Published as Physical Review A 103, 023305 [2021]

# Generalized gravity-gradient mitigation scheme

Christian Ufrecht<EMAIL_ADDRESS>Institut für Quantenphysik and
Center for Integrated Quantum Science and Technology (IQST), Universität Ulm,
Albert-Einstein-Allee 11, D-89069 Ulm, Germany

###### Abstract

A major challenge in high-precision light-pulse atom interferometric
experiments such as in tests of the weak equivalence principle is the
uncontrollable dependency of the phase on initial velocity and position of the
atoms in the presence of inhomogeneous gravitational fields. To overcome this
limitation, mitigation strategies have been proposed, however, valid only for
harmonic potentials or only for small branch separations in more general
situations. Here, we provide a mitigation formula for anharmonic perturbation
potentials including local gravitational effects that vary on length scales
much smaller than the spatial extent probed by the atoms and originate e.g.
from buildings that surround the experiment. Furthermore, our results are
applicable to general interferometer geometries with arbitrary branch
separation and allow for compensation of Coriolis effects in rotating
reference frames.

##  I Introduction

The high sensitivity of light-pulse atom interferometry with promising
applications as inertial sensor in gravimetry [1, 2], gradiometry [3, 4, 5]
and tests of the weak equivalence principle (WEP) [6, 7, 8] has led to
ambitious proposals on ground [9, 10, 11, 12] and in space [13, 14].

A serious challenge for next-generation atom interferometric high-precision
measurements is posed by non-linearities in the gravitational potential and
Coriolis forces which lead to non-perfect overlap of the trajectories in both
momentum and position after the final laser pulse. As a consequence, the phase
contains the initial position and velocity of the atoms and the contrast drops
dramatically in long-time interferometry [13, 15]. Since control over the
initial conditions is limited [16], mitigation strategies had to be developed.
Interferometer schemes insensitive to initial kinematics in the presence of
homogeneous gradients and rotations can be constructed by folding the
interferometer geometry symmetrically [17, 18]. However, in these schemes also
the dominant part of the phase from linear gravity cancels, including a
possible violation signal in a test of the WEP.

Similar to the mitigation strategies developed for Coriolis effects by using
tip-tilt mirrors [19, 20, 21], the crucial insight to achieve compensation of
initial-condition-dependent phases in the presence of homogeneous gravity
gradients was a modification of the pulse timing [15] or the momentum transfer
of the central pulse e.g. in a Mach-Zehnder (MZ) interferometer as a function
of the gradients [22, 23, 24]. This method reduces the mismatch of the
trajectories at the end of the interferometer sequence while leaving the phase
from linear gravity unaffected. Besides other successive work to mimic an
inertial frame [25, 26], gravity-gradient compensation was extended to
spatially inhomogeneous gravity gradients [24] based on an additional
modification of the momentum transfer of the final laser pulse. However, the
derivation in Ref. [24] is based on the midpoint theorem [27], which becomes
less accurate with increasing branch separation when applied to anharmonic
potentials. Therefore, future proposals including large-momentum-transfer
techniques [28, 29] to increase the space-time area of the interferometer or
long-time interferometry with large branch separation require a further
generalization of the method to such situations.

In this article particular emphasis is put on the perturbative character of
the description which allows the application of the mitigation scheme to
arbitrary anharmonic perturbations in the gravitational potential as for
example present in the experimental setups of Refs. [24, 30]. The formula
derived in the present article is furthermore valid for general interferometer
geometries and arbitrary branch separation. Our derivation within a full
quantum-mechanical framework also allows to consistently include contributions
from wave-packet dynamics.

In Sec. II we briefly review the perturbative formalism employed to derive the
compensation formula for inhomogeneous gravity gradients in Sec. III. In Sec.
IV we generalize the formula to include rotations and finally discuss
conditions for validity of our derivation in Sec. V.

Figure 1: Gravity-gradient compensation in a small cubic potential. In an MZ
interferometer a $\pi/2$ pulse at $t=0$ splits the initial wave function into
two components and transfers the momentum $\hbar k$ on one of them. The atoms
are then redirected by a $\pi$ pulse at $t=T$ and finally the wave function is
recombined by a second $\pi/2$ pulse at $t=2T$. In absence of a non-linear
gravitational potential the atoms follow the unperturbed trajectories (thin
solid lines). These trajectories are modified (dashed lines) in presence of a
small perturbation potential so that the branches do not overlap perfectly in
both momentum and position at the final laser pulse, the interferometer is
then referred to as open. Note that this deviation is displayed strongly
exaggerated in the figure. If the wave vector of the second and the final
pulse is modified appropriately by $\Delta\bm{k}_{\ell}$ (decreased in the
example shown in the figure), the interferometer can be closed (thick solid
lines) and dependence of the phase on the initial conditions is eliminated to
first order.

