# Atom interferometry using $\sigma^{+}-\sigma^{-}$ Raman transitions between
$\ket{F=1,m_{F}=\mp 1}$ and $\ket{F=2,m_{F}=\pm 1}$

J. Bernard<EMAIL_ADDRESS>DPHY,ONERA, Université Paris Saclay,
F-91123 Palaiseau, France  LCM-CNAM, 61 rue du Landy, 93210, La Plaine Saint-
Denis, France  Y. Bidel  DPHY,ONERA, Université Paris Saclay, F-91123
Palaiseau, France  M. Cadoret  LCM-CNAM, 61 rue du Landy, 93210, La Plaine
Saint-Denis, France  C. Salducci  DPHY,ONERA, Université Paris Saclay, F-91123
Palaiseau, France  N. Zahzam  DPHY,ONERA, Université Paris Saclay, F-91123
Palaiseau, France  S. Schwartz  DPHY,ONERA, Université Paris Saclay, F-91123
Palaiseau, France  A. Bonnin  DPHY,ONERA, Université Paris Saclay, F-91123
Palaiseau, France  C. Blanchard  DPHY,ONERA, Université Paris Saclay, F-91123
Palaiseau, France  A. Bresson  DPHY,ONERA, Université Paris Saclay, F-91123
Palaiseau, France

###### Abstract

We report on the experimental demonstration of a horizontal accelerometer
based on atom interferometry using counterpropagative Raman transitions
between the states $\ket{F=1,m_{F}=\mp 1}$ and $\ket{F=2,m_{F}=\pm 1}$ of
87Rb. Compared to the $\ket{F=1,m_{F}=0}\leftrightarrow\ket{F=2,m_{F}=0}$
transition usually used in atom interferometry, our scheme presents the
advantages to have only a single counterpropagating transition allowed in a
retroreflected geometry, to use the same polarization configuration than the
magneto-optical trap and to allow the control of the atom trajectory with
magnetic forces. We demonstrate horizontal acceleration measurement in a
close-to-zero velocity regime using a single-diffraction Raman process with a
short-term sensitivity of $25\times 10^{-5}$ m.s-2.Hz-1/2. We discuss specific
features of the technique such as spontaneous emission, light-shifts and
effects of magnetic field inhomogeneities. We finally give possible
applications of this technique in metrology or for cold-atom inertial sensors
dedicated to onboard applications.

###### pacs:

##  I Introduction

During the last two decades, light-pulse atom interferometers (LPAIs) Kasevich
and Chu (1991) which exploit the wave-like interference of atoms, have become
unique instruments for precision measurements of inertial forces, with
applications in both applied and fundamental science. For example, atom
interferometric techniques have been employed in measurements of gravitational
Fixler  _et al._ (2007); Rosi  _et al._ (2014) and fine structure constants
Parker  _et al._ (2018); Morel  _et al._ (2020), test of the equivalence
principle Asenbaum  _et al._ (2020), searches for dark sector particles
Hamilton  _et al._ (2015); Elder  _et al._ (2016); Jaffe  _et al._ (2018), and
even proposed for use in gravitational wave detection Dimopoulos  _et al._
(2008); Canuel  _et al._ (2018). They have also enabled the realization of
high performance accelerometers Canuel  _et al._ (2006); Geiger  _et al._
(2011); Xu  _et al._ (2017); Perrin  _et al._ (2019a), gyroscopes Gustavson
_et al._ (1997); Durfee  _et al._ (2006); Gauguet  _et al._ (2009); Stockton
_et al._ (2011); Tackmann  _et al._ (2012); Berg  _et al._ (2015); Dutta  _et
al._ (2016); Savoie  _et al._ (2018) and gravimeters Peters  _et al._ (2001);
Gillot  _et al._ (2014); Hu  _et al._ (2013); Hauth  _et al._ (2013); Karcher
_et al._ (2018) demonstrating great promise for fielded inertial sensors based
on atom interferometry Bidel  _et al._ (2018, 2020); Wu  _et al._ (2019). In
addition, they can also be utilized for probing the field gradient of an
external field such as gravity McGuirk  _et al._ (2002); Sorrentino  _et al._
(2014); Duan  _et al._ (2014) or magnetic fields Hu  _et al._ (2011); Hardman
_et al._ (2016).

