Signatures of linearized gravity in atom interferometers: A simplified
computational framework
Leonardo Badurina ,
*
Yufeng Du ,
†
Vincent S. H. Lee ,
‡
Yikun Wang,
§
and Kathryn M. Zurek
∥
Walter Burke Institute for Theoretical Physics,
California Institute of Technology
,
Pasadena, California 91125, USA
(Received 9 September 2024; accepted 6 December 2024; published 5 February 2025)
We develop a general framework for calculating the leading-order, general relativistic contributions to
the gravitational phase shift in single-photon atom interferometers within the context of linearized gravity.
We show that the atom gradiometer observable, which only depends on the atom interferometer
propagation phase, can be written in terms of three distinct contributions: the Doppler phase shift, which
accounts for the tidal displacement of atoms along the baseline, the Shapiro phase shift, which accounts for
the delay in the arrival time of photons at atom-light interaction points, and the Einstein phase shift, which
accounts for the gravitational redshift measured by the atoms. For specific atom gradiometer configu-
rations, we derive the signal and response functions for two physically motivated scenarios: (i) transient
gravitational waves in the transverse-traceless gauge and, for the first time, in the proper detector frame, and
(ii) transient massive objects sourcing weak and slow-varying Newtonian potentials. We find that the
Doppler contribution of realistic Newtonian noise sources (e.g., a freight truck or a piece of space debris) at
proposed atom gradiometer experiments, such as AION, MAGIS, and AEDGE, can exceed the shot noise
level and thus affect physics searches if not properly subtracted.
DOI:
10.1103/PhysRevD.111.042002
I. INTRODUCTION
Atom interferometry is a versatile and rapidly-developing
experimental technique that can be used for a wide variety
of precision measurements
[1]
. For instance, atom interfer-
ometers (AIs) have been used to measure fundamental
constants
[2
–
4]
, probe the foundational principles of general
relativity
[5
–
9]
and quantum mechanics
[10
–
12]
, and test
models of dark energy and modified gravity
[13
–
16]
.Atom
gradiometers (AGs), which consist of two spatially sepa-
rated AIs that are referenced by common lasers, have also
been proposed to detect gravitational waves (GWs) in the
unexplored
“
midfrequency band
”
[17
–
22]
, search for vio-
lations of the universality of free fall
[23]
and measure time-
varying corrections to atomic transition energies induced by
scalar ultralight dark matter
[24
–
26]
.
In recent years, a number of ambitious AG experiments
have been proposed as quantum sensors for fundamental
physics (see Ref.
[27]
for a recent review). These include
large-scale terrestrial experiments, such as AION
[28]
,
MAGIS
[29]
,MIGA
[30]
,ELGAR
[31]
, and ZAIGA
[32]
, and futuristic space-based experiments, such as
STE-QUEST
[33]
and AEDGE
[34]
. By overcoming a
number of experimental systematics, these experiments are
expected to operate at the shot-noise level. However,
fluctuations in an atom
’
s local gravitational field, e.g.,
due to seismic waves
[35]
or Newtonian noise
[36]
,may
significantly reduce the projected reach in the key
ð
10
−
3
–
1
Þ
Hz frequency window. If left unmitigated, these
effects will dramatically limit the physics potential of these
ambitious experiments.
Instead of cutting large sections of an experiment
’
stime
series
[36]
, it may be possible to subtract the phase shift
from transient sources of Newtonian noise and recover an
experiment
’
s shot-noise limited sensitivity provided that this
phase shift is known with sufficient precision. Depending
on an experiment
’
s projected reach and type of physics
search, such a strategy may require a fully relativistic
calculation. For example, a massive object traveling with
speed
v
s
will induce a phase shift by accelerating the atoms,
given by
Δ
φ
nonrel
∼
k
eff
aT
2
[37]
,with
k
eff
being the
maximum momentum difference between the two arms
of an interferometer,
T
being the interrogation time, and
a
being the acceleration of the atoms. This is typically
calculated utilizing only classical Newtonian mechanics.
However, relativistic corrections are expected. Using dimen-
sional analysis and the invariance of general relativity under
parity and time reversal, the dominant relativistic phase shift
is expected to scale as
Δ
φ
rel
∼
v
s
Δ
φ
nonrel
. Although useful
to estimate the size of the effect,
Δ
φ
rel
does not inform us of
*
Contact author: badurina@caltech.edu
†
Contact author: yfdu@caltech.edu
‡
Contact author: szehiml@caltech.edu
§
Contact author: yikunw@jhu.edu
∥
Contact author: kzurek@caltech.edu
PHYSICAL REVIEW D
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=
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=
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=
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© 2025 American Physical Society
(i) the coefficient of this phase shift term, which may be
sequence dependent, and (ii) the shape of the power
spectrum associated with this phase shift, both of which
can only be inferred from a general-relativistic (i.e., general
coordinate-invariant) phase shift calculation.
Several coordinate-invariant formalisms have been pro-
posed
[7,38
–
41]
and have been used to calculate, e.g., the
phase shift induced by gravitational waves
[17,18,42]
and static potentials (e.g., the Earth
’
s gravitational field)
[7,38,40,41]
. Notably, the formalism proposed in Ref.
[38]
defines all tunable experimental parameters in a frame-
independent manner, solves for the geodesics of the freely
falling atomic wave packets and laser pulses, and determines
the momentum transferred to the atoms as a result of atom-
light interactions in the atom
’
s local interial frame. Although
crucial for correctly predicting the size of relativistic con-
tributions to the phase shift, these formalisms are computa-
tionally cumbersome when the number of interaction points
exceeds
O
ð
1
Þ
. This is especially relevant in proposed atom
gradiometer experiments employing large momentum trans-
fer (LMT) such as AION and MAGIS, which plan up to
O
ð
10
4
Þ
atom-light interactions per cycle.
