CALT-TH/2023-030

# Contrast Loss from Astrophysical Backgrounds in Space-Based Matter-Wave
Interferometers

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  Department of Physics & Astronomy,
University of Southern California, Los Angeles, CA, 90089, USA  Walter Burke
Institute for Theoretical Physics, California Institute of Technology,
Pasadena, CA 91125, 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

Atom and matter interferometers are precise quantum sensing experiments that
can probe differential forces along separated spacetime paths. Various atom
and matter interferometer experiments have been proposed to study dark matter,
gravitational waves, and exotic new physics. Increasingly, these experimental
concepts have proposed space-based designs to maximize interrogation times and
baselines. However, decoherence and phase shifts caused by astrophysical
backgrounds could largely undermine or destroy the target sensitivity of the
experiments. We calculate the decoherence effects induced by solar photons,
the solar wind, cosmic rays, solar neutrinos and zodiacal dust on space-based
atom and matter interferometers. We find that, in future space-based atom and
matter interferometers, the solar wind generically produces decoherence beyond
the quantum noise limit, without proper shielding. In addition, solar photons
are also an important background for matter interferometers.

††preprint: APS/123-QED

###### Contents

  1. I Introduction
  2. II Overview and Summary of Results
    1. II.1 Observables in Matter-Wave Interferometers
    2. II.2 Particle Backgrounds
    3. II.3 Space-based Matter-Wave Interferometer Experiments
    4. II.4 Summary of Results
  3. III General Formalism
  4. IV Solar Photons
  5. V Cosmic Rays and the Solar Wind
    1. V.1 Charged Particle Scattering with Low $q$
    2. V.2 Charged Particle Scattering with High $q$
    3. V.3 Combined Rate
  6. VI Other Backgrounds
    1. VI.1 Zodiacal Dust
    2. VI.2 Solar Neutrinos
  7. VII Discussion & Conclusions
  8. A Cross Section Formalism
  9. B The Angular Phase Space Integration

##  I Introduction

Atom and matter interferometers are becoming increasingly precise quantum
sensing experiments. Atom interferometers use coherent atomic wavepackets to
measure the differential forces along two “arms” (for a general review of atom
interferometers, see Ref. [1]). For concreteness, consider a Mach-Zender atom
interferometer [2]. Analogously to the conventional Mach-Zender
interferometer, here the de-Broglie wave nature of the atoms plays the role of
light, and laser pulses act as the beamsplitters and mirrors. The atoms start
in a coherent wavepacket, then are split by a laser pulse into two different
wavepackets, which travel along two different paths separated by macroscopic
distances. Two more laser pulses reverse the motions of these wavepackets and
recombine them. The state of the recombined wavepacket is then read, with the
relative populations depending on the initially prepared state, the timing of
the laser pulses, and any differential forces felt along the two paths. Matter
interferometers work similarly, but the spatial superpositions occur at the
level of the material object. Matter-wave interferometers111We use “matter-
wave interferometers” to refer to both atom and matter interferometers. have
proven to be powerful tests of the equivalence principle, e.g. [3], and
promising probes of gravitational waves and dark matter [4, 5, 6, 7, 8, 9].

The current and next generation stages of matter-wave interferometers are all
terrestrial [3, 10, 11]. Because the sensitivity of matter-wave
interferometers for most of their science objectives is set by their phase
sensitivity, matter-wave interferometers seek long interrogation times. This
has led to tall towers on Earth, or deep mine shafts. However, the best
interrogation times and longest baselines can only be achieved in
microgravity. This reality has led to many proposals for space-based matter-
wave interferometers, e.g. [12, 13, 14, 15]. Although many of these proposals
are futuristic, their promise as powerful and multi-purpose scientific
instruments warrants their careful consideration.

An important assumption that has been made in all space-based experimental
proposals is that they can be properly shielded from any particle backgrounds.
Scattering of particles on the atomic clouds (or solid objects) in these
experiments risks decohering the system, becoming potential backgrounds in new
physics searches. For matter interferometers, some backgrounds could decohere
the system to a degree precluding a fringe measurement. In the solar system,
these experiments would be buffeted by solar photons, solar wind particles,
cosmic rays, solar neutrinos, and zodiacal dust. There are terrestrial,
experimental constraints on the rate of decoherence from gas particles [16],
as well as a long literature regarding theoretical calculations of these
decoherence rates (see, e.g., [17, 18, 19, 20]). In addition, recent work has
computed the decoherence from long-range forces [21]. However, there are no
detailed, particle-based calculations of the rates of these space-based
backgrounds.

In this paper, we derive the decoherence effects of various particle
backgrounds on space-based matter-wave interferometers. We show that some
backgrounds, like solar photons and the solar wind, could prove to be
important without proper shielding. In Section II, we provide an overview of
the different backgrounds and experiments that we consider in this paper, and
we give a brief summary of our results. In Section III, we introduce our
theoretical formalism. In Sections IV & V, we discuss the most important
backgrounds: photons and charged particles. In Section VI, we consider other
particle backgrounds. We review our results and conclude in Section VII.

##  II Overview and Summary of Results

In this work, we consider the decoherence effects of background particles on
matter-wave interferometers in space. In this section, we first describe the
observable effects that these particles produce (II.1). We then give an
overview of the four main backgrounds we consider: solar photons, charged
particles (i.e., the solar wind and galactic cosmic rays), zodiacal dust
particles, and solar neutrinos. We discuss the typical energies and fluxes,
and provide a summary of their decoherence effects on matter-wave
interferometers (II.2). A full discussion of each process is relegated to
Sections IV, V & VI. We briefly describe the various matter-wave
interferometer experiments we consider in this paper (II.3). Finally, in
Section (II.4), we summarize our results.

###  II.1 Observables in Matter-Wave Interferometers

First, let us discuss the main observable effects that background particles
would induce on space-based matter-wave interferometers. The two “arms” of
matter-wave interferometers probe the spacetime and differential forces along
two paths. Matter-wave interferometers begin with a coherent state, which is
then split such that wavepackets in an excited state separate from the ground
state wavepackets and follow physically separated paths. When the wavepackets
are recombined, the amount of decoherence and any coherent phase shifts
induced by environmental effects can be measured. Particle scattering can lead
to both decoherence and phase shifts in the system [17, 19, 22, 23, 24].
Decoherence measurements can be used to measure particle dark matter
scattering [24], and quantum gravity signatures [25]. The phase is the
relevant observable for gravity, gravitational wave, and ultralight dark
matter measurements [3, 4, 5, 6, 7, 8, 9]. However, if any background
decoherence effects are sufficiently robust, the fringe would be destroyed and
phase shifts could not be measured. In this paper, we focus on the decoherence
effects only and leave the phase effects to future work.

To quantify the amount of decoherence, we must discuss the actual observables
further. For concreteness, consider an unentangled atom interferometer (e.g.,
a cold atom interferometer) with $N_{\rm atoms}$ independent atoms. In what
follows, we will use $N_{\rm{ind}}$ to mean the number of independent objects
that are measured at the read-out port. So, a cold atom interferometer has
$N_{\rm{ind}}=N_{\rm{atoms}}$. The density matrix parametrizing the two-level
system of each of these independent objects (atoms in a cold atom
interferometer) is:

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

where $\gamma=\exp(-s+i\phi)$ is the decoherence factor, which can be
decomposed into a dimensionless decoherence, $s$ (real, positive), and a
phase, $\phi$ (real). Eq. (1) is displayed in the basis of the state in which
the object is initially prepared and the higher energy level that gives the
macroscopic wavepacket separation $\Delta x$.

At the end of the interrogation time, the two wavepackets are recombined and
the atom states are read. Atom interferometer experiments measure the relative
populations of atoms in the final, recombined wavepackets [1, 26]222 Most
modern matter-wave interferometers produce a fringe measurement, similar to
those produced in light interferometers, that allows them to measure both the
visibility and phase of each measurement. This can be achieved either via the
cloud preparation, modifications to the detection scheme, or through the use
of multiple matter-wave interferometers [27, 28, 29, 30]. However, we find the
original measurement scheme simpler to use for our explanations here.. The
measurement is performed in the states defining the basis of Eq. (1): the
state in which they were initially prepared, which we call port I (or the
bright port), and in the higher energy level state, which we call port II (or
the dim port). The probability of measuring an atom in port I is given by:

$p_{I}=\frac{1}{2}\left(1+{\cal R}{\rm
e}\\{\gamma\\}\right)=\frac{1}{2}\left(1+e^{-s}\cos\phi\right)\;.$ (2)

In the absence of decoherence ($s=0$), each atom is a pure quantum state and,
with the proper final phase $(\phi=0)$, has a 100% probability of being
measured in port I. Therefore, counts in port II suggest that the system
experienced decoherence. For an ensemble of $N_{\rm ind}$ measurements, the
number of expected counts in port I, $N_{I}$, corresponds to the expectation
value of the combined probability distribution of the individual atoms in Eq.
(2). Assuming all of the atoms are independently measured, which is the case
for unentangled systems like cold atom interferometers, the combined
probability is given by the binomial distribution [31],

$P_{I}=\begin{pmatrix}N_{\rm{ind}}\\\
N_{I}\end{pmatrix}p_{I}^{N_{I}}(1-p_{I})^{N_{II}}\;,$ (3)

where $N_{II}=N_{\rm ind}-N_{I}$. Its expectation value and variance are given
by:

$\langle P_{I}\rangle=(N_{I}+N_{II})\,p_{I}=N_{\rm
ind}\,\,p_{I},\qquad\sigma^{2}=N_{\rm ind}\,\,p_{I}(1-p_{I})\;.$ (4)

In order to claim that the system has decohered, one needs to compare the
number of atoms measured in port II, $N_{II}$, with the variance of the
distribution (obtained by marginalizing the probability $p_{I}$ over $\phi$
with $s=0$), $\bar{\sigma}=\sqrt{N_{\rm ind}}/2$, which can be identified as
the one-sided shot noise in the large $N_{\rm ind}$ limit.333The binomial
distribution and its variance apply to classical independent measurements of
independent states. Ref. [31] shows that the same results can be derived for a
$2^{N_{\rm ind}}$-dimensional Hilbert space of $N_{\rm ind}$ atoms,
constructed from the direct product states of the form:
$|\Psi_{1},\Psi_{2},\cdots,\Psi_{N_{\rm
in}}\rangle=\prod_{\otimes}^{i=1,\cdots,N_{\rm ind}}|\Psi_{i}\rangle\;,$ (5)
where $|\Psi_{i}\rangle$ are the individual atom states.

The atom interferometer experiment literature parameterizes the impact of the
decoherence on the relative number counts as follows:

$\frac{N_{I}}{N_{I}+N_{II}}=\frac{1}{2}\left(1+V\cos\Delta\phi\right)\ ,$ (6)

where the left-hand side is the measured number of counts in port I,
normalized by the total counts.444Note that this is not necessarily the
initial number of atoms in an experiment. Atoms that escape the system do not
affect the final results, except to reduce the statistical power. The main
observables are therefore the amplitude $V$, called visibility or contrast,
and the phase, $\Delta\phi$. Visibility is a measure of the decoherence of the
system, and the phase measures the path differences between the two arms. In
an ideal atom interferometer experiment, there would be no decoherence
($V=1$). However, a loss of visibility, i.e. $V\neq 1$, is common in realistic
experiments due to environmental effects and systematics. The visibility $V$
and its quantum noise-limited (QNL) error $\sigma_{V}^{\rm QNL}$ are obtained
by normalizing the expectation value and the variance of the binomial
distribution, shown in Eq. (4), as: $V=e^{-s}$ and:

$\sigma_{V}^{\rm QNL}\equiv\frac{\bar{\sigma}}{N_{\rm
ind}}=\frac{1}{2\sqrt{N_{\rm ind}}}\;,$ (7)

reflecting the statistics of shot noise fluctuations.

Entangled atom interferometers, matter interferometers, and quantum resonators
also exhibit decoherence and phase effects. These experiments are similar to
the cold atom interferometers, except that only one independent measurement is
made per shot: the quantum state where the single macroscopic object ends. The
probability of the object to be measured in port I is also given by Eq. (6),
except with $N_{\rm{ind}}=1$. Note that $s$, in this case, is not the
decoherence rate per atom, but rather the decoherence rate over the full
macroscopic object. In this case, both the real and imaginary part of $\gamma$
should be inferred over many trials [23, 32].

In this paper, we compute the reduction in the fringe amplitude, parametrized
by $V$ in Eq. (6), for the cloud (or solid). We note that if the fringe is
destroyed by any of these backgrounds, then it is impossible to measure the
phase.

Current matter-wave interferometers have demonstrated detection sensitivity
that is limited by the quantum noise limit (see, e.g., [33]). For the
backgrounds studied here, we will give both the absolute reduction in the
amplitude of the fringe, and the corresponding signal-to-noise ratio (SNR)
when compared to the QNL, $\sigma_{V}^{\rm QNL}$ (Eq. (7)). The SNR determines
whether a reduction in amplitude can be detected, given that the QNL
quantifies the sensitivity to the amplitude. If we define the amplitude
reduction as $\Delta V=V_{0}-V$, where $V_{0}=1$ is the amplitude in the
absence of decoherence, then the SNR per shot is:

$\left.\text{SNR}\right|_{\rm shot}\equiv\frac{|\Delta V|}{\sigma_{V}}\;,$ (8)

where in this work we take $\sigma_{V}=\sigma_{V}^{\rm{QNL}}$. This SNR
compares the loss of visibility from a background to shot noise fluctuations.
Note that for matter interferometers (e.g., MAQRO) or entangled clouds of
atoms, where the target enters the detector as a single object, $N_{\rm
ind}=1$, and $\sigma_{V}^{\rm QNL}=0.5$.

