coherent axion scattering with the atom
cloud, which causes an overall decoherence similar to the nuclear recoil case
with heavy mediators. The reference cross section in this case is then:

$\bar{\sigma}=\frac{\mu^{2}}{256\pi}{\left(\frac{\delta
m_{N}}{f_{a}^{2}}\right)}^{2}\frac{1}{m_{a}^{2}}\;,$ (46)

with a trivial mediator form factor. Recasting the heavy mediator limit shown
in Fig. 4, this probes $f_{a}$ only up to $100\text{ GeV}$ and thus is not a
promising target.

##  IV Comparison with Other Constraints

In this section, we put the limits we derive above into context. We first
describe how they compare to previous atom interferometer dark matter limits.
Then, we compare our results to other direct detection methods, and relevant
astrophysical and cosmological constraints.

###  IV.1 Comparison to previous calculations

Refs. [60, 61] were the first to suggest using atom interferometers to probe
dark matter through the decoherence effect we consider in this paper. They
solely consider nuclear recoil calculations and their results are equivalent
to ours if we choose the same statistical methods from each paper. Ref. [61]
only considers the decoherence observable. While [60] does consider the phase
observable, it is not clear how they set their statistics for the phase.
Several other works [57, 58, 59] have considered the phase effects in atom
interferometers from ultralight dark matter (ULDM). However, these works rely
on phase effects produced by varying the fine structure constant and electron
mass, which are quite distinct from the effects we consider here.

###  IV.2 Comparison to other direct detection methods

Most direct detection experiments cannot probe the small DM masses to which
atom interferometers will be sensitive. The notable exceptions will be
superfluid Helium experiments [49] and phonon excitations in semiconductors
[44], as shown by the grey lines in Fig. 2. These experiments will outperform
the atom interferometers we discuss in this paper for mediator masses
$m_{\phi}>10^{-7}m_{\chi}$; however, none of these experiments can probe below
$m_{\chi}\lesssim 10~{}\rm{keV}$. Another recent work has proposed using
optomechanical sensors to measure DM scattering [85]. The most ambitious of
these would produce comparable limits in the heavy mediator case to MAQRO in
the $0.01-1~{}\rm{MeV}$ range.

Atom interferometers are so far the only way to effectively probe
$m_{\chi}\lesssim 10~{}\rm{keV}$ DM. Note however that atom interferometers
can only effectively probe spin-independent DM scattering – any spin-dependent
effects will break the coherence that is used to get such effective limits.
Thus, some combination of atom interferometers and the other direct detection
experiments described above are necessary to rule out this parameter space.

Although not expressly shown in the figures, we want to stress that the space
missions we discuss in this paper would also be sensitive to large cross
section DM. If DM had very large nucleon cross sections, then current,
terrestrial direct detection experiments would not be able to detect DM due to
shielding by the atmosphere. The space-based missions we describe here would
be complementary to proposed balloon and low-Earth-based orbit satellites
[41].

###  IV.3 Astrophysical & Cosmological Constraints

In Fig. 2, we show in gray the parameter space ruled out by the existing
astrophysical bounds on the reference cross section described in Eqn. 4. We
briefly address these and other astrophysical and cosmological limits in this
section, and refer the reader to [63] for further details.

There are two couplings entering in the reference cross section,
$\bar{\sigma}$:

  1. 1.

$y_{n}$, which quantifies the interaction between the mediator, $\phi$, and
the nucleons. Light bosonic mediators can be emitted in stars, leading to
their potential rapid cooling. Constraints on new forms of energy loss in
stars then translate into a bound for the coupling $y_{n}$ [74, 75].
Particularly stringent for the light mediator masses that we are considering
are the fifth force constraints. The most relevant bounds come from Casimir
force, cantilever, and torsion balance experiments [73], which constrain the
attractive Yukawa potential generated by the $\phi$-nucleon interaction,

$V(r)=-\frac{y_{n}^{2}}{4\pi}\frac{1}{r}e^{-m_{\phi}r},$ (47)

which translates into a bound on $y_{n}$ for a given $m_{\phi}$. The solid
gray shaded area in Fig. 2 corresponds to the parameter space in the
$\bar{\sigma}$ \- $m_{\chi}$ plane excluded by the above constraints when
assuming a large $\phi$-$\chi$ coupling, $y_{\chi}=\sqrt{4\pi}$.

  2. 2.