Aside from these computational considerations, out-
standing questions remain about the physical interpretation
of existing gauge-invariant frameworks for computing
gravitational phase shifts, such as the interpretation of
AGs as gravitational antennas, or equivalently the mapping
between AG and laser interferometer observables. For
example, Ref.
[38]
decomposes the gauge-invariant phase
shift for a single AI (and consequently for an AG) into
three contributions: the phase shift associated with the free
evolution of atomic wave packets in spacetime (i.e., the
propagation phase), the phase shift imprinted by the laser
pulses during atom-light interactions (i.e., the laser phase)
and the phase shift associated with the degree to which
the two spatially separated wave packets do not overlap at
the application of the final beam splitter pulse (i.e., the
separation phase). As shown for GWs in Ref.
[43]
and
recently for more general metric perturbations via a proper
time treatment in Ref.
[44]
, the observable in laser
interferometers can be written as a sum of three distinct
(and separately not diffeomorphism-invariant) contribu-
tions: the time delay caused by the tidal displacement of
the mirrors along the baseline (i.e., the Doppler time
delay), the delay in the arrival time of photons at the
mirrors (i.e., the Shapiro time delay), and the time-delay
due to the gravitational redshift measured by the beam
splitter (i.e., the Einstein time delay). Since AGs have been
proposed as gravitational wave interferometers, it should
be possible to extract these three contributions from the
gauge-invariant AG phase shift. This endeavor would
clarify the interpretation of AGs as exquisite accelerom-
eters
[45]
and time-keeping devices
[46]
, and elucidate the
origin of the relativistic phase shift contributions in
different frames.
In this work, we address these points by developing from
first principles a simplified general coordinate-invariant
framework for calculating phase shifts in single-photon
AGs. Since the metric perturbation in the problems of
interest is very small, we will work within the context of
linearized gravity. Additionally, as the motion of the atoms
relative to the laser sources is highly nonrelativistic, we
work to leading order in the atom velocity. Notably, for the
case of AGs, we express the differential phase shift in terms
of contributions which are in one-to-one correspondence
with the time-delays that enter the laser interferometer
observable: the phase shifts associated with the Doppler,
Shapiro, and Einstein time delays. Equipped with this
formalism, we compute the signal and response function
induced by two well-motivated physical scenarios: tran-
sient GWs and weak and slow-varying Newtonian poten-
tials sourced by transient massive objects. Since the form of
the response function depends on the pulse sequence, we
will perform explicit calculations for gradiometers employ-
ing Mach-Zehnder and LMT configurations. Importantly,
our examples highlight the gauge-invariance of our frame-
work (as explicitly shown in the GW calculation, which is
performed in both the transverse-traceless and the proper
detector frame) and the accuracy of our formalism in
reproducing existing results in the literature in a more
physically and computationally transparent fashion.
This paper is structured as follows. After reviewing the
basics of atom interferometry, in Sec.
II
we introduce our
general coordinate-invariant framework for computing
gradiometer phase shifts in linearized gravity. After deriv-
ing the basis for our formalism in Sec.
II A
,inSec.
II B
we
introduce the gradiometer observable and provide expres-
sions for Mach-Zehnder and LMT gradiometer configu-
rations. As example applications, in Sec.
III
we compute
the phase shifts induced by gravitational waves and slow-
varying weak Newtonian potentials. In Sec.
IV
we sum-
marize the key results of this paper. Appendices
A
–
D
support the calculations in Secs.
II
–
III
.
II. DERIVATION OF THE LEADING-ORDER
GRADIOMETER PHASE SHIFT
IN LINEARIZED GRAVITY
Schematically, AIs utilize matter wave interference to
detect the phase difference between two coherent atomic
states in a spatial superposition. In order for an atom to be
prepared in a spatial superposition and then measured via
matter wave interference, the trajectories of the atomic
wave packets are manipulated using laser pulses. Most
experiments rely on a two-level system where the external
momentum and the internal energy state of the atoms can
be manipulated via Rabi oscillations
[47]
. A laser pulse
that interacts with the atom over a quarter of the Rabi cycle
(a
π
=
2
pulse) takes an atom in one state to an equal
superposition of two states, thus acting as a
“
beam
splitter
”
; a pulse over half of the Rabi cycle (a
π
pulse)
BADURINA, DU, LEE, WANG, and ZUREK
PHYS. REV. D
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reverses the state of the atom, thus acting as a
“
mirror
”
[48]
. In this paper, we focus on experiments that rely on
two spatially separated AIs operating common lasers and
single-photon transitions with energy separation
ω
a
.We
refer to these configurations as single-photon AGs.
In order to correctly capture relativistic effects, which
manifest through the dependence of dynamics on the
curvature of spacetime, it is of paramount importance to
describe the interferometer sequence and all experimental
quantities in a frame-independent manner. Importantly, this
guarantees that the observable is free of gauge artifacts. Let
us consider the description of atomic fountains (which we
assume in this work) in the language of general relativity
(e.g., Ref.
[38]
). In these AI experiments, the atoms are in
free fall. Provided that the radius of curvature is much larger
than the wave packet size, a semiclassical treatment is
sufficient. In this regime, atom trajectories are described in
terms of timelike geodesics, while photon trajectories are
described in terms of null geodesics. The spacetime points
at which atom-light interactions occur can be solved in
coordinate time and position. Furthermore, for the phase
shift to be frame independent, the coordinate times at which
the laser pulses are emitted must be related to the established
time difference measured by an observer traveling along the
laser
’
s worldline. Finally, the four momentum of a pulse
must be related to the frequency of the pulse at emission;
this four momentum is then evolved from emission to the
designated atom-light interaction point, where the atom
’
s
recoil is computed in a local inertial frame.