The sensitivity to the signal, and therefore the total SNR, will increase
linearly with the exposure time (provided $s\ll 1$) for a background-free
experiment. If this holds, $\left.\text{SNR}\right|_{T_{\rm
exp}}=\left.\text{SNR}\right|_{\text{shot}}N_{\rm meas}$, where $N_{\rm
meas}=T_{\rm exp}/t_{\rm shot}$ is the number of measurements affected per
total running time of the experiment, $T_{\rm exp}$. In the presence of a
well-characterized background, sensitivity to a small signal can still be
achieved over a large background, and in this case, the total SNR scales as
$\sqrt{N_{\rm meas}}$.

Figure 1: Summary of backgrounds relevant for space-based matter-wave
interferometer experiments. In the left panel, we show the spectral fluxes
($d\Phi/dK$) and relevant kinetic energies ($K$) of the solar photons (red,
[34, 35]), solar wind (brown, [36]), solar neutrinos (orange, [37, 38, 39]),
and galactic cosmic rays (turquoise, [40]). In the right panel, we show the
“resolution” ($\lambda=q^{-1}$) of each background to the range of momenta,
$q$, that the background can imprint on the target. For elastic scatterings,
$q$ can go from zero to its maximum value, given by kinematics (see Eq. (9)),
which determines the shortest wavelength that each background can resolve at a
given energy. For charged particles, such as those in the solar wind and
cosmic rays, the maximum momentum transferred depends on the mass of the
target, which we assume to be protons. The maximum resolution power of the
solar photons and neutrinos solely depends on their incoming energy. We do not
include the Zodiacal dust due to its inherently different behavior (see
Section VI.1).

###  II.2 Particle Backgrounds

Let us return to the phenomenological picture of atom interferometry, and
again consider the decoherence effect. Decoherence from particle scattering in
matter-wave interferometers occurs when particles scatter off of one of the
wavepackets in an interferometer, and thus “resolve” the two paths. Let the
distance between the two interferometer arms be $\Delta x$, and let the radius
of each “cloud” (i.e., wavepacket) be $r_{\rm{cloud}}$. The resolving power of
an incoming particle is related to its momentum transfer magnitude, $q$. When
$q>1/\Delta x$, the particle is capable of decohering the system (see Fig. 1
of Ref. [24] for a visual explanation of this relationship). However, if the
momentum of the incoming particle is too high, then the particle will begin to
resolve individual atoms within the cloud. When these individual atoms are not
resolved, $q<1/r_{\rm{cloud}}$, the rate of interactions will be boosted by
the Born enhancement effect – the rate will scale with the number of targets
squared (if they share the same charge) [41]. Thus, the most dangerous
particle backgrounds are those that allow for momentum transfers with
$1/\Delta x<q<1/r_{\rm{cloud}}$. Thus, it is low momentum, high flux particles
that could provide the highest backgrounds to our experiments. In addition, we
note that we only consider elastic scattering processes in this work.
Inelastic processes are not coherent, so their rates are very suppressed
compared to the coherent elastic processes we consider.

In this paper, we consider a wide variety of possible particle backgrounds for
space-based experiments. In Fig. 1, we provide an overview of the most salient
features for each background. As can be seen from Fig. 1, the flux of the
background particles is highest for particles with low kinetic energies. Each
of our particle backgrounds has vastly different fluxes and interaction
mechanisms. The left panel of Fig. 1 shows the spectral fluxes for the
backgrounds we consider, except dust which is treated separately in Section
VI.1. Solar photons dominate the flux. Next, the solar wind and solar
neutrinos have similar fluxes, which are many orders of magnitude smaller than
the photon flux. Galactic cosmic rays have the smallest flux of the
backgrounds we consider.

The energy ranges for each of these backgrounds is set by the processes that
produce them and environmental effects. The solar photon spectrum is, roughly,
a blackbody with effective temperature set by the temperature of the Sun’s
photosphere ($T_{\rm{eff},\odot}=5772~{}\rm{K}$ [42]). Small deviations from
the blackbody are introduced by the chemical composition and physical motions
of the Sun’s chromosphere. The energy limits are based on the instruments used
to measure the solar spectrum, rather than any intrinsic energy limits. The
solar wind is composed of charged particles (mostly protons and electrons)
that are produced and accelerated by the Sun and its magnetic field. The
energy range is set by the velocities and number densities of the particles,
but the flux itself is roughly constant [43, 44, 45]. Solar neutrinos are
byproducts of nuclear fusion in the core of the Sun. As we describe in further
detail in Section VI.2, the maximum energy is set by the mass difference of
the other final and initial particles involved in the reactions. For the most
relevant reaction chain, $p+p\to d+e+\nu_{e}$, the endpoint energy of the
neutrino is $420.22$ keV, but including the thermal kinetic energy of the
protons in the solar plasma raises it to $423.41$ keV [37]. Solar neutrinos
have no minimum energy, but their flux falls according to a power law for low
frequencies because of the low energy neutrino phase-space [46]. Galactic
cosmic rays are charged particles that have been accelerated to high energies
by shocks (e.g., from supernovae). The flux of galactic cosmic rays is
attenuated by the Sun’s solar wind – this barrier blocks lower energy galactic
cosmic rays from entering the solar system. We arbitrarily cut the high end of
the galactic cosmic ray spectrum at 100 GeV, which is set by the dataset we
use [40]. However, since the flux only decreases further and these higher
energies do not contribute to any scattering rates that are boosted by
coherent enhancement, the exact cut-off should not affect our results. Note
that this means that we do not consider extragalactic cosmic rays in this
paper.

In the right panel of Fig. 1, we show the comparison between the range of the
elastic scattering $\lambda$ (i.e. the inverse momentum transfer $q^{-1}$),
with various characteristic length scales of the matter-wave interferometer
systems. The maximum momentum transfer is determined by kinematics, and for
elastic scattering it ranges from zero to:

$q_{\rm max}=2M\frac{\sqrt{K^{2}+2mK}(M+m+K)}{(M+m)^{2}+2MK}~{},$ (9)

in the laboratory frame. Here $K$ is the kinetic energy of the background
particle, $m$ is its mass, and $M$ is the mass of the target. When the
interaction range is longer than the size of the cloud, $r_{\rm cloud}$,
individual atoms in the cloud are indistinguishable to a coherent scattering
process, and thus the cross-section is Born enhanced by the large number of
atoms in the cloud. At the same time, if the scattering does not resolve the
two interfering clouds, which are separated by a distance $\Delta x$, the
decoherence is suppressed. This provides a cutoff in the long-range limit of
scattering decoherence. For each background, the interaction mechanism depends
on the incoming energy of the particles. For example, the scattering range for
solar photons is never smaller than the Bohr radius, and thus the solar photon
only scatters with the matter-wave interferometer clouds through the
polarizability. This is known as Rayleigh scattering for long-wavelength
photons. However, because of their heavy mass and relatively high kinetic
energy, cosmic ray ion scattering could resolve the atom in some regimes, and
thus its interactions would occur through the Coulomb potential. Solar
neutrinos scatter with the cloud through the neutrino-nucleon interaction, and
because of their low energies are always in the coherent regime. However, the
decoherence effect is very suppressed due to the low solar neutrino flux and
low interaction strength. For zodiacal dust, we do not consider specific known
interactions, and we instead just take the geometric limit. We provide a
detailed discussion of the specific interactions we consider in the following
sections.

We note that all of these backgrounds are highly anisotropic for a spacecraft
in an Earth-like orbit. However, we assume they are isotropic for the purposes
of our calculations in this paper (see appendix in Ref. [24] to gauge the
effect of the daily modulation on the fringe reduction). We revisit and
discuss this assumption further in Section VII.

###  II.3 Space-based Matter-Wave Interferometer Experiments

In this paper, we consider only space-based matter-wave interferometer mission
concepts. We note that all of the experiments are futuristic and do not have
finalized experimental parameters. Here we give an overview of each of the
experiments and the parameters we assume for each. A summary of this
information is given in Table 1.

We note that the flux of background particles will depend heavily on the
orbits of the spacecrafts. While some of the experiments have proposed orbits
(e.g., BECCAL on the International Space Station [12], and AEDGE on a medium
Earth orbit [14]), most of the orbits are unknown. For all of these
experiments, we assume an Earth-like orbit. We do not calculate any
attenuation or reflection effects from the Earth itself and focus solely on
solar and galactic backgrounds that would be present on a spacecraft in a 1 AU
orbit about the Sun. In addition, as mentioned above, we do not consider any
angular effects – all backgrounds are taken to be isotropic. We discuss these
points further in Section VII.

The Bose-Einstein Condensate and Cold Atom Laboratory (BECCAL) is a proposed
upgrade [12] to the Cold Atom Lab (CAL), which is currently running on the
International Space Station (ISS) [47, 48]. This upgrade would improve the
current atom interferometer capabilities of CAL, including a larger number of
atoms in the condensates. Of the experiments we consider, this is the most
likely to occur on a short timescale. BECCAL would study a few different atom
types; 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 (shot) time of 2.6 s [12]. The
separation of the atom interferometer arms is not given. We assume $\Delta
x=3~{}\rm{mm}$, which is larger than the cloud sizes and smaller than the
total size of the experimental setup.

The Gravity Probe and Dark energy Detection mission (GDM) is a NASA Innovative
Advanced Concepts (NIAC) Phase II mission concept [49, 15]. It would consist
of a constellation of four spacecraft, each with six atom interferometers. In
this paper, we consider the effects on just one of these interferometers. This
is a futuristic mission concept and many of the parameters are not concrete
yet. We take the same values assumed by the GDM team in their benchmark
forecasts [50]. We assume an atom interferometer baseline of $\Delta
x=25~{}\rm{m}$, a shot time of $20~{}\rm{s}$, a cloud size of
$r_{\rm{cloud}}=1~{}\rm{mm}$, and $10^{8}$ 87Rb atoms.

The Atomic Experiment for Dark Matter and Gravity Exploration in Space (AEDGE)
is another futuristic mission concept [14]. It is envisaged as a successor to
the long-baseline terrestrial interferometers MAGIS [11] and AION [51]. Like
these experiments and GDM, AEDGE would rely on a network of atom
interferometers that are separated by very long ($>10^{7}~{}\rm{m}$)
baselines. As with GDM, we consider just one of AEDGE’s atom interferometers
in this paper. AEDGE will use 87Sr atoms that are dropped with a measurement
time of $600$ s [14]. The maximum separation of the atom clouds will be
$\Delta x=90~{}\rm{cm}$ [14]. The number of atoms and size of the atomic
clouds is not reported in their whitepaper. We assume that their reported
phase sensitivity, $10^{-5}~{}\rm{rad~{}Hz}^{-1/2}$, is the QNL, and thus
scales with the number of atoms via $\delta\phi=1/\sqrt{N_{\rm{atoms}}}$ [31,
52]. For AEDGE, the atom clouds would be interleaved and the measurements
would occur every $1$ s [6]. Thus, this trivially converts to the total phase
sensitivity per shot, $\delta\phi=10^{-5}~{}\rm{rad}$. This then gives us
$N_{\rm{atoms}}=10^{10}$. Likewise, the radius of the cloud is not specified
in their whitepaper. We estimate it by assuming that the cloud has a density
similar to the atom clouds used in the drop tower experiment at Stanford [53,
54], which is also a diffuse cold atom interferometer. This then gives a cloud
radius of: $r_{\rm cloud}^{\rm AEDGE}=r_{\rm cloud}^{\rm Stanford}(N_{\rm
atoms}^{\rm AEDGE}/N_{\rm atoms}^{\rm Stanford})^{1/3}\simeq 3$ mm.

Macroscopic Quantum Resonators (MAQRO) is a proposed space mission to perform
high-mass matter interferometry [55, 13]. This is the only matter
interferometer that we consider in this paper, and thus the only experiment
with $N_{\rm{ind}}=1$. The experiment would use SiO2 molecules [23] in a solid
sphere with $10^{10}$ nucleons and a radius of $120~{}\rm{nm}$. The baseline
separation would be $100~{}\rm{nm}$ and the measurement time per shot would be
$100~{}\rm{s}$ [13].

Experiment | Type of Exp. | Target | $N_{\rm{nucleon}}$ |  $r_{\rm{cloud}}$ [m] |  $\Delta x$ [m] |  $t_{\rm{shot}}$ [s]
---|---|---|---|---|---|---
MAQRO [55, 13] | Solid | SiO2 | $10^{10}$ | $1.2\times 10^{-7}$ | $10^{-7}$ | $100$
BECCAL [12, 47, 48] | BEC |  87Rb | $8.7\times 10^{7}$ | $1.5\times 10^{-4}$ | $3\times 10^{-3}$ | 2.6
GDM [15, 49, 50] | BEC |  87Rb | $8.7\times 10^{9}$ | $10^{-3}$ | $25$ | $20$
AEDGE [14] | Diffuse cloud |  87Sr | $8.7\times 10^{11}$ | $3\times 10^{-3}$ | 0.9 | 600
Table 1: Mission concept parameters assumed in this paper. The parameters are
taken from the references shown next to each experiment. The details are given
in Section II.3.