$y_{\chi}$, which weights the interaction between the mediator, $\phi$, and
the dark matter, $\chi$. If $\chi$ composes all of the DM in the universe,
dark matter self-interactions (DMSI) are constrained by cluster mergers and
halo shaped observations to satisfy $\sigma_{\text{DMSI}}/m_{\chi}<1-10\text{
cm}^{2}/\text{g}$ [87], which together with the previous bounds on $y_{n}$,
impose severe constraints on $\bar{\sigma}$. This would rule out the
possibility of testing DM with atom interferometers. However, the DMSI bound
can be considerably relaxed if $\chi$ is a subcomponent of the total DM. In
this paper, we assume that $\chi$ composes $5\%$ of the total DM.

For the light DM particles we are considering, cosmological observations can
also impose relevant bounds on the couplings $y_{\chi}$ and $y_{\phi}$, and
thus indirectly on $\bar{\sigma}$, by constraining the contribution of the
light particles from the dark sector to the number of effective relativistic
degrees of freedom ($N_{\text{eff}}$). The contribution of distinct
relativistic fields from the dark sector (each denoted with subscript $i$) to
$N_{\text{eff}}$ is given by:

$\Delta
N_{\text{eff}}=\frac{4}{7}\sum_{i}\,g_{i}\,\left(\frac{g(T_{\nu_{L}}^{\text{dec}})}{g(T_{i})}\right)^{4/3},$
(48)

where $g_{i}$ are the number of degrees of freedom of the particle $i$,
$T_{\nu_{L}}^{\text{dec}}$ is the decoupling temperature of the active
neutrinos, and $T_{i}$ is the temperature of the dark sector.

Whatever UV interaction that generates the coupling of the mediator with the
nucleons ($y_{n}$) will also induce an effective interaction between $\phi$
and the gluons. Then, above the QCD phase-transition, thermal scatterings of
the mediator with gluons, such as $gg\to\phi g$, can bring the mediator into
thermal equilibrium with the SM bath. Once thermalized, the mediator will
decouple as the universe cools down, depending on the strength of $y_{n}$.

For a light mediator ($m_{\phi}\propto m_{\chi}$ or fixed $m_{\phi}=1\text{
eV}$), for all the parameter space that we consider, $y_{n}<10^{-9}$ because
of the stringent $5^{\text{th}}$ force and stellar constraints. In this case,
$\phi$ decouples from the SM bath before the QCD phase transition, as derived
in Ref. [63]. After the decoupling of $\phi$, the dark matter $\chi$ cannot be
thermally produced and also decouples from the SM bath. The contribution to
the number of relativistic degrees of freedom from the dark sector then
depends on the number of relativistic particles that were in thermal
equilibrium at the time the dark sector decouples. Hence, $\Delta
N_{\text{eff}}\simeq 0.06\sum_{i}g_{i}$, where $g_{i}$ depends on the nature
of the dark matter and the mediator. Note that such a small contribution is
far from being in tension with the current $N_{\text{eff}}$ measurements, but
will be easily probed by upcoming experiments, such as CMB-S4 [88], PICO [89],
CORE [90], and CMB-HD [91].

Finally, we want to address the particular range in $m_{\chi}$ that we choose
for our plots. The upper limit is set by $m_{\chi}\ll m_{N}$, which allows us
to make several simplifying assumptions in the nuclear recoil case. Normally,
the lower bound would be set by warm DM constraints (typically at
$m_{\chi}\sim 1~{}\rm{keV}$, see for example Ref. [92]); however, we consider
$\chi$ to be a small subcomponent of the DM. Thus, we set our lower bound by
mandating that the DM velocity should be less than the escape velocity of the
Milky Way. This sets the reasonable limit that we should have DM to detect in
our galaxy. If, as we discuss above, the DM decouples above the QCD phase
transition, this sets a temperature for the DM of $T_{\rm{DM}}\sim
0.1T_{\gamma}\sim 10^{-5}~{}\rm{eV}$ today, where $T_{\gamma}$ is the
temperature of the cosmic microwave background today. Approximating
$v\sim\sqrt{T_{\rm{DM}}/m_{\chi}}$ and setting $v\lesssim 0.001$ gives a lower
limit on the DM mass of $m_{\chi}\gtrsim 10\mbox{ eV}$.

