# Bose-Einstein condensates in microgravity and fundamental tests of gravity

Christian Ufrechta, Albert Rourab and Wolfgang P. Schleicha,b,c,d CONTACT C.
Ufrecht. Email<EMAIL_ADDRESS>aInstitut für Quantenphysik and
Center for Integrated Quantum Science and Technology (IQST), Universität Ulm,
Albert-Einstein-Allee 11, D-89069 Ulm, Germany; bInstitute of Quantum
Technologies, German Aerospace Center (DLR), Söflinger Straße 100, D-89077
Ulm, Germany; cHagler Institute for Advanced Study and Department of Physics
and Astronomy, Institute for Quantum Science and Engineering (IQSE), Texas A&M
University, College Station, Texas 77843-4242, USA; dTexas A&M AgriLife
Research, Texas A&M University, College Station, Texas 77843-4242, USA

###### Abstract

Light-pulse atom interferometers are highly sensitive to inertial and
gravitational effects. As such they are promising candidates for tests of
gravitational physics. In this article the state-of-the-art and proposals for
fundamental tests of gravity are reviewed. They include the measurement of the
gravitational constant $G$, tests of the weak equivalence principle, direct
searches of dark energy and gravitational-wave detection. Particular emphasis
is put on long-time interferometry in microgravity environments accompanied by
an enormous increase of sensitivity. In addition, advantages as well as
disadvantages of Bose-Einstein condensates as atom sources are discussed.

###### keywords:

Tests of gravity; Bose-Einstein condensates; Atom interferometry; High-
precision measurements; Microgravity

##  1 Introduction

The wave nature of matter [1] enables interferometry with massive particles
such as electrons, neutrons, atoms, or molecules. In a matter-wave
interferometer the wave function of a particle is split into two components
that follow different paths, are subsequently redirected and finally
recombined. As in conventional interferometry with light, the difference
between the phases accumulated along the two branches of the interferometer
can be obtained from the interference of the two components, and the quantity
responsible for this phase shift can be inferred.

In contrast to interferometry with light, matter-wave interferometers are far
more sensitive to gravitational effects due to the much longer interrogation
times that are possible, which makes matter-wave interferometers very
promising instruments for fundamental tests of gravity.

This review presents a selection of proposals and experiments for testing
gravitational physics with light-pulse atom interferometers that can not only
surpass classical tests in accuracy, but also constitute promising ways for
probing regions of parameter space inaccessible to classical experiments. Over
the years a number of excellent reviews [2, 3, 4, 5, 6, 7, 8] with different
emphasis and scope have been published. The present article focuses on a non-
technical introduction to the theory and experimental techniques for
fundamental tests of gravity with light-pulse atom interferometry.

The article is organized as follows. In Sec. 2 the basic diffraction
mechanisms used to realize beam splitters and mirrors are introduced. From the
interferometer phase shift, derived in Sec. 3, the high sensitivity to
inertial effects and gravity is subsequently inferred, and the main challenges
as well as possible solutions are investigated. In Sec. 4 various approaches
to long-time atom interferometry are presented. The resulting high sensitivity
is crucial for the fundamental tests of gravity outlined in Sec. 5. In this
respect, the extraordinary coherence properties and low expansion rate of
Bose-Einstein condensates (BECs) make them an ideal atom source for long-time
interferometry. The advantages and disadvantages of their use for atom
interferometry are discussed in Sec. 6. Moreover, key mitigation techniques
for major systematic effects are presented in Sec. 7. Finally, we conclude
with a brief outlook in Sec. 8.

##  2 Beam splitters and mirrors made out of light

Historically, evidence for wave-like behavior of matter was first found in
electron experiments [9] but soon also for molecules [10] and neutrons [11].
Interferometry with the latter allowed to measure the phase shift induced by
the gravitational field of Earth [12, 13] for the first time with matter-wave
interferometers. The next step was to perform interferometry with neutral
atoms, which in contrast to neutrons are much easier to produce and in
contrast to electrons are less sensitive to electromagnetic stray fields.
Initially, experiments were performed with double slits [14] and material
gratings [15], but it was soon realized that the internal structure of atoms
provides excellent means for precise control with laser light. Profiting from
these pioneering experiments and from advance in quantum optics, the beam
splitters and mirrors were replaced by light gratings made out of standing
waves [16, 17, 18] or laser beams driving optical transitions [19, 20].

Alternatively, employing short laser pulses [21] provided more flexibility and
pioneered the field of light-pulse atom interferometry, which will be at the
focus of this review. As shown in Figure 1 a), in a light-pulse atom
interferometer beam splitters and mirrors are typically realized by subjecting
the atoms to counterpropagating laser beams, driving a two-photon transition
via a far-detuned auxiliary state. Intuitively, the process can be understood
as the absorption of a photon associated with a recoil in direction of the
first laser beam and a subsequent recoil in the same direction by stimulated
emission in direction of the second laser. Consequently, the atom gains the
momentum $\hbar k_{\mathrm{eff}}$ with $k_{\mathrm{eff}}=k_{1}+k_{2}$ and the
energy $\hbar\Delta\omega$ with $\Delta\omega=\omega_{1}-\omega_{2}$ where
$k_{1}$, $k_{2}$ and $\omega_{1}$, $\omega_{2}$ are the wave numbers and
frequencies of the laser pair. Therefore, during a pulse at time $t$ that
transfers momentum $\hbar k_{\mathrm{eff}}$ to the atomic wave packet, the
following laser phase is imprinted:

$\phi_{\mathrm{L}}(t)=k_{\mathrm{eff}}z(t)-\Delta\omega\,t-\Delta\varphi,$ (1)

where $z(t)$ is the mean position of the wave packet and the relative phase
$\Delta\varphi=\varphi_{1}-\varphi_{2}$ involves the difference between the
phase terms of the two counter-propagating laser beams that are spatially and
time-independent.

Figure 1: a) The diffraction process is realized by two counterpropagating
laser beams of frequency $\omega_{1}$ and $\omega_{2}$ and the respective wave
numbers $k_{1}$ and $k_{2}$. b) In Bragg diffraction a photon from the blue
laser is absorbed and subsequently emitted by stimulated emission in direction
of the red laser. The atom therefore gains the momentum $\hbar
k_{\mathrm{eff}}=\hbar(k_{1}+k_{2})$. The process is driven via a far-detuned
virtual auxiliary state (black curve) back to the initial state of the atom.
c) In Raman diffraction the frequencies of the lasers are chosen so that the
atom is excited into a higher-energy state. Therefore, in contrast to Bragg
diffraction in Raman diffraction the momentum $\hbar k_{\mathrm{eff}}$ is
transferred together with a change of the internal state.

There are two different diffraction mechanisms commonly employed: Bragg
diffraction [22, 23, 16, 24] and Raman diffraction [21, 19]. See Ref. [25] for
a comparison in terms of efficiency. In Bragg diffraction (Figure 1 b)) the
atom returns via stimulated emission back to the original internal state while
in Raman diffraction (Figure 1 c)) a different internal state is selected
(typically another hyperfine state). The frequency difference required between
the laser pair depends on which of these two processes is driven and is
determined by energy and momentum conservation. By choosing the interrogation
time and laser intensity appropriately, beam splitters and mirrors can be
realized. A $\pi/2$ pulse, or beam splitter, sets the initial wave packet with
mean momentum $p_{0}$ into an equal superposition of two components, one with
mean momentum $p_{0}$ and another one with $p_{0}+\hbar k_{\mathrm{eff}}$. In
contrast, a $\pi$ pulse, or mirror, inverses the momenta of the two
components. In case of Raman diffraction each momentum transfer is
additionally associated with a change of the internal state.

A light-pulse interferometer consists of an alternating sequence of beam
splitters, mirrors and free evolution in between. One common type of
interferometer scheme used for many applications due to its simplicity is the
Mach-Zehnder (MZ) interferometer displayed in Figure 2. In an MZ
interferometer a $\pi/2$ pulse coherently splits the initial wave function
into two components which subsequently spatially separate due to their
momentum difference $\hbar k_{\mathrm{eff}}$, constituting the two branches of
the interferometer. After time $T$ the two components are redirected by a
$\pi$ pulse and finally interfered by a second $\pi/2$ pulse after another
time $T$ has passed. Subsequently, the phase shift between the two branches is
read out from the fraction of atoms detected at each exit port of the
interferometer through fluorescence or absorption imaging.

Figure 2: Space-time diagram of a Mach-Zehnder (MZ) interferometer. An MZ
interferometer consists of two $\pi/2$-pulse beam splitters and a $\pi$ pulse
(mirror) separated by the free interrrogation time $T$. Initially, the wave
function of the atoms is split into two components with momentum transfer
$\hbar k_{\mathrm{eff}}$ on one of them. Due to the momentum difference, the
two components subsequently spatially separate, constituting the two branches
of the interferometer. At the final $\pi/2$ pulse the two components are
interfered and the phase shift between the two branches can be inferred by
measuring the number of atoms leaving each exit port. In a gravitational field
with acceleration $g$ the atoms are in free fall between the laser pulses, the
trajectories are shown by the blue solid parabolas and are compared to the
grey dashed lines corresponding to zero gravity. As shown in Sec. 3.1, the
phase shift due to the gravitational field in a MZ gravimeter is
$\delta\phi=-k_{\mathrm{eff}}gT^{2}$.