###  II.4 Summary of Results

The loss of visibility per shot, $\Delta V$, for each of the backgrounds
considered in this work is summarized in Table 2. The decoherence effects
caused by solar neutrinos and zodiacal dust are negligible and is not shown
explicitly in the table. Among these backgrounds, the solar wind induces the
most significant visibility loss. In particular, the solar wind causes a
decoherence background in matter-wave interferometers above the QNL,
$\sigma_{V}^{\rm{QNL}}$, within a single experimental shot. For the matter
interferometer mission considered in this work, the solar wind would also
destroy the fringe in a single shot and thus preclude any possible
measurements. This can be understood due to its relatively high number density
(as opposed to galactic cosmic rays), stronger coupling to the target (as
compared to solar photons), and relatively low kinetic energy that allows for
Born enhancement in some kinematic regimes. Solar photons, with the highest
flux, could also destroy the fringe of matter interferometers. For atom
interferometers, while the induced contrast loss is below the QNL for each
shot, over the full experimental run time, solar photon backgrounds
statistically accumulate over the number of total shots (or measurements),
$N_{\rm meas}$, and exceed the QNL, thus mimicking a signal without shielding
or background rejection. Lastly, the high energy galactic cosmic rays are not
a relevant background through elastic scatterings as considered in this work.
This is because of their relatively low flux and high kinetic energy,
excluding Born enhancement in most of their kinematic phase space. We
acknowledge that these high energy particles could have an impact through
inelastic processes such as ionization, but this is beyond the scope of this
work.

To aid with the intuition of how these processes affect matter-wave
interferometers, we show in Fig. 2 how the SNRshot scales with $\Delta x$ and
$r_{\rm{cloud}}$ for the two most relevant backgrounds: the solar wind and
solar photons. For illustration purposes, we have chosen $t_{\rm shot}=1$ s,
87Rb atoms, and different values of $N_{\rm atoms}$ for both atom
interferometers ($N_{\rm ind}=N_{\rm atoms}$) and matter interferometers
($N_{\rm ind}=1$). Atoms are assumed to be uniformly distributed in the
target. Note that the $\Delta x<r_{\rm cloud}$ region (below the solid, black
line) is generally not implemented experimentally, since the clouds would be
overlapping. For missions we consider in this work, only MAQRO has $\Delta
x\sim r_{\rm cloud}$. There exist simple scaling laws in some of the regions
of the parameter space.

For solar photons with a typical wavelength $\lambda_{\gamma{\odot}}$, in the
region where $\Delta x\gg r_{\rm cloud}\gg\lambda_{\gamma{\odot}}$ (the upper
right region in the left panel of Fig. 2), the differential rate scales
roughly as:

$d\Gamma_{\rm{tot},\gamma\odot}/d\omega\propto
N_{\rm{atoms}}\lambda_{\gamma{\odot}}^{-4}\,[N_{\rm{atoms}}\,(r_{\rm{cloud}}/\lambda_{\gamma{\odot}})^{-4}+\mathcal{O}(1)]\quad\mathrm{for}\>\Delta
x\gg r_{\rm cloud}\gg\lambda_{\gamma{\odot}}\;,$ (10)

where $\omega$ is the photon frequency. In the region where $r_{\rm
cloud}<\lambda_{\gamma{\odot}}$ or $r_{\rm cloud}\gg N_{\rm
atoms}^{1/4}\,\lambda_{\gamma{\odot}}$, the rate only depends on $\Delta x$.
Elsewhere, the rate depends on both $r_{\rm cloud}$ and $\Delta x$. Overall,
one observes that larger clouds and smaller cloud separations are less
affected.

For the solar wind, the rate for Coulomb potential scattering (high momentum
transfer) does not depend on either $\Delta x$ or $r_{\rm cloud}$, provided
both are greater than the size of an atom. This constant contribution only
depends on the type of atom and scales linearly with $N_{\rm atoms}$. On the
other hand, the rate for the scattering through polarizability (low momentum
transfer), in general, depends on both $\Delta x$ and $r_{\rm cloud}$.
However, again considering both scales are super-atomic, for $\Delta x>r_{\rm
cloud}$, the rate only depends on $r_{\rm cloud}$ and scales as:

$\Gamma_{\rm{tot,SW}}^{{\rm low\,}q}\propto N_{\rm{atoms}}r_{\rm
atom}^{-4}[N_{\rm{atoms}}\,(r_{\rm{cloud}}/r_{\rm
atom})^{-4}\log(r_{\rm{cloud}}/r_{\rm
atom})^{4}+\mathcal{O}(1)]\quad\mathrm{for}\>\Delta x>r_{\rm cloud}\;,$ (11)

which describes the vertical behavior of the contours on the right panel of
Fig. 2. Notice that the rate is independent of $r_{\rm cloud}$ and $\Delta x$
when $r_{\rm cloud}\gg N_{\rm atoms}^{1/4}\,r_{\rm atom}$. As can be read from
the scaling and the right panel of Fig. 2, larger clouds are less affected by
the solar wind, independent of their $\Delta x$.

The overall picture is that adequate shielding of both the solar wind and
solar photons will be important for the success of future space-based matter-
wave interferometers. We now turn to describe our methods and results in
detail.

Figure 2: The SNR per shot for the solar photon (left) and the solar wind (right) backgrounds as a function of matter-wave interferometer cloud size, $r_{\rm cloud}$, and cloud separation $\Delta x$ for different numbers of atoms, $N_{\rm atoms}$, in matter-wave interferometers. We have used $t_{\rm shot}=1\,$s and the 87Rb atom for illustration purposes. Some of the experiments we consider in this paper are shown as benchmark points in the plane, marked by ’x’; we do not mark GDM and AEDGE whose cloud separation is meter scale and hence not in the plot region. We discuss the limiting behavior governing these curves in Section II.4. Exp./Bkgs. | Solar Photons | Solar Wind | Galactic Cosmic Rays | $\sigma_{V}^{\rm QNL}$ | $N_{\rm meas}$
---|---|---|---|---|---
MAQRO |  $1$ [$2$] |  $1$ [$2$] |  $3\times 10^{-7}$ [$6\times 10^{-7}$] | $0.5$ | $3\times 10^{5}$
BECCAL |  $1\times 10^{-6}$ [$2\times 10^{-3}$] |  $2\times 10^{-4}$ [$0.4$] |  $1\times 10^{-16}$ [$2\times 10^{-13}$] | $5\times 10^{-4}$ | $10^{7}$
GDM |  $1\times 10^{-5}$ [$0.2$] |  $1\times 10^{-3}$ [$2\times 10^{1}$] |  $7\times 10^{-16}$ [$1\times 10^{-11}$] | $5\times 10^{-5}$ | $2\times 10^{6}$
AEDGE |  $8\times 10^{-4}$ [$2\times 10^{2}$] |  $3\times 10^{-2}$ [$6\times 10^{3}$] |  $2\times 10^{-14}$ [$4\times 10^{-9}$] | $5\times 10^{-6}$ | $5\times 10^{4}$
Table 2: Loss of visibility $\Delta V\equiv 1-V$ over a single experimental
interrogation time $t_{\rm shot}$ as caused by scatterings with various
particle backgrounds in the space environment. The decoherence caused by solar
neutrinos and Zodiacal dust is negligible. The numbers are obtained assuming
no shielding. For comparison with the experimental sensitivity, we list in
squared brackets the SNRshot defined in Eq. (8), where we assume the noise is
dominated by quantum shot noise $\sigma_{V}^{\rm QNL}$ (second to last column,
Eq. (7)). Depending on the measurement scheme, different statistics on the
number of measurements $N_{\rm meas}=T_{\rm exp}/t_{\rm shot}$ (last column)
over the experiment run time should be further applied to
$\left.\text{SNR}\right|_{\rm shot}$, as discussed below Eq. (8). We have
assumed $T_{\rm exp}=1$ year and an isotropic flux consistent with a 1 AU
orbit for all missions. The details of each experiment are given in Table 1
and Section II.3.

##  III General Formalism

In this section, we summarize the formalism that we use to calculate the
decoherence induced by the particle backgrounds we consider. We largely follow
the formalism outlined in Ref. [24], which builds on the formalism in Refs.
[56, 57]. However, unlike in the dark matter case, we do not assume a constant
density of particles at all energies. In addition, we must modify the mediator
and velocity phase space terms for the particular particles and interactions
we consider.

We begin with the rate of decoherence-inducing interactions. This is a
function of the number density and energies of the incoming particles, their
scattering interactions with the cloud555In the case of atom interferometers,
the relevant object for the scattering rate is the cloud of atoms. For matter
interferometers or resonators, the relevant object is the solid. In the
following, we refer to both as “clouds”., and the response of the cloud
itself. We can write the differential decoherence rate for the entire cloud
as:

$\begin{split}\frac{d\,\Gamma_{\rm tot}}{d\omega}=V\int
d\Omega\frac{dn(\omega,\Omega)}{d\omega\,d\Omega}\Gamma(\omega,\Omega),\end{split}$
(12)

where $n$ is the number density of the incoming particle, and $\omega$ and
$\Omega$ are the energy and solid angle of the incoming particle,
respectively. All of the physics of the interactions lies in the decoherence
rate per interaction, $\Gamma(\omega,\Omega)$. Applying Fermi’s Golden rule,
this reads:

$\begin{split}\Gamma(\omega,\Omega)&=\frac{1}{V}\int\frac{d^{3}{\bf
q}}{(2\pi)^{3}}\overline{|\sf{M}|^{2}}\,p_{\rm{dec}}({\bf q\cdot\Delta
x})\sum_{f}\left|\langle f|\mathcal{F}_{\rm T}({\bf
q})|i\rangle\right|^{2}2\pi\delta(E_{f}-E_{i}-\omega_{\bf q})\;,\\\
\end{split}$ (13)

where the momentum transfer is ${\bf q}={\bf p^{\prime}}-{\bf p}$, and the
energy deposition is $\omega_{\bf q}=\omega^{\prime}-\omega$.
$\omega^{\prime}$ denotes the energy of the outgoing particle (see the diagram
below for the definition of the quantities relevant for the kinematics of the
scattering process).

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$\overline{|\sf{M}|^{2}}$ is the spin-averaged scattering matrix element
squared, with the quantum states unit normalized, e.g.
$\langle\mathbf{p}|\mathbf{p}\rangle=\langle i|i\rangle=1$ [57].666We use the
matrix element for unit normalized quantum states, $\sf{M}$, to follow the
conventions used in Ref. [24]. It relates to the matrix element, ${\cal M}$,
in the standard normalization ($\langle p_{k}|p_{k}\rangle=2E_{k}$) by
$\overline{|\sf{M}|^{2}}=\overline{|{\cal
M}|^{2}}/(2E_{i}\,2E_{f}\,2\omega\,2\omega^{\prime})$. See Appendix A for more
details. The probability of producing decoherence is given by [22, 58, 19, 17,
24]:

$p_{\rm dec}={\cal R}{\rm e}\\{1-{\rm exp}[i\mathbf{q}\cdot\mathbf{\Delta
x}]\\}\;.$ (15)

We define the static structure function, which gives the cloud response to the
scattering, as $S(q)\equiv\sum_{f}\left|\langle f|\mathcal{F}_{\rm T}({\bf
q})|i\rangle\right|^{2}$, where $\mathcal{F}_{\rm T}({\bf q})$ is the cloud
form factor [59]. Here the sum runs over the constituents of the cloud, which
we refer to as “targets” of the interaction (e.g. atoms, nucleons, electrons
inside the cloud, or even the cloud itself, depending on the interaction),
that scatter with the background particles. The Born enhancement effect and
other effects that scale with $N_{\rm{target}}$ are contained within this
term. Note that since the directional information of the target is included in
the decoherence form factor $p_{\rm dec}({\bf q\cdot\Delta x})$, the structure
function only depends on the magnitude of the momentum transfer.

Throughout this paper, we adopt the laboratory frame and choose the
$\hat{\mathbf{z}}$ direction to be aligned with the momentum of the incoming
particle $\mathbf{p}$. In such a frame, the target is initially at rest
($E_{i}=M$, $\mathbf{k}=\mathbf{0}$) and is kicked by some background
particle, which transfers a momentum $\mathbf{q}$ to the target, making it
recoil with a final energy $E_{f}=\sqrt{q^{2}+M^{2}}$. Notice that the recoil
energy $E_{f}-M$ is suppressed, since $q\ll M$. Thus, we assume that the
targets are non-recoiling in this formalism.

According to momentum and energy conservation, the magnitude of the momentum
transfer, $q\equiv\|{\bf q}\|$, can be determined by the background particle’s
mass, $m$, the target particle’s mass $M$, the kinetic energy $K$, and the
angle between $\mathbf{p}$ and $\mathbf{q}$. In the lab frame, $0\leq q\leq
q_{\rm max}$, where $q_{\rm max}$ is defined in Eq. (9). As discussed in Sec.
II.2, $q_{\rm max}$ determines the shortest scattering scale allowed by the
kinematics.