##  II Path-dependent description

In this article, we rely on the perturbative formalism recently developed in
Refs. [31, 32]. The Hamiltonian

$\hat{H}^{(\alpha)}=\hat{H}_{0}^{(\alpha)}+V(\hat{\bm{r}},t)$ (1)

describes the evolution through the interferometer along the upper
($\alpha=u$) and the lower ($\alpha=l$) branch. It consists of a dominant part
$\hat{H}_{0}^{(\alpha)}$ with respect to which the interferometer is closed
(that is perfect overlap after the final pulse) and a perturbation
$V(\hat{\bm{r}},t)$ which slightly opens the interferometer and renders the
phase dependent on initial position and velocity of the atoms. As an example
we illustrate in Fig. 1 the case of an MZ interferometer where a small cubic
potential leads to a slight mismatch of the trajectories at the final laser
pulse and explain how custom-tailored laser pulses can mitigate this effect.

In a gravimeter configuration the natural choice for the unperturbed
Hamiltonian is

$\hat{H}_{0}^{(\alpha)}=\frac{\hat{\bm{p}}^{2}}{2m}-m\bm{g}\hat{\bm{r}}+V_{\mathrm{em}}^{(\alpha)}(\hat{\bm{r}},t)$
(2)

where $m$ is the mass of the atoms and $\bm{g}$ is the local linear
acceleration. Furthermore,

$V_{\mathrm{em}}^{(\alpha)}(\hat{\bm{r}},t)=-\hbar\sum_{\ell}[\bm{k}^{(\alpha)}_{\ell}\hat{\bm{r}}+\varphi_{\ell}^{(\alpha)}]\delta(t-t_{\ell})$
(3)

is the effective laser interaction potential, imprinting the momentum
$\hbar\bm{k}_{\ell}$ and the laser phase $\varphi_{\ell}^{(\alpha)}$ at time
$t_{\ell}$ on branch $\alpha$. Additional contributions to the gravitational
potential such as gradients of Earth’s gravitational potential, gravitational
fields of the local environment such as from buildings surrounding the
experiment [30] etc., are incorporated into the perturbation potential
$V(\hat{\bm{r}},t)$ and treated perturbatively.

The phase $\varphi$ and contrast $C$ of a matter-wave interferometer is
defined as

$\langle\hat{U}^{(l)\dagger}\hat{U}^{(u)}\rangle=\langle\mathrm{e}^{\mathrm{i}\hat{\phi}}\rangle=C\mathrm{e}^{\mathrm{i}\varphi}$
(4)

where $\hat{U}^{(\alpha)}$ is the time-evolution operator with respect to
Hamiltonian (1) for the respective branch and the expectation value is taken
with respect to the initial wave function. In Refs. [32, 31] we merged the two
time-evolution operators on the left-hand side in favour of the operator
$\hat{\phi}$, which reads to first order in the perturbation

$\hat{\phi}=\phi_{0}-\frac{1}{\hbar}\oint\\!\mathrm{d}t\,V(\hat{\bm{r}}(t))$
(5)

and where $\phi_{0}$ is the interferometer phase for vanishing perturbation.
The integral runs along the upper branch and returns along the lower. It is
taken over the perturbation potential evaluated at the branch-dependent and
operator-valued Heisenberg trajectories $\hat{\bm{r}}^{(\alpha)}(t)$ generated
by the unperturbed Hamiltonian (2).

The Heisenberg trajectories can be decomposed into the sum of the classical
trajectory with the initial conditions given by the initial mean position and
velocity of the wave packet and a fluctuation operator of the form
$\hat{\overline{\bm{r}}}(t)=\hat{\bm{r}}-\langle\hat{\bm{r}}\rangle+[\hat{\bm{p}}-\langle\hat{\bm{p}}\rangle]t/m$
where the expectation values are taken with respect to the initial wave
function [32]. This decomposition is valid as the unperturbed Hamiltonian is
only linear in the position operator. The standard deviation of the
fluctuation operator is a measure for the size of the expanding wave packet
and characterizes wave-packet effects. In a modification to the form from Ref.
[32] we define the fluctuation operator

$\delta\hat{\overline{\bm{r}}}(t)=\delta\bm{r}_{0}(t)+\hat{\bm{r}}-\langle\hat{\bm{r}}\rangle+\frac{\hat{\bm{p}}-\langle\hat{\bm{p}}\rangle}{m}t$
(6)

which includes the deviation of the trajectories $\delta\bm{r}_{0}(t)$ due to
uncertainties in the initial conditions. Thus, the Heisenberg trajectory reads