In a LPAI, sequences of laser pulses are used to split, deflect and recombine
matter-waves to create atom interference. In inertial sensors, these sequences
of light pulses commonly use counterpropagating two-photon Raman transitions
with large one-photon detuning Kasevich and Chu (1991) between hyperfine
ground states of alkali atoms (e.g $\ket{F=1}$ and $\ket{F=2}$ for 87Rb). They
form the basic atom-optics elements by finely controlling the external degree
of freedom of the atoms through the generation of coherent superposition of
momentum states. In a counterpropagating configuration, the transfer between
the two internal ground states is always accompanied with a change of
$\pm\hbar k_{\textrm{eff}}$ of the momentum state, where $k_{\textrm{eff}}$ is
the effective wave vector.

In order to achieve high precision measurements, the two counterpropagating
Raman lasers are usually obtained thanks to a retroreflected geometry where a
single laser beam with two laser frequencies is retroreflected off a mirror.
This geometry allows to mitigate parasitic effects induced by wave front
distortions which are critical to achieve good accuracy and long term
stability Schkolnik  _et al._ (2015); Karcher  _et al._ (2018). It also
reduces interferometer phase noise as most vibration effects are common to the
two laser fields. In addition, in order to avoid systematic errors induced by
first order Zeeman effect, the atoms are commonly manipulated in the
magnetically-insensitive $m_{F}=0$ sublevels in the interferometer.

In this work, we report on the experimental realization of a Raman transition-
based LPAI between magnetically-sensitive internal states in a Mach-Zehnder
type geometry. Using the supplementary internal degree of freedom of atoms
manipulated in sensitive magnetic sublevels, we realize a sensor which
simultaneously measures inertial and magnetic accelerations. Our work focuses
on the specific case of 87Rb. Using a $\sigma^{+}-\sigma^{-}$ polarized light
arrangement, we manipulate the atoms in the interferometer between the two
magnetically-sensitive ground states $5S_{1/2}$ $\ket{F=1,m_{F}=\mp
1}\to\ket{F=2,m_{F}=\pm 1}$, also used as atomic clock transition for
magnetically trapped 87Rb Harber  _et al._ (2002), taking benefit of their
similar first-order Zeeman shift. Using this technique we perform the
measurement of the horizontal component of acceleration, in a close-to-zero
velocity regime, using a single-diffraction Raman scheme Perrin  _et al._
(2019a), without need for alternative techniques to lift the degeneracy of the
double diffraction process Lévèque  _et al._ (2009). We demonstrate a short-
term sensitivity of $25\times 10^{-5}$ m.s-2.Hz-1/2 for absolute acceleration
measurement. We then discuss some specifics of our technique in comparison
with usual magnetically-insensitive Raman-based atom interferometers, such as
spontaneous emission rate, additional sensitivity to magnetic field
inhomogeneity and light-shifts. Finally, in light of the advantages of this
technique, we propose atom interferometer designs which could be of interest
in metrology, as well as for improving the performances of cold-atom inertial
sensors in operational field conditions Bidel  _et al._ (2018); Wu  _et al._
(2019); Bidel  _et al._ (2020); Geiger  _et al._ (2011).