##  V Discussion & Conclusions

In this paper, we consider the effects of dark matter scattering on atom
interferometers. We calculate the decoherence and phase shifts produced in
atom interferometers from several different channels: nuclear recoils, hidden
photon processes, and coherent axion scattering. The resulting forecasted
limits for various proposed atom interferometer experiments are given in Figs.
2, 3, and 4. In this final section, we discuss some caveats and opportunities
for improvement, both theoretically and experimentally.

We make various assumptions and approximations in this paper. For example, we
neglect to include the mass of the targets in the kinematic matching equations
(e.g., Eqn. 24). However this will only have an effect at $m_{\chi}\sim
m_{N}$, which is at the very upper end of the masses we consider here. We also
provide a simplistic statistical formalism and do not include the DM flux
modulation for setting the decoherence limits (although a discussion of this
effect is included in Appendix A). For all of these cases, we have taken the
more conservative approach to our approximations. One final caveat we should
note: we do not consider any specific backgrounds that could affect these
results. A recent study showed that long-distance forces are negligible for
decoherence, especially for the short ($\lesssim 10$ s) measurement timescales
we consider here [71]. As we discuss in Section II, we expect these processes
to be suppressed relative to the light DM models with light mediators.
However, for the space-based experiments we consider, we plan to carry out a
careful study of the cosmic ray and solar photon backgrounds in the future.

Future work could address the above simplifications in more detail. Exploring
how atom interferometers could probe other models, such as large bound states
of DM (i.e., DM nuggets) could also be interesting [84]. These models are also
coherent at low momentum transfer, making them prime targets for atom
inteferometers. Another avenue for probing DM could examine the collective
excitations produced in Bose-Einstein condensates. While these could not be
probed by traditional atom interferometers, other BEC experiments could be
used.

In this paper, we only considered a few of the many proposed atom
interferometer experiments [see, e.g., 76, 77, 78]. The experiments we chose
have a wide spread in $\Delta x$, $N$, $r_{\rm{cloud}}$, and $t_{\rm{exp}}$,
so that our results span much of the parameter space that other
interferometers could probe. We note that in this paper we only consider how a
single atom interferometer could probe DM; however, some of the experiment
concepts, such as GDM and MAQRO, will have networks of atom interferometers,
and will be especially sensitive to differential phase measurements between
two atom interferometers, potentially improving the reach. We leave a detailed
investigation of atom interferometer networks to future work.

We would now like to address how atom interferometers could improve their
bounds on DM physics. As we discuss in Section III.1, we find that the best
constraint for a given experiment can be approximated by:
$m_{\chi}\sim(\sqrt{\Delta x\,r_{C}}\,m_{\phi}/m_{\chi})^{-1}$, if $\Delta x$
is of the same order or larger than $r_{C}$. Thus, experiments can target
specific mass ranges by properly tuning the cloud separation and size. To
probe the weakest part of the current astrophysical constraints,
$10~{}\rm{eV}\lesssim m_{\chi}\lesssim 1~{}\rm{MeV}$ with
$m_{\phi}=10^{-5}~{}m_{\chi}$, atom interferometers would want
$1~{}\rm{cm}\gtrsim\Delta x\gtrsim 100~{}\rm{nm}$, with similarly sized cloud
radii. The cross section limit sensitivity is then mostly set by the number of
nucleons – the more, the better. The most important design change that these
experiments could consider is to increase their nucleon count.

As we show in this paper, atom interferometers would be a complementary probe
to current direct detection efforts. They would uniquely probe light dark
matter, including improving current astrophysical bounds by up to 10 orders of
magnitude in some cases. These and other quantum sensing missions should
continue to be studied for use as DM probes.

## Acknowledgements

The authors would like to thank Sheng-wey Chiow, Curt Cutler, Marianna
Safronova, and Tanner Trickle for useful discussions. Part of this work was
done at the Jet Propulsion Laboratory, California Institute of Technology,
under a contract with the National Aeronautics and Space Administration. The
work of KZ is supported by the DoE under contract DE-SC0011632, and by a
Simons Investigator award. This work is also supported by the Walker Burke
Institute for Theoretical Physics.

Software: astropy [93], matplotlib [94], numpy [95], PhonoDark [96], scipy
[97]

##  Appendix A Daily Modulation

As has been shown in the main text, e.g. through Eqn. 7, both the decoherence
rate and the phase are sensitive to the orientation of $\mathbf{\Delta x}$.
This directional signal will modulate during a day (orbit period), as the
Earth (space-based experiment), rotates around the Earth’s axis (Sun). In this
appendix, we present a daily modulation formalism and calculation to show the
significance of the time-varying signal. For space-based experiments, the
modulation will depend explicitly on the orbit. We expect this to have a
similarly-sized effect as the daily modulation, since they originate from the
same directional information of $\mathbf{\Delta x}$. We leave a specific orbit
study to future work.