To relate the measurement result to a theoretical model, the interferometer
phase shift can be calculated within several levels of approximation. In the
first step, the far-detuned auxiliary state can be adiabatically eliminated
[26, 27, 28], leading to a reduced description in which the atom couples to an
effective light field with wave number $k_{\mathrm{eff}}$ and frequency
$\Delta\omega$. Next, the center-of-mass motion of the atom and internal
degrees of freedom corresponding to the two remaining internal states in case
of Raman diffraction are separated. In contrast, for Bragg diffraction well
separated momentum classes are introduced [29] centered around multiples of
$\hbar k_{\mathrm{eff}}$ which are coupled only during the lasers are on.
Assuming near-resonance diffraction and negligible occupation of higher
momentum classes in case of Bragg diffraction (two-level approximation),
further reduction to a path-dependent description is possible [30], where the
probability for an atom to leave at the first exit port is given by

$P=\frac{1}{2}\big{(}1+\mathfrak{Re}\big{\langle}\hat{U}^{(l)\dagger}\hat{U}^{(u)}\big{\rangle}\big{)}=\frac{1}{2}(1+C\cos{\delta\phi})\,.$
(2)

In this expression the expectation value with respect to the initial wave
function defines the phase shift $\delta\phi$ and contrast $C$ of the
interferometer. The center-of-mass time-evolution operators
$\hat{U}^{(\alpha)}$ including the momentum transfer and laser-phase imprint
generate the motion along the upper and lower branch ($\alpha=u,l$).
Similarly, the probability for the second exit port is given by an analogous
expression but with the opposite sign for the term proportional to
$\cos\delta\phi$. Therefore, by measuring the fraction of atoms detected at
each exit port, one can infer the phase shift $\delta\phi$.

Many different methods have been developed to calculate phase shift and
contrast at this stage, each of them with particular advantages and
disadvantages regarding range of application and computational ease.
Noteworthy are path-integral methods [31, 32], calculations in phase space
[33, 6] and representation-free descriptions on the operator level [34, 30, 4]
in terms of comoving frames [7, 35, 36, 37] or perturbation theory [37, 38] as
well as a relativistic description in curved spacetime [39, 40]. In the
following sections the description given so far is extended and the details
needed to understand atom-interferometric fundamental test of gravity are
elaborated.

##  3 Atom interferometry for precision gravitational measurements

Since the first application of light-pulse atom interferometers as gravity
sensor in 1991 [41] enormous effort has been devoted to increasing their
sensitivity and characterizing systematic effects. Based on these
improvements, today’s light-pulse atom interferometers are highly sensitive
inertial sensors. Indeed, absolute gravimeters [42, 43, 44, 45, 46, 47, 48]
used to measure the local gravitational acceleration have demonstrated
sensitivities of $4.2\times
10^{\text{-}8}\,\mathrm{m}\,\mathrm{s}^{\text{-}2}\,\mathrm{Hz}^{\text{-}1/2}$
[46]. In gyroscopy [49, 50, 51, 52, 53, 54, 55, 56] a sensitivity of $3\times
10^{\text{-}8}\mathrm{rad}\,\mathrm{s}^{\text{-}1}\mathrm{Hz}^{\text{-}1/2}$
has been reported [55]. These sensitivities are comparable to those of the
best classical gravimeters [57] and gyroscopes [58]. The decisive benefit of
atom interferometers, however, is their high stability due to low drift rates.
Indeed in gyroscopy a record stability of $3\times
10^{\text{-}10}\,\mathrm{rad}\,\mathrm{s}^{\text{-}1}$ [55] was reached. For
applications in navigation, geodesy [59, 60], seismometry and other
geophysical studies, compact and mobile devices have been developed [61, 62,
63] designed to leave the laboratories. In the following, in Sec. 3.1 the
phase shifts measured by a gravimeter and a gyroscope is motivated for an
idealized experiment. In Sec. 3.2 challenges in more realistic situations and
possible solutions are discussed.

###  3.1 Inertial sensors based on atom interferometry

In this section the principles underlying the operation of light-pulse atom
interferometers as gravimeters and gyroscopes are illustrated. These devices
measure the acceleration and rotation rate of the frame defined by the laser
beams. They rely on the fact that neutral atoms in magnetically insensitive
states are excellent inertial references and on the interferometer phase shift
being directly related to the relative acceleration between the atoms and the
laser wave fronts. The latter point can be easily understood by considering a
freely falling frame, where the atoms move with constant velocity between the
laser kicks. In particular, for a linear gravitational potential the phase
difference between upper and lower branch can be attributed to the laser phase
imprinted on the wave packets at each laser pulse. Consequently, by comparing
to Figure 2, the phase shift for an MZ interferometer reads
$\delta\phi=\phi_{\mathrm{L}}(0)-2\phi_{\mathrm{L}}(T)+\phi_{\mathrm{L}}(2T)$.
In the freely falling frame the frequency $\Delta\omega$ appearing in Equation
(1) for the laser phase is Doppler shifted so that
$\phi_{\mathrm{L}}(t)=k_{\mathrm{eff}}z(t)-\Delta\omega\,t+\bm{k}_{\mathrm{eff}}\cdot\bm{a}\,t^{2}/2-\Delta\varphi$
where $\bm{a}$ is the acceleration of the atoms with respect to the reference
frame of the laser. For a MZ interferometer the spatial part of the laser
phases cancels out and so does $\Delta\varphi$ provided that the relative
phase between the counter-propagating laser beams is properly stabilized
during the whole interferometer sequence. Thus, the phase shift becomes
$\delta\phi=\bm{k}_{\mathrm{eff}}\cdot\bm{a}\,T^{2}$. As the atoms accelerate
relative to the laser wave fronts, they quickly leave resonance which can be
compensated in an accelerometer by dynamically modulating the effective laser
frequency with a chirp rate $\alpha$, so that
$\Delta\omega\,t\rightarrow\Delta\omega\,t-\alpha t^{2}/2$. At resonance
$\alpha=-\bm{k}_{\mathrm{eff}}\cdot\bm{a}$ and the interferometer phase shift
vanishes. However, due to Heisenberg’s time-energy uncertainty relation and
the finite pulse duration, this resonance has a finite width, so that beam
splitters and mirrors are still possible close to the resonance condition. The
acceleration can then be read out from the phase shift

$\delta\phi=(\bm{k}_{\mathrm{eff}}\cdot\bm{a}+\alpha)T^{2}$ (3)

by scanning over $\alpha$ at constant $T$ in repeated measurements. In a
linear gravitational field an absolute gravimeter measures the local
gravitational acceleration $\bm{a}=\bm{g}$. Note that Equation (3) holds in
the limit of vanishing pulse duration and small corrections areise when taking
into account their finite duration [64, 65].

The measurement of rotations can be understood as follows. If the experimental
set-up, including the laser beams, rotates with an angular velocity
$\bm{\Omega}$, the atoms will experience in that frame a Coriolis acceleration
$\bm{a}=2(\bm{v}_{0}\times\bm{\Omega})$, where $\bm{v}_{0}$ is the initial
velocity of the atoms. This acceleration gives rise to the following Sagnac
phase shift:

$\delta\phi=2\bm{k}_{\mathrm{eff}}\cdot\left(\bm{v}_{0}\times\bm{\Omega}\right)T^{2},$
(4)

which is obtained by substituting the Coriolis acceleration into Equation (3).
When both linear acceleration and rotations are present, there are further
phase-shift contributions coupling $\bm{a}$ and $\bm{\Omega}$ [4].

###  3.2 Challenges and possible solutions

In the previous section the origin of the phase shift in a measurement of
inertial effects was explained in an ideal experiment. Of course in reality
the situation is less idealized and several noise sources limit the precision.
In this subsection laser phase noise, impact of optical elements and vibration
noise is briefly discussed.

As discussed in the previous section each laser pulse imprints the phase
$-\Delta\omega t$ on the wave packet. Consequently, random fluctuations of the
laser phase directly translate into the interferometer phase. By phase locking
the lasers, their frequency difference can be controlled by referencing to an
external stable frequency source [66, 43, 21].

Additional noise from collimators, beam shapers, optical fibers etc. during
preparation of the beams can be strongly suppressed by feeding both beams into
the vacuum chamber via the same optical components and retroreflecting them on
a mirror [43]. The polarization of the two beams is rotated by $\pi/2$ by
passing two times through a quarter-wave plate in front of the mirror.
Choosing the appropriate initial polarization, only pairs of beams with
different laser frequency interact and further undesired scattering processes
are excluded. In a gravimeter configuration only one pair of beams drives
transitions as the other is strongly Doppler detuned due to the free fall of
the atoms. In zero-gravity environment, in contrast, both laser pairs interact
with the atoms, resulting in Double diffraction [67, 29, 68, 69].

The remaining noise from vibrations severely limits state-of-the-art absolute
gravimetry. While vibrations of the optical components and the laser system
are rejected when the laser beams travel together to the injection point,
parasitic vibrations of the mirror constitute a major source of noise (which
is not rejected since only one beam is reflected off the mirror). The distance
travelled by this beam varies between different pulses due to the vibrational
motion of the mirror. In the quasistationary approximation [70] this shifts
the laser-phase contribution $\Delta\varphi_{j}$ for each pulse by
$-k_{\mathrm{eff}}z_{\mathrm{m}}(t_{j})$, where $z_{\mathrm{m}}(t_{j})$ is the
position of the mirror at the time $t_{j}$ of the $j$th laser pulse, and
results in the following additional contribution to the total phase-shift:

$\delta\phi_{\text{vib}}=-k_{\mathrm{eff}}[z_{\mathrm{m}}(0)-2z_{\mathrm{m}}(T)+z_{\mathrm{m}}(2T)]\,.$
(5)

The effect of vibrations can be mitigated with the help of vibration isolation
platforms [71, 72] in combination with additional active feedback control
[73]. The residual effect of vibrations imposes the current limit on the
sensitive for absolute measurements of gravity.

Instead, when two gravity measurements are performed simultaneously with two
interferometers and the laser beams are reflected off the same mirror, the
phase shifts from vibrations cancel in the differential phase shift. This
suppression in a differential measurement can be exploited for fundamental
tests of gravity. Here, two experimental arrangements suggest themselves. (i)
In gradiometry the local gravitational acceleration is measured at two
different locations allowing the detection of spatial variations of gravity
with applications for the measurement of Newton’s gravitational constant $G$
as discussed in Sec. 5.1. (ii) In tests of the universality of free fall,
discussed in Sec. 5.2, the two interferometers are superimposed and the
differential acceleration of two different atomic species is measured.

Before the experimental setups are explained in more detail, possible
strategies to increase the sensitivity of atom interferometers are illustrated
in the next section.