##  II Method

###  II.1 Principle and advantages of the method

We implement our method using a horizontal Mach-Zehnder type LPAI based on
counterpropagative stimulated two-photon Raman transitions between the
$\ket{F=1,m_{F}=\mp 1}$ and $\ket{F=2,m_{F}=\pm 1}$ hyperfine levels of the
5S1/2 ground state of 87Rb. These two states are coupled via an intermediate
state, using lasers of frequencies $\omega_{1(2)}$ detuned from the
$\ket{5P_{3/2},F^{\prime}=1(2)}$ state by the one-photon detuning
$\Delta_{1(2)}$ (see Fig.1(a)). A static bias magnetic field of magnitude $B$
aligned with the Raman lasers is applied to define a quantization axis for the
atoms. This field shifts the $\ket{F,m_{F}}$ ground state magnetic sublevels
by $\Delta E=\mu_{B}g_{F}m_{F}B$ as a first approximation, where $F$ is the
atomic total angular momentum, $m_{F}=0,\pm 1,...,\pm F$ are its projections
on the quantization axis, $\mu_{B}$ is the Bohr magneton, and $g_{F}$ is the
Landé factor, equal to $-(+)1/2$ for the $F=1(2)$ states respectively. The two
counterpropagating Raman beams are generated using a retroreflective setup.
Contrary to many LPAI experiments using a lin$\perp$lin polarization
configuration and a large one-photon detuning allowing to exclusively drive
counterpropagative Raman transitions between the magnetic-insensitive
$m_{F}=0$ sublevels, we implement here a $\sigma^{+}-\sigma^{-}$ configuration
(see Fig.1(b)): the Raman beams have a $\sigma^{+}$ polarization in one
direction, and a $\sigma^{-}$ polarization in the retroreflected direction.
The quantum state at the input of the interferometer is prepared to be in one
single Zeeman sublevel $\ket{F=1,m_{F}}$ ($m_{F}=+1$ or $m_{F}=-1$). Thus,
according to the electric dipole transition selection rules, only one
counterpropagating transition is possible. Indeed the $\sigma^{+}-\sigma^{-}$
Raman laser configuration only allows $\Delta m_{F}=\pm 2$ transitions.
Consequently, the two-photon Raman transition couples the magnetically-
sensitive hyperfine states $\ket{F=1,m_{F}=\pm
1}\leftrightarrow\ket{F=2,m_{F}=\mp 1}$ with an effective Rabi frequency Steck
(data):

$\Omega_{\textrm{eff}}=\Gamma^{2}\frac{\sqrt{I_{1}I_{2}}}{4I_{\textrm{sat}}}\frac{1}{4\sqrt{3}}\left(\frac{1}{\Delta_{1}}-\frac{1}{\Delta_{2}}\right)$
(1)

where $\Gamma=2\pi\times 6.07$ MHz is the natural line width,
$I_{\textrm{sat}}=c\pi h\Gamma/3\lambda^{3}=1.67$ mW.cm-2 is the saturation
intensity (with $c$ the speed of light, $h$ the Planck’s constant and
$\lambda=780$ nm), $I_{1(2)}$ are the Raman laser intensities and
$\Delta_{1(2)}$ are the one-photon detunings with respect to the hyperfine
levels $\ket{5P_{3/2},F^{\prime}=1(2)}$ (see Fig.1(a)). The Rabi frequency of
the two-photon Raman transition constrains us to tune the one-photon
transition in between $\ket{F^{\prime}=1}$ and $\ket{F^{\prime}=2}$ in order
to avoid destructive interferences between the transition probability
amplitudes for each excited states $\ket{F^{\prime}=1}$ and
$\ket{F^{\prime}=2}$. Consequences of such a close-to-resonance detuning are
discussed in Section IV.1.

Figure 1: (a) Scheme of the $\sigma^{+}-\sigma^{-}$ Raman transitions between
Zeeman sublevels of the two hyperfine ground states of 87Rb in the presence of
a magnetic field. Solid lines represent the $\sigma^{+}-\sigma^{-}$ polarized
beams performing the $\ket{F=1,m_{F}=-1}\leftrightarrow\ket{F=2,m_{F}=1}$
transition. Dashed lines represent the $\sigma^{-}-\sigma^{+}$ polarized beams
performing the $\ket{F=1,m_{F}=1}\leftrightarrow\ket{F=2,m_{F}=-1}$
transition. (b) Schematic setup of two-photon Raman transitions in the
commonly used retroreflected geometry. A two-level atom is interacting with
two pairs of counterpropagating light fields with $\sigma^{+}-\sigma^{-}$
polarizations. This polarization arrangement allows for only one pair of
couterpropagating light fields to drive the Raman coupling leading to a single
diffraction Raman process despite zero Doppler shift.