In general, the flux of incoming particles is time-dependent, due to the
motion of the spacecraft. For simplicity, in this work, we omit this time
dependence, and thus the daily/orbit modulation of the decoherence. As we
discuss in Section II.2, we further assume that the background particles are
isotropic. Thus, the angular part of the phase space can be integrated out. We
refer the reader to Appendix B for the explicit integration of the angular
part. With these assumptions, the decoherence per measured object, $s$, over a
measurement time per shot, $t_{\rm shot}$, in the lab frame is:

$\begin{split}s&=t_{\rm shot}\,\frac{\Gamma_{\rm tot}}{N_{\rm ind}}\\\
&=\frac{t_{\rm shot}}{2\pi\,N_{\rm ind}}\int
d\omega\frac{dn(\omega)}{d\omega}\int
dq\,q\frac{M+\omega-\sqrt{M^{2}+q^{2}}}{\sqrt{\omega^{2}-m^{2}}}\,\overline{|{\sf
M}(q,\omega)|^{2}}\,S(q)\,\Big{(}1-{\rm sinc}(q\Delta x)\Big{)}\;,\end{split}$
(16)

where $N_{\rm ind}$ is the number of independent objects that are measured.
For a solid, $N_{\rm ind}=1$, and for a cold atom cloud, $N_{\rm ind}=N_{\rm
atoms}$. Notice that, in the mean-field approximation, the right-hand side of
Eq. (16) is always real, implying that $\phi_{\rm iso}=0$ for isotropic
sources (see Ref. [24] for a longer discussion of this point). In addition,
note that the spectral density can also be written in terms of the flux
(number of particles per area per time). In Appendix A, we give the derivation
of the general rate formula in terms of the cross section and the flux of
incoming particles.

##  IV Solar Photons

As the right panel of Fig. 1 shows, solar photons have the highest flux of the
backgrounds that we consider, and their energies are not high enough to
resolve the atoms in the clouds. In this section, we compute the decoherence
associated with solar photon scattering in matter-wave interferometers. We
first discuss the solar flux data, and we then derive the relevant matrix
element for photon scattering with atoms.

For our flux calculations, we use the solar irradiance (flux/surface area)
data from Ref. [34], accessed via Ref. [35]. This spectrum gives the spectral
irradiance of solar photons at 1 AU for zero air mass (e.g., outside of
Earth’s atmosphere). It is calculated by combining measurements from various
space-based, balloon-borne, sounding rocket, and ground-based experiments. The
integrated solar irradiance is normalized to the solar constant value of
$1366.1~{}\rm{W/m}^{2}$. Note that there are other solar spectra that have
been compiled, but they are mostly within 5% of this spectrum (see, e.g.,
[60]). The data is given as a spectral irradiance in
$\rm{W/(m}^{2}\mu\rm{m})$, which we convert to
$\rm{MeV}^{-1}\rm{cm}^{-2}\rm{s}^{-1}$. The resulting flux is shown in the
left panel of Fig. 1.

Although atoms have neutral electromagnetic charge, they can acquire an
electric dipole moment proportional to an applied electromagnetic field. The
tendency of atoms to acquire a dipole moment is parameterized by their
polarizability. The scattering process of photons with atoms through this
polarizability is known as Raleigh scattering. Therefore, we compute the
interaction of a solar photon with the cloud atoms through the following
Hamiltonian:

$H_{I}=-\frac{1}{2}\int
d^{3}\mathbf{r}\,\mathbf{P}(\mathbf{r})\cdot\mathbf{E}_{\rm
out}(\mathbf{r})=-\frac{1}{2}\alpha_{N}\int
d^{3}\mathbf{r}\,n(\mathbf{r})\,\mathbf{E}_{\rm
in}(\mathbf{r})\cdot\mathbf{E}_{\rm out}(\mathbf{r})\;,$ (17)

where $\mathbf{E}_{\rm in}(\mathbf{r})$ ($\mathbf{E}_{\rm out}(\mathbf{r})$)
represents the incoming (outgoing) solar photon. We assume linear
polarizability induced solely by the electric field of the incoming photon,
thus $\mathbf{P}(\mathbf{r})=\alpha_{N}\mathbf{E}_{\rm in}(\mathbf{r})$, where
$\alpha_{N}=4\pi r_{\rm
atom}^{3}\epsilon_{0}(\epsilon_{r}-1)/(\epsilon_{r}+2)$. Here, $r_{\rm atom}$
is the size of the scattered atom, $\epsilon_{0}$ is the vacuum permittivity,
and $\epsilon_{r}$ is the dielectric of the target. The electric fields are
quantized as follows:

$\displaystyle\mathbf{E}_{\rm in}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2V}}\sum_{\mathbf{p}}\sum_{\lambda}\sqrt{\omega}\left(\mathbf{\epsilon}(\mathbf{p},\lambda)a_{\mathbf{p},\lambda}e^{i\mathbf{p}\cdot\mathbf{r}}-\mathbf{\epsilon}^{*}(\mathbf{p},\lambda)a^{\dagger}_{\mathbf{p},\lambda}e^{-i\mathbf{p}\cdot\mathbf{r}}\right),$
(18) $\displaystyle\mathbf{E}_{\rm out}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2V}}\sum_{\mathbf{p^{\prime}}}\sum_{\lambda}\sqrt{\omega^{\prime}}\left(\mathbf{\epsilon}(\mathbf{p^{\prime}},\lambda)a_{\mathbf{p^{\prime}},\lambda}e^{i\mathbf{p^{\prime}}\cdot\mathbf{r}}-\mathbf{\epsilon}^{*}(\mathbf{p^{\prime}},\lambda)a^{\dagger}_{\mathbf{p^{\prime}},\lambda}e^{-i\mathbf{p^{\prime}}\cdot\mathbf{r}}\right)\;.$
(19)

The target density can be quantized as:

$n(\mathbf{r})=n_{0}+V^{-1/2}\sum_{\mathbf{q}}n_{\mathbf{q}}e^{i\mathbf{q}\cdot\mathbf{r}}\;.$
(20)

Applying the quantized electric fields and density to Eq. (17),

$\begin{split}\sum_{f,\mathbf{p}^{\prime}}|\langle\mathbf{p}^{\prime},f|H_{I}|\mathbf{p},i\rangle|^{2}&=\frac{1}{V}\int\frac{d^{3}\mathbf{q}}{(2\pi)^{3}}\frac{1}{16}\alpha_{N}^{2}\omega\,\omega^{\prime}\sum_{f}\left|\langle
f|\sum_{l=1,\cdots,N_{\rm
atoms}}e^{i\mathbf{q}\cdot\mathbf{r}_{l}}|i\rangle\right|^{2},\end{split}$
(21)

we obtain the matrix element
$\overline{|\sf{M}|^{2}}=\alpha_{N}^{2}\omega\omega^{\prime}/16$. Finally, the
total decoherence rate is:

$\begin{split}\left.\Gamma_{\rm
tot}\right|_{\gamma_{\odot}}&=\frac{\alpha_{N}^{2}}{32\pi}\int
d\omega\frac{dn(\omega)}{d\omega}\,\omega\int
dq\Big{(}\omega+M-\sqrt{M^{2}+q^{2}}\Big{)}\,q\,S(q)\,\Big{[}1-{\rm
sinc}(q\Delta x)\Big{]}\;,\end{split}$ (22)

where the structure-function reflects the sum over all $N_{\rm atoms}$ atoms
in the cloud:

$\begin{split}S(q)=\sum_{f}\left|\langle f|\sum_{l=1,\cdots,N_{\rm
atoms}}e^{i\mathbf{q}\cdot\mathbf{r}_{l}}|i\rangle\right|^{2}=N_{\rm
atoms}+N_{\rm atoms}^{2}F_{\rm cloud}(qr_{\rm cloud})^{2}\;.\end{split}$ (23)

$F_{\rm cloud}$ will depend on the nature of the cloud, i.e. the wavefunction
of the initial and final states, $\psi_{i,f}\equiv\langle
i,f|\mathbf{r}_{l}\rangle$. If the atom cloud is initially prepared as a Bose-
Einstein Condensate (BEC), such as BECCAL or GDM, the form factor is given by
(see Appendix B from Ref. [24]):

$F_{\rm BEC}(qr_{\rm cloud})=\exp\left[-(qr_{\rm cloud}/2)^{2}\right]\;.$ (24)

For all other types of targets, we assume the atoms are uniformly distributed,
and thus use the form factor [56]:

$F(qr_{\rm cloud})=\frac{3j_{1}(qr_{\rm cloud})}{qr_{\rm cloud}}\;,$ (25)

where $j_{1}$ is the spherical Bessel function of the first kind. The
polarizability values for the relevant materials are: $\alpha_{\rm
Rb}=47.39\,(8)\,$Å3 [61], $\alpha_{{\rm SiO}_{2}}=4.84\,$Å3 [62], and
$\alpha_{\rm Sr}=192.40\,a_{0}^{3}$ [63], where $a_{0}$ is the Bohr radius.

$\gamma_{\odot}$ | MAQRO | BECCAL | GDM | AEDGE
---|---|---|---|---
Rate [$\text{s}^{-1}$] | $\ \ 1.2\times 10^{7}\ \ $ | $\ \ 0.46\ \ $ | $\ \ 5.4\times 10^{1}\ \ $ | $1.4\times 10^{4}$
Table 3: Total decoherence rate $\Gamma_{\rm tot}$ caused by solar photons, in
$s^{-1}$.

In Table 3, we show the decoherence rate caused by solar photons for the
different space missions considered in this work. The visibility for most of
the experiments receives a reduction similar to the QNL error from the action
of solar photons. In particular, MAQRO is severely affected compared to the
other proposals. This enhancement is expected due to the smaller cloud radius
and the larger number of atoms contained in MAQRO. This can be understood by
having a close look at the rate in Eq. (22). The $q$ integral is dominated by
the Born $N_{\rm atoms}^{2}$ enhancement since for most of the missions
$\Delta x\gg r_{\rm cloud}$ (for MAQRO $\Delta x\sim r_{\rm cloud}$, but the
same reasoning applies). Due to the $q$ scaling, the largest contribution
occurs at $q\sim 1/r_{\rm cloud}$. Hence, one can estimate the relative
strength of the rate between different experiments by taking the ratio of the
scaling of the rate with the parameters of the experiment,
$ds/dt\propto\alpha^{2}N_{\rm atoms}^{2}/r_{\rm cloud}^{2}$, which roughly
justifies the results shown in Table 3.

##  V Cosmic Rays and the Solar Wind

Charged particle backgrounds in space at energies lower than
${\sim}1~{}\rm{GeV}$ are mainly sourced by the Sun and galactic processes.
Charged particles from the Sun, i.e. the solar wind, have a narrow energy
spectrum with low kinetic energy, $0.3-3\,$keV. The spectral density of the
solar wind is approximately constant in this kinetic energy window; however,
it can have large time-dependent fluctuations because of solar activity.
Galactic cosmic ray particles, on the other hand, have relatively high kinetic
energies, in the MeV - TeV range.

Galactic cosmic rays are composed of $\gtrsim 90\%$ protons, $\lesssim 10\%$
alpha particles, and $\ll 1\%$ other particles. The solar wind has about an
equal number of electrons as protons, where they have approximately the same
velocity [64]. Galactic cosmic rays can also include electrons, and their
spectral flux is higher than protons at low kinetic energies [40]. However, we
get similar or weaker decoherence rates for electrons than for the protons.
Therefore, we only explicitly calculate the proton background, which should
approximately capture the impact of these backgrounds. For a given kinetic
energy and scattering angle, there are two separate interaction types that can
dominate. These cases will depend on whether the proton resolves the atom
(momentum transfer $q>1/r_{\rm atom}\sim{\cal O}(\text{MeV})$) or not. We
first discuss the flux data we use for both types of particles. We then
discuss the interactions that dominate these two momentum transfer limits.
Finally, we discuss how we combine these limits to calculate a total rate.

The solar wind has a roughly constant energy flux [43, 44, 45]. We use the
solar wind parameters for the most recent epoch of 2009–2013, provided in Ref.
[36]. This gives proton velocities of $v=398~{}\rm{km/s}$, and proton number
densities of $n=5.74~{}\rm{cm}^{-3}$. Assuming the solar wind being all
protons at low kinetic energies (keV), we convert this to an irradiance of:
$343~{}\rm{W/m}^{2}$, which is within an order of magnitude of the commonly
accepted value, $70~{}\rm{W/m}^{2}$. We assume that the flux is constant over
the range 0.3–3 keV [64]. We note that these numbers roughly agree with the
values that can be found using real-time data from the DSCOVR satellite [65]
via Ref. [66]. In addition, we ignore any coronal mass ejection events in
setting these numbers. These would produce much higher energy particles, but
these are short events that could be excluded from matter-wave interferometer
data collection.

For the flux of galactic cosmic rays, we use theoretical models fit to the
Voyager $2$ data [40], as plotted in the right-handed panel of Fig. 1. We note
that this data is taken from beyond the heliopause (e.g., outside of the
protective influence of the Sun’s magnetic field). Thus, the flux is slightly
higher than the flux found within the solar system (see e.g., Ref. [67] for a
discussion of this point). However, this is the highest quality data available
at these energy levels that is outside of the Earth’s magnetic field, which
would also distort the measured galactic cosmic rays. Depending on the orbit
of each mission, the galactic cosmic ray flux would vary according to these
considerations.