Inspired by this description, in the following sections we
provide a detailed derivation from first principles of the
gauge-invariant gradiometer phase shift. Importantly, our
framework is valid to leading order in a generic metric
perturbation and correctly predicts the coefficients of phase
shift terms that are linear in the atom
’
s recoil velocity.
Starting from the phase shift for a single AI, we show that
the single-photon AG observable can be reinterpreted in
terms of coordinate time delays that are well understood in
the context of laser interferometers: the Doppler, Shapiro,
and Einstein time delays.
A. Propagation phase shift
Let us consider the semiclassical evolution of an atomic
wave packet
j
ψ
i
in spacetime. In this regime, the atom
’
s
dynamics can be described in terms of the evolution of the
atom
’
s c.m. Consequently, solving the Schrödinger equa-
tion,
j
ψ
i
∝
exp
ð
i
S
Þ
,where
S
is the action of the atom
’
s
c.m.
[49]
. In general relativity, the action of a massive
pointlike particle can be expressed in terms of the proper
time elapsed along particle
’
s timelike worldline
[50]
.
Therefore, the phase difference associated with the space-
time propagation of two spatially separated wave packets
corresponds to the difference in the actions evaluated along
the worldlines of the two wave packets
’
c.m. from state
initialization to measurement. For a spacetime with metric
g
μν
, the propagation phase is
Δ
φ
¼
I
C
md
τ
¼
I
C
m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
−
g
μν
dx
μ
dt
dx
ν
dt
r
dt;
ð
1
Þ
where
m
is the path-dependent mass of the atom, with
m
¼
m
o
and
m
¼
m
o
þ
ω
a
for ground and excited states,
respectively.
1
The proper time along the atom
’
s worldline
is defined as
τ
, which is not to be confused with the
coordinate time
x
0
¼
t
, and the atom
’
s path-dependent four
coordinate velocity is defined as
dx
μ
=dt
. We parametrize
the atom
’
s evolution with respect to
t
and perform the loop
integral over the closed semiclassical path
C
.Thispath
depends on the following: (i) the free evolution of the
atomic wave packets between atom-light interaction
points, (ii) the arrival time of laser pulses at atom-light
interaction points, and (iii) the metric-dependent correc-
tion to the laser beam
’
s wave vector, which leads to a
correction in the recoil of the atoms after atom-light
interactions. Here, we assume that
C
is initiated when
the superposition of states is created (i.e., at the initial
beam splitter pulse) and closes when the interference
pattern is measured. The latter should not be confused
with the final beam splitter used to redirect the atomic
states to the measurement ports.
Note that the full AI phase observable,
Δ
φ
j
AI
, also
depends on the laser phase
Δ
φ
laser
, which is imprinted onto
the atoms during atom-light interaction and arises from the
linear coupling between the photon field and the atom
’
s
electric dipole
[48]
:
Δ
φ
j
AI
¼
Δ
φ
þ
Δ
φ
laser
:
ð
2
Þ
In gradiometer setups featuring more than one AI refer-
enced by the same laser, the AIs experience a common laser
phase which cancels out in a differential measurement.
2
Since we focus on gradiometer observables, we do not
consider this effect further, though it is important to note
that the laser phase contribution is generally nonzero in
single-AI setups.
For the relativistic effects considered in this work, it is
sufficient to linearize the metric tensor, i.e.,
g
μν
¼
η
μν
þ
h
μν
,
with
h
μν
≪
1
. By working at leading order in
h
μν
and
1
We work in natural units (i.e.,
ℏ
¼
c
¼
1
) and we use the
convention
η
μν
¼
diag
ð
−
1
;
þ
1
;
þ
1
;
þ
1
Þ
for the flat spacetime
metric.
2
Since the radius of curvature considered in this work is much
larger than the wavelength of the electromagnetic waves driving
the Rabi oscillations, it suffices to work in the limit of geometric
optics. In this limit, the phase of electromagnetic waves is
constant along null geodesics and can be defined with respect
to any timelike observer
[50,51]
. For instance, Ref.
[38]
defines
this phase with respect to an observer traveling along the laser
’
s
worldline.
SIGNATURES OF LINEARIZED GRAVITY IN ATOM
...
PHYS. REV. D
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relating
C
to ancillary paths, which we schematically depict
in Fig.
1
, a number of simplifications will arise. First, in
the absence of metric perturbations,
g
μν
¼
η
μν
, we assume
that the atomic wave packets travel along trajectories that
recombine at the application of the final beam splitter pulse;
we will refer to this path as
C
o
. Second, as alluded to above,
we define
C
as the closed path traced by the atomic wave
packets in the presence of a nonzero metric perturbation
h
μν
;
however, while
C
is closed at measurement, it is no longer
closed at the final beam splitter pulse. Finally, we define a
third and fictitious path
̃
C
, constructed from
C
assuming the
same metric perturbation
h
μν
; the initial conditions (e.g.,
atom velocity) are adjusted for the path to close at the final
beam splitter pulse. Hence, generally
̃
C
≠
C
.