LPAIs usually manipulate atoms in the magnetically-insensitive $m_{F}=0$
sublevels. For zero-velocity atoms the use of retroreflected Raman beams leads
naturally to a double-diffraction scheme: two stimulated Raman transitions
with opposite momentum transfer $\pm\hbar\vec{k}_{\textrm{eff}}$ are
simultaneously resonant Lévèque  _et al._ (2009). Our scheme has the advantage
of having only one counterpropagating Raman transition allowed despite the
retroreflection. In addition, we can very easily implement the
$k_{\textrm{eff}}$-reversal technique Peters  _et al._ (2001) to eliminate
some systematics by alternatively preparing the atoms in the
$\ket{F=1,m_{F}=\mp 1}$ states. Moreover the Raman beams have the same
polarization as the magneto-optical trap beams, enabling a more compact and
simple sensor.

###  II.2 Experimental apparatus and lasers

The experiment was carried out in the LPAI setup described in Perrin  _et al._
(2019b, a). The usual steps of atom interferometry (preparation,
interferometry and population detection) were performed with the laser system
described in Theron  _et al._ (2017). On the one hand an Erbium distributed
feedback fiber laser (DFB-FL) at 1.5 $\mu$m, locked to a rubidium transition
through a saturated absorption setup Theron  _et al._ (2015), is used to cool
and detect the atoms. On the other hand a DFB laser diode at 1.5 $\mu$m,
frequency controlled by a beat-note with the fiber laser, provides the LPAI
laser source. The two Raman frequencies are generated using a fibered phase
modulator Carraz  _et al._ (2012). Both lasers are finally combined at 1.5
$\mu$m through an electro-optical modulator acting like a continuous optical
switch between each laser, before seeding a 5 W Erbium-doped fiber amplifier
(EDFA). The output of the EDFA is sent to a second harmonic generation bench.
The complete laser setup and optical bench description can be found in Perrin
_et al._ (2019a).

###  II.3 State preparation and Raman spectroscopy

We investigate our method by selecting the atomic input state and implementing
Raman spectroscopy. A cold 87Rb atom sample is produced in a three-dimensional
magneto-optical trap (MOT) loaded from a background vapor in 840 ms. An
optical molasses cools the atoms down to 2 $\mu$K in 8 ms. After turning off
the cooling beams, the atoms are in free fall. Then a horizontal bias magnetic
field $B\sim 400$ mG is switched on and a microwave $\pi$-pulse is applied,
followed by a blow-away beam, allowing to select the atoms in the
$\ket{F=1,m_{F}=-(+)1}$ state. Raman spectroscopy is performed using a Raman
pulse of duration $\tau=10~{}\mu$s. Finally an internal state-selective
vertical light-induced fluorescence detection is used to measure the
proportion of atoms in each hyperfine state $\ket{F=1}$ and $\ket{F=2}$. The
cycling time of the experiment is $T_{\textrm{cycle}}=1$ s.

Figure 2(a) displays the measured transition probability as a function of the
Raman frequency difference $(\omega_{1}-\omega_{2})/2\pi$. The atoms being
prepared in the $\ket{F=1,m_{F}=-1}$ state, the electric dipole transition
selection rules state that only two transitions are possible with our Raman
beam polarization configuration: a copropagating transition
$\ket{F=1,m_{F}=-1}\rightarrow\ket{F=2,m_{F}=-1}$ (almost insensitive to
Doppler effect and therefore narrower) and a counterpropagating transition
$\ket{F=1,m_{F}=-1}\rightarrow\ket{F=2,m_{F}=1}$ (see Fig.1(a)). A third
transition can be observed in Fig.2(a) due to spontaneous emission: a fraction
of the atoms are transferred by spontaneous emission to the $m_{F}=0,1$
magnetic sublevels of $F=1$ and undergoes one of the two degenerate
transitions $\ket{F=1,m_{F}=1}\rightarrow\ket{F=2,m_{F}=1}$,
$\ket{F=1,m_{F}=0}\rightarrow\ket{F=2,m_{F}=2}$. Spontaneous emission will be
further discussed in Section IV.1.

One can notice not only that the $\sigma^{+}-\sigma^{-}$ transition frequency
is independent of the magnetic field magnitude $B$ at first order (see
Fig.2(c)), but also that the $\pm k_{\textrm{eff}}$ transition frequencies
(corresponding to $\ket{F=1,m_{F}=\mp 1}\leftrightarrow\ket{F=2,m_{F}=\pm 1}$)
are the same (see Fig.2(b)), which is very useful when implementing the
$k_{\textrm{eff}}$-reversal technique (see Section III.1).