###  V.1 Charged Particle Scattering with Low $q$

When $q\ll 1/r_{\rm atom}$, the scattering of charged particle backgrounds
does not resolve individual atoms. Therefore, similarly to the case of solar
photons, charged particles interact with the neutral atoms in the matter-wave
interferometers through their polarizability $\alpha_{N}$. For protons, since
the flux drops exponentially above $1\,$GeV, we assume that the incoming and
outgoing charged particles are non-relativistic, and so can then be described
by classical waves according to the Born approximation:

$\begin{split}\psi_{i,f}=\frac{1}{\sqrt{V}}e^{i\mathbf{p}\cdot\mathbf{r}},\frac{1}{\sqrt{V}}e^{i\mathbf{p}^{\prime}\cdot\mathbf{r}}\;.\end{split}$
(26)

Atoms in the cloud will be polarized by the Coulomb field of the charged
particles, which has an interaction potential of:

$\begin{split}U(\mathbf{r})=-\sum_{l=1,\cdots,N_{\rm
atoms}}\frac{\alpha_{N}\,Z_{\rm{ion}}^{2}\,e^{2}}{2(4\pi)^{2}\,\|\mathbf{r}-\mathbf{r}_{l}\|^{4}}\;,\\\
\end{split}$ (27)

where $Z_{\rm{ion}}$ is the charge number of the charged particle, and
$\mathbf{r}_{l}$ is the location of the atoms. According to Fermi’s golden
rule:

$\displaystyle\sum_{f,\mathbf{p}^{\prime}}|\langle
f,\psi_{f}|H_{I}|i,\psi_{i}\rangle|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{V}\frac{\alpha_{N}^{2}Z_{\rm{ion}}^{4}e^{4}}{4(4\pi)^{4}}\int\frac{d^{3}q}{(2\pi)^{3}}\sum_{f}\left|\langle
f|\int d^{3}\mathbf{r}\sum_{l=1,\cdots,N_{\rm
atoms}}e^{i\mathbf{q}\cdot\mathbf{r}}\frac{1}{\|\mathbf{r}-\mathbf{r}_{l}\|^{4}}|i\rangle\right|^{2}$
(28) $\displaystyle=$
$\displaystyle\frac{1}{V}\frac{\pi^{2}Z_{\rm{ion}}^{4}\alpha_{N}^{2}\alpha^{2}}{64}\int\frac{d^{3}q}{(2\pi)^{3}}q^{2}S(q)\;,$

where $\alpha=e^{2}/4\pi$ is the fine-structure constant, we obtain the matrix
element
$\overline{|\sf{M}|^{2}}=\pi^{2}Z_{\rm{ion}}^{4}\alpha_{N}^{2}\alpha^{2}q^{2}/64$.
The structure-function $S(q)$ reflects the sum of all atoms in the cloud and
would take the same form as given by Eqs. (23), (24), & (25), depending on the
experiment type. Using Eq. (16), the total decoherence rate is then:

$\begin{split}\left.\Gamma_{\rm tot}\right|_{\textsc{cpb},{\rm
low}\,q}&=\frac{\pi}{128}Z_{\rm{ion}}^{4}\alpha^{2}\alpha_{N}^{2}\int
d\omega\frac{dn}{d\omega}\frac{1}{\sqrt{\omega^{2}-m^{2}}}\\\ &\quad\times\int
dq\,\Big{(}\omega+M-\sqrt{M^{2}+q^{2}}\Big{)}q^{3}\,S(q)\,\Big{[}1-{\rm
sinc}(q\Delta x)\Big{]}\;,\end{split}$ (29)

where the subscript cpb refers to charged particle backgrounds.

###  V.2 Charged Particle Scattering with High $q$

We now describe the interactions of cosmic ray charged particles that resolve
individual atoms ($q\gg 1/r_{\rm atom}$) within the matter-wave interferometer
clouds. The matrix element squared of this process is given by:

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{20.11916pt}{-32.9229pt}\pgfsys@lineto{40.23831pt}{-44.50623pt}\pgfsys@stroke\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
} {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@transformcm{0.86662}{-0.49895}{0.49895}{0.86662}{30.18971pt}{-38.72089pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}
{{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{{}}} }
{}\pgfsys@moveto{20.11916pt}{-11.58333pt}\pgfsys@curveto{20.11916pt}{-12.83333pt}{22.61916pt}{-13.45833pt}{22.61916pt}{-14.70833pt}\pgfsys@curveto{22.61916pt}{-15.95833pt}{20.11916pt}{-16.58333pt}{20.11916pt}{-17.83333pt}\pgfsys@lineto{20.11916pt}{-32.9229pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{gathered}\quad\begin{gathered}\quad\overline{|{\cal
M}|^{2}}=2(4\pi)^{2}\alpha^{2}Z_{\rm{ion}}^{2}\frac{2(m^{2}+M^{2}-s)^{2}+2st+t^{2}}{t^{2}}\;.\end{gathered}$
(30)

In this regime of $q$, the target can be either an electron or a proton. In
the following we focus on protons as targets. Similar results are obtained
when considering the electrons instead.

In the laboratory frame, we can write the Mandelstam variables as functions of
the masses and the momentum transfer:

$t=2M^{2}(1-\sqrt{1+q^{2}/M^{2}})\quad\text{ and }\quad s=m^{2}+M^{2}+2\omega
M\;.$ (31)

From Fig. 1, we can see that the highest flux is dominated by non-relativistic
cosmic rays. Thus, assuming protons are the target, $t\sim-q^{2}$. Expanding
in small $q$ ($q\ll m,M$), the matrix element simplifies to:

$\overline{|{\cal
M}|^{2}}\simeq\frac{4^{4}\pi^{2}\alpha^{2}Z_{\rm{ion}}^{2}\omega^{2}M^{2}}{q^{4}}\;,$
(32)

which matches the scattering amplitude that would be obtained from the cosmic
rays interacting with a Coulomb potential $U(r)=e/(4\pi r)$, whose Fourier
transform is $\tilde{U}(q)=e/q^{2}$. Converting Eq. (32) to $|{\sf M}|^{2}$ by
unity normalizing the states (see Eq. (54) in Appendix A) and applying it to
Eq. (16), we obtain the total decoherence rate caused by cosmic rays
interacting with protons when $q\gg 1/r_{\rm atom}$:

$\begin{split}\left.\Gamma_{\rm tot}\right|_{\textsc{cpb},{\rm
high}\,q}&\simeq 8\pi Z_{\rm ion}^{2}\alpha^{2}\int
d\omega\frac{d\Phi(\omega)}{d\omega}\frac{\omega^{2}}{\omega^{2}-m^{2}}\int_{0}^{q_{\rm
max}(\omega)}dq\,\frac{S(q)}{q^{3}}\Big{[}1-{\rm sinc}(q\Delta
x)\Big{]}\;,\end{split}$ (33)

where $d\Phi/d\omega$ is the spectral flux. The structure-function reflects
the sum of all protons in the target, and thus takes the form:

$\begin{split}S(q)=N_{\rm
atoms}Z_{\rm{target}}\Big{(}1+Z_{\rm{target}}{F}^{2}_{\rm
N}(q\,r_{N})\Big{)}\;,\end{split}$ (34)

where $Z_{\rm{target}}$ is the atomic number of the target, and
$F_{N}(q\,r_{N})$ is the nucleus form factor, which is given by the Helm form
factor [68]:

$F_{\rm N}(q\,r_{N})=\frac{3j_{1}(qr_{N})}{qr_{N}}e^{-q^{2}s^{2}/2}\;,$ (35)

where $r_{N}=1.14\,A^{1/3}$ fm [69] is the effective radius of the target
nucleus, and $s_{p}=0.9$ fm [69] is the skin thickness of the nucleus.

###  V.3 Combined Rate

The total rate involves integrating over $q$ from $0$ to the $q_{\rm max}$,
set by Eq. (9). Thus, the rate always involves the low $q$ region, and will
involve the high $q$ region if $q_{\rm max}\gg 1/r_{\rm atom}$. There is also
an intermediate regime we must address: when $q\sim 1/r_{\rm atom}$, i.e., the
transition between where the charged particles see individual nucleons versus
the full atom. The full rate can be written parametrically as:

$\left.\Gamma_{\rm tot}\right|_{\textsc{cpb}}=\int_{0}^{q_{\rm
max}}\frac{d\left.\Gamma_{\rm
tot}\right|_{\textsc{cpb}}}{dq}=\int_{0}^{q_{\text{low}}}\frac{\left.d\Gamma_{\rm
tot}\right|_{\textsc{cpb}\text{,low
q}}}{dq}+\int_{q_{\text{low}}}^{q_{\text{high}}}\frac{\left.d\Gamma_{\rm
tot}\right|_{\textsc{cpb}\text{,inter
q}}}{dq}+\int_{q_{\text{high}}}^{q_{\text{max}}}\frac{d\left.\Gamma_{\rm
tot}\right|_{\textsc{cpb}\text{,high q}}}{dq}\;.$ (36)

For a cosmic ray - proton interaction, the intermediate regime, $q\sim
1/r_{\rm atom}$, could be addressed by adding to the Coulomb potential the
impact of the negative charge of the electrons screening the positive charge
of the nucleus [70]:

$\tilde{U}(\mathbf{q})=\frac{e}{4\pi
q^{2}}\left(Z+F(\mathbf{q})\right)\quad\text{ where }\quad F(\mathbf{q})=\int
d^{3}\mathbf{r}\,n(\mathbf{r})\,e^{i\mathbf{q}\mathbf{r}}\;,$ (37)

where $n(\mathbf{r})$ is the density of the cloud of electrons. For $q\gg
1/r_{\rm atom}$, $F(q)\to 0$, due to the oscillatory behavior of the
exponential. For $q\ll 1/r_{\rm atom}$, the coherent interactions of the
charged particles with the dipole induced in the neutral atom dominate. Both
low-q and high-q regimes can be well-separated from the intermediate regime by
choosing $q_{\rm low}\ll 1/r_{\rm atom}$ and $q_{\rm high}\gg 1/r_{\rm atom}$,
respectively. For the numbers shown in this section, we choose the threshold
to be $q_{\rm low}=1/(10\ r_{\rm atom})$ and $q_{\rm high}=10/r_{\rm atom}$,
where we take $r_{\rm atom}$ to be the Bohr radius, i.e. $r_{\rm atom}=5\times
10^{-11}\text{ m}$ [71]. We summarize these regions in Fig. 3.

$\begin{gathered}\leavevmode\hbox to361.8pt{\vbox
to41.84pt{\pgfpicture\makeatletter\hbox{\hskip
85.75827pt\lower-19.15837pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{}{{}}{}{}{}{}{{}}{}\pgfsys@moveto{28.45276pt}{-5.69046pt}\pgfsys@moveto{28.45276pt}{-5.69046pt}\pgfsys@lineto{28.45276pt}{5.69046pt}\pgfsys@lineto{142.2638pt}{5.69046pt}\pgfsys@lineto{142.2638pt}{-5.69046pt}\pgfsys@closepath\pgfsys@moveto{142.2638pt}{5.69046pt}\pgfsys@fill\pgfsys@invoke{
} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
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{\pgfsys@beginscope\pgfsys@setlinewidth{0.64pt}\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{}
{}{}{} {}{}{}
\pgfsys@moveto{-1.55997pt}{2.07996pt}\pgfsys@curveto{-1.42996pt}{1.29997pt}{0.0pt}{0.12999pt}{0.38998pt}{0.0pt}\pgfsys@curveto{0.0pt}{-0.12999pt}{-1.42996pt}{-1.29997pt}{-1.55997pt}{-2.07996pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{-85.35828pt}{0.0pt}\pgfsys@lineto{255.36482pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{255.36482pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-71.7962pt}{8.88092pt}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-39.892pt}{-13.88092pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{low-$q$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.57616pt}{11.97083pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$q_{\rm
low}=\frac{1}{10\,r_{\rm atom}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{27.06387pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$|$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{41.8582pt}{-13.88092pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{intermediate regime}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{185.67862pt}{8.88092pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$A_{0}(r)=1/r$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{124.55383pt}{12.42648pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$q_{\rm
high}=\frac{10}{r_{\rm atom}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{high-$q$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{140.87491pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$q$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{gathered}$ Figure
3: Integration regimes in $q$ have been considered by employing different
approaches.

As we show in Table 4, the low-q and high-q contributions are already enough
for the solar wind to erase the fringe for MAQRO, and to dominate over the QNL
error for all of the other experiments. The contribution from the intermediate
regime can only worsen the visibility, so we do not need to consider it to
make the claim that the solar wind will be an important background for matter-
wave interferometers in space.

We note that Ref. [21] also calculated the effect of cosmic rays on atom
interferometers. However, they only calculated the effect on the phase, and
they ignored the effect of the solar wind. As in our case, they find that
cosmic rays produce a negligible effect on atom interferometers.