The advantage of introducing these ancillary paths is
that, at leading order in the metric perturbation, the phase
shift along the true path
C
is equivalent to the phase shift
computed along the unphysical path
̃
C
. This can be seen
simply as follows. Let
ξ
μ
ð
λ
Þ
be the coordinate separation
between the atom geodesics in
C
and
̃
C
, where
λ
∈
½
0
;
1
parameterizes the atom worldlines and
ξ
μ
ð
0
Þ¼
ξ
μ
ð
1
Þ¼
0
,
since both
C
and
̃
C
are closed with respect to state
initialization and state measurement. Expanding in the
coordinate deviation
ξ
and defining
L
¼
m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g
μν
̇
x
μ
̇
x
ν
p
,
where
̇
x
μ
¼
dx
μ
=d
λ
, the phase shift over the true path
C
takes the form
3
Δ
φ
¼
I
C
md
τ
¼
I
̃
C
md
τ
þ
Z
1
0
ξ
μ
d
dt
∂
L
∂
̇
x
μ
−
∂
L
∂
x
μ
R
d
λ
−
Z
1
0
ξ
μ
d
dt
∂
L
∂
̇
x
μ
−
∂
L
∂
x
μ
L
d
λ
þ
O
ð
ξ
2
Þ
¼
I
̃
C
md
τ
þ
O
ð
ξ
2
Þ
;
ð
3
Þ
where the left and right trajectories are marked by the
L
and
R
subscripts, respectively. The second and third terms
vanish because the atom worldlines in
̃
C
are geodesics, and
hence extrema of the action.
We next expand Eq.
(3)
to leading order in the metric
perturbation, utilizing
̃
C
¼
C
o
þ
δ
C
with
δ
C
¼
O
ð
h
Þ
:
Δ
φ
¼
I
̃
C
m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
−
η
μν
dx
μ
dt
dx
ν
dt
r
dt
−
1
2
I
C
o
m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
−
η
μν
dx
μ
dt
dx
ν
dt
q
h
μν
dx
μ
dt
dx
ν
dt
dt
þ
O
ð
h
2
Þ
:
ð
4
Þ
Writing the phase in Eq.
(4)
to leading order in
O
ð
hv
Þ
,
we find
4
Δ
φ
¼
I
̃
C
mdt
−
I
C
o
mv
i
δ
v
i
dt
−
1
2
I
C
o
mh
00
dt
−
I
C
o
mh
0
i
v
i
dt
þ
O
ð
h
2
;v
2
Þ
;
ð
5
Þ
where
δ
v
i
is the
O
ð
h
Þ
correction to the atom three velocity
arising from free-falling atoms, obtained by solving geo-
desic equations
5
FIG. 1. Schematic spacetime diagram showing the atom
trajectories used in deriving the framework. The dotted yellow
lines denote hypersurfaces defined by the initial beam splitter
pulse, the final beam splitter pulse, and the measurement. The
unperturbed atom trajectory
C
o
is schematically shown as
the dashed black lines. In the presence of a metric perturbation,
the atom follows a geodesic
C
(purple), which can be deformed
into another geodesic
̃
C
(blue) that closes at the final beam
splitter.
Δ
x
μ
is the separation of path
C
at the final beam splitter
pulse.
λ
parametrizes the atom geodesics, and the deformation
ξ
ð
λ
Þ
is of order
O
ð
h
Þ
and only enters into the observable at
O
ð
h
2
Þ
.At
O
ð
hv
Þ
,
̃
C
is a good approximation for
C
in calculating
the propagation phase shift.
3
Note that, while we assume here that
C
is an exactly closed
path at measurement in the presence of metric perturbations,
C
is
generally not precisely closed even at measurement. This leads to
a correction to the phase shift which appears at leading order in
the metric perturbation
h
and
v
. Nevertheless, as discussed in
Appendix
A
, the separation phase can be neglected by choosing
an appropriate ancillary path.
4
Although this approximation is frame dependent, we can
always choose to work in a frame where
v
≪
1
. The only possible
transformation from a frame where the atoms are nonrelativistic
to one where the atoms are moving at relativistic speeds, and
which does not spoil the gauge transformation in the metric in
linearized gravity, is a Lorentz transformation. Such a trans-
formation is global and therefore boosts the entire laboratory
apparatus, namely the atoms and the laser sources.
5
Recall that the coordinate four velocity is defined as
dx
μ
=dt
,
where
t
is coordinate time. Hence, metric perturbations can
only affect the
μ
∈
f
1
;
2
;
3
g
components of this four vector,
dx
μ
=dt
¼ð
1
;v
i
þ
δ
v
i
Þ
, where
δ
v
i
¼
O
ð
h
Þ
.
BADURINA, DU, LEE, WANG, and ZUREK
PHYS. REV. D
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δ
v
i
ð
t; x
Þ¼
−
Z
t
t
0
dt
0
Γ
i
00
ð
x; t
0
Þ
;
¼
−
η
ij
Z
t
t
0
dt
0
∂
0
h
j
0
−
1
2
∂
j
h
00
þ
O
ð
h
2
;v
Þ
;
ð
6
Þ
where
Γ
μ
αβ
is the Christoffel symbol in linearized gravity
and
t
0
is the time at which the experiment is initialized. We
neglect the contribution to the atom
’
s recoil velocity after
atom-light interactions that arises from the
O
ð
h
Þ
correction
to the photon
’
s wave vector, since the leading-order
correction is proportional to
v
and therefore enters the
observable at next-to-leading order.
The first term on the rhs of Eq.
(5)
corresponds to phase
shift due to the deformation of the atom trajectories in
going from
C
o
to
̃
C
. Because we are interested in signatures
of linearized gravity, we only consider
O
ð
h
Þ
effects, which
are attributed to
δ
C
. Since multiple pulses divide each
interferometer arm into path segments, which we label by
k
,
the first term on the rhs of Eq.