##  4 Long-time atom interferometry and associated challenges

In a measurement of the local gravitational acceleration $g$ with a gravimeter
the noise in the value of $g$ follows from Equation (3) by error propagation
as

$\Delta g=\frac{\Delta\phi}{k_{\mathrm{eff}}T^{2}}$ (6)

if $\bm{k}_{\mathrm{eff}}$ orthogonal to $\bm{a}=\bm{g}$ and assuming the
values of $k_{\mathrm{eff}}$ and $T$ to be known. The phase-shift uncertainty
$\Delta\phi$ consists of two sources: (i) Uncontrollable phase shifts, for
instance from laser phase noise, vibration noise etc. and (ii) the uncertainty
by which the phase shift itself can be extracted from the interferometer
signal. For uncorrelated particles the latter is limited by shot noise which
is given by $1/\sqrt{N}$ where $N$ is the number of particles per shot. This
noise term originates from the fluctuation of $\sqrt{N}$ particles over the
two exit ports in case of independent particles. For subsequent measurements
uncorrelated random fluctuations with zero bias additionally decrease with
$1/\sqrt{n}$ where $n$ is the number of shots.

The inverse scaling with $k_{\mathrm{eff}}T^{2}$ in Equation (6) suggests two
strategies to decrease $\Delta g$: (i) Large-momentum transfer techniques
(LMT) resulting in large effective wave vectors $\bm{k}_{\mathrm{eff}}$ and
(ii) increasing the interrogation time $T$. With the former strategy
considerable progress has been achieved over recent years with higher-order
Bragg and Raman diffraction [74, 75], sequential Bragg diffraction [76, 77],
Bloch oscillations [78, 79, 80] and combinations [81] which has resulted in an
enormous increase of the space-time area enclosed by the branches of the
interferometer [82, 83, 84, 76]. However, spurious diffraction phases [85] and
parasitic interferometers [86] arise which are hard to control.

The second strategy is based on increasing the interrogation time $T$ which
might even be more effective due to the quadratic scaling of $T$ in Equation
(6). On Earth the free-fall time of the atoms is limited by the size of the
apparatus. Still, total interferometer times of up to 2.3s have been realized
in the Stanford 10-meter tower [87] by launching the atoms initially.

This limitation can be overcome by relocating the setup to microgravity
environments which will be briefly discussed in the following.

### Microgravity platforms

Experiments in several different microgravity facilities have been performed.
In parabola flights 20s of zero $g$ were realized [60, 88] and a first proof-
of-principle test of the universality of free fall in microgravity was
performed in which the influence of the strong vibrational background on a
plane could be rejected to some extent. Much quieter environments are provided
in droptower facilities as for example in Bremen [89] where experiment times
of about 5s in the drop mode and almost 10s in the catapult mode are available
[90, 91]. However, repetition rates are low in the droptower. Much higher
repetition rates and free-fall times of about $4$s can be reached in the newly
commissioned Einstein Elevator in Hanover [92]. The proof-of-principle
experiment in a sounding rocket [93, 94] offers valuable insight into
challenges associated with future experiments in space such as on the
International Space Station (ISS) [95, 96, 97] or even in dedicated satellite
missions [98, 99] with in principle unlimited interferometer time.

The combination of differential measurements discussed in Sec. 3.2 which large
common-mode suppression and experiments in microgravity with large
interferometer times allows to reach high sensitivities for experimental tests
of fundamental properties of gravity which will be discussed in the following
section.

Figure 3:  The drop tower at the Center for Applied Space Technology and
Microgravity (ZARM) in Bremen provides a platform for high-quality
weightlessness of almost 10 seconds inside an evacuated tube. In the 2017
sounding-rocket mission MAIUS-I 6 min of in-space flight have been realized
and the production and manipulation of BECs as well as basic interferometric
principles have been demonstrated. The International Space Station in
principle provides unlimited interferometer times in microgravity. In NASA’s
Cold Atom Laboratory (CAL) first interferometer experiments have been
performed. Based on the experiences from CAL, the Bose-Einstein Condensate and
Cold Atom Laboratory (BECCAL) will be installed in the coming years and
operated by a collaboration between NASA and DLR.

##  5 Fundamental gravitational measurements

In the previous sections the basic ideas of light-pulse atom interferometry
were introduced and challenges together with possible solutions were
illustrated. Building on these principles, in this section measurements and
proposals for probing fundamental gravitational physics are illustrated with
focus placed on experiments particularly suited for operation in microgravity
and in a differential mode. This includes measurements of Newton’s constant
$G$, tests of the weak equivalence principle, dark energy and gravitational-
wave detection.

Other tests of fundamental physics not discussed in this review are
measurements of the fine structure constant with unprecedented accuracy [100,
101, 102, 103] as well as measurements of special-relativistic time dilation
[104, 105] and the gravitational redshift [39, 106, 107, 108] using quantum-
clock interferometry [109, 110].

###  5.1 Measurements of Newton’s gravitational constant

Newton’s constant $G$ is one of the least accurately determined constants of
nature but even at the present level of accuracy conflicting values have been
reported [111]. While classical measurements have been performed mainly with
torsion balances or pendulums [111] and their working principle is very
similar to the first measurement of $G$ performed by Cavendish more than two
centuries ago, an atom-interferometric measurement provides a complementary
experiment of completely different type which might help reveal unaccounted
systematic effects and resolve the dissatisfying inconsistencies. The
important role $G$ takes in astrophysical and geophysical calculations makes a
more accurate value of the constant an important goal.

The gravitational constant $G$ can be determined by measuring the
gravitational field generated by a well-characterized source mass, which can
in turn be obtained from the acceleration experienced by a test mass close to
the source mass. In particular, one can make use of atom interferometry to
measure the acceleration of the atoms caused by a nearby macroscopic mass.
However, a precise measurement is very challenging because the gravitational
force is rather weak and experiments cannot be shielded from the gravitational
environment. This means that one needs to separate the small signal of
interest from the much larger acceleration associated with Earth’s
gravitational field, which can be achieved by performing a differential
measurement of two spatially separated atom interferometers. In that way, the
uniform part of Earth’s gravity field cancels out. Besides the inhomogeneous
gravitational field generated by the source mass, Earth’s gravity gradient
will also contribute to the differential measurement, but the two
contributions can be separated by repeating the measurement for different
positions of the source mass.

Figure 4: a) Gradiometry configuration. Two gravimeters (see Sec. 3) separated
by a distance $\Delta z_{0}$ are simultaneously interrogated with the same
laser pair. In contrast to absolute gravimetry, where the absolute value of
the gravitational field is detected, in this gradiometry setup spatial
variations over the distance $\Delta z_{0}$ can be extracted from the
differential phase shift. Note that the branch separation is not to scale in
the figure. Indeed, as an example, for an interferometer with 87Rb atoms,
$T=1$s and a momentum transfer $\hbar k_{\mathrm{eff}}$ corresponding to twice
the single-photon recoil, the branch separation would be of the order of one
centimeter while $\Delta z_{0}$ may typically be of the order of a meter. b)
Test of the weak equivalence principle. Two interferometers with different
atomic species e.g. 87Rb and 41K are superimposed. The relative acceleration
between the atoms can be obtained from the differential phase shift with
nonzero result in case the WEP is violated. Note the separation between the
potassium and rubidium trajectories due to their different mass and different
laser frequencies required to drive the respective internal transitions. The
trajectories in the figure are to scale for the parameters in the space-borne
proposal of Ref. [98] with $T=20$s. The parameters lead to a maximal branch
separation of about $1$m for 41K and $0.5$m for the 87Rb atoms.

More specifically, a differential measurement of this kind can be achieved in
a gradiometry configuration where two interferometers are placed at different
locations [112, 113, 114, 115, 116] as shown in Figure 4a. Both
interferometers are interrogated with the same laser beam retroreflected off a
common mirror. As a consequence, vibration and laser phase noise, which would
otherwise mask the signal of interest, are common to both interferometers and
largely cancel out in the differential measurement. The gravitational
potential on Earth’s surface can be decomposed into the dominant linear part
$mgz$ with nonlinear perturbations on top. Since the acceleration exerted on a
test particle by the former is constant over the whole gradiometry setup, the
differential phase shift between the interferometers only depends on the
latter, the spatial variations of the gravitational field. This phase-shift
difference can be extracted by plotting the signal of the interferometers
against each other. As a function of the chirp rate of the Raman lasers the
signals parametrize an ellipse [117] from which the differential phase shift
can be extracted by a least-square fit even in noisy environments. In fact,
due to vibration noise, the phase-shift contribution
$\delta\phi_{\mathrm{vib}}$ given by Equation (5) will typically take values
randomly distributed in the whole range $(-\pi,\pi)$ for this kind of
experiments, giving rise to data points randomly distributed along the ellipse
even for a fixed chirp rate.

The practicability of the experimental setup explained above for a measurement
of Newton’s constant was shown in proof-of-principle experiments [118, 119,
120]. In the first high-precision measurement of $G$ using atom
interferometry, a relative uncertainty of 150ppm was reported [121] deviating
1.5 combined standard deviations from the CODATA recommended value [122] at
that time but also was the first value measured by a new type of experiment.
In Ref. [121] the atoms of both interfermeters are launched simultaneously
close to two annular test masses at variable position made out of 516kg of
tungsten. The limiting systematics of the experiment were identified as
mounting errors and inhomogeneities in the density of the test mass as well as
uncertainties in the initial position of the atoms and the large cloud size
with further improvement expected in the future [123]. Dependence of the phase
shift on initial kinematics can be strongly suppressed by the mitigation
techniques explained in Sec. 7.1 and Sec. 7.2. The spread of the wave function
can be drastically reduced in both position and momentum space by employing
BECs as atom sources as explained in Sec. 6.

In a similar setup Yukawa-type fifth forces were constrained [124] with great
potential improvement by local gravity measurements using Bloch oscillations
in optical lattices close to a source mass [125, 126, 127].

###  5.2 Tests of the weak equivalence principle

The weak equivalence principle, also referred to as universality of free fall
(UFF), is an essential ingredient for the construction of general relativity
(GR). It states that the acceleration experienced by a test particle in a
gravitational field is independent of the particles’s composition as long as
it is kept together only by non-gravitational forces [128]. Being a necessary
condition for GR, any violation of the weak equivalence principle would prove
that GR is not the ultimate gravitational theory but needs to be modified in
some respect. There are various possible origins of violations for example
from modified gravity-matter coupling or standard-model extensions [129, 130].