$\hat{\bm{r}}^{(\alpha)}(t)=\bm{r}^{(\alpha)}_{0}(t)+\delta\hat{\overline{\bm{r}}}(t)$
(7)

where $\bm{r}^{(\alpha)}_{0}(t)$ is the classical unperturbed trajectory
without this deviation. Consequently, we find for the expectation value that
$\langle\delta\hat{\overline{\bm{r}}}(t)\rangle=\delta\bm{r}_{0}(t)$. Note
that all expectation values are taken with respect to the initial wave
function. In summary, in the real classical unperturbed trajectories
$\bm{r}_{0}^{(\alpha)}(t)+\delta\bm{r}_{0}(t)$ we choose
$\bm{r}_{0}^{(\alpha)}(t)$ as a fixed reference while $\delta\bm{r}_{0}(t)$
describes their uncertainty due to e.g. limited initial characterization time
[16].

Making use of the decomposition from Eq. (7), a small value of
$\delta\bm{r}_{0}(t)$ and a small wave-packet size will allow for a Taylor
expansion of the perturbation potential around the classical unperturbed
trajectories in the next section. The dominant linear contribution of
$\delta\hat{\overline{\bm{r}}}(t)$ in $\hat{\phi}$ not only introduces a
dependence on the initial conditions but also leads to a loss of contrast [13,
15] when calculating the expectation value in Eq. (4). Following Refs. [22],
slightly modifying the momentum transfer of the laser pulses will eliminate
$\delta\hat{\overline{\bm{r}}}(t)$ to leading order and therefore strongly
mitigate these two effects.

In our fully quantum-mechanical treatment phase contribution that arise from
the square and higher powers of $\delta\hat{\overline{\bm{r}}}(t)$ include
both the residual, strongly suppressed dependence on the initial conditions
and wave-packet effects due to e.g. different dynamics along the
interferometer branches.

##  III Gravity Gradients

As anticipated in the previous section, we start by replacing

$\bm{k}_{\ell}^{(\alpha)}\rightarrow\bm{k}_{\ell}^{(\alpha)}+\Delta\bm{k}_{\ell}^{(\alpha)}$
(8)

for each laser pulse and modify the effective laser-atom interaction potential
in Eq. (3) accordingly. As the validity of the perturbative approach requires
a closed unperturbed interferometer [32], we keep the unperturbed Hamiltonian
(2) unchanged and therefore consider

$\Delta
V_{\mathrm{em}}^{(\alpha)}(\bm{\hat{r}},t)=-\hbar\sum_{\ell}\Delta\bm{k}^{(\alpha)}_{\ell}\hat{\bm{r}}\delta(t-t_{\ell})$
(9)

as part of the perturbation potential $V(\hat{\bm{r}},t)$. In the following we
will omit the branch index $\alpha$ whenever possible. Furthermore, quantities
without operator hat are understood to be evaluated at the unperturbed
trajectories $\bm{r}_{0}(t)$ and we will omit the explicit time dependence.

Inserting Eq. (7) into the perturbation potential followed by Taylor expansion
about the classical trajectories, we find
$V(\hat{\bm{r}})=V-m\bm{a}\delta\hat{\overline{\bm{r}}}+...$ with the
acceleration $\bm{a}=-\nabla V/m$ and consequently to first order in
$\delta\hat{\overline{\bm{r}}}$

$\hat{\phi}=\phi_{0}-\frac{1}{\hbar}\oint\\!\mathrm{d}t\,\left(V-m\bm{a}\delta\hat{\overline{\bm{r}}}\right)+\Delta\phi_{k}+\sum_{\ell}\Delta\bm{k}_{\ell}\delta\hat{\overline{\bm{r}}}(t_{\ell})\,.$
(10)

The last contribution in Eq. (10) and the phase

$\Delta\phi_{k}=\sum_{\ell}\Delta\bm{k}_{\ell}^{(u)}\bm{r}_{0}^{(u)}(t_{\ell})-\Delta\bm{k}_{\ell}^{(l)}\bm{r}_{0}^{(l)}(t_{\ell})$
(11)

originate from the perturbation $\Delta V_{\mathrm{em}}$ evaluated at
Heisenberg trajectories and we abbreviated
$\Delta\bm{k}_{\ell}=\Delta\bm{k}^{(u)}_{\ell}-\Delta\bm{k}^{(l)}_{\ell}$ to
alleviate notation.