(5)
can be written as
I
̃
C
mdt
⊃
I
δ
C
mdt
¼
X
k
∈
E
ω
a
Δ
t
ð
k
Þ
!
R
−
X
k
∈
E
ω
a
Δ
t
ð
k
Þ
!
L
:
ð
7
Þ
Notice that the leading order term is proportional to
ω
a
since
the rest mass contribution vanishes under a loop integral,
i.e.,
H
̃
C
m
o
dt
¼
0
6
; hence we only sum over the excited
segments, denoted as the set
E
. As a result, only excited
state path segments contribute to the phase shift in Eq.
(7)
;
this will be very important in the calculations below. In
Eq.
(7)
,
Δ
t
ð
k
Þ
is the
O
ð
h
Þ
perturbation to the coordinate time
duration of the
k
th path segment, and path segments on both
the right (
R
) and left arm (
L
) are summed over. The overall
minus sign in the left arm contributions originates from
the loop integral. The coordinate time corrections can be
computed by solving for the intersection of atom and photon
worldlines. Importantly, these atom-light interaction points
define the path segments. We label each path segment
k
by
the laser pulse that starts the sequence. The
k
th segment is
ended by the laser pulse labeled as
k
þ
1
, which starts the
subsequent segment
k
þ
1
. Denoting the perturbed initial
and final times of the
k
th path segment as
̄
t
ð
k
Þ
þ
δ
t
ð
k
Þ
and
̄
t
ð
k
þ
1
Þ
þ
δ
t
ð
k
þ
1
Þ
, the perturbed atom worldline as
̄
x
i
ð
t
Þþ
δ
x
i
ð
t
Þ
, the worldlines of the photons as
̄
x
i
γ
ð
k
Þ
ð
t
Þþ
δ
x
i
γ
ð
k
Þ
ð
t
Þ
and
̄
x
i
γ
ð
k
þ
1
Þ
ð
t
Þþ
δ
x
i
γ
ð
k
þ
1
Þ
ð
t
Þ
, with unperturbed quan-
tities denoted by overbars, and
n
i
as the unit vector pointing
along the baseline (considering the motion of both the
photons and atoms in a single spatial direction), the
corresponding equation to solve is
n
i
½
̄
x
i
ð
̄
t
ð
k
Þ
þ
δ
t
ð
k
Þ
Þþ
δ
x
i
ð
̄
t
ð
k
Þ
Þ
¼
n
i
½
̄
x
i
γ
ð
k
Þ
ð
̄
t
ð
k
Þ
þ
δ
t
ð
k
Þ
Þþ
δ
x
i
γ
ð
k
Þ
ð
̄
t
ð
k
Þ
Þ
:
ð
8
Þ
Expanding Eq.
(8)
to
O
ð
h
Þ
and neglecting the unperturbed
atom velocity [as the prefactor
ω
a
in Eq.
(7)
is parametri-
cally
O
ð
v
Þ
], we find
δ
t
ð
k
Þ
¼ðÞ
ð
k
Þ
n
i
½
δ
x
i
ð
t
ð
k
Þ
Þ
−
δ
x
i
γ
ð
k
Þ
ð
t
ð
k
Þ
Þ
;
ð
9
Þ
where
ðÞ
ð
k
Þ
is taken to be
þ
1
or
−
1
for outgoing or
incoming photons (i.e., parallel or antiparallel to the base-
line) interacting with the atom, respectively. Here we drop
the overbars, which is valid to
O
ð
h
Þ
. The perturbed atom
positions,
δ
x
i
ð
t
Þ
, are given by integrating Eq.
(6)
,i.e.,
δ
x
i
ð
t; x
Þ¼
Z
t
t
0
δ
v
i
ð
t
0
;x
Þ
dt
0
;
ð
10
Þ
and the perturbed photon trajectories,
δ
x
i
γ
ð
k
Þ
ð
t
Þ
, are given by
solving the null geodesic condition,
ds
2
¼
g
μν
dx
μ
dx
ν
¼
0
,
leading to
n
i
d
dt
δ
x
i
γ
ð
k
Þ
ð
t
Þ¼ð
∓
Þ
ð
k
Þ
H
ðÞ
ð
k
Þ
;
ð
11
Þ
where we defined
H
≡
1
2
ð
h
00
2
h
0
i
n
i
þ
h
ij
n
i
n
j
Þ
:
ð
12
Þ
Putting Eqs.
(10)
and
(11)
into Eq.
(9)
, the
O
ð
h
Þ
perturba-
tion to the coordinate time duration of the
k
th path segment,
namely
Δ
t
ð
k
Þ
, can thus be expressed as the sum of Doppler
and Shapiro time delays,
Δ
t
ð
k
Þ
¼
δ
t
ð
k
þ
1
Þ
−
δ
t
ð
k
Þ
≡
Δ
t
ð
k
Þ
D
þ
Δ
t
ð
k
Þ
S
:
ð
13
Þ
The Doppler term is due to the atom
’
s motion under the
metric perturbation as derived in Eq.
(10)
:
Δ
t
ð
k
Þ
D
¼
n
i
½ðÞ
ð
k
þ
1
Þ
δ
x
i
ð
t
ð
k
þ
1
Þ
;x
ð
k
þ
1
Þ
Þ
−
ðÞ
ð
k
Þ
δ
x
i
ð
t
ð
k
Þ
;x
ð
k
Þ
Þ
;
ð
14
Þ
with
x
ð
k
Þ
and
x
ð
k
þ
1
Þ
being the unperturbed initial and final
atom positions of the
k
th and (
k
þ
1
)th path segment. The
Shapiro term, which corresponds to the time delay accrued
by the photon along its geodesic as derived in Eq.