Comparing the accelerations $g_{1}$ and $g_{2}$ experienced by two test
particles, a natural measure of the size of possible violations is given by
the Eötvös parameter

$\eta_{1,2}=2\frac{g_{1}-g_{2}}{g_{1}+g_{2}}$ (7)

which vanishes if no violation is present. UFF tests aim at confining the
value of $\eta_{1,2}$ and by that allow to set bounds on parameters of
alternative gravitational theories. Vice versa once the rough form of
couplings in a modified theory is evaluated, the best choice of test masses
that maximize the Eötvös parameter can be predicted [131].

By comparing the acceleration of macroscopic test masses with torsion balances
on Earth, the violation parameter was confined to the $10^{-13}$ level [132,
133, 134] and recently $\eta=2\times 10^{-14}$ consistent with no violation of
the UFF was reported by the MICROSCOPE mission [135, 136] in a satellite-based
measurement in an orbit around Earth.

The advent of atom interferometry has enabled quantum tests of the weak
equivalence principle that contribute to broadening the range of possible
materials employed as test masses. Furthermore, in contrast to macroscopic
objects, all atoms of the same isotope and in the same internal state are
identical and are therefore less prone to certain kinds of systematic errors
associated with macroscopic test masses in classical tests.

Using effective-absorption gratings of light, the first atom interferometric
test was performed in 2004, which comparied the acceleration of 87Rb and 85Rb
isotopes [137]. A few years later approximately the same accuracy was reached
in the first light-pulse atom interferometer based test [138], later pushed
further by another order of magnitude [139]. In addition, tests for probing
gravity-spin coupling have been performed [140]. The first test with different
atomic species validated the weak equivalence principle at the $10^{-7}$ level
by comparing the accelerations of 87Rb and 39K atoms [141]. At about the same
time, the acceleration of two Sr isotopes was compared by measuring the Bloch
frequency in an optical lattice with the result $\eta<10^{-7}$ [142].
Recently, the UFF was verified at the $10^{-12}$ [143] level for rubidium
isotopes using light-pulse atom interferometers, therefore improving on
earlier results by more than four orders of magnitude.

In a WEP test, as illustrated in Figure 4 b), two interferometers involving
different atomic species are operated simultaneously. However, in general not
only the internal transitions used for the diffraction but also the mass of
the two atomic species are different, leading to different recoil velocities.
The resulting separation between the trajectories of the two species can range
from few millimeters in experiments on ground with different isotopes of the
same chemical element up to about half a meter in proposals for satellite
missions with different elements.

As explained in Sec. 3.1, the phase shift accumulated by species $j$ is
$\delta\phi_{j}=k_{\mathrm{eff}}^{(j)}\,g_{j}T^{2}$+$\Delta\phi_{j}$. The
first term originates from the linear gravitational potential where the
species-dependent acceleration $g_{j}$ models possible violations of the UFF.
Any additional phase-shift contribution is comprised in $\Delta\phi_{j}$. The
differential effective acceleration

$\bar{g}_{1}-\bar{g}_{2}=\frac{\delta\phi_{1}}{k_{\mathrm{eff}}^{(1)}T^{2}}-\frac{\delta\phi_{2}}{k_{\mathrm{eff}}^{(2)}T^{2}}=g_{1}-g_{2}+\left(\frac{\Delta\phi_{1}}{k_{\mathrm{eff}}^{(1)}T^{2}}-\frac{\Delta\phi_{2}}{k_{\mathrm{eff}}^{(2)}T^{2}}\right)$
(8)

is proportional to the Eötvös parameter defined in Equation (7) plus
additional terms dependent on the spurious phase shifts $\Delta\phi_{j}$. The
theoretical accuracy of the WEP test is then determined by the difference
inside the brackets. The differential nature of the scheme leads to
cancellation of many systematic effects like leading-order phase shifts from
Coriolis effects and gravity gradients. Also note that the position of the
mirror is common to both species. Consequently, phase shifts from vibration
noise which are proportional to $k_{\mathrm{eff}}^{(j)}$ as shown in Equation
(5), largely cancel in Equation (8) .

Other systematic effects that act differently on both species and hence do not
cancel in the differential phase shift need to be well controlled such as
magnetic field gradients (second-order Zeeman effect [144]) and temperature
gradients in the setup (black-body radiation [145, 146]). The most precise
test performed so far with atom interferometry [143] is currently limited by
two systematic effects. The first is due to kinematic differences of the
initial atomic clouds. Mitigation strategies to reduce the impact of this
effect on the phase shift are discussed in detail in Sec. 7 and already are a
major ingredient in the experimental setup of Ref. [143]. But even stronger
mitigation will be required for tests beyond the current accuracy. The second
uncertainty is due to the shift of the atomic levels during the lasers are
turned on (AC-Stark shift). While in case of Bragg diffraction this effect
ideally cancels separately for each species, laser intensity fluctuations over
the separation of the branches induce a non-negligible phase shift associated
with pulses that are resonant with only one of the two arms, which arise when
employing multi-pulse sequences for larger momentum transfer. Moreover, the
inhomogeneous gravitational potentials induced by local mass sources
surrounding the experiment lead to nonvanishing differential accelerations on
the two species as they follow different trajectories. This remaining bias in
Equation (8) might limit future WEP tests on ground if the gravitational
environment is not precisely characterized.

Ambitious proposals for further improving the current limits have been put
forward for experiments on ground [147, 148] and in space [99, 98, 149].
Experiments performed in satellites orbiting Earth [98] can in particular
profit from the quiet environment and a further mitigation technique, also
discussed in Sec. 7.

###  5.3 Dark energy

Cosmological observations have revealed that about 70% of the energy density
of the cosmos corresponds to _dark energy_ , a form of energy that cannot be
accounted for by the Standard Model of particle physics and drives the current
accelerated expansion of the universe [150, 151]. Its defining feature is the
large negative pressure, comparable in absolute value to its energy density,
but its underlying nature is still an open question. A more detailed
investigation of its properties is being pursued with more refined
cosmological observations together with efforts towards direct detection with
high-precision measurements at low energies.

An important class of dark-energy theories that can naturally address the so-
called cosmic coincidence problem are _quintessence_ models [152, 153, 154]
involving a scalar field with particular kinds of potentials, such as inverse
power laws, that imply very light masses at the present cosmological epoch.
However, since they are expected to couple to regular matter with a strength
comparable to the gravitational interaction, they would mediate a long-range
interaction acting as a “fifth force” that would have been detected in tests
of the equivalence principle and solar system measurements. Interestingly,
_chameleon_ models [155] can evade detection through a screening mechanism:
whenever the matter density is not too small, its coupling to the chameleon
field leads to a larger effective mass of the field and the interaction that
it mediates becomes a short range one. In this way, only a thin layer on the
surface of macroscopic objects would source the long-range interaction and
even air’s density would be enough to screen it. Similar conclusions apply
also to _symmetron_ fields [156].

It has been argued that the sensitivity to chameleon and symmetron fields
could be enhanced by employing microscopic masses because in that case the
interaction with the test mass would not be screened (although the screening
mechanism would still apply to the source mass) [157]. Atom interferometry is
therefore a natural candidate for this kind of measurements and since the
experiments are performed in a vacuum chamber, any screening due to air is
also absent [157, 158, 159]. Indeed, ground-based experiments with atom
interferometers have substantially improved the bounds on such models [160,
161, 162], but they are limited by the fact that the atoms are freely falling.

Under microgravity conditions, on the other hand, it is possible for the atoms
to spend long times close to the macroscopic source mass, which increases the
atom interferometric signal associated with chameleon and symmetron fields.
Nevertheless, it is crucial to mitigate the noise sources and systematic
effects that also grow with time and can become very important for long
interferometer times. In this respect, a particularly interesting scheme,
displayed in Figure 5, was proposed in Ref. [163]. The source mass is a long
cylinder with a series of equidistant disks perpendicular to the cylinder
walls and separated by a distance $l$. Each disk has a small hole at the
center, so that the cylinder’s symmetry axis, along which the atoms will move,
goes through all of them as depicted in Figure 5a. The scheme involves a
differential measurement of two identical atom interferometers separated along
the symmetry axis by a semi-integer multiple of $l$ and interrogated by common
laser beams propagating along this axis. The atoms start at rest in both
interferometers and a sequence of laser pulses is applied at various times as
shown in Figure 5b.

Figure 5: Scheme for dark-energy detection with atom interferometry. The left
panel displays a section of the cylindrical source mass with equidistant disks
separated by a distance $l$ as well as a schematic representation of the
motion of the atoms in the two arms of the interferometer (blue and orange)
along the symmetry axis that goes through the holes at the center of each
disk. The right panel is a spacetime diagram of the wave-packet trajectories
along the two interferometer arms. A second identical interferometer separated
by a semi-integer of $l$ is simultaneously operated and interrogated by the
same laser pulses. The signal of interest is obtained from a differential
phase-shift measurement.

In addition to the initial and final beam-splitter pulses, all the
intermediate mirror pulses guarantee that the maximal spatial separation
between the interferometer arms, which is proportional to $T$, remains
relatively small even for long total interferometer times: one simply needs to
consider a large number $n$ of intermediate pulses, which leads to a total
time $2n\,T$. The small arm separation and the periodic configuration of the
arm trajectories and laser pulses minimize the impact of laser wave-front
distortions and inhomogeneities of the magnetic field (which give rise to
spurious forces through the quadratic Zeeman effect) because their effects are
then similar on both interferometer arms.