Thanks to the simple form of the unperturbed Hamiltonian we find
$\delta\bm{r}_{0}=\delta\bm{r}_{\mathrm{i}}+\delta\bm{v}_{\mathrm{i}}t$ where
$\delta\bm{r}_{\mathrm{i}}$ and $\delta\bm{v}_{\mathrm{i}}$ are the
uncertainties of initial position and velocity. Thus, making additionally use
of the explicit form of the fluctuation operator shown in Eq. (6), all terms
in Eq. (10) linear in $\delta\hat{\overline{\bm{r}}}$ vanish if we require
that

$\bm{J}_{0}=-\sum_{\ell}\Delta\bm{k}_{\ell}\;\;\;\text{and}\;\;\;\bm{J}_{1}=-\sum_{\ell}\Delta\bm{k}_{\ell}t_{\ell}$
(12)

with the abbreviations

$\bm{J}_{0}=\frac{m}{\hbar}\oint\\!\mathrm{d}t\,\bm{a}(t)\;\;\;\text{and}\;\;\;\bm{J}_{1}=\frac{m}{\hbar}\oint\\!\mathrm{d}t\,\bm{a}(t)t\,.$
(13)

After eliminating the operator-valued terms in Eq. (10), the operator
$\hat{\phi}$ has become a $c$-number (to the order considered here) and we
find

$\varphi=\phi_{0}-\frac{1}{\hbar}\oint\\!\mathrm{d}t\,V+\Delta\phi_{k}$ (14)

where we stress again that $V$ is evaluated at the classical unperturbed
trajectories $\bm{r}_{0}^{(\alpha)}$. The linear set of equations in Eq. (12)
can be solved in general if we slightly change the wave vectors of the laser
at two different times, say $t_{1}$ and $t_{2}$, so that we find

$\Delta\bm{k}_{1}=-\frac{\bm{J}_{1}-\bm{J}_{0}t_{2}}{t_{1}-t_{2}}\;\;\;\;\;\text{and}\;\;\;\;\;\Delta\bm{k}_{2}=\frac{\bm{J}_{1}-\bm{J}_{0}t_{1}}{t_{1}-t_{2}}\,.$
(15)

The functions $\bm{J}_{0}$ and $\bm{J}_{1}$ in Eq. (13) allow an intuitive
interpretation. The former is proportional to the integrated differential
acceleration between the branches, while the latter corresponds to its average
over time. As shown in Appendix A and B, the dependence of $\Delta\bm{k}_{1}$
and $\Delta\bm{k}_{2}$ on these quantities allows to design interferometer
geometries for which the mitigation scheme simplifies. Furthermore, in Eq.
(15) any two distinct laser pulses can be chosen that not necessarily have to
correspond to the second and final laser pulse.

Once $\Delta\bm{k}_{1}$ and $\Delta\bm{k}_{2}$ are known, the shifts in
momentum transfer can be distributed between the two branches satisfying
$\Delta\bm{k}_{\ell}=\Delta\bm{k}_{\ell}^{(u)}-\Delta\bm{k}_{\ell}^{(l)}$. For
example we find in case of a laser pulse imprinting opposite momentum on the
two branches that
$\Delta\bm{k}_{\ell}^{(u)}=-\Delta\bm{k}_{\ell}^{(l)}=\Delta\bm{k}_{\ell}/2$
as for instance in case of the central pulse in Fig. 1.

Before we generalize Eq. (15) to rotating frames in the next section, we
conclude by the following remarks.

The key advantage of the mitigation scheme is that the accuracy to which
initial momentum and position of the atoms need to be determined is
significantly relaxed, which might otherwise take longer than the experiment
itself in future satellite-based WEP tests [16].

After mitigation of the initial kinematics the phase still depends on the
local gradients [33] (second and third term on the right-hand side of Eq.
(14)) and it was discussed whether it is meaningful to extent the mitigation
scheme to also compensate these contributions [34, 35]. While in principle
compensation of the initial conditions can be achieved without knowledge of
the gravitational background by calibrating the interferometer prior to the
measurement [8, 24], extension of the mitigation scheme to cancel all
gradient-dependent phases would still require a precise characterization of
the gravitational background as in classical tests with torsion balances [36].
It therefore seems more practical to postcorrect these phases. This
postcorrection of course can only be done to the accuracy by which the gravity
gradients are known. Therefore, it needs to be guaranteed that the remaining
phase shifts only influence the measurement result below the target accuracy.

In atom interferometric tests of the WEP the phases of two interferometers
operated with different atomic isotopes or atomic species are compared. In the
latter case the wave numbers of the lasers are generally different. In order
to compare the differential effective gravitational acceleration, the phases
have to be rescaled by the respective wave numbers (assuming the same
interferometer time $T$) before taking the difference. In ground-based tests
this procedure is only meaningful as long as the uncertainty in the wave
vectors is smaller than the target accuracy of the WEP violation parameter. In
microgravity, however, this constraint is significantly relaxed. If the
mitigation scheme is applied, remaining gradient-induced phase shifts
independent of the initial kinematics cancel differentially in case of
homogeneous gradients. In case of locally varying gravitational potentials,
however, the atoms feel different local potentials along the species-dependent
trajectories. As a consequence, these phase contributions are only suppressed
in the differential rescaled phase but not cancelled. While this remaining
differential phase is small, it might nevertheless impose limits on future
tests of the WEP on ground if the gravitational background is not known
precisely.