Solar wind |  MAQRO |  BECCAL |  GDM |  AEDGE
---|---|---|---|---
low-$q$ rate [$\text{s}^{-1}$] | $2.2\times 10^{1}$ | $2.0$ | $1.9\times 10^{2}$ | $7.0\times 10^{3}$
high-$q$ rate [$\text{s}^{-1}$] | $4.4\times 10^{3}$ | $5.5\times 10^{1}$ | $5.5\times 10^{3}$ | $5.5\times 10^{5}$

Cosmic rays | MAQRO | BECCAL | GDM | AEDGE
---|---|---|---|---
low-$q$ rate [$\text{s}^{-1}$] | $1.7\times 10^{-11}$ | $1.4\times 10^{-12}$ | $1.4\times 10^{-10}$ | $5.2\times 10^{-9}$
high-$q$ rate [$\text{s}^{-1}$] | $\ 2.8\times 10^{-9}\ $ | $\ 3.5\times 10^{-11}\ $ | $\ 3.5\times 10^{-9}\ $ | $3.5\times 10^{-7}$

Table 4: Total decoherence rate $\Gamma_{\rm tot}$ in $s^{-1}$ caused by solar
wind (upper table) and by cosmic rays (lower table).

##  VI Other Backgrounds

In this section, we discuss the final two particle backgrounds that we
consider in this paper: zodiacal dust and solar neutrinos. Unlike the previous
two backgrounds, these particles produce negligible amounts of decoherence in
the experiments that we consider. We include these calculations for
completeness.

###  VI.1 Zodiacal Dust

Interplanetary dust in the inner solar system mostly comes from the
fragmentation of comets and meteoroids. The mass (and size) distribution of
zodiacal dust grains is inferred from the frequency of micro-craters on lunar
samples and meteorites, as well as data from experiments such as Pioneers 8,
9, 10 and 11, Helios, and HEOS-II, amongst others [72]. Interstellar dust also
flows into the solar system as the Sun moves through the galaxy. However, we
assume that this is subdominant at 1 AU [73]. We take the interplanetary
meteoroids model [72], which estimates the dust at 1 AU, for our zodiacal dust
mass distribution.

Since these dust grains have sizes $a\sim 10^{-4}\,\mu{\rm m}-1\,{\rm cm}$
(masses $10^{-8}\,{\rm g}<m<1\,{\rm g}$) , which is comparable to the size of
the clouds in the AIs, we use the geometric cross section to quantify the
probability of interaction:

$\sigma_{\rm geo}=\pi(r_{\rm cloud}+a)^{2}\;,$ (38)

where the radius of the dust grain is related to its mass by adopting a
material density of $\rho_{\rm dust}\sim 3\text{ g/cm}^{3}$ [74]. The total
rate between Zodiacal dust and the AI clouds is given by:

$\begin{split}\left.\Gamma_{\rm tot}\right|_{\rm dust}&=\int
dm\,\frac{d\Phi_{\rm dust}}{dm}\sigma_{\rm geo}\,N_{\rm clouds}\;\\\
&=\int\frac{dm}{m}\,\left(\frac{dn_{\rm dust}}{d\log m}\right)\frac{v_{\rm
dust}}{k\,\ln 10}\,\pi\left(r_{\rm cloud}+\left(\frac{3m}{4\pi\rho_{\rm
dust}}\right)^{1/3}\right)^{2}N_{\rm clouds}\;,\end{split}$ (39)

where $dn/d\log m$ is the spatial number density, which we take from [72],
$N_{\rm clouds}=2$ is the number of clouds involved in the experiment, and $k$
is a constant (in particular, $k=4$ for an isotropic flux). Notice that such a
contact interaction always resolves the two paths, thus $p_{\rm dec}=1$. As
with all of our backgrounds, we assume the zodiacal dust is isotropic. Above,
$v_{\rm dust}=v_{0}\sqrt{r/r_{0}}$ is the orbital velocity of the dust
particles, with $r$ referring to the radial distance from the Sun, $r_{0}=1$
AU, and $v_{0}=20$ km/s, taken from meteor observations and satellite
measurements [72]. We take $r=1$ AU for the missions we consider. Table 5
shows the decoherence rate for the experiments we consider. Overall, these are
negligible.

Dust | MAQRO | BECCAL | GDM | AEDGE
---|---|---|---|---
Rate [$\text{s}^{-1}$] | $2.2\times 10^{-14}$ | $2.4\times 10^{-8}$ | $1.1\times 10^{-6}$ | $9.6\times 10^{-6}$
Table 5: Total decoherence rate $\Gamma_{\rm tot}$ in $s^{-1}$ caused by
Zodiacal dust.

###  VI.2 Solar Neutrinos

Solar neutrinos are byproducts of nuclear fusion processes in the core of the
Sun. Their fluxes depend on the solar temperature, opacity, chemical
composition, nuclear cross sections, and other solar features. Typically,
solar models are employed to calculate the solar neutrino fluxes [37, 38, 39].
Solar neutrinos are created throughout the “pp-chain” whereby the Sun fuses
hydrogen ions into helium ions in a number of steps. The first reaction, which
we call “pp” here, is: $p+p\to d+e^{+}+\nu_{e}$, $E_{\nu}\leq 0.42\text{
MeV}$. This step composes 90% of the solar neutrino flux [37]. Other
components of the solar neutrino flux are the pep-neutrinos ($p+e+p\to
d+\nu_{e}$, $E_{\nu}=1.44\text{ MeV}$ -monochromatic),
$\phantom{I}{}^{7}\text{Be}$-neutrinos
($e+\\!\\!\\!\phantom{I}^{7}\text{Be}\to\\!\\!\\!\phantom{I}^{7}\text{Li}+\nu_{e}$,
$E_{\nu}=0.86\text{ MeV (90\%)},\,0.38\text{ MeV (10\%)}$),
$\\!\phantom{I}{}^{8}B$-neutrinos
($\\!\phantom{I}{}^{8}B\to\\!\\!\\!\phantom{I}^{8}\text{Be}^{*}+e^{+}+\nu_{e}$,
$E_{\nu}<15\text{ MeV}$), and hep-neutrinos
($\\!\phantom{I}{}^{3}\text{He}+p\to\\!\\!\\!\phantom{I}^{4}\text{He}+e^{+}+\nu_{e}$,
$E_{\nu}<18.8\text{ MeV}$) [75]. The CNO (carbon-nitrogen-oxygen) cycle has a
subdominant contribution (about $1\%$) [75]. In Fig. 1, we show the total flux
of solar neutrinos from different production channels.

The spin-averaged modulus squared of the scattering amplitude for a neutrino
interacting with a nucleon through the vector coupling of the $Z$ boson is
given by:

$\begin{gathered}\leavevmode\hbox to75.01pt{\vbox
to61.48pt{\pgfpicture\makeatletter\hbox{\hskip 16.16696pt\lower-52.0772pt\hbox
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}{}\pgfsys@moveto{21.33957pt}{-8.53583pt}\pgfsys@moveto{23.33125pt}{-8.53583pt}\pgfsys@curveto{23.33125pt}{-7.43584pt}{22.43956pt}{-6.54414pt}{21.33957pt}{-6.54414pt}\pgfsys@curveto{20.23958pt}{-6.54414pt}{19.34789pt}{-7.43584pt}{19.34789pt}{-8.53583pt}\pgfsys@curveto{19.34789pt}{-9.63582pt}{20.23958pt}{-10.52751pt}{21.33957pt}{-10.52751pt}\pgfsys@curveto{22.43956pt}{-10.52751pt}{23.33125pt}{-9.63582pt}{23.33125pt}{-8.53583pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{-8.53583pt}\pgfsys@fillstroke\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{1,.5,0}\pgfsys@color@rgb@fill{1}{.5}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{21.33957pt}{-34.1433pt}\pgfsys@moveto{23.33125pt}{-34.1433pt}\pgfsys@curveto{23.33125pt}{-33.0433pt}{22.43956pt}{-32.15161pt}{21.33957pt}{-32.15161pt}\pgfsys@curveto{20.23958pt}{-32.15161pt}{19.34789pt}{-33.0433pt}{19.34789pt}{-34.1433pt}\pgfsys@curveto{19.34789pt}{-35.24329pt}{20.23958pt}{-36.13498pt}{21.33957pt}{-36.13498pt}\pgfsys@curveto{22.43956pt}{-36.13498pt}{23.33125pt}{-35.24329pt}{23.33125pt}{-34.1433pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{-34.1433pt}\pgfsys@fillstroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{gathered}\qquad\begin{gathered}\overline{|{\cal
M}|^{2}}\simeq 8|g_{\nu A}|^{2}|g_{Z\nu
V}|^{2}\frac{2(M^{2}-s)^{2}+2st+t^{2}}{m_{Z}^{4}}\;,\end{gathered}$ (40)

where the target mass $M$ is the mass of the nucleon (N). In the laboratory
frame and using that $q\ll M$ (see Fig. 1), $s=M^{2}+2\omega M$, and
$t=-q^{2}$. Hence:

$\overline{|{\cal M}|^{2}}\simeq 8\frac{|g_{\nu A}|^{2}|g_{Z\nu
V}|^{2}}{m_{Z}^{4}}\left(8\omega^{2}M^{2}-2Mq^{2}(2\omega+M)+q^{4}\right)\;.$
(41)

The vector coupling between the Standard Model $Z$ boson and a left-handed
neutrino is given by $|g_{Z\nu}|=e/2\sin 2\theta_{W}$, where the electroweak
angle is $\sin^{2}\theta_{W}=0.23$ [76].

Neutrinos from the sun are always below 20 MeV, and therefore their
interactions with nuclei are within the coherent regime. Hence, the neutrino-
atom coupling reads [41]:

$\begin{split}|g_{\nu{\rm A}}|^{2}&=\frac{\alpha\,\pi}{(\sin
2\theta_{W})^{2}}|Z(4\sin^{2}\theta_{W}-1)+(A-Z)|^{2}|F_{N}(qr_{N})|^{2}\simeq\frac{\alpha\,\pi}{(\sin
2\theta_{W})^{2}}(A-Z)^{2}\;,\end{split}$ (42)

where $A$ and $Z$ are the numbers of nucleons and protons in the nucleus of
the target, respectively. In the last equality above, we have used that
$\sin^{2}\theta_{W}\sim 1/4$ and that in the coherent regime
$F_{N}(qr_{N})\sim 1$, where $F_{N}(qr_{N})$ is the nucleus form factor (see
Eq. (35)).

Furthermore, if the de-Broglie wavelength of the interaction is larger than
the size of the cloud, i.e. $q<1/r_{\rm cloud}$, there is another coherent
enhancement across all of the atoms, and the probability amplitude scales
as777Note that such an enhancement does not occur in experiments for direct
detection of neutrinos based on nuclear recoil. This is because the lowest
recoil energy these experiments are sensitive to ($E_{\rm rec}\sim 1$ meV)
sets a lower threshold on the momentum transfer, i.e. $q_{\rm min}\gtrsim
1/r_{\rm atom}$. $N_{\rm atoms}^{2}$. Thus, the scattering amplitude reads
[77]:

$\begin{split}\overline{|{\cal
M}|^{2}}&=8G_{F}^{2}M^{2}\omega^{2}\left(1-\frac{q^{2}}{4\omega^{2}}\left(\frac{2\omega}{M}+1\right)+\frac{q^{4}}{8\omega^{2}M^{2}}\right)\\\
&\quad\times(A-Z)^{2}\left(|F_{N}(qr_{\rm N})|^{2}+N_{\rm atoms}|F(qr_{\rm
cloud})|^{2}\right)\;,\end{split}$ (43)

where $G_{F}=4\pi\alpha/(\sqrt{2}M_{Z}^{2}\sin^{2}2\theta_{W})$ is the Fermi
constant. Converting $\overline{|{\cal M}|^{2}}$ to $|{\sf M}|^{2}$ (see Eq.
(54) in Appendix A) and plugging this into Eq. (16), we find:

$\begin{split}\left.\Gamma_{\rm
tot}\right|_{\nu_{\odot}}&\simeq\frac{G_{F}^{2}}{4\pi}\int_{0}^{\infty}d\omega\frac{d\Phi}{d\omega}\int_{0}^{2\omega}dq\,q\left(1-\frac{q^{2}}{4\omega^{2}}\left(\frac{2\omega}{M}+1\right)+\frac{q^{4}}{8\omega^{2}M^{2}}\right)\\\
&\qquad\qquad\times N_{\rm atoms}(A-Z)^{2}\left(|F_{N}(qr_{N})|^{2}+N_{\rm
atoms}F(qr_{\rm cloud})|^{2}\right)\,\Big{[}1-{\rm sinc}(q\Delta
x)\Big{]}\;,\end{split}$ (44)

where the nuclear contribution is neglected because the incoming energy of
solar neutrinos is not high enough to resolve the nucleus of the atom, see
Fig. 1. In Table 6, we show the results for the decoherence rate due to solar
neutrinos for the space-based matter-wave interferometers that we consider. As
with the zodiacal dust case, these rates are negligible.

$\nu_{\odot}$ | MAQRO | BECCAL | GDM | AEDGE
---|---|---|---|---
Rate [$\text{s}^{-1}$] | $\ 1.0\times 10^{-23}\ $ | $\ 1.3\times 10^{-25}\ $ | $\ 1.3\times 10^{-23}\ $ | $\ 1.3\times 10^{-25}\ $
Table 6: Total decoherence rate $\Gamma_{\rm tot}$ in $s^{-1}$ caused by solar
neutrino fluxes.