In conclusion, once the Raman frequency is properly tuned, it is only the
state preparation in the magnetic sublevel $m_{F}=\mp 1$ that defines which
transition $\pm k_{\textrm{eff}}$ will be addressed during the interferometer.
This means that reversing the effective wave vector is different from LPAIs
using the $\ket{F=1,m_{F}=0}-\ket{F=2,m_{F}=0}$ transition, where the Raman
frequency needs to be changed in order to reverse $k_{\textrm{eff}}$.

Figure 2: Raman resonance spectra obtained by scanning the Raman frequency
$(\omega_{1}-\omega_{2})/2\pi$ across the resonances. (a) Raman spectrum with
the atoms prepared in the $\ket{F=1,m_{F}=-1}$ sublevel. The magnetic field
amplitude is $B=411$ mG. Three resonance peaks are observed and correspond to
four transitions: $\ket{F=1,m_{F}=-1}\rightarrow\ket{F=2,m_{F}=-1}$,
$\ket{F=1,m_{F}=-1}\rightarrow\ket{F=2,m_{F}=1}$ and two degenerate
transitions $\ket{F=1,m_{F}=1}\rightarrow\ket{F=2,m_{F}=1}$ and
$\ket{F=1,m_{F}=0}\rightarrow\ket{F=2,m_{F}=2}$. (b) Raman spectrum with the
atoms prepared in the $\ket{F=1,m_{F}=+1}$ sublevel. The magnetic field
amplitude is $B=411$ mG. (c) The atoms being prepared in the
$\ket{F=1,m_{F}=-1}$ sublevel, the magnetic field amplitude is tuned from 200
mG to 400 mG. The transition $\ket{F=1,m_{F}=-1}\rightarrow\ket{F=2,m_{F}=1}$
is insensitive at first order to a magnetic field amplitude variation.

##  III Atom interferometry with $\sigma^{+}-\sigma^{-}$ transitions

Our Mach-Zehnder type LPAI in a horizontal configuration consists of a
$\pi/2-\pi-\pi/2$ Raman pulse sequence, with each pulse separated by a time
$T$. Due to free fall of atoms across the laser beams of waist 5.5 mm
($1/e^{2}$ radius), the interrogation time is limited to $2T=33$ ms. At the
output of the interferometer the phase shift is the sum of two terms:
$\Delta\phi=\Delta\phi_{\textrm{propagation}}+\Delta\phi_{\textrm{laser}}$
where $\Delta\phi_{\textrm{propagation}}$ is the difference in the action
computed along the classical trajectory of each interferometer arm, and
$\Delta\phi_{\textrm{laser}}$ is the phase difference imprinted on the atoms
by the Raman lasers at different locations Storey and Cohen-Tannoudji (1994).
The complete calculation of $\Delta\phi_{\textrm{propagation}}$ is done in
Section IV.2 and shows that
$\Delta\phi_{\textrm{propagation}}=\vec{k}_{\textrm{eff}}\cdot\vec{a}_{B}T^{2}~{}+$
smaller terms (see Eq.(12)) with $\vec{a}_{B}$ an acceleration due to a
magnetic force. This magnetic acceleration depends on the transition $\pm
k_{\textrm{eff}}$ and is expressed as
$\vec{a}_{B}=-\frac{\mu_{B}}{m}g_{F}m_{F}\vec{\nabla}B=-\frac{\hbar}{m}\alpha_{m}\vec{\nabla}B$,
where $m$ is the 87Rb atomic mass, $\hbar$ is the reduced Planck constant,
$\alpha_{m}=\pm 2\pi\times 0.70$ MHz/G is the Zeeman shift of
$\ket{F=1,m_{F}=\mp 1}$ respectively Steck (data), and $\vec{\nabla}B$ is the
gradient of the magnetic field magnitude $B$. In the infinitely short,
resonant-pulse limit, the second phase term of $\Delta\phi$ is given by
$\Delta\phi_{\textrm{laser}}=\vec{k}_{\textrm{eff}}\cdot\vec{a}_{\text{inertial}}~{}T^{2}$
where $\vec{a}_{\text{inertial}}$ is the acceleration of the atoms due to
gravito-inertial effects. This interferometer geometry thus exhibits at its
output an atomic phase shift sensitive to the combined acceleration of the
atoms due to gravito-inertial effects and a force due to a magnetic field
gradient:

$\displaystyle\Delta\phi$
$\displaystyle=\vec{k}_{\textrm{eff}}\cdot\vec{a}~{}T^{2}$ (2)
$\displaystyle=\vec{k}_{\textrm{eff}}\cdot\left(\vec{a}_{\textrm{inertial}}-\frac{\hbar}{m}\alpha_{m}\vec{\nabla}B\right)T^{2}$

At the end of the interferometric sequence we measure the proportion of atoms
in each output port of the interferometer $\ket{F=1}$ and $\ket{F=2}$ by
fluorescence. The normalized population in the state $\ket{F=2}$ at the LPAI
exit is a sinusoidal function of the interferometer phase shift:

$P=P_{m}-\frac{C}{2}\cos\Delta\phi=P_{m}-\frac{C}{2}\cos(\vec{k}_{\textrm{eff}}\cdot\vec{a}~{}T^{2})$
(3)

where $P_{m}$ is the fringe offset and $C$ is the fringe contrast. In the
following we neglect the smaller terms from
$\Delta\phi_{\textrm{propagation}}$ (see Eq.(12)) which will be studied in
Section IV.2.

The force responsible of the magnetic acceleration depends on the magnetic
field inhomogeneities. To evaluate this force in our setup, we proceed as
follows: just as in a LPAI gravimeter we apply a radio frequency chirp $\beta$
(Hz/s) to the effective Raman frequency to scan the interference fringes. The
sinusoidal dependence of the probability (see Eq.(3)) leads to an ambiguity in
the acceleration measurement. We solve this issue by measuring interference
fringes for different interrogation times $T$. Reversing the sign of the
effective wave vector (i.e. preparing the atoms in $\ket{F=1,m_{F}=\mp 1}$
alternatively), the magnetic acceleration changes sign and the phase shifts
are respectively:

$\displaystyle\Delta\phi_{\pm}$
$\displaystyle=\left[\pm\vec{k}_{\textrm{eff}}\cdot\left(\vec{a}_{\textrm{inertial}}+\vec{a}_{B\pm}\right)-2\pi\beta\right]T^{2}$
(4)
$\displaystyle=\left[\pm\vec{k}_{\textrm{eff}}\cdot\vec{a}_{\textrm{inertial}}+\vec{k}_{\textrm{eff}}\cdot\vec{a}_{B+}-2\pi\beta\right]T^{2}$

where

$\left\\{\begin{aligned} \vec{a}_{B\pm}&=\vec{a}_{B}\left(F=1,m_{F}=\mp
1\right)=\vec{a}_{B}\left(F=2,m_{F}=\pm 1\right)\\\
\vec{a}_{B+}&=-\vec{a}_{B-}\end{aligned}\right.$

This means that whatever the pulse separation $T$, the phase shift is zero
when the chirp reaches $\beta_{0\pm}=\frac{k_{\textrm{eff}}}{2\pi}\cdot(\pm
a_{\textrm{inertial}}+a_{B+})$. From this we easily extract the magnetic
acceleration
$a_{B+}=\frac{2\pi}{2k_{\textrm{eff}}}\cdot(\beta_{0+}+\beta_{0-})$. Its
numerical value is $-7.79\times 10^{-3}$ m.s-2, i.e. several interfringes
$i=\lambda/2T^{2}=1.43\times 10^{-3}$ m.s-2 ($T=16.5$ ms). The corresponding
magnetic field gradient is $\partial_{x}B=-24.2$ mG.cm-1 and is therefore
responsible for a non negligible bias on the inertial acceleration
measurement. The $k_{\textrm{eff}}$-reversal technique (i.e. preparing the
atoms in the $\ket{F=1,m_{F}=\mp 1}$ states alternatively) is essential to
eliminate such a bias.