(11)
,is
given by
6
In symmetric configurations,
H
C
o
mdt
¼
0
, so that
H
̃
C
mdt
¼
H
δ
C
mdt
.
SIGNATURES OF LINEARIZED GRAVITY IN ATOM
...
PHYS. REV. D
111,
042002 (2025)
042002-5
Δ
t
ð
k
Þ
S
¼
Δ
T
ðÞ
ð
k
þ
1
Þ
S
ð
t
ð
k
þ
1
Þ
L
;x
ð
k
þ
1
Þ
L
;x
ð
k
þ
1
Þ
Þ
−
Δ
T
ðÞ
ð
k
Þ
S
ð
t
ð
k
Þ
L
;x
ð
k
Þ
L
;x
ð
k
Þ
Þþ
Δ
t
ð
k
Þ
laser
;
ð
15
Þ
where we defined
Δ
T
S
ð
t; x
1
;x
2
Þ
≡
Z
x
2
x
1
H
ð
t
ð
x
0
−
x
1
Þ
;x
0
Þ
dx
0
:
ð
16
Þ
Here, the photons which interact with the atom at the
k
th
and (
k
þ
1
)th intersection are emitted from the lasers at
x
ð
k
Þ
L
and
x
ð
k
þ
1
Þ
L
, and
t
ð
k
Þ
−
t
ð
k
Þ
L
≡
ðÞ
ð
k
Þ
ð
x
ð
k
Þ
−
x
ð
k
Þ
L
Þ
with
t
ð
k
Þ
L
the
corresponding photon emission time. The quantity
Δ
t
ð
k
Þ
laser
depends on the
O
ð
h
Þ
correction to the photon
’
s emission
spacetime points and cancels out in a differential measure-
ment, as discussed in Appendix
B
.
The remaining three terms on the rhs of Eq.
(5)
only
depend on the unperturbed path
C
o
. By using Eq.
(6)
, these
terms can be rewritten as
Δ
φ
E
≡
−
I
C
o
mv
i
δ
v
i
dt
−
1
2
I
C
o
mh
00
dt
−
I
C
o
mh
0
i
v
i
dt;
¼
−
1
2
I
C
o
dtm
1
þ
v
i
Z
t
t
0
dt
0
∂
i
h
00
:
ð
17
Þ
These contributions only depend on
h
00
, and are thus
identified as the Einstein term (i.e., time dilation as
measured by the atoms). We note that this phase shift
can be significantly simplified if the metric perturbation is
spatially slow varying over
C
o
, which allows it to be
expanded as
h
μν
ð
t; x
AI
þ
x
a
Þ¼
h
μν
ð
t; x
AI
Þþ
x
i
a
∂
i
h
μν
ð
t; x
AI
Þþ
O
ð
v
2
Þ
:
ð
18
Þ
Here, the distance traveled by the atoms from their initial
position is
x
i
a
ð
t
Þ
∝
v
i
and
x
AI
is the unperturbed position of
the AI at the start of the sequence. The expansion is valid
when
ð
vT
Þ
∂
i
h
μν
≪
h
μν
, where
T
is the interrogation time,
and holds for all examples considered in this work.
7
With
this approximation and integrating by parts, Eq.
(17)
simplifies to
Δ
φ
E
¼
X
k
∈
E
ω
a
Δ
t
ð
k
Þ
E
!
R
−
X
k
∈
E
ω
a
Δ
t
ð
k
Þ
E
!
L
;
ð
19
Þ
where the nonvanishing contribution comes from the
excited state segments, as the ground state contribution
sums to zero over the closed loop, and
Δ
t
ð
k
Þ
E
¼
−
1
2
Z
C
ð
k
Þ
o
h
00
ð
t; x
AI
Þ
dt
þ
O
ð
v
2
Þð
20
Þ
defines the Einstein time delay.
In summary, the propagation phase shift in Eq.
(5)
can be
written schematically as
Δ
φ
¼
Δ
φ
D
þ
Δ
φ
S
þ
Δ
φ
E
þ
O
ð
h
2
;v
2
Þ
;
ð
21
Þ
where the Doppler and Shapiro phase shifts originate from
Eq.
(7)
with
Δ
t
ð
k
Þ
given in Eq.
(13)
, while the Einstein
phase shift originates from Eq.
(17)
. The expressions of
the Doppler, Shapiro, and Einstein time delays are given
in Eqs.
(14)
,
(15)
, and
(20)
, respectively. Together with
Eqs.
(19)
and
(21)
facilitates the direct comparison between
atom interferometer experiments, such as AION and
MAGIS, and laser interferometers, such as LIGO
[52]
and GQuEST
[53]
. Indeed, the time delays that enter the
atom interferometer propagation phase also appear in laser
interferometer calculations. The only difference between
the two propagation phase shifts is the physical origin of
the frequency: in the case of atom interferometers, the
phase shift is proportional to the energy difference between
the ground and excited state; in the case of laser interfer-
ometers, the phase shift is proportional to the frequency of
the laser pulse traveling along the baseline. More broadly,
the appearance of the Doppler and Shapiro time delays
facilitates the analogy between the atom interferometers
and the mirrors of a one-dimensional laser interferometer.
Furthermore, the appearance of the Einstein time delay
clarifies the analogy between an atom interferometer and
the beam splitter in a laser interferometer. Finally, since we
started with the action for massive pointlike particles
[cf. Eq.
(1)
], the phase shift is manifestly invariant under
diffeomorphisms.