##  VII Discussion & Conclusions

In this paper, we calculated the decoherence from solar photons, the solar
wind, galactic cosmic rays, zodiacal dust, and solar neutrinos on space-based
matter-wave interferometer experimental concepts. We developed the formalism
necessary to treat both relativistic and non-relativistic particles with non-
uniform spectral density. As summarized in Tab. 2, the solar wind and solar
photons are important backgrounds to these proposed experiments. In addition,
solar photons, along with cosmic rays, could erase the fringe of matter
interferometers like MAQRO.

Given the decoherence rates we find for solar photons and the solar wind, it
is likely that these experiments will need to consider shielding carefully.
While shielding against solar photons is relatively simple, shielding against
the solar wind could be quite difficult and costly. Given some of the large
cloud separations, especially for GDM, this could prove to be especially
prohibitive.

Another possible resolution to dealing with these backgrounds is to change the
orbit of the spacecraft. In this work, we considered an Earth-like orbit. To
mitigate against both solar photons and the solar wind, a spacecraft could be
positioned in a relatively low Earth orbit. This would allow the spacecraft to
be in Earth’s shadow for part of its orbit and within the protective envelope
of Earth’s magnetic field. BECCAL would already be in such a position, given
that it is proposed to be on the ISS. Solar orbits at larger semi-major axes
and outside of the ecliptic plane could also be possible. However, a detailed
study of the decoherence given different orbital configurations is left to
future work.

Our work also assumes that these backgrounds are isotropic. However, all of
these backgrounds are actually highly directional. Including the directional
information would not have a large effect on the total rate – a 50%
enhancement or reduction at most (see Appendix A of Ref. [24]). Nevertheless,
this directional information could help with separation from other signals,
such as decoherence from dark matter particles [24].

In this work, we solely considered the decoherence effects of these background
particles. However, we expect these particles to also induce phase effects in
these matter-wave interferometer experiments. The highly anisotropic nature of
these backgrounds will likely mean that these phase shifts could be subtracted
out of any science analysis, given sufficient knowledge of the spacecraft
orbit and background flux directions. However, we leave a detailed analysis of
this topic to future work.

Matter-wave interferometers are sensitive experiments and have the potential
to be important probes of gravitational waves and dark matter. However,
careful modeling and mitigation of backgrounds is needed for them to reach
their potential. In this paper, we have shown that cosmic rays, in particular,
will need to be shielded by all of the proposed experiments. In addition, we
have developed the formalism needed to calculate rates for varying fluxes of
these particles. The inclusion of shielding and orbit mitigation in space-
based matter-wave interferometer designs is recommended.

###### Acknowledgements.

The authors would like to thank Leonardo Badurina, Sheng-wey Chiow, Ryan
Plestid and Jess Riedel for especially helpful discussions. We would also like
to thank Haiming Deng, Gerard Fasel, Brandon Hensley, Ed Rhodes, and Roger
Ulrich. KP was supported by the U.S. Department of Energy, Office of Science,
Office of High Energy Physics, under Award No. DE-SC0021431, the Quantum
Information Science Enabled Discovery (QuantISED) for High Energy Physics
(KA2401032). This work is supported by the U.S. Department of Energy, Office
of Science, Office of High Energy Physics, under Award Number DE-SC0011632 and
by the Walter Burke Institute for Theoretical Physics. This work was performed
in part at Aspen Center for Physics, which is supported by National Science
Foundation grant PHY-2210452.

##  Appendix A Cross Section Formalism

The number of events contributing to the decoherence can also be obtained from
the scattering cross section of the background particle with the matter-wave
interferometer cloud, as opposed to using the formalism based on Fermi’s
Golden rule, described in Section III. The cross section is defined as the
number of events per effective area ($A_{T}$) per incoming ($N_{\rm back}$)
and target ($N_{\rm target}$) particles: $\sigma=\\#_{\rm
events}\,A_{T}/(N_{\rm back}\,N_{\rm target})$. The rate of events causing
decoherence can then be written as a function of the decoherence cross section
and the flux of incoming particles ($\Phi=N_{\rm
back}/(A_{T}\times\text{time})$ ),

$\frac{d\Gamma_{\rm tot}}{dt}=N_{\rm target}\int
d\omega\left(\frac{1}{4\pi}\frac{d\Phi}{d\omega}\right)\int
d\Omega\int_{0}^{q_{\rm max}(\omega)}\,dq\,\frac{d\sigma}{dq}.$ (45)

Above we have assumed that the flux of incoming particle is isotropic. The
decoherence form factor, $p_{\rm dec}(\mathbf{q}\cdot\mathbf{\Delta x})$, is
included in the cross section. In the laboratory frame, with
$\hat{\mathbf{z}}$ axis aligned with the momentum of the incoming particle
$\mathbf{p}$ (see Section III), the momentum transfer can be written as:

$\mathbf{q}=(q\sin\theta_{\mathbf{q}\mathbf{p}}\cos\phi_{\mathbf{q}\mathbf{p}},q\sin\theta_{\mathbf{q}\mathbf{p}}\sin\phi_{\mathbf{q}\mathbf{p}},q\cos\theta_{\mathbf{q}\mathbf{p}})\;,$
(46)

where $q\equiv|\mathbf{q}|$. In this frame, the differential cross-section is
given by:

$\begin{split}\frac{d\sigma}{dq}&=\frac{1}{2\omega
2\omega^{\prime}|v-v^{\prime}|}\frac{q^{2}}{(2\pi)^{2}}\\\ &\times\int
d\cos\theta_{\mathbf{q}\mathbf{p}}\frac{\overline{|{\cal
M}|^{2}}}{2E_{i}2E_{f}}\left(1+N_{\rm target}|{\cal
F}_{T}(q)|^{2}\right)\,\delta^{(0)}(\omega+\omega^{\prime}-E_{i}-E_{f})\times\int
d\phi_{\mathbf{q}\mathbf{p}}\left(1-e^{i\mathbf{q}\cdot\mathbf{\Delta
x}}\right)\;,\end{split}$ (47)

where the relative velocity between the incoming particle ($v$) and the target
($v^{\prime}$) can be expressed as
$\omega\,\omega^{\prime}|v-v^{\prime}|=M\sqrt{\omega^{2}-m^{2}}$. The term
$\left(1+N_{\rm target}|{\cal F}_{T}(q)|^{2}\right)$ accounts for the
potential Born enhancement due to elastic scattering with the target, which is
parameterized by the Fourier transform of the energy density of the target’s
wavefunction, ${\cal F}_{T}(q)$. Above, ${\cal M}$ refers to the scattering
matrix element with the standard normalization, i.e., quantum states
normalized with their energy. We have explicitly written the form of $p_{\rm
dec}(\mathbf{q}\cdot\mathbf{\Delta x})$. Furthermore, we have exploited the
fact that the matrix element in the lab frame does not depend on the azimuthal
angle, $\phi_{\mathbf{q}\mathbf{p}}$, since the Mandelstam variables in the
laboratory frame only depend on $\omega$, $q$, and the incoming particle and
target masses. In other words:

$s=(p+k)^{2}=M^{2}+m^{2}+2\omega M,\qquad\text{and}\qquad
t=(k-k^{\prime})^{2}=2M^{2}\left(1-\sqrt{1+\frac{q^{2}}{M^{2}}}\right)\;.$
(48)

The decoherence form factor, however, does depend on
$\phi_{\mathbf{q}\mathbf{p}}$. The vector parameterizing the separation of the
two clouds, written in the laboratory frame, is:

$\mathbf{\Delta x}=(\Delta x\,\sin\theta_{\mathbf{\Delta
x}\mathbf{p}}\cos\phi_{\mathbf{\Delta x}\mathbf{p}},\Delta
x\,\sin\theta_{\mathbf{\Delta x}\mathbf{p}}\sin\phi_{\mathbf{\Delta
x}\mathbf{p}},\Delta x\,\cos\theta_{\mathbf{\Delta x}\mathbf{p}})\;,$ (49)

where $\Delta x\equiv|\mathbf{\Delta x}|$, and $\theta_{\mathbf{\Delta xp}}$
is the angle between $\mathbf{\Delta x}$ and $\mathbf{p}$. Therefore,

$\begin{split}&\int_{0}^{2\pi}d\phi_{\mathbf{q}\mathbf{p}}\left(1-e^{iq\Delta
x(\sin\theta_{\mathbf{q}\mathbf{p}}\sin\theta_{\mathbf{\Delta
x}\mathbf{p}}\cos(\phi_{\mathbf{q}\mathbf{p}}-\phi_{\mathbf{\Delta
x}\mathbf{p}})+\cos\theta_{\mathbf{q}\mathbf{p}}\cos\theta_{\mathbf{\Delta
x}\mathbf{p}}}\right)\\\ &\qquad=(2\pi)\left(1-J_{0}(q\Delta
x\,\sin\theta_{\mathbf{q}\mathbf{p}}\sin\theta_{\mathbf{\Delta
x}\mathbf{p}})e^{iq\Delta
x\,\cos\theta_{\mathbf{q}\mathbf{p}}\cos\theta_{\mathbf{\Delta
x}\mathbf{p}}}\right)\,.\end{split}$ (50)

Since the angle between the incoming particle and the transferred momentum is
a function of the incoming energy, the two variables can be exchanged
according to the Jacobian:

$d\cos\theta_{\mathbf{q}\mathbf{p}}=-\frac{\omega}{q\sqrt{\omega^{2}-m^{2}}}d\omega\;,$
(51)

and use the energy delta function to perform the integral over $\omega$,
leading to the following expression:888The integration of Eq. (50) over
$d\Omega$ gives $8\pi^{2}(1-\text{sinc}(q\Delta x))$. Note that the result is
independent of $\theta_{\mathbf{q}\mathbf{p}}$, which becomes a function of
$q$ after employing the energy delta function. See Appendix B for an explicit
derivation.

$\int
d\Omega\,\frac{d\sigma}{dq}=\frac{q}{8M(\omega^{2}-m^{2})}\frac{\overline{|{\cal
M}|^{2}}}{\sqrt{M^{2}+q^{2}}}\left(1+N_{\rm target}|{\cal
F}_{T}(q)|^{2}\right)\left(1-\frac{\sin(q\Delta x)}{q\Delta x}\right)\;,$ (52)

where we have commuted the integrals over $q$ and $\Omega$, since the limits
of $q$ do not depend on the angular phase space of the incoming particles (see
the explicit form of $q_{\rm max}$ in Eq. (9)). Plugging the differential
cross section into Eq. (45), we obtain the master equation for the decoherence
rate:

$\frac{d\Gamma_{\rm tot}}{dt}=\frac{N_{\rm target}}{32\pi M}\int
d\omega\frac{d\Phi}{d\omega}\frac{1}{\omega^{2}-m^{2}}\int_{0}^{q_{\rm
max}(\omega)}dq\,q\frac{\overline{|{\cal
M}(\omega,q)|^{2}}}{\sqrt{M^{2}+q^{2}}}\left(1+N_{\rm target}|{\cal
F}_{T}(q)|^{2}\right)\left(1-\frac{\sin(q\Delta x)}{q\Delta x}\right)\;.$ (53)

From the equation above one can reproduce Eq. (16) in Section A by unity
normalizing the states in the matrix element:

$\overline{|{\cal
M}|^{2}}=(2\omega)(2\omega^{\prime})(2E_{i})(2E_{f})\overline{|{\sf
M}|^{2}}=16\omega\left(M+\omega-\sqrt{M^{2}+q^{2}}\right)M\sqrt{M^{2}+q^{2}}\
\ \overline{|{\sf M}|^{2}}\;,$ (54)

and writing the spectral flux as the spectral density, which is related by the
velocity of the incoming particle $v=\sqrt{1-\gamma^{-2}}=p/\omega$ (with
$\gamma=\sqrt{1+p^{2}/m^{2}}$):

$\frac{d\Phi}{d\omega}=\frac{dn}{d\omega}v=\frac{dn}{d\omega}\frac{\sqrt{\omega^{2}-m^{2}}}{\omega}\;.$
(55)

Finally, the term $N_{\rm target}(1+N_{\rm target}|{\cal F}_{T}(q)|^{2})$ can
be identified as $S(q)$ (see Eq. (16) in Section III).

##  Appendix B The Angular Phase Space Integration

In this section, we show how to reduce the six-dimensional phase space
integration $d\omega d\Omega d^{3}\mathbf{q}$ by integrating out the angular
phase space. We first derive a general formalism applicable to any spectral
density of the incoming particle with directional dependence, where we reduce
the phase space integration to $d\omega d\Omega dq$. Then we consider the
special case of an isotropic incoming source, which further reduces the
integration to be two-dimensional, over $d\omega dq$.