The loss of contrast and systematic effects caused by rotations and gravity
gradients, discussed in Sec. 7, can be naturally overcome with the mitigation
techniques presented there. Indeed, by considering interferometers with an
even number of loops in the spacetime diagram of Figure 5b (corresponding to
an even number of laser pulses), the effects of rotations with a constant
rotation rate cancel out, as pointed out in Sec. 7.1. Similarly, the effects
of gravity gradients can be compensated by adjusting the momentum transfer of
all the intermediate pulses according to Equation (12), as explained in Sec.
7.2. Furthermore, the effects of laser phase noise, vibration noise or any
residual accelerations of the experimental set-up, will cancel out in the
differential measurement of the two interferometers shown in Figure 5b, which
are operated simultaneously and interrogated by common laser pulses.

To calculate the phase shift for this kind of interferometers, one can proceed
as follows. First, one starts by taking into account that the phase
accumulated along each interferometer arm is given by the classical action
evaluated along the mean trajectory of the atomic wave packet [31, 32, 7, 39]
and including the effect of the laser pulses in the potential [34]. The phase
shift is thus given by $S[\bm{x}^{(u)}(t)]/\hbar-S[\bm{x}^{(l)}(t)]/\hbar$,
where $\bm{x}^{(u)}(t)$ and $\bm{x}^{(l)}(t)$ correspond to the mean
trajectories depicted in Figure 5b (orange and blue lines respectively). Next,
one separates the contribution of the potential $V_{\mathrm{src}}$ generated
by the source mass, which includes the gravitational interaction and the
chameleon or symmetron field:
$S[\bm{x}^{(\alpha)}(t)]=S_{0}[\bm{x}^{(\alpha)}(t)]+\int
dt\,V_{\mathrm{src}}\big{(}\bm{x}^{(\alpha)}(t)\big{)}$. Since
$V_{\mathrm{src}}$ is rather small, it can be treated perturbatively and the
trajectories $\bm{x}^{(\alpha)}(t)$, which are classical solutions of the full
action, can be expanded as
$\bm{x}^{(\alpha)}(t)=\bm{x}^{(\alpha)}_{(0)}(t)+\bm{x}^{(\alpha)}_{(1)}(t)+\cdots$,
where $\bm{x}^{(\alpha)}_{(0)}(t)$ is the solution of the unperturbed action
$S_{0}$. As shown for example in Appendix A of Ref. [164], when evaluating the
full action to first order, it is actually sufficient to employ the
unperturbed solution $\bm{x}^{(\alpha)}_{(0)}(t)$ because the contributions of
the perturbations $\bm{x}^{(\alpha)}_{(1)}(t)$ cancel out. (The same
conclusions can be reached with operator-based methods [38].)

The unperturbed actions for the two arms are identical except for the laser
phases. Hence, their contribution to the phase shift reduces to
$S_{0}\big{[}\bm{x}^{(u)}_{(0)}(t)\big{]}/\hbar-
S_{0}\big{[}\bm{x}^{(l)}_{(0)}(t)\big{]}/\hbar=\delta\phi_{\mathrm{L}}$ and
the total phase shift is given by

$\delta\phi=\delta\phi_{\mathrm{L}}+\int dt\
V_{\mathrm{src}}\left(\bm{x}^{(u)}_{(0)}(t)\right)-\int dt\
V_{\mathrm{src}}\left(\bm{x}^{(l)}_{(0)}(t)\right).$ (9)

When considering the differential phase shift between the two interferometers
displayed in Figure 5b, the term $\delta\phi_{\mathrm{L}}$, associated with
the laser phases, cancels out and the same would happen if we had considered
small residual accelerations or vibrations. Therefore, one is left with the
last two terms on the right-hand side of Equation (9) multiplied by two
because these two terms give the same result for the two interferometers but
with the upper and lower paths exchanged.

Considering the potential $V_{\mathrm{src}}(z)$ along the symmetry axis and
taking the origin a the center of a disk, the differential phase shift can be
well approximated by
$\delta\phi_{2}-\delta\phi_{1}=n\,T\,\big{(}V_{\mathrm{src}}(l/2)-V_{\mathrm{src}}(0)\big{)}$,
where finite-size effects of the source mass have been neglected. This result
reflects the fact that the atoms in one interferometer arm spend half the time
at a maximum of the potential (closest to the source), whereas in the other
arm they spend half of the time at a minimum. In this way, one can measure the
potential $V_{\mathrm{src}}(z)$, whose $z$ dependence can be determined by
repeating the measurements keeping the initial position of one of the
interferometers fixed but scanning the initial position of the other. This
potential includes the contributions of both the chameleon (or symmetron) and
the gravitational field. In order to extract the former, one needs to subtract
the gravitational potential, which can be accurately modeled for a well-
characterized density distribution of the source mass. Additionally, one can
employ more sophisticated source designs that suppress the gravitational
contribution to the differential phase shift compared to that of the chameleon
(or symmetron) field [163].

Such an experiment can be naturally performed in a space mission, but given
its immunity to rotations, gravity gradients and residual accelerations, it
could also be carried out in Hannover’s Einstein Elevator facility [92], which
enables a fairly high repetition rate (up to 300 launches per day). The main
limitation in this case would come from the maximum microgravity time of
$4\,\mathrm{s}$ for each launch.

###  5.4 Gravitational waves

Gravitational waves are ripples in spacetime that propagate at the speed of
light and are generated by rapidly changing mass distributions such as pairs
of astrophysical objects orbiting at large velocities. The first confirmed
detection by the LIGO observatory half a decade ago [165] marked the birth of
gravitational-wave astronomy and opened a whole new window into the universe.
Besides LIGO [166] and Virgo [167], additional ground-based gravitational-wave
detectors are being pursued in Japan, Australia and India. Furthermore, after
a successful Pathfinder mission [168] the launch of LISA [169], a space-borne
gravitational-wave observatory, is planned for 2034.

A monochromatic plane wave of gravitational radiation gives rise to the
simultaneous dilation and contraction, respectively, of distances along two
perpendicular directions transverse to the direction of propagation that
alternate in time as the wave propagates. In gravitational-wave detectors
based on laser interferometry the simultaneous dilation and contraction along
perpendicular directions give rise to opposite changes in the time of flight
along the two arms of a Michelson interferometer, which can be detected in the
interference signal at the exit port. However, since these waves are so weak
and the change in length is proportional to the arm length, very long
interferometer arms are necessary. Indeed, even for LIGO’s 4-km arms, the
resulting change of length is still several orders of magnitude smaller than
the proton radius.

Gravitational waves also affect the propagation of matter waves and it was
argued in Ref. [170] that fundamental differences compared to light
propagation could be exploited to build table-top atom interferometers with
sensitivities comparable to LIGO. A more careful analysis, however, revealed
that this was not the case [164] and that the corrected results could be
reconciled with those obtained in other studies [171, 172, 173, 174]. Much
more viable proposals put forward after that [175, 176] involve pairs of atom
interferometers separated by a long distance $L$ and interrogated by common
laser beams propagating along the baseline as shown in Figure 6a. Similarly to
laser interferometers, these proposals rely upon the effect of gravitational
waves on laser light propagating along the long baseline, whereas the atom
interferometers at each end act as inertial references (instead of freely
suspended mirrors) and are also employed to read out the small changes in the
relative laser phase $\Delta\varphi_{j}$ of the different pulses caused by
gravitational waves; see Ref. [177] for a review.

The potential advantages of these proposals are connected with the use of atom
interferometers as inertial references. Firstly, in contrast to freely
suspended mirrors, atom interferometers do not suffer from (low-frequency)
suspension noise. Secondly, besides the atom interferometers at the two ends
of the baseline, a number of additional atom interferometers distributed along
the baseline can be included to help discriminate and subtract the
contributions of Newtonian noise [178, 179, 180]. Both suspension noise and
Newtonian noise are responsible for the limited sensitivity at tens of Hz and
below of ground-based laser-interferometric gravitational-wave detectors.
Therefore, atom-interferometric gravitational antennas are a potential
candidate for detecting gravitational waves whose frequencies lie between the
two frequency ranges accessible to LISA and LIGO, as shown in Figure 6.

Figure 6: (a) Spacetime diagram of the wave-packet trajectories for a pair of
atom interferometers separated by a long distance $L$ along the $x$ direction
(not to scale) and interrogated by common laser beams propagating along this
baseline. Gravitational waves affect their propagation and the phase that they
imprint on the atomic wave packets, which can be read out from a differential
measurement of the two interferometers. The effects of rotations and gravity
gradients are respectively mitigated by using a two-loop geometry and suitably
adjusting the frequency of the 2nd and 3rd pulses. (b) This scheme has been
proposed for the European Laboratory for Gravitation and Atom-interferometric
Research (ELGAR) [179], which would develop a gravitational antenna based on
atom interferometry targeting the detection of gravitational waves with
frequencies lying between the frequency ranges accessible to LIGO/Virgo and
LISA, as shown by the strain sensitivity curves plotted in the right panel.

On the other hand, one of the main challenges for this kind of detectors is
the low flux of ultracold atoms that can be currently achieved in atom
interferometers compared to the much higher flux of photons in laser
interferometers such as LIGO or Virgo, which drastically limits the phase-
shift resolution that can be achieved by the former. Besides increasing the
atom flux, quantum metrology techniques making use of entangled atoms can be
exploited to push the phase-shift resolution limit several orders of magnitude
below the shot-noise limit $1/\sqrt{N_{\mathrm{at}}}$, with the ultimate limit
being set by the Heisenberg limit, which scales as $1/N_{\mathrm{at}}$.
However, these techniques are still at a rather early development stage.
Another strategy is to enhance the atom-interferometric phase shift caused by
the gravitational wave by replacing each one of the diffraction pulses with a
sequence of many pulses, possibly combined with the use of higher diffraction
orders. The resulting phase shift scales with the total momentum transfer
$k_{\mathrm{eff}}$, which is proportional to the number of pulses in the pulse
sequence, and momentum transfers corresponding to thousands of pulses or more
are being considered [179]. (Note, however, that laser interferometers on
ground also get an enhancement factor from the storage time in the Fabry-Perot
cavities, which roughly corresponds to bouncing back and forth several hundred
times between the two mirrors.)