B. Doppler, Shapiro, and Einstein phase
shifts for atom gradiometers
In an AG, the observable is the difference between
the phase shifts measured by each AI, i.e.,
Δ
φ
grad
≡
Δ
φ
j
AI
1
−
Δ
φ
j
AI
2
. Since the transitions in each AI are driven
by common laser pulses, the gradiometer observable exclu-
sively depends on the difference between the propagation
phase shifts of each AI and, subsequently, on the difference
between the coordinate time corrections between path
segments, which we refer to as gradiometer time delays.
Without loss of generality, we can write the gradiometer
time delays for each segment
k
because the pair of laser
pulses that start and end the path segment in a given AI also
start and end a path segment in the other AI. Restricting our
attention to such configurations, the
O
ð
h
Þ
correction to the
7
As an example, for gravitational waves with angular fre-
quency
ω
, the condition is satisfied when
ω
≪
ð
v
a
T
Þ
−
1
∼
1
GHz
ð
10
−
9
=v
a
Þð
1
s
=T
Þ
. This requirement is met by many
orders of magnitude, as most proposed AI experiments are not
sensitive to gravitational waves beyond 1 Hz.
BADURINA, DU, LEE, WANG, and ZUREK
PHYS. REV. D
111,
042002 (2025)
042002-6
laser
’
s motion cancels, as we carefully show in Appendix
B
.
Pulling all parts of the derivation together, the gradiometer
phase shift for experiments using single-photon transitions
can be schematically expressed as
Δ
φ
grad
¼
Δ
φ
grad
;
D
þ
Δ
φ
grad
;
S
þ
Δ
φ
grad
;
E
þ
O
ð
h
2
;v
2
Þ
;
Δ
φ
grad
;
D
;
S
;
E
¼
X
k
Δ
φ
ð
k
Þ
grad
;
D
;
S
;
E
¼
X
k
∈
E
ω
a
Δ
t
ð
k
Þ
grad
;
D
;
S
;
E
!
R
−
X
k
∈
E
ω
a
Δ
t
ð
k
Þ
grad
;
D
;
S
;
E
!
L
;
ð
22
Þ
where the gradiometer time delay is expressed in terms
of the single AI time delays
Δ
t
ð
k
Þ
D
;
S
;
E
[cf. Eqs.
(14)
,
(15)
,
and
(20)
]as
Δ
t
ð
k
Þ
grad
;
D
;
S
;
E
≡
Δ
t
ð
k
Þ
D
;
S
;
E
j
AI
1
−
Δ
t
ð
k
Þ
D
;
S
;
E
j
AI
2
:
ð
23
Þ
Since the ground state contribution vanishes over the loop,
we only sum over the path segments where the atoms are in
the excited state.
Explicit expressions for the Doppler, Shapiro, and
Einstein phase shifts can be extracted for particular con-
figurations. In what follows, we provide analytical phase
shift expressions for two popular single-photon gradiometer
designs: the Mach-Zehnder (MZ) and the LMT gradiom-
eters. For convenience, we will describe both experiments in
a frame where the lasers are at rest in the absence of a metric
perturbation.
8
By convention, the right arm receives the
initial momentum deposition.
1. Mach-Zehnder gradiometer
A MZ gradiometer employs a
“
π
=
2
−
π
−
π
=
2
”
sequence. At time
t
0
, which is also the time at which the
atoms are released from their initial positions, a
π
=
2
-pulse is
emitted from the laser to create an equal superposition of
ground and excited state in both interferometers. As a result
of atom-light interactions, the excited state wave packet
recoils with external momentum
k
eff
¼
ω
a
.
9
At time
t
0
þ
T
,
the laser emits a
π
pulse, which swaps the atomic states in
the arms of each AI, and reverts the relative momentum
between the two wave packets in each AI. The two arms of
the interferometer are redirected before the application of a
final beam splitter pulse, which is emitted by the laser at
time
t
0
þ
2
T
and splits each arm into an equiprobable
superposition of ground and excited states. This is followed
by the interference measurement. The corresponding space-
time diagram is schematically depicted in the left panel
of Fig.
2
.
In this configuration, the atom spends half of the time in
the excited state. Denoting the locations of the two AIs as
x
AI
1
and
x
AI
2
¼
x
AI
1
þ
L
and neglecting the distance
between the laser source and the first AI, the gradiometer
time delays for pairs of excited state paths initiated by a
pulse emitted at unperturbed coordinate time
t
can be
written as
Δ
t
MZ
grad
;
E
ð
t
Þ¼
−
1
2
Z
t
þ
T
t
h
00
ð
t
0
;x
AI
1
Þ
dt
0
−
Z
t
þ
T
þ
L
t
þ
L
h
00
ð
t
0
;x
AI
2
Þ
dt
0
;
Δ
t
MZ
grad
;
D
ð
t
Þ¼
n
i
½
δ
x
i
ð
t
þ
T; x
AI
1
Þ
−
δ
x
i
ð
t; x
AI
1
Þþ
δ
x
i
ð
t
þ
L; x
AI
2
Þ
−
δ
x
i
ð
t
þ
T
þ
L; x
AI
2
Þ
;
Δ
t
MZ
grad
;
S
ð
t
Þ¼
Z
x
AI
2
x
AI
1
H
þ
ð
t
ð
x
0
Þ
;x
0
Þ
dx
0
−
Z
x
AI
2
x
AI
1
H
þ
ð
t
ð
x
0
Þþ
T; x
0
Þ
dx
0
;
ð
24
Þ
where
δ
x
i
is given in Eq.
(10)
, and
t
ð
x
0
Þ
≡
t
þ
x
0
−
x
AI
1
is the coordinate time parametrizing the initiating photon pulse.