In Eq. (10) the Taylor expansion is truncated at first order in
$\delta\hat{\overline{\bm{r}}}$. Corrections to the phase from higher powers
in the fluctuation operator can be calculated with the cumulant expansion [37,
31, 32], however, are often negligible [32]. Corrections of this kind will be
discussed in more detail in Sec. V. As anharmonic potentials are treated
quantum mechanically in this work rather than within a semiclassical
approximation, our results also cover the application of large-momentum
transfer techniques where the branch separation can become comparable to the
spatial extent probed by the atoms. In Appendix A we explain the
approximations needed to obtain the expressions derived in the Supplemental
material of Ref. [24] and discuss their validity. We furthermore show how Eq.
(15) reduces to the result originally derived in Ref. [22] for an MZ
interferometer in presence of homogeneous gravity gradients. In Appendix B we
investigate simplifications of our general results in case of trajectories
symmetric in time. We stress the importance of treating the perturbation
potential locally [38]. For instance the gravitational profile reported in
Ref. [30] cannot be Taylor expanded over the extent of the interferometer due
to its variations on short lengths scales. A numerical integration of Eq. (13)
for this example shows that these local perturbations influence the value of
$\Delta\bm{k}_{1}$ at the ten-percent level and above.

##  IV Rotations

For experiments in a rotating reference frame Coriolis and centrifugal forces
need to be considered additionally. Fortunately, it is straightforward to
extend our result to such situations as will be shown next. The Hamiltonian in
a rotating frame is obtained by adding

$\hat{H}_{\Omega}=\bm{\Omega}\cdot(\bm{\hat{p}}\times\bm{\hat{r}})$ (16)

to Hamiltonian (1) where $\bm{\Omega}$ is the rotation frequency. Additional
centrifugal forces present for example in a reference frame fixed on Earth’s
surface only redefine the direction and absolute value of $\bm{g}$. In
complete analogy to Sec. III we integrate $\hat{H}_{\Omega}$ along the
Heisenberg trajectories shown in Eq. (7) and recall that
$\hat{\bm{p}}(t)=m\,\mathrm{d}/\mathrm{d}t\,\hat{\bm{r}}(t)$. Consequently,
Eq. (10) is extended by the term

$-\frac{1}{\hbar}\oint\\!\mathrm{d}t\,\bm{\Omega}\cdot[\bm{\hat{p}}(t)\times\bm{\hat{r}}(t)]=\phi_{\Omega}+\frac{2m}{\hbar}\oint\\!\mathrm{d}t\,[\bm{v}_{0}(t)\,\times\,\bm{\Omega}]\cdot\delta\hat{\overline{\bm{r}}}(t)$
(17)

where we neglected terms quadratic in the fluctuation operator and made use of
partial integration for which we appreciated that the unperturbed
interferometer is closed. Furthermore, $\bm{v}_{0}(t)$ is the velocity of the
atoms on the unperturbed trajectories and we abbreviated

$\phi_{\Omega}=-\frac{1}{\hbar}\oint\\!\mathrm{d}t\,\bm{\Omega}\cdot[\bm{p}_{0}(t)\times\bm{r}_{0}(t)]\,.$
(18)

Consequently, by comparing to Eq. (10), the mitigation schemes can be
generalized to rotating reference frames with the replacement

$\bm{a}(t)\rightarrow\bm{a}(t)+2\bm{v}_{0}(t)\times\bm{\Omega}$ (19)

in Eq. (13) and by adding $\phi_{\Omega}$ to Eq. (14). Alternatively, the
effects of rotations can be analyzed in a non-rotating frame, where the laser
is rotating instead [39].

##  V Validity of perturbative treatment

In the previous section we developed a general mitigation scheme based on a
perturbative treatment, covering both rotations and gravity gradients. In the
following we discuss the validity of this approach and the approximations made
in the derivation.

In a perturbative calculation of the phase in powers of the potential $V$
subsequent orders are suppresses by $\epsilon=\Delta VT^{2}/(m\xi^{2})$ [32]
where $\Delta V$ is the characteristic change of the potential over the
interferometer size, $\xi$ is the typical length scale on which the potential
changes and $T$ the characteristic interferometer time. The parameter
$\epsilon$ can be understood as the deviation of the trajectories caused by
the perturbation compared to the length $\xi$. For gravity gradients on
Earth’s surface corresponding to the potential
$V=m\bm{r}^{\mathrm{T}}\Gamma\bm{r}/2$, one would choose $\xi$ as the extent
of the interferometer, given approximately by $\xi=gT^{2}/2$ in a gravimeter
configuration. Thus, $\Delta V\sim m\Gamma\xi^{2}/2$ and consequently
$\epsilon=\Gamma T^{2}$, leading to a value $\epsilon<10^{-5}$ for typical
interferometer times. Local variations as in the gravitational potential of
Ref. [30], in contrast, can lead to values of $\xi$ much smaller than the
spatial extent probed by the atoms. A similar suppression factor for rotations
takes the form $\epsilon_{\Omega}=\Omega T$ with $\epsilon_{\Omega}<10^{-4}$
for the rotation of Earth. Consequently, the relative uncertainty in the phase
achieved by a first-order calculation already is of the order of $\epsilon$.