For the reasons mentioned above, atom-interferometric gravitational antennas
are far from being competitive with the shot-noise limited sensitivity of
laser interferometers. Nevertheless, their potential domain of applicability
could be the detection of gravitational waves with low frequencies in the Hz
and dHz regime, where the sensitivity of laser interferometers on ground is
severely limited by suspension noise and Newtonian noise and is orders of
magnitude worse than the shot-noise limit. Furthermore, in order to reduce
aliasing effects due to the finite sampling rate, it is necessary to operate
concurrently a number of interferometers with atom clouds launched at a rate
significantly higher than the targeted detection frequencies, and this is
easier for lower frequencies.

Alternatively, for space detectors with very long baselines of millions of
kilometers or more, the photon flux at the receiver is much smaller and
feasible atom fluxes are no longer a major limitation. Therefore, atom-
interferometric gravitational antennas using heterodyne laser links [181] and
targeting gravitational-wave frequencies in the mHz regime and below could
eventually be competitive with laser interferometers. Moreover, by employing
the atom-interferometry scheme based on single-photon clock transitions [182]
discussed below, such a space-borne gravitational-wave detector could be
implemented with a single baseline and without the need for drag-free
satellites, which could significantly reduce the complexity and costs of the
mission. Maximizing the sensitivity to such low frequencies implies rather
long interferometer times of several hundred seconds, which require extremely
narrow momentum widths [183] and the use of atomic lensing techniques,
discussed in Sec. 6.2, to achieve that.

Reaching phase-shift sensitivities in the $\mu\mathrm{rad}$ regime with a
$k_{\mathrm{eff}}$ corresponding to thousands of photon recoils will require a
great control of laser intensity fluctuations and wave-front distortions.
Other important noise sources are the coupling of initial-position and
-velocity jitter to rotations, gravity gradients and wave-front distortions
[180]. Mitigation techniques addressing the effects of rotations and gravity
gradients will be discussed in Sec. 7. As shown in Figure 6, one possibility
is to use a two-loop interferometer geometry that mitigates the effects of
rotations together with a suitable frequency adjustment of the 2nd and 3rd
pulses for the compensation of static gravity gradients. This scheme was
identified as the most suitable for a recently proposed large-scale facility
[179, 180] involving two horizontal baselines with two beams at different
heights per baseline, and could be tested in a 150-m single-baseline prototype
already commissioned [184] in a low-noise underground laboratory.
Alternatively, one could use three-loop geometries with different momentum
transfers per pulse [$k_{\mathrm{eff}}$, $(9/4)k_{\mathrm{eff}}$ and
$(5/2)k_{\mathrm{eff}}$] which simultaneously suppress the effects of
rotations and gravity gradients [176]. Such interferometer geometries can be
implemented in space and in ground experiments with a single vertical
baseline. However, they cannot be employed in horizontal configurations with
two beams per baseline unless the atoms are relaunched during the
interferometer sequence [185].

The proposals for gravitational-wave detection with atom interferometry
mentioned above involve diffraction pulses based on two-photon transitions
such as Raman or Bragg diffraction. An alternative atom interferometry scheme
using a diffraction mechanism based on single-photon transitions between the
two clock states in atoms such as Sr or Yb, which are commonly employed in
optical atomic clocks, was proposed in Ref. [182] (see also Ref. [186] for a
related suggestion). The great advantage of this scheme is that differential
measurements with pairs of atom interferometers are immune to laser phase
noise even for very long baselines and a single baseline is therefore
sufficient. This is advantageous for space missions, where it can lead to
substantial simplifications [182, 181], and it enables vertical configurations
for ground experiments. In fact, projects for ground-based prototypes with a
vertical 100-m baseline [187, 188], which can also be exploited to search for
ultralight dark matter [189], are already underway and plans for future space
missions are being considered too [190, 191, 192].

It should be noted that the clock transition is forbidden for bosonic isotopes
unless a strong magnetic field is applied, which does not seem a viable option
for high-precision measurements. Hence, one will need to employ fermionic
isotopes instead. For long interferometer times it is crucial to use ultracold
atoms, as discussed in the next section. However, Bose-Einstein condensation
is not possible for fermions and one will have to use fermionic clouds close
to quantum degeneracy, which will not be discussed in this review.

##  6 BECs and atomic lensing

The proposals of the previous sections suggest interferometer times of tenth
of seconds in microgravity. While large interrogations times are associated
with high sensitivity, problems related to the size of the atom cloud are
implied as well. Initially the atoms are confined by e.g. a dipole trap.
However, due to the cloud’s finite momentum spread, the it will start to
expand drastically after release. Indeed, an order of magnitude estimation for
cold rubidium atoms indicates a size of several tenth of centimeters after ten
seconds of interferometer time [193]. Atom clouds of large extent in position
and momentum space make long-time interferometry experimentally difficult if
not impossible for the following reasons: A cloud with broad momentum
distribution cannot be addressed resonantly with the laser beams, leading to
velocity selectivity [23] and consequently inefficient diffraction. In
addition, at the exit port of the interferometer the diffraction orders cannot
be discriminated if the momentum width of the cloud is of the order of $\hbar
k_{\mathrm{eff}}$. In position space the Rabi frequency should be uniform over
the size of the cloud for efficient diffraction, thus its spatial extent
should be small compared to the beam width (of the order of a cm).
Furthermore, the small atom density of a large sample cannot be resolved
accurately at the detector. Finally, large clouds are strongly susceptible to
wave-front distortions [194, 66]. During the diffraction process the spatially
dependent phase of the effective laser beam is imprinted on the wave packet.
Since the phase of a realistic laser beam deviates from the ideal planar wave
fronts, phase shifts dependent on the shape of the atomic wave packet are
introduced at each laser pulse. While this bias can be partially corrected for
[194, 195], wave-front distortions are one of the limiting systematic effects
in absolute gravimetry [45, 196] but can be significantly decreased with
smaller atom clouds [197, 193].

An obvious strategy to achieve smaller cloud sizes is cooling to ultra-low
temperatures since the expansion rate scales with the spread in momentum space
which in turn decreases with the temperature of the sample. In case of bosonic
atoms, the system will eventually reach quantum degeneracy and undergo Bose-
Einstein condensation with a momentum width given approximately by the
Heisenberg uncertainty relation [198]. To provide some intuition, the momentum
spread of a BEC in a trap of $2\pi\cdot 50\text{Hz}$ is comparable to the one
of a Maxwell-Boltzmann distribution at the pico-Kelvin range. A second
strategy to achieve small cloud sizes is atomic lensing, where the clouds are
collimated after some free expansion time. This method, which is described in
more detail in Sec. 6.2, is applicable to both ultra-cold non-degenerate
samples and BECs. A combination of BECs as atom source and atomic lensing for
further reducing the expansion rate, overcomes the technical problems
described above and significantly reduces the impact of wave-front
distortions. For an overview over error sources and a comparison between the
properties of ultra-cold non-degenerate samples and BECs see Ref. [193].

Some further intuition into the properties of BECs is provided in the
subsequent section.

###  6.1 Bose-Einstein condensation

Bose-Einstein condensation was first observed experimentally in 1995 [199,
200] but today is reproduced routinely in many laboratories all over the
world, for a review see [201]. What makes BECs the preferable source for
interferometry in many applications compared to cold non-degenerate samples is
the extraordinary coherence length and small momentum spread.

In a thermal ensemble many modes are occupied according to the Bose-Einstein
distribution. Consequently, the coherence length of this mixture is much
smaller than the spatial extent of the cloud. At the critical temperature the
system undergoes Bose-Einstein condensation, arising from combinatorial
properties of the indistinguishable bosonic particles. Below the critical
temperature a major fraction of the atoms occupies a single mode and the
coherence length becomes comparable to the width of the condensate. As a
consequence of the single-mode occupation, a BEC in a harmonic trap
approximately exhibits a momentum width given by the Heisenberg uncertainty
relation [198]. Mathematically, the intuition given above can be formalized by
the Penrose-Onsager criterion [202], serving as a rigorous definition of BEC.

The ground state and time evolution of the BEC mode function is described by
the Gross-Pitaevskii equation [203, 204], which takes the form of a nonlinear
Schrödinger equation. The additional self-interaction term accounts for
interactions among the particles which is much more pronounced in BECs than in
much more dilute thermal samples with the same number of particles. The Gross-
Pitaevskii equation can be derived invoking symmetry breaking [205, 206] or
number-conserving approaches [207, 208].

Light-pulse interferometers with BECs as atom source [23, 209, 24] have shown
nearly perfect contrast [209]. BECs have been employed in microgravity [90,
91] reaching several seconds of free-evolution time and are a basic element of
proposals for long-time interferometry testing fundamental properties of
gravity.

###  6.2 Atomic lensing

Figure 7 compares the expansion rates after release from a harmonic trap
between a BEC and a thermal cloud. As can be seen from the figure, the
expansion rate of the BEC is much smaller than the one of the thermal ensemble
but it still reaches a size of several centimeters after an expansion time of
10 seconds (dashed lines in the figure). To reach experimentally acceptable
final cloud sizes, a collimation step known as delta-kick collimation [210,
211] or alternatives [212] is applied. After an initial free expansion time,
the atom trap is switched on again for a specific amount of time. By that,
kinetic energy is extracted from the system and the subsequent expansion can
be drastically reduced. Theoretically, the correct timing and duration of the
delta-kick pulse can be calculated using approximate scaling solutions
available for the Gross-Pitaevskii equation [213, 214, 215] and the Liouville
equation [216] that can be used to model a thermal non-degenerate quantum gas.
Today, delta-kick collimation is routinely applied in experiments on ground
[80, 48] and in drop-tower facilities [91] in microgravity.

Experiments involving two different atomic species, as for example in tests of
the WEP, are sourced by mixtures of BECs which are first cooled to a common
ground-state configuration [217, 218, 219]. Alternatively, a mixtures of a BEC
and a degenerate Fermi gases [220] could be employed for which a rich class of
groundstate configurations has been predicted [221]. In case of different
trapping frequencies of the species and the resulting different expansion
rates after release, more sophisticated collimation techniques are required.
For Boson-Boson mixtures particularly tailored collimation sequences [222] can
achieve small expansion rates for both components.