Daily modulation effects can be parameterized through the time-dependence of
the Earth’s velocity $\mathbf{v}_{e}(t)$ in the lab frame, where the phase-
space distribution of the dark matter takes the boosted Maxwell-Boltzmann
distribution as in Eqn. 11. In the lab frame where ${\bf\Delta x}$ is chosen
to be aligned with the ${\bf\hat{z}}$ axis, the Earth’s velocity at time $t$
reads:

${\bf
v_{\text{e}}}(t)=\|\mathbf{v}_{e}\|\begin{pmatrix}\text{s}\theta_{\text{e}}\,\text{c}\theta_{\text{x}}\,\text{s}\phi(t)-\text{s}\theta_{\text{x}}\,\text{s}\theta_{\text{e}}\,\text{c}\theta_{\text{l}}\,\text{c}\phi(t)+\text{s}\theta_{\text{x}}\,\text{c}\theta_{\text{e}}\,\text{s}\theta_{\text{l}}\\\
\text{c}\theta_{\text{e}}\,\text{s}\theta_{\text{g}}\,\text{s}\theta_{\text{l}}\,\text{c}\theta_{\text{x}}-\text{s}\theta_{\text{e}}\,\text{s}\theta_{\text{g}}\,\text{c}\theta_{\text{l}}\,\text{c}\theta_{\text{x}}\,\text{c}\phi(t)-\text{s}\theta_{\text{e}}\,\text{c}\theta_{\text{g}}\,\text{s}\theta_{\text{l}}\,\text{c}\phi(t)-\text{c}\theta_{\text{e}}\,\text{c}\theta_{\text{g}}\,\text{c}\theta_{\text{l}}-\text{s}\theta_{\text{e}}\,\text{s}\theta_{\text{g}}\,\text{s}\theta_{\text{x}}\,\text{s}\phi(t)\\\
\text{s}\theta_{\text{e}}\,\text{c}\theta_{\text{g}}\,\text{c}\theta_{\text{l}}\,\text{c}\theta_{\text{x}}\,\text{c}\phi(t)-\text{c}\theta_{\text{e}}\,\text{c}\theta_{\text{g}}\,\text{s}\theta_{\text{l}}\,\text{c}\theta_{\text{x}}-\text{s}\theta_{\text{e}}\,\text{s}\theta_{\text{g}}\,\text{s}\theta_{\text{l}}\,\text{c}\phi(t)-\text{c}\theta_{\text{e}}\,\text{s}\theta_{\text{g}}\,\text{c}\theta_{\text{l}}+\text{s}\theta_{\text{e}}\,\text{c}\theta_{\text{g}}\,\text{s}\theta_{\text{x}}\,\text{s}\phi(t)\end{pmatrix},$
(49)

where the abbreviations “s” and “c” refer to sine and cosine, respectively. In
the above matrix, $\theta_{\text{e}}\approx 42^{\circ}$ is the angle between
the Earth’s velocity ${\bf v_{\text{e}}}(t=0)$ and the north pole,
$\theta_{\text{l}}$ is the angle between the location of the experiment and
the north pole, $\phi(t)=2\pi\times t/$24h accounts for the rotation of the
Earth and introduces the time dependence, $\theta_{\text{x}}$ represents the
orientation of ${\bf\Delta x}$ in the plane perpendicular to the free-falling
direction $\mathbf{\hat{g}}$, while the angle $\theta_{\text{g}}$ is the angle
between ${\bf\Delta x}$ and such plane.

We show the daily modulation for Pino in Fig. 5, taking three different
orientations for $\mathbf{\Delta x}$. Although the final modulation is
sensitive to the orientation, as well as the orbit information for space-based
experiments, the different curves shown in Fig. 5 give an estimate of the
range of modulations we expect. By taking $\theta_{\text{x}}=0$ and
$\theta_{\text{g}}=0$, we set ${\bf\Delta x}\perp{\bf\hat{g}}$ and ${\bf\Delta
x}\perp({\bf\hat{L}}\times{\bf\hat{g}})$, shown in the blue lines of Fig. 5,
where ${\bf\hat{L}}$ is the unit vector for the Earth’s spin axis ($\vv{\rm
SN}$). Taking $\theta_{\text{x}}=\pi/2$ and $\theta_{\text{g}}=0$, we are
considering the experiment to be ${\bf\Delta x}\perp{\bf\hat{g}}$ and
${\bf\Delta x}\parallel({\bf\hat{L}}\times{\bf\hat{g}})$, represented with
orange lines in the same figure. Finally, taking $\theta_{\text{g}}=\pi/2$ we
are considering that the experiment is (anti)-aligned with the free-falling
direction, represented by the green lines in Fig. 5.