The term $m\oint\\!\mathrm{d}t\,\bm{a}\delta\hat{\overline{\bm{r}}}/\hbar$ in
Eq. (10) introduces the dominant dependence on the initial conditions.
Estimating $\nabla V\sim\delta V/\xi$ [32] where $\delta V$ is the change of
the potential over the branch separation and introducing the abbreviation
$\eta=\delta VT/\hbar$, this phase contribution scales as $\eta\delta
r_{0}/\xi$.

Application of the mitigation scheme requires prior knowledge of the
gravitational background which can be obtained by measurement, numerical
simulation of the gravitational sources surrounding the experiment, or a
combination of both. However, determination of deviations from linear gravity
will only be possible to some relative uncertainty $\kappa$, which then also
constitutes the suppression factor for initial-condition dependent phases.
Estimations suggest that at least $\kappa=10^{-3}$ seems plausible [24, 14],
thereby considerably relaxing the requirements on determination of initial
position and velocity of the atoms. As described in the beginning of this
section, further terms linear in the initial conditions which would result
from the second-order calculation in the perturbation potential are suppressed
by $\epsilon$ compared to the first-order terms. Consequently, an extension of
the mitigation scheme to second order in the perturbation [31] is only
necessary if $\kappa<\epsilon$ since otherwise initial-condition-dependent
phases that are compensated only partially are still larger than contributions
from the second-order calculation. Higher-order corrections to the dominant
phase in Eq. (14), in contrast, can be obtained as shown in detail in Ref.
[32].

Moreover, Taylor expansion of Eq. (5) around the classical unperturbed
trajectories to first order neglects terms scaling as
$\eta(\delta\hat{\overline{\bm{r}}}/\xi)^{2}$. Comparing to the residual
contribution $\kappa\,\eta\delta\hat{\overline{\bm{r}}}/\xi$ from the first-
order calculation, these terms and further corrections can be disregarded if
$\delta r_{\mathrm{i}}/\xi<\kappa$ and $\delta v_{\mathrm{i}}T/\xi<\kappa$.
For $\delta r_{\mathrm{i}}\sim 1$µm as well as $\delta v_{\mathrm{i}}\sim
1$µm/s and $\xi\sim 1\,\mathrm{m}$ the latter requirements are satisfied well.

In addition, wave-packet effects originating from different expansion dynamics
along the branches in an anharmonic potential are generally small but
straightforward to include if necessary.

##  VI Discussion

Finally, we conclude by the following remarks. The acceleration in Eq. (13)
not necessarily points in direction of momentum transfer. Consequently,
gravity-gradient compensation might also require a tilt of the mirrors in
order to adapt the direction of $\bm{k}$ appropriately. However, generally for
experiments on Earth’s surface the required modification of momentum transfer
orthogonal to the sensitive axis ($\bm{k}$ pointing in direction of $\bm{g}$)
often is much smaller than the parallel component due to symmetry in the mass
distribution surrounding the apparatus [30].

Obviously, the compensation method is equally applicable to perturbations of
non-gravitational origin. However, initial condition-dependent phases from
e.g. magnetic field gradients [40], black-body radiation [41], etc. are
generally much smaller than those from gravity and can be neglected.

Moreover, to avoid the necessity of a precise characterization of the
gravitational background, the compensation scheme can be implemented
experimentally through calibration prior to the measurement by introducing
artificial large deviations of the initial conditions [8, 24].

In the reference frame of an inertial-pointing satellite orbiting Earth the
gravitational potential is time dependent. In a WEP test the varying
projection of a possible violation signal on the sensitive axis can be
utilized to demodulate systematic effects [14]. This technique also might
relax the required accuracy [42] to which the gravity gradients have to be
measured for application of the mitigation scheme. Note that formula (15) also
applies to time-dependent gravitational potentials as in this situation.