For convenience, let us define:

$f(\omega,\Omega)\equiv\omega\frac{dn}{d\omega d\Omega}\;.$ (56)

Rewriting the master equation in case of the (angular) phase space of the
incoming source and the momentum transfer:

$\begin{split}\Gamma_{\rm
tot}&=\frac{1}{(2\pi)^{2}}\int\frac{d\omega}{\omega}\int
dq\,q^{2}\overline{|{\sf M}|^{2}}S(q)\int
d\Omega_{\mathbf{p}}d\Omega_{\mathbf{q}}f(\omega,\Omega_{\mathbf{p}},t)\left(1-e^{i{\bf
q}\cdot{\bf\Delta x}}\right)\delta(E_{f}-E_{i}-\omega_{\bf q})\;,\end{split}$
(57)

where we specifically label the solid angle with the corresponding momentum.
Notice that, for all of the angles, the energy deposition,
$\omega_{\mathbf{q}}$, only depends on the angle between $\mathbf{p}$ and
$\mathbf{q}$, and that:

$\begin{split}\frac{d\omega_{\mathbf{q}}}{d\cos\alpha_{\mathbf{p}\mathbf{q}}}=q\frac{\sqrt{\omega^{2}-m^{2}}}{\omega-
E_{\rm rec}}\;,\end{split}$ (58)

where $E_{\rm rec}=\sqrt{M^{2}+q^{2}}-M$. Let us define $\alpha_{0}$ as
$\left.E_{\rm
rec}=\omega_{\mathbf{q}}\right|_{\alpha_{\mathbf{p}\mathbf{q}}=\alpha_{0}}$.
Then we can solve for:

$\begin{split}\cos\alpha_{0}=\frac{1}{2\sqrt{\omega^{2}-m^{2}}q}\left(q^{2}+2\omega
E_{\rm rec}-E_{\rm rec}^{2}\right)\;.\end{split}$ (59)

Accordingly, the Kronecker delta function enforcing energy conservation can be
written as a function of the angle $\alpha_{\mathbf{p}\mathbf{q}}$:

$\begin{split}\delta(E_{f}-E_{i}-\omega_{\bf q})=\frac{\omega-E_{\rm
rec}}{q\sqrt{\omega^{2}-m^{2}}}\delta(\cos\alpha_{\mathbf{p}\mathbf{q}}-\cos\alpha_{0})\;.\end{split}$
(60)

The angular part of the integration can be done generically:

$\begin{split}&\int
d\Omega_{\mathbf{p}}d\Omega_{\mathbf{q}}f(\omega,\Omega_{\mathbf{p}},t)\left(1-e^{i{\bf
q}\cdot{\bf\Delta
x}}\right)\delta(\cos\alpha_{\mathbf{p}\mathbf{q}}-\cos\alpha_{0})\\\
=&(2\pi)\int
d\Omega_{\mathbf{p}}f(\omega,\Omega_{\mathbf{p}},t)\Big{[}1-J_{0}(q\Delta
x\sin\alpha_{0}\sin\theta_{\mathbf{p}\mathbf{\Delta x}})e^{iq\Delta
x\cos\alpha_{0}\cos\theta_{\mathbf{p}\mathbf{\Delta x}}}\Big{]}\;.\\\
\end{split}$ (61)

In deriving the above, we have used changes of variables, e.g.,

$\begin{split}d\theta_{\mathbf{p}}d\phi_{\mathbf{p}}=|J|\,d\alpha_{\mathbf{p}\mathbf{q}}d\gamma_{\mathbf{p}\mathbf{q}}\;,\end{split}$
(62)

with

$\begin{split}\cos\alpha_{\mathbf{p}\mathbf{q}}&=\cos\theta_{\mathbf{p}}\cos\theta_{\mathbf{q}}+\sin\theta_{\mathbf{p}}\sin\theta_{\mathbf{q}}\cos(\phi_{\mathbf{q}}-\phi_{\mathbf{p}})\\\
\sin\gamma_{\mathbf{p}\mathbf{q}}&=\frac{\sin\theta_{\mathbf{q}}\sin(\phi_{\mathbf{q}}-\phi_{\mathbf{p}})}{\sin\alpha_{\mathbf{p}\mathbf{q}}}\;.\end{split}$
(63)

The Jacobian of the transformation is
$|J|=\sin\alpha_{\mathbf{p}\mathbf{q}}/\sin\theta_{\mathbf{p}}$. The phase
space integration is now reduced to be over $d\omega d\Omega_{\mathbf{p}}dq$,
independent of the angular phase space of $\mathbf{q}$.

In the case of an isotropic source, the density spectrum does not depend on
the solid angle, i.e.,

$\begin{split}f(\omega,\Omega_{\mathbf{p}},t)=\omega\frac{dn(\omega,\Omega_{\mathbf{p}})}{d\omega
d\Omega_{\mathbf{p}}}\to\frac{\omega}{4\pi}\frac{dn(\omega)}{d\omega}\;.\end{split}$
(64)

The angular part of the integration simplifies to be:

$\begin{split}&(2\pi)\int
d\Omega_{\mathbf{p}}f(\omega,\Omega_{\mathbf{p}},t)\Big{[}1-J_{0}(q\Delta
x\sin\alpha_{0}\sin\theta_{\mathbf{p}\mathbf{\Delta x}})e^{iq\Delta
x\cos\alpha_{0}\cos\theta_{\mathbf{p}\mathbf{\Delta x}}}\Big{]}\\\
=&\frac{\omega}{2}\frac{dn}{d\omega}\int d\phi_{\mathbf{p}\mathbf{\Delta
x}}d\cos\theta_{\mathbf{p}\mathbf{\Delta x}}\Big{[}1-J_{0}(q\Delta
x\sin\alpha_{0}\sin\theta_{\mathbf{p}\mathbf{\Delta x}})e^{iq\Delta
x\cos\alpha_{0}\cos\theta_{\mathbf{p}\mathbf{\Delta x}}}\Big{]}\\\
=&(2\pi)\omega\frac{dn}{d\omega}\Big{[}1-{\rm sinc}(q\Delta
x)\Big{]}\;.\end{split}$ (65)

Now the phase space integration has been reduced to be two-dimensional,
$d\omega dq$, as has been used in the main text.

## References

  * Geiger  _et al._ [2020] R. Geiger, A. Landragin, S. Merlet, and F. Pereira Dos Santos, AVS Quantum Science 2, 024702 (2020), arXiv:2003.12516 [physics.atom-ph] .
  * Kasevich and Chu [1991] M. Kasevich and S. Chu, Phys. Rev. Lett. 67, 181 (1991).
  * Asenbaum  _et al._ [2020] P. Asenbaum, C. Overstreet, M. Kim, J. Curti, and M. A. Kasevich, Phys. Rev. Lett. 125, 191101 (2020), arXiv:2005.11624 [physics.atom-ph] .
  * Dimopoulos  _et al._ [2008] S. Dimopoulos, P. W. Graham, J. M. Hogan, M. A. Kasevich, and S. Rajendran, Phys. Rev. D 78, 122002 (2008), arXiv:0806.2125 [gr-qc] .
  * Arvanitaki  _et al._ [2015] A. Arvanitaki, J. Huang, and K. Van Tilburg, Phys. Rev. D 91, 015015 (2015), arXiv:1405.2925 [hep-ph] .
  * Graham  _et al._ [2016] P. W. Graham, J. M. Hogan, M. A. Kasevich, and S. Rajendran, Phys. Rev. D 94, 104022 (2016), arXiv:1606.01860 [physics.atom-ph] .
  * Arvanitaki  _et al._ [2018] A. Arvanitaki, P. W. Graham, J. M. Hogan, S. Rajendran, and K. Van Tilburg, Phys. Rev. D 97, 075020 (2018), arXiv:1606.04541 [hep-ph] .
  * Geraci and Derevianko [2016] A. A. Geraci and A. Derevianko, Phys. Rev. Lett. 117, 261301 (2016), arXiv:1605.04048 [physics.atom-ph] .
  * Badurina  _et al._ [2021] L. Badurina, O. Buchmueller, J. Ellis, M. Lewicki, C. McCabe, and V. Vaskonen, Phil. Trans. A. Math. Phys. Eng. Sci. 380, 20210060 (2021), arXiv:2108.02468 [gr-qc] .
  * Overstreet  _et al._ [2022a] C. Overstreet, P. Asenbaum, J. Curti, M. Kim, and M. A. Kasevich, Science 375, 226 (2022a), https://www.science.org/doi/pdf/10.1126/science.abl7152 .
  * Abe  _et al._ [2021] M. Abe _et al._ (MAGIS-100), Quantum Sci. Technol. 6, 044003 (2021), arXiv:2104.02835 [physics.atom-ph] .
  * Frye  _et al._ [2021] K. Frye, S. Abend, W. Bartosch, A. Bawamia, D. Becker, H. Blume, C. Braxmaier, S.-W. Chiow, M. A. Efremov, W. Ertmer, and et al., EPJ Quantum Technology 8, 1–38 (2021).
  * Kaltenbaek  _et al._ [2016] R. Kaltenbaek, M. Aspelmeyer, P. F. Barker, A. Bassi, J. Bateman, K. Bongs, S. Bose, C. Braxmaier, Č. Brukner, B. Christophe, M. Chwalla, P.-F. Cohadon, A. M. Cruise, C. Curceanu, K. Dholakia, L. Diósi, K. Döringshoff, W. Ertmer, J. Gieseler, N. Gürlebeck, G. Hechenblaikner, A. Heidmann, S. Herrmann, S. Hossenfelder, U. Johann, N. Kiesel, M. Kim, C. Lämmerzahl, A. Lambrecht, M. Mazilu, G. J. Milburn, H. Müller, L. Novotny, M. Paternostro, A. Peters, I. Pikovski, A. Pilan Zanoni, E. M. Rasel, S. Reynaud, C. J. Riedel, M. Rodrigues, L. Rondin, A. Roura, W. P. Schleich, J. Schmiedmayer, T. Schuldt, K. C. Schwab, M. Tajmar, G. M. Tino, H. Ulbricht, R. Ursin, and V. Vedral, EPJ Quantum Technology 3, 5 (2016).
  * Abou El-Neaj  _et al._ [2020] Y. Abou El-Neaj, C. Alpigiani, S. Amairi-Pyka, H. Araujo, A. Balaz, A. Bassi, L. Bathe-Peters, B. Battelier, A. Belic, E. Bentine, J. Bernabeu, A. Bertoldi, R. Bingham, D. Blas, V. Bolpasi, K. Bongs, S. Bose, P. Bouyer, T. Bowcock, W. Bowden, O. Buchmueller, C. Burrage, X. Calmet, B. Canuel, L.-I. Caramete, A. Carroll, G. Cella, V. Charmandaris, S. Chattopadhyay, X. Chen, M. L. Chiofalo, J. Coleman, J. Cotter, Y. Cui, A. Derevianko, A. De Roeck, G. Djordjevic, P. Dornan, M. Doser, I. Drougkakis, J. Dunningham, I. Dutan, S. Easo, G. Elertas, J. Ellis, M. El Sawy, F. Fassi, D. Felea, C.-H. Feng, R. Flack, C. Foot, I. Fuentes, N. Gaaloul, A. Gauguet, R. Geiger, V. Gibson, G. Giudice, J. Goldwin, O. Grachov, P. W. Graham, D. Grasso, M. van der Grinten, M. Gundogan, M. G. Haehnelt, T. Harte, A. Hees, R. Hobson, B. Holst, J. Hogan, M. Kasevich, B. J. Kavanagh, W. von Klitzing, T. Kovachy, B. Krikler, M. Krutzik, M. Lewicki, Y.-H. Lien, M. Liu, G. Gaetano Luciano, A. Magnon, M. Mahmoud, S. Malik, C. McCabe, J. Mitchell, J. Pahl, D. Pal, S. Pandey, D. Papazoglou, M. Paternostro, B. Penning, A. Peters, M. Prevedelli, V. Puthiya-Veettil, J. Quenby, E. Rasel, S. Ravenhall, H. Rejeb Sfar, J. Ringwood, A. Roura, D. Sabulsky, M. Sameed, B. Sauer, S. Alaric Schaffer, S. Schiller, V. Schkolnik, D. Schlippert, C. Schubert, A. Shayeghi, I. Shipsey, C. Signorini, M. Soares-Santos, F. Sorrentino, Y. Singh, T. Sumner, K. Tassis, S. Tentindo, G. M. Tino, J. N. Tinsley, J. Unwin, T. Valenzuela, G. Vasilakis, V. Vaskonen, C. Vogt, A. Webber-Date, A. Wenzlawski, P. Windpassinger, M. Woltmann, M. Holynski, E. Yazgan, M.-S. Zhan, X. Zou, and J. Zupan, EPJ Quantum Technology 7, 6 (2020), arXiv:1908.00802 [gr-qc] .
  * Amini  _et al._ [2021] R. Amini, S.-W. Chiow, C. Cutler, O. Doré, J. Jewell, K. Pardo, J. Rhodes, A. Singh, and N. Yu,  _Gravity Probe and Dark Energy Detection Probe (GDM)_, NASA BPS 2021 Research Campaign White Paper (2021).
  * Hornberger  _et al._ [2003] K. Hornberger, S. Uttenthaler, B. Brezger, L. Hackermüller, M. Arndt, and A. Zeilinger, Phys. Rev. Lett. 90, 160401 (2003), arXiv:quant-ph/0303093 [quant-ph] .
  * Joos and Zeh [1985] E. Joos and H. D. Zeh, Z. Phys. B 59, 223 (1985).
  * Gallis and Fleming [1990] M. R. Gallis and G. N. Fleming, Phys. Rev. A 42, 38 (1990).