Figure 7: Comparison of the expansion dynamics of a BEC and a thermal cloud
above the critical temperature. Initially a zero-temperature BEC and a thermal
cloud with a temperature of $50\text{\textmu K}$ are confined in a harmonic
trap with frequency $\omega=2\pi\cdot 50\text{Hz}$. Upon release both states
start to expand. After the initial free evolution time of
$t_{\mathrm{DKC}}=25$ms the expansion is decelerated by delta-kick
collimation. The large size of about $10$cm of the thermal state after $10$s
expansion indicates a clear advantage of using BECs for long-time
interferometry which reaches a final cloud size of only $1$mm. The parameters
for each state are chosen for $10^{6}$ ${}^{87}\mathrm{Rb}$ atoms.

###  6.3 Interaction-related systematics

The collimated small atom clouds with relatively high atom density bring about
new parasitic effects arising from particle-particle interaction. As the
interaction strength quickly decreases with the size of the cloud, an initial
expansion time before the interferometer sequence and the collimation step
[99] is crucial. There are two negative impacts of particle-particle
interactions briefly explained in the following. The first is referred to as
mean-field phase shifts which are captured by the non-linear term in the
Gross-Pitaevskii equation. In case of a particle imbalance between the
branches a differential systematic phase shift is accumulated [223].
Therefore, in ambitious tests of gravity, high control over the beam-splitting
process is necessary [99]. Mean-field phase shifts are also introduced during
separation of the cloud that follows the beam splitter [224]. On a theoretical
level, BECs in interferometers with large branch separation can be treated
from the perspective of observers comoving [225, 37] with the clouds on the
interferometer branches. By that, the system becomes numerically tractable.
Note that interaction effects could also be canceled by tuning the particle-
particle interaction strength [226] via Feshbach resonances. However, the
fluctuations of the required strong magnetic fields would lead to large
systematic effects.

The second detrimental effect is due to the quantum state after the first beam
splitter which is a superposition of different particle-number states
distributed over the two branches of the interferometer with fixed total
number of atoms. The fluctuation of size $\sqrt{N}$ of the distribution after
the first beam splitter is responsible for the shot noise in case of non-
interacting particles. In case of interacting BECs the mean-field phase shift
depends on the atom density, thus each of the states in the superposition
acquires a different phase shift. The resulting effect is a derogation of the
relative phase between the branches and is known as phase diffusion [227, 228,
229, 230, 231] which leads to larger atom-number fluctuations after the final
beam splitter. An estimation of this effect in the Thomas-Fermi limit [231]
shows that its impact can be disregarded for small interferometer times [209]
but drastically reduces the single-shot sensitivity in long-time
interferometry [37].

##  7 Mitigation techniques

A relative displacement between the mean positions of the two interfering wave
packets at each exit port, or a difference between their mean momenta, leads
to a reduction of the interferometer contrast in Equation (2), which lowers
the strength of the interference signal and the sensitivity of the
interferometer [35]. Furthermore, such displacements between the interfering
wave packets also imply a dependence of the interferometer phase shift on the
mean position and velocity of the initial state [35, 232]. Both rotations and
gravity gradients give rise to open interferometers with non-vanishing
relative displacements that grow with the interferometer time and pose a major
challenge for long-time atom interferometry. Indeed, the resulting sensitivity
to the initial kinematics of the atomic wave packet is a major source of
systematic effects that imposes very demanding requirements on the control of
the initial position and velocity in high-precision measurements. Mitigation
techniques that successfully address the effects associated with rotations and
gravity gradients will be presented in the next two subsections. In addition,
we will discuss in a third subsection how orbital modulation can be exploited
to mitigate these and other systematic effects in space-based atom-
interferometry measurements.

###  7.1 Rotations

Describing the evolution of the atomic wave packets is simpler in a non-
rotating frame, but then one needs to take into account the rotations of the
experimental set-up and the laser beams with respect to that frame [4]; in
particular, for ground experiments the laboratory frame will be co-rotating
with the Earth. As a consequence, the momentum transfer
$\bm{k}_{\mathrm{eff}}$ associated with each laser pulse will be rotated by a
different angle and to lowest order amounts to a change
$(\bm{\Omega}\,t_{n})\times\bm{k}_{\mathrm{eff}}$ for the $n$th laser pulse,
where $\bm{\Omega}$ is the angular velocity and $t_{n}$ is the time at which
the pulse is applied. Altogether the changes for the pulses in a MZ
interferometer lead to relative displacement between the mean positions of the
interfering wave packets perpendicular to $\bm{k}_{\mathrm{eff}}$ and given by
$-2(\bm{\Omega}\,T)\times(\bm{k}_{\mathrm{eff}}/m)\,T$ [4, 35]. This
displacement implies a dependence of the phase shift on the initial velocity
according to Equation (4). Moreover, for sufficiently long interferometer
times (or large angular velocities) the mismatch between the interfering wave
packets can cause a substantial loss of contrast. In what follows, we present
two mitigation techniques that successfully overcome these difficulties.

(i) One method for compensating the effects of rotations relies on the use of
a _tip-tilt mirror_ for retro-reflection of the laser beams and works as
follows. The two counter-propagating laser beams with wave vectors
$\bm{k}_{1}$ and $\bm{k}_{2}$ in Figure 1, which give rise to a total momentum
transfer $\bm{k}_{\mathrm{eff}}=\bm{k}_{1}-\bm{k}_{2}$, are obtained by
injecting two co-propagating beams with frequencies $\omega_{1}$ and
$\omega_{2}$ that are reflected back by a retro-reflection mirror. Bragg or
Raman diffraction can be understood as absorption of a photon with momentum
$\bm{k}_{1}$ from the incoming beam and the stimulated emission of another
photon with momentum $\bm{k}_{2}$ induced by the reflected beam with frequency
$\omega_{2}$. The two incoming beams are parallel with wave vectors
$\bm{k}_{1}\parallel\bm{k}^{\prime}_{2}$, and for a mirror perpendicular to
them one has $\bm{k}_{2}=-\bm{k}^{\prime}_{2}$ for the reflected beam.
However, if one slightly tilts the mirror so that it is no longer exactly
perpendicular to the incoming beams, one has
$\bm{k}_{2}\neq-\bm{k}^{\prime}_{2}$, but the resulting
$\bm{k}_{\mathrm{eff}}$ is still perpendicular to the mirror. Hence, by
tilting the mirror by a small angle, one can change the direction of
$\bm{k}_{\mathrm{eff}}$. In particular, one can choose the tilt angle (and
direction) for every pulse so that it compensates the effect of rotations.
Indeed, although rotations of the experimental set-up would lead to a change
of direction of the incoming beams with respect to the non-rotating frame,
tilting the mirror accordingly in each case so that its orientation remains
the same for all pulses guarantees that the direction of
$\bm{k}_{\mathrm{eff}}$ is left unchanged in this frame for the whole
interferometer sequence.

The use of a tip-tilt mirror for compensation of rotations, which was proposed
in Ref. [7] and experimentally implemented in Refs. [233, 87], has by now
become a standard technique.

(ii) A second method for compensating the effects of rotations involves
employing atom interferometers with _multi-loop geometries_ [30, 33, 4]. The
simplest case is the two-loop interferometer consisting of 4 pulses shown in
Figure 6. For the choice of times between pulses displayed there and
corresponding to $(T,2T,T)$ the contributions to the relative displacement
between interfering wave packets cancel out to first order in
$(\bm{\Omega}\,T)$ [30, 33, 4]. This interferometer geometry has been employed
in applications to high-precision measurements of the rotation rate [234, 54,
55] that benefit from the insensitivity to the initial velocity $\bm{v}_{0}$
and the absence of contrast loss due to rotations despite the long
interferometer times. Indeed, the best cold-atom gyroscopes to date, with a
long-term stability of $3\times 10^{-10}\,\mathrm{rad/s}$, rely on this kind
of interferometers [55]. Note that the phase-shift contributions of the same
form as the right-hand side of Equation (4) cancel out in this case and cannot
be used to determine the rotation rate. Nevertheless, there is still the
following contribution that depends on the angular velocity:

$\delta\phi=4\,\bm{k}_{\mathrm{eff}}\cdot\left(\bm{\Omega}\times\bm{g}\right)T^{3}.$
(10)

Therefore, when combined with an independent measurement of the local
gravitational acceleration $\bm{g}$, the projection of the angular velocity on
the direction perpendicular to $\bm{k}_{\mathrm{eff}}$ and $\bm{g}$ can be
inferred.

In atom interferometers with multi-loop geometries that compensate the effects
of rotations the contributions to the leading term associated with the
gravitational acceleration, proportional to
$\bm{k}_{\mathrm{eff}}\cdot\bm{g}\,T^{2}$, cancel out too [30]. Thus, this
kind of interferometers are not suitable for gravimetric measurements or UFF
tests. On the other hand, they can still be useful for gradiometric
measurements of small time-dependent variations of the gravitational field
with frequencies comparable to $1/T$, which have interesting geophysical
applications and are also being considered as gravitational antennas for the
detection of gravitational waves in the infrasound regime [179, 180], as shown
in Figure 6. It can also be applied to the dark-energy measurements discussed
in Sec. 5.3 and illustrated in Figure 5 as long as an even number of loops is
employed.

###  7.2 Gravity gradients

Deviations from a uniform gravitational field can be characterized in terms of
the gravity-gradient tensor $\Gamma_{ij}=-\partial^{2}U/\partial x^{i}\partial
x^{j}$, where $U$ is the gravitational potential, and give rise to tidal
forces $f_{i}=\Gamma_{ij}\,x^{j}$. Assuming spherical symmetry, the gravity-
gradient tensor associated with Earth’s gravitational field is given by
$\Gamma=\mathrm{diag}(\Gamma_{xx},\Gamma_{yy},\Gamma_{zz})=(GM_{\oplus}/2r^{2})\,\mathrm{diag}(-1,-1,2)$,
where $z$ corresponds to the radial direction and $x,y$ lie on the orthogonal
plane. Near Earth’s surface $\Gamma_{zz}\approx 3\times
10^{-6}\,\mathrm{s}^{-2}$.