Notice that for large DM mass, where $m_{\chi}v_{0}\gg 1/\|\mathbf{\Delta
x}\|$, there is no daily modulation. In the limit of low DM mass, where
$m_{\chi}v_{0}\ll 1/\|\mathbf{\Delta x}\|$, the amount of daily modulation
does not depend on the DM mass. However, the total decoherence factor is
suppressed in this limit, as explained in the main text. Unexpectedly, the
mediator mass $m_{\phi}$ does not play a major role in the daily modulation.
Although large momentum transfer is preferred for heavy mediator mass
$m_{\phi}\gg m_{\chi}$, the anisotropy of the target still picks up daily
modulation in the low to medium DM mass range, similar to other directional
detection scenarios [98]. Overall, we estimate the daily/orbit modulation to
affect the signal by a factor of a few.

Figure 5: Daily modulation of the decoherence and phase signals for Pino, at
latitude $47^{\circ}$N, assuming three different orientations of
$\mathbf{\Delta x}$: $\mathbf{\Delta x}\perp{\bf\hat{g}}$ and $\mathbf{\Delta
x}\perp({\bf\hat{L}}\times{\bf\hat{g}})$ (blue); $\mathbf{\Delta
x}\perp{\bf\hat{g}}$ and $\mathbf{\Delta
x}\parallel({\bf\hat{L}}\times{\bf\hat{g}})$ (orange); $\mathbf{\Delta
x}\parallel{\bf\hat{g}}$ (green). ${\bf\hat{g}}$ is the direction of free-fall
and ${\bf\hat{L}}$ is aligned with the Earth’s spin axis ($\vec{\rm SN}$).
Left panel shows the normalized decoherence effect $s(t)/s_{0}$ and the right
panel shows the normalized phase effect $\phi(t)/\phi_{0}$. The normalization
factor $s_{0}=s_{\mathbf{\Delta x}\parallel{\bf\hat{g}}}(t=0)$ and
$\phi_{0}=\phi_{\mathbf{\Delta x}\parallel{\bf\hat{g}}}(t=0)$. We use
$m_{\chi}=1$ MeV and $m_{\phi}=10^{-3}m_{\chi}$ for the plots.

##  Appendix B Derivation of the Hidden Photon Process through Kinematic
Mixing

Here we derive the dynamic structure factor for the DM scattering mediated by
a hidden photon through kinematic mixing, which we consider in Section III.2.
This largely follows the derivation in Ref. [49]. Consider a dark photon,
$A^{\prime}$, that couples to the SM photon through a kinetic mixing operator,

${\cal L}\supset\frac{\kappa}{2}F^{\mu\nu}F^{\prime}_{\mu\nu}\;,$ (50)

where $\kappa$ is the kinetic mixing parameter, $F_{\mu\nu}$ is the photon
field strength, and $F^{\prime}_{\mu\nu}$ is that for the dark photon. The SM
photon couples to the target material via its polarizability. To leading
order, this gives a polarization
$\mathbf{P}(\mathbf{r})=\alpha\,n(\mathbf{r})\mathbf{E}(\mathbf{r})$, where
$\mathbf{E}$ is the total electric field in the medium, $n(\mathbf{r})$ is the
number density of the medium. The polarizability can be normalized as
$\alpha=(\varepsilon^{(r)}-1)/n_{0}$, where $\varepsilon^{(r)}$ is the
relative linear dielectric constant, and $n_{0}$ is the average number
density. The Hamiltonian for the polarization then reads:

$H_{\rm I}=-\frac{1}{2}\alpha\int
d^{3}\mathbf{r}\,n(\mathbf{r})\mathbf{E}(\mathbf{r})\cdot\mathbf{E}(\mathbf{r})\;.$
(51)

After a field re-definition in the presence of a hidden photon,
$A_{\mu}\rightarrow A_{\mu}+\kappa A^{\prime}_{\mu}$, this gives

$H_{\rm I}=-\alpha\kappa\int
d^{3}\mathbf{r}\,n(\mathbf{r})\mathbf{E}(\mathbf{r})\cdot\mathbf{E}^{\prime}(\mathbf{r}),$
(52)

where $\mathbf{E}^{\prime}$ is the electric field for the dark photon, that
describes the interaction between the dark photon, the SM photon and the
target medium.