Using Eqs.
(22)
–
(24)
, the Einstein, Doppler, and Shapiro gradiometer phase shifts for an experiment initiated at time
t
0
may
be compactly rewritten as
Δ
φ
MZ
grad
;
E
;
D
;
S
ð
t
0
Þ¼
ω
a
ð
Δ
t
MZ
grad
;
E
;
D
;
S
ð
t
0
Þ
−
Δ
t
MZ
grad
;
E
;
D
;
S
ð
t
0
þ
T
ÞÞ
:
ð
25
Þ
Equivalently, the three contributions may be expressed in the frequency domain. As we show in Appendix
C1
, the
Fourier transforms
10
of Eq.
(25)
can be compactly expressed as
8
One could also choose to describe the experiment in the midpoint trajectory frame
[54]
.
9
Here, we assume that the laser is on resonance with the atom
’
s transition frequency.
10
Throughout the paper, we use the Fourier transform convention commonly adopted by the gravitational wave community:
̃
x
ð
ω
Þ¼
R
∞
−∞
dtx
ð
t
Þ
e
−
i
ω
t
and
x
ð
t
Þ¼ð
2
π
Þ
−
1
R
∞
−∞
d
ω
̃
x
ð
ω
Þ
e
i
ω
t
, where
ω
≡
2
π
f
[55]
.
SIGNATURES OF LINEARIZED GRAVITY IN ATOM
...
PHYS. REV. D
111,
042002 (2025)
042002-7
Δ
̃
φ
MZ
grad
;
E
ð
ω
Þ¼
ω
a
T
2
ω
2
K
MZ
ð
ω
Þ
−
1
2
i
ω
̃
h
00
ð
ω
;x
AI
Þ
e
i
ω
ð
x
AI
−
x
AI
1
Þ
x
AI
1
x
AI
2
;
Δ
̃
φ
MZ
grad
;
D
ð
ω
Þ¼
ω
a
T
2
ω
2
K
MZ
ð
ω
Þ
h
n
i
δ
e
x
i
ð
ω
;x
AI
Þ
e
i
ω
ð
x
AI
−
x
AI
1
Þ
i
x
AI
1
x
AI
2
;
Δ
̃
φ
MZ
grad
;
S
ð
ω
Þ¼
ω
a
T
2
ω
2
K
MZ
ð
ω
Þ
h
Δ
̃
T
þ
S
ð
ω
;x
AI
1
;x
AI
Þ
i
x
AI
1
x
AI
2
;
ð
26
Þ
where
ω
is the angular frequency and
Δ
̃
T
þ
S
is defined
according to Eq.
(16)
. For convenience, we define the
response kernel for the MZ pulse sequence as
K
MZ
ð
ω
Þ¼
e
i
ω
T
sinc
2
ω
T
2
;
ð
27
Þ
which is suppressed at angular frequencies
ω
≳
1
=
2
T
and
asymptotically becomes a pure phase factor at low frequen-
cies, as expected. Importantly, Eqs.
(26)
and
(27)
can be
directly used to calculate the power spectrum density of the
total AG phase shift in a MZ gradiometer induced by an
arbitrary metric perturbation.
2. Large-momentum-transfer gradiometer
The LMT sequence is a type of gradiometer that employs
multiple
“
kicks
”
to increase an AI
’
s spacetime area, and
therefore an AI
’
s sensitivity. In this configuration, the
“
beam
splitter sequence
”
and
“
mirror sequence
”
consist of
alternating laser pulses that are emitted from laser sources
located at opposite ends of the baseline. Using the con-
vention of Refs.
[35,42]
, a typical LMT sequence consists of
4
n
−
1
pulses, where
n
is even. The first
“
beam splitter
sequence
”
starts with a
π
=
2
pulse emitted at
t
0
by the laser
source closest to the first AI; this is immediately followed
by
n
−
1
consecutive
π
pulses that are emitted alternatingly
from each laser and only interact with the right arm of each
AI, thus incrementally depositing momentum onto the
atoms. This is achieved by tuning the laser
’
s frequency
to the Doppler-shifted narrow transition frequency of the
target atomic wave packet. After a time
∼
nL
, both paths are
in the ground state with relative velocity
n
ω
a
=m
o
.The
“
mirror sequence
”
consists of
2
n
−
1
consecutive
π
pulses,
the first of which is emitted at time
t
0
þ
T
−
ð
n
−
1
Þ
L
by
the laser closest to the second AI; the first set of
n
−
1
pulses
interact with the right arm of each AI; the
π
pulse emitted at
time
t
0
þ
T
from the laser closest to the first AI interacts
with both arms of each AI; the remaining
n
−
1
pulses
interact with the left arm of each AI, thus bringing the
FIG. 2. Spacetime diagram of a gradiometer with Mach-Zehnder pulse sequence (left) and
n
¼
4
LMT pulse sequence (right). The
laser worldlines are shown with gray solid lines, while the pulses they emit are shown as green beam splitter
π
=
2
or yellow
π
pulses;
π
pulses that affect one (two) arm(s) of the AI are dashed (solid). Atoms in the ground state are shown with blue worldlines, while in the
excited state with red. The interaction with the atoms (marked with dots) of the
π
=
2
pulses cause half of the atoms (corresponding to one
arm or the other) to flip from ground to excited (or vice versa), while the
π
pulses cause all the atoms to flip (from ground to excited or
vice versa). Note the LMT sequence features a total of
4
n
−
1
laser pulses.
BADURINA, DU, LEE, WANG, and ZUREK
PHYS. REV. D
111,
042002 (2025)
042002-8