##  VII Acknowledgements

The author thanks É. Wodey and S. Loriani for fruitful discussions and W. P.
Schleich, A. Roura, A. Friedrich, F. Di Pumpo and E. Giese for a careful
reading of the manuscript. This work is supported by the German Aerospace
Center (Deutsches Zentrum für Luft- und Raumfahrt, DLR) with funds provided by
the Federal Ministry for Economic Affairs and Energy (Bundesministerium für
Wirtschaft und Energie, BMWi) due to an enactment of the German Bundestag
under Grant Nos. DLR 50WM1556 and 50WM1956. The author thanks the Ministry of
Science, Research and Art Baden-Württemberg (Ministerium für Wissenschaft,
Forschung und Kunst Baden-Württemberg) for financially supporting the work of
IQST.

## Appendix

##  Appendix A Weakly varying potential

In this appendix we start from the general result in Eq. (15) and rederive the
result of Ref. [24] in case of small branch separation. The modified wave
vector in Eq. (15) is a function of the atom’s mass as the gravitational
potential $\Phi$ with $V=m\Phi$ is evaluated at the mass-dependent
trajectories. However, if the local acceleration varies only moderately over
the branch separation (of the order of centimeter for a few $\hbar k$ momentum
transfer and a 10-m baseline) this dependence cancels out and we will find the
result of Ref. [24] for an MZ interferometer.

###  A.1 Local gravity gradients

To prove this statement, we first decompose the trajectories

$\bm{r}_{0}^{(\alpha)}(t)=\overline{\bm{r}}(t)+\bm{r}^{(\alpha)}_{*}(t)$ (20)

into a suitably chosen branch-independent mean trajectory
$\overline{\bm{r}}(t)$ and the deviation $\bm{r}^{(\alpha)}_{*}(t)$. Thus, we
Taylor expand

$\bm{a}^{(\alpha)}(\bm{r}_{0})=\bm{a}(\overline{\bm{r}})-\Gamma(\overline{\bm{r}})\bm{r}^{(\alpha)}_{*}+...$
(21)

where the gradient tensor is defined as
$\Gamma_{ij}=\partial_{i}\partial_{j}V/m$. Substituting Eq. (21) into Eq.
(13), we find

Figure 2: Compensation for uniform gravity gradients. a) In an MZ
interferometer the position of the geometric center $t_{c}$ on the time axis
of the space-time area $\bm{A}$ enclosed by the two branches coincides with
the position of the central pulse. For this reason, as shown in the main text,
global gravity gradients can be compensated by adapting the momentum transfer
of the central pulse only. The required modification is proportional to the
space-time area enclosed by the trajectories. b) In a Ramsey-Bordé
interferometer the geometric center is situated exactly in between the two
central pulses at $t=T_{1}$ and $t=T_{1}+T_{2}$. In this case compensation can
be achieved by modifying the momentum transfer of these pulses equally.

$\bm{J}_{0}=-\frac{m}{\hbar}\oint\\!\mathrm{d}t\,\Gamma(\overline{\bm{r}})\bm{r}_{*}\;\;\;\text{and}\;\;\;\bm{J}_{1}=-\frac{m}{\hbar}\oint\\!\mathrm{d}t\,\Gamma(\overline{\bm{r}})\bm{r}_{*}\,t$
(22)

since $\bm{a}(\overline{\bm{r}})$ is independent of the branch and therefore
cancels in the looped integrals. If the mean trajectory $\overline{\bm{r}}$
only contains the mass-independent part of the trajectory generated by linear
gravity while $\bm{r}_{*}$ is the additional contribution from the laser
pulses, Eq. (22) becomes mass independent since $\bm{r}_{*}$ is inversely
proportional to the mass through $\bm{v}_{r}=\hbar\bm{k}/m$. To connect with
previous results, we specify the case of an MZ gravimeter where the atoms are
launched initially in $z$ direction so that
$\overline{\bm{r}}=gt(T-t/2)\bm{\mathrm{e}}_{z}$. Consequently, with

$\displaystyle\bm{r}_{*}^{(u)}=\bm{v}_{r}t\,,\quad\quad\quad$
$\displaystyle\bm{r}_{*}^{(l)}=0\quad\quad\quad\quad\quad\quad$ $\displaystyle
0\leq t<T$ (23) $\displaystyle\bm{r}_{*}^{(u)}=\bm{v}_{r}T\,,$
$\displaystyle\bm{r}_{*}^{(l)}=\bm{v}_{r}(t-T)$ $\displaystyle T\leq t\leq 2T$
(24)

we find from Eq. (22) the expressions

$\bm{J}_{0}=-\int\limits_{0}^{T}\\!\mathrm{d}t\,t\Gamma(\overline{\bm{r}})\bm{k}-\int\limits_{T}^{2T}\\!\mathrm{d}t\,(2T-t)\Gamma(\overline{\bm{r}})\bm{k}$
(25)

and

$\bm{J}_{1}=-\int\limits_{0}^{T}\\!\mathrm{d}t\,t^{2}\Gamma(\overline{\bm{r}})\bm{k}-\int\limits_{T}^{2T}\\!\mathrm{d}t\,(2T-t)t\Gamma(\overline{\bm{r}})\bm{k}$
(26)

derived in the Supplemental material of Ref. [24] after appropriate
resummation of the integrals.