In the atom interferometers that we consider in this paper, there are no
external B-fields. Thus, we consider a scattering process mediated by a dark
photon with mass $m_{A^{\prime}}$ converted to a SM photon with momentum
$\mathbf{k}_{1}$ and energy $\omega_{1}$. Following Ref. [49], where the
electric field of the photon is quantized while the non-relativistic hidden
photon field is sourced by a Coulomb potential, the polarization-averaged
squared matrix element of:

$\displaystyle|\langle\mathbf{p}_{i}|H_{I}|\mathbf{p}_{f};\mathbf{k}_{1}\rangle|^{2}$
$\displaystyle=\left.\frac{\alpha^{2}\kappa^{2}}{2V^{2}}\omega_{1}\frac{q^{2}}{\left(q^{2}+m_{A^{\prime}}\right)^{2}}\left|\langle\Phi_{0}|n_{-\mathbf{k}_{2}}|\Phi_{0}\rangle\right|^{2}\right|_{\mathbf{k}_{2}=\mathbf{q}-\mathbf{k}_{1}}$
(53)
$\displaystyle=\left.\frac{\alpha^{2}\kappa^{2}}{2V^{2}}\omega_{1}\frac{q^{2}}{\left(q^{2}+m_{A^{\prime}}\right)^{2}}\frac{1}{V}S(\mathbf{k}_{2})\right|_{\mathbf{k}_{2}=\mathbf{q}-\mathbf{k}_{1}}$
(54)
$\displaystyle=\left(\frac{\alpha}{a_{0}^{3}}\right)^{2}\frac{1}{V^{2}}\frac{\pi\bar{\sigma}}{\mu^{2}}\mathcal{F}_{\rm
med}^{2}(q)\;a_{0}^{6}\frac{1}{V}\omega_{1}\frac{q^{2}}{2}S(\mathbf{q}-\mathbf{k}_{1})$
(55)

where $\mathbf{q}$ is the momentum transfer, $|\Phi_{0}\rangle$ is the ground
state, and $n_{-\mathbf{k}_{2}}$ is the Fourier transform of the number
density operator $n(\mathbf{r})$:

$\displaystyle
n_{-\mathbf{k}_{2}}=\frac{1}{\sqrt{V}}\sum_{i=1,\cdots,N_{A}}e^{-i\mathbf{k}_{2}\cdot\mathbf{y}_{i}},$
(56)

and the resulting static structure factor
$S(\mathbf{k}_{2})\equiv\sum_{i,j=1,\cdots,N_{A}}\langle
e^{-i\Delta\mathbf{y}_{i,j}\cdot\mathbf{k}_{2}}\rangle$ has the same
functional dependence as the one derived for the nuclear recoil case in Eqn.
21, which reflects that this process receives the same coherent scattering
enhancement as the nuclear recoil case. Notice that in the last line of the
derivation we have rearranged according to the reference cross section and
mediator form factor, defined in Eqns. 40 and 41. Factoring out the model-
dependent dimensionless coupling $\alpha n_{0}$, the model-independent
structure function (convoluted with the kinematic function) for this process
is:

$\begin{split}\sum_{f}|\langle
f|\mathcal{F}_{T}(\mathbf{q})|i\rangle|^{2}g(\mathbf{q},\omega_{1})&=a_{0}^{6}\frac{1}{V}\sum_{\mathbf{p}_{f},\mathbf{k}_{1}}\omega_{1}\frac{q^{2}}{2}S(\mathbf{q}-\mathbf{k}_{1})g(\mathbf{q},\omega_{1})\\\
&=a_{0}^{6}\frac{q^{2}}{2}\int\frac{d^{3}\mathbf{k}_{1}}{(2\pi)^{3}}\omega_{1}S(\mathbf{q}-\mathbf{k}_{1})g(\mathbf{q},\omega_{1}).\;\end{split}$
(57)

Notice that due to the target’s non-trivial polarization response, this form
factor does not have a closed form. Thus, we numerically evaluate the
decoherence effect, as stated in the main text.

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