In order to understand the effects of gravity gradients in an atom
interferometer, it is convenient to consider a suitable freely falling frame,
as shown in the spacetime diagram of Figure 8b. In this frame the atomic wave
packets would follow straight lines (dashed lines) if the field were uniform,
but the tidal forces tend to open up the trajectories (continuous lines) and
lead to and open interferometer. The position and momentum mismatch between
the interfering wave packets can cause a substantial loss of contrast for long
interferometer times and gives rise to the following phase-shift contributions
that depend on the initial position and momentum [232, 35]:

$\delta\phi=\bm{k}_{\mathrm{eff}}^{\mathrm{T}}\left(\Gamma\,T^{2}\right)\big{(}\bm{x_{0}}+\bm{v}_{0}T\big{)},$
(11)

where the superindex $\mathrm{T}$ denotes matrix transposition. As a result,
small discrepancies in the initial kinematics of the two atomic species can
mimic violations of the equivalence principle in tests of UFF. For example,
reaching a precision of $\eta_{1,2}\lesssim 10^{-15}$ would require a control
of the relative initial position of the two atomic species at the level of
$1\,\mathrm{nm}$ and of the initial relative velocity at the level of
$0.1\,\mathrm{nm/s}$ for $T=5\,\mathrm{s}$, which are extremely demanding
requirements. The sensitivity to the initial kinematics of the two atomic
clouds is also the main systematic effect in measurements of the gravitational
constant $G$ and an important aspect driving the design of the experimental
set-up in order to minimize it [121, 235].

A very effective technique for overcoming these difficulties was proposed in
Ref. [232]. The key idea is to change slightly the momentum transfer of the
second pulse by

$\Delta\bm{k}\approx\frac{1}{2}\left(\Gamma\,T^{2}\right)\bm{k}_{\mathrm{eff}},$
(12)

so that for each arm the trajectory of the atomic wave packet after the
central pulse is the time reversal of the trajectory before the pulse, as
illustrated in Figure 8b, and one is left with a closed interferometer with no
relative displacement between the interfering wave packets. This change of the
momentum transfer can be accomplished by increasing the two frequencies
$\omega_{1}$ and $\omega_{2}$ appearing in Figure 1 by $\Delta k/c$, which
amounts to increasing the detuning $\Delta$ with respect to the auxiliary
state while keeping the resonance condition for the two-photon process.

Figure 8: Orbital modulation (a) and gravity-gradient compensation (b) are two
examples of effective mitigation techniques. (a) In an inertial-pointing
spacecraft $\bm{k}_{\mathrm{eff}}$, and hence the sensitivity axis, remains
constant with respect to a non-rotating frame, whereas the direction of the
local gravitational acceleration $\bm{g}$ changes along the orbit. The signal
of interest (possible UFF violations) is therefore modulated with a frequency
$\omega_{\mathrm{orb}}$ and can be separated from systematic effects whose
modulation frequencies are different multiples of $\omega_{\mathrm{orb}}$. (b)
Spacetime diagram of the wave-packet trajectories in an atom interferometer
for a freely falling frame. For a uniform field these trajectories would be
straight lines (dashed), but tidal forces open them up (continuous lines) and
would lead to an open interferometer. This can be overcome by suitably
adjusting the momentum transfer from the central pulse.

The experimental implementation of this method for gravity-gradient
compensation was demonstrated in Refs. [236, 237] and has been instrumental in
improving by four orders of magnitude the bounds on violations of UFF based on
atom interferometry [143], with prospects for further improvement in the near
future. Moreover, by exploiting this technique, it should be possible to
perform new measurements of $G$ that are competitive with the best
measurements using macroscopic test masses and even outperform them [236,
235]. It is also an important ingredient in the search of certain kinds of
dark-energy using atom interferometry proposed in Ref. [163] and illustrated
in Figure 5. Finally, the technique can also be employed to compensate the
effects of static gravity gradients in gravitational antennas as illustrated
in Figure 6, where the effect of the tidal forces is the opposite of Figure 8b
because Earth’s gravity gradient along the vertical and horizontal directions
have opposite sign.

Note that while the method outlined above applies to quadratic gravitational
potentials (i.e. approximately uniform gravity gradients), it can be
straightforwardly extended to anharmonic potentials by changing appropriately
the frequencies of two laser pulses instead of one [237, 238].

###  7.3 Orbital modulation

In laboratories on Earth the experimental set-up typically co-rotates with
Earth. In contrast, experiments in orbit around Earth offer more flexibility
in this respect. In fact, one can take advantage of the orbital modulation to
separate the signal of interest from various kinds of systematic effects [239,
240]. A simple choice is a spacecraft with inertial pointing, i.e. vanishing
angular velocity with respect to the non-rotating frame, where the direction
of $\bm{k}_{\mathrm{eff}}$ is fixed and unwanted effects associated with
rotations are minimized. For a UFF test with a differential measurement of two
atomic species, as described in Sec. 5.2, the signal is given by
$\bm{k}_{\mathrm{eff}}\cdot(\bm{g}_{2}-\bm{g}_{1})\,T^{2}=\eta_{2,1}\,\bm{k}_{\mathrm{eff}}\cdot\bm{g}\,T^{2}$
and for a circular orbit with inertial pointing
$\bm{k}_{\mathrm{eff}}\cdot\bm{g}\propto\cos(\omega_{\mathrm{orb}}t)$.
Instead, the systematic effects due to Earth’s gravity gradient and given by
Equation (11) are proportional to $\cos(2\,\omega_{\mathrm{orb}}t)$, as can be
seen from the fact that the gravity gradient tensor is the same for any pair
of antipodal points on the orbit. On the other hand, the contribution from the
gravity gradient caused by the spacecraft itself is constant along the orbit
and the same is true for higher orders in a multipole expansion of the
gravitational field around the location of the atoms111Earth’s gravitational
field gives a much smaller contribution to higher orders in this multipole
expansion which does change along the orbit, but is typically suppressed by
seven orders of magnitude or more compared to the gravity-gradient term.,
which are dominated by the local mass distribution (as explained for instance
in Appendix A of Ref. [39]).

It is therefore possible to separate those contributions with different
modulation frequencies when demodulating the measured signal through Fourier
analysis. In particular, while the signal of interest is modulated at a
frequency $\omega_{\mathrm{orb}}$, systematic effects due to Earth’s gravity
gradient are modulated at $2\,\omega_{\mathrm{orb}}$ and those associated with
the spacecraft’s local mass distribution are time-independent (zero
frequency). In addition, an elliptic (rather than circular) orbit as well as
multipoles of Earth’s gravitational field characterizing the deviations from
spherical symmetry will lead to small contributions with modulation
frequencies corresponding to higher multiples of $\omega_{\mathrm{orb}}$. This
separation of the signal from systematic effects with different orbital
modulation frequencies has been successfully employed by the MICROSCOPE
mission [135, 136], which tested the UFF at the $10^{-14}$ level with a pair
of freely falling macrosopic masses in orbit.

Besides systematic effects associated with gravity gradients, one can exploit
the orbital modulation to address other systematic effects too, such as those
that are entirely connected with the experimental apparatus and can be
maintained stable for times longer than the orbital period. In particular,
this could include systematic effects associated with small gradients in the
black body radiation [146, 145] provided that it is not affected by changes of
the solar radiation along the orbit.

Interestingly, these demodulation methods can be combined with the gravity-
gradient compensation technique proposed in Ref. [232] and described in the
previous subsection. In fact, it has been argued that the systematic effects
associated with gravity gradients could then be suppressed to the $10^{-18}$
level [240].

##  8 Outlook

Atom interferometers have enabled the most accurate measurements of the fine
structure constant to date [102, 103], but have also shown their great
potential in fundamental tests of gravity. Indeed, after improving the results
of previous experiments by several orders of magnitude, tests of UFF based on
atom interferometry have recently reached relative uncertainties of the order
of $10^{-12}$ [143] and with prospects of further improvement in the near
future, which brings them very close to the best results obtained on ground
experiments with macroscopic masses. Furthermore, plans for future space
missions exploiting the mitigation techniques described in Sec. 7 and striving
for sensitivities at the $10^{-17}$ level are currently under consideration
[98]. Similarly, atom interferometers have been employed in measurements of
the gravitational constant $G$ with an accuracy of 150 p.p.m. [121], and it
has been convincingly argued [235] that new designs relying on the gravity-
gradients compensation technique [232, 236] of Sec. 7.2 will become
competitive with the best measurements using macroscopic test masses and even
outperform them.

In addition, atom interferometers constitute an ideal tool for constraining
dark-energy models that exhibit screening mechanisms capable of evading
detection with macroscopic test masses. In fact, they have already contributed
to excluding large parts of their parameter space [160, 161, 162], but
considerable improvements are expected by performing the experiments in
microgravity [163], where the atoms can spend much longer times close to the
source mass.

Finally, promising proposals for detection of low-frequency gravitational
waves and searches for ultralight dark matter have not reached the same level
of maturity yet, but ground prototypes with 100-m baselines [187, 188] are
already being commissioned. Moreover, longer-term plans for dedicated space
missions [190, 191, 192] are under consideration as well.

## Acknowledgements

We thank E. Giese, F. Di Pumpo and A. Friedrich for interesting discussions
and the whole QUANTUS consortium for a fruitful collaboration on these topics
over the years.

## Disclosure statement

No potential conflict of interest was reported by the authors.

## Funding

This work is partly supported by the German Aerospace Center (Deutsches
Zentrum für Luft- und Raumfahrt, DLR) with funds provided by the Federal
Ministry of Economic Affairs and Energy (Bundesministerium für Wirtschaft und
Energie, BMWi) due to an enactment of the German Bundestag under Grants No.
50WM1556 (QUANTUS IV) and 50WM1956 (QUANTUS V). The work of IQST is
financially supported by the Ministry of Science, Research and Art Baden-
Württemberg (Ministerium für Wissenschaft, Forschung und Kunst Baden-
Württemberg). W. P. S. is grateful to Texas A&M University for a Faculty
Fellowship at the Hagler Institute for Advanced Study at Texas A&M University
and to Texas A&M AgriLife for the support of this work.

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