In Eq. (21) corrections from the next order of the Taylor expansion are
suppressed by $v_{r}T/\xi$ where $\xi$, the characteristic length scale on
which the potential changes. For the local variations in the gravitational
profile from Ref. [30] the factor $v_{r}T/\xi$ might approach unity in future
experiments involving large-momentum transfer techniques and therefore limits
the validity of Eqs. (25) and (26). Instead, using the midpoint theorem [27]
without the approximation in Eq. (21), the result still deviates from the
exact expressions in Eq. (13) but only by a factor $(v_{r}T/\xi)^{2}$ which
justifies its application in many cases but care has to be taken when
employing LMT techniques or the potential changes on short length scales. This
deviation results from the semiclassical approximation in the derivation of
the midpoint theorem which limits its application to anharmonic potentials.

###  A.2 Global gravity gradients

So far we have discussed general anharmonic perturbations which might even
change rapidly over the branch separation. In this paragraph we assume that
the deviations from linear gravity are smooth enough to be accurately
described over the extent of the whole interferometer by a global gradient.
Correspondingly,

$V=\frac{1}{2}m\bm{r}^{\mathrm{T}}\Gamma\bm{r}$ (27)

where the position-independent gradient tensor $\Gamma$ is chosen fully
symmetric. In case all laser pulses are aligned, we define the vector
$\bm{A}=\oint\\!\mathrm{d}t\,\bm{r}_{0}$, whose modulus corresponds to the
space-time area enclosed by the two branches of the unperturbed
interferometer. With $\bm{a}=-\Gamma\bm{r}_{0}$ the expression

$\bm{J}_{0}t_{c}=\bm{J}_{1}$ (28)

defines the position of the geometric center $t_{c}$ of this area on the time
axis and we can distinguish two different situations corresponding to the two
classes of interferometer geometries displayed in Fig. 2: a) Suppose the
interferometer exhibits a laser pulse at $t_{1}=t_{c}$ as for example in the
MZ interferometer visualized in Fig. 2 a). From Eq. (15) together with Eq.
(28) we find

$\Delta\bm{k}_{1}=\frac{m}{\hbar}\Gamma\bm{A}\,,\;\;\;\text{and}\;\;\;\Delta\bm{k}_{2}=0$
(29)

which agrees with the result of Ref. [22] for an MZ interferometer with
$\bm{A}=\bm{v}_{r}T$ where the modification of momentum transfer is
distributed equally over both branches. Thus, in case of uniform gradients,
compensation is particularly simple if the interferometer exhibits a laser
pulse at the geometric center of the space-time area enclosed by the branches.
b) In contrast if $t_{c}$ is located exactly in between two pulses at $t_{1}$
and $t_{2}$, that is $t_{c}=(t_{1}+t_{2})/2=T_{1}+T_{2}/2$, we find

$\Delta\bm{k}_{1}=\Delta\bm{k}_{2}=\frac{m}{2\hbar}\Gamma\bm{A}\,.$ (30)

This situation is for example found in a Ramsey-Bordé interferometer shown in
Fig. 2 b) for which we find $\bm{A}=\bm{v}_{r}T_{1}(T_{1}+T_{2})$.

##  Appendix B Symmetric trajectories

In this appendix we consider interferometer schemes symmetric around a pulse
at time $t_{s}$, that is $\bm{r}_{0}(t_{s}+t)=\bm{r}_{0}(t_{s}-t)$ for both
branches. Since a function $q(t)$ symmetric around time $t_{s}$ satisfies

$\int_{0}^{2t_{s}}\\!\mathrm{d}t\,q(t)t=t_{s}\int_{0}^{2t_{s}}\\!\mathrm{d}t\,q(t)\,,$
(31)

we conclude from Eq. (13) that

$\bm{J}_{0}t_{s}=\bm{J}_{1}\,.$ (32)

Consequently, with the help of Eq. (15) the compensation scheme simplifies to

$\Delta\bm{k}_{1}=-\bm{J}_{0}\quad\quad\text{and}\quad\quad\Delta\bm{k}_{2}=0$
(33)

for e.g. $t_{1}=t_{s}$, so that only the pulse at this time must be modified.
A geometry satisfying this symmetry requirement is realized for example in an
MZ interferometer with initial velocity $v_{0z}=gT-v_{r}/2$ of the atoms in
direction of the subsequent momentum transfer. This result remains a good
approximation if the branches are only approximately symmetric [24].

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