###  III.1 Correlation fringes

When scanning the interference fringes by varying the frequency chirp $\beta$,
the fringes are washed out because of vibration noise (typically as soon as
$T\geq 6$ ms). To recover the interference fringes, we perform a correlation-
based technique Lautier  _et al._ (2014) which combines both measurements of
the LPAI output signal $P$ and of the classical accelerometer fixed to the
Raman mirror. Figure 3 shows the typical fringe pattern obtained by plotting
the transition probability $P$ at the LPAI output versus the acceleration
measured by the classical accelerometer. The fringe contrast obtained from a
sinusoidal least-squares fit of the data is $C=13~{}\%$, which is the best
contrast that we obtained when adjusting the Raman laser intensity at a fixed
Raman pulse duration of $\tau=10~{}\mu$s. We demonstrated in Perrin  _et al._
(2019a) a horizontal hybrid accelerometer with a contrast of $40~{}\%$ on the
same experimental setup. In Section IV we investigate the loss of contrast
associated to the $\sigma^{+}-\sigma^{-}$ technique.

Figure 3: Horizontal atom interferometer fringe pattern. The total
interferometer time is $2T=32$ ms and the Raman pulse duration is
$\tau=10~{}\mu$s. The solid line is a sinusoidal least-squares fit using
Eq.(3). The estimated fringe contrast is $C\sim 13~{}\%$. The fringes are
shifted from zero because of the magnetic acceleration and the bias of the
classical accelerometer.

###  III.2 Accelerometer sensitivity

We analyze the sensitivity and stability of the horizontal atom accelerometer
by hybridizing the classical and the atomic sensors Lautier  _et al._ (2014);
Bidel  _et al._ (2018). The sign of the effective Raman wave vector
$\vec{k}_{\textrm{eff}}$ is reversed every measurement cycle, i.e. the atoms
are alternatively prepared in the $\ket{F=1,m_{F}=\mp 1}$ states. We calculate
the inertial acceleration by computing the half sum of the phase shifts
measured on each correlation fringe pattern $\pm k_{\textrm{eff}}$. The fringe
ambiguity is removed by assuming that the magnetic acceleration has the same
value as calculated for smaller interrogation times, i.e. $a_{B}=-7.79\times
10^{-3}$ m.s-2. Figure 4 displays the Allan standard deviation (ADEV) of the
hybridized atom interferometer signal. We achieve a short-term sensitivity of
$25\times 10^{-5}$ m.s−2/$\sqrt{\textrm{Hz}}$, which is not as good as the
state of the art for horizontal configurations Xu  _et al._ (2017); Perrin
_et al._ (2019a). In Section IV we discuss several arguments to explain this
sensitivity. The ADEV of the atomic sensor scales as $\tau^{-1/2}$ and reaches
$3.8\times 10^{-6}$ m.s−2 after 3300 s integration time. No conclusion can be
drawn regarding the long-term stability of the atom accelerometer because of
angular drifts of the Raman mirror. An auxiliary tilt sensor could be used to
monitor the angle between the Raman beam and the horizontal direction during
the measurements.

Figure 4: Allan standard deviation (ADEV) of the hybridized horizontal atom
accelerometer (blue line). The dashed line illustrates the $\tau^{-1/2}$
scaling.

##  IV Specifics of the method

###  IV.1 Spontaneous emission

As shown in Eq.(1) the transition probability amplitudes for each excited
state $\ket{F^{\prime}=1}$ and $\ket{F^{\prime}=2}$ interfere destructively.
In order to address the counterpropagating transition, the laser detuning is
set between the $\ket{F^{\prime}=1}$ and $\ket{F^{\prime}=2}$ levels and
therefore induces spontaneous emission and coherence loss. Assuming that atoms
which undergo spontaneous emission do not interfer anymore, the contrast at
the atom interferometer output is reduced, and so is the sensor sensitivity.
We experimentally tuned the Raman laser frequency to minimize the spontaneous
emission rate. The detuning $\Delta_{2}$ from the excited state
$\ket{F^{\prime}=2}$ is adjusted via a beat-note between the fiber laser and
the Raman laser diode. The probability of transfer by spontaneous emission in
$\ket{F=2}$ is estimated by measuring the transfer probability during a 10
$\mu$s out-of-resonance Raman pulse. Figure 5 shows the experimental result
(red dots) as a function of the detuning $\Delta_{2}$, along with the
theoretical prediction of the probability of transfer by spontaneous emission
in the $\ket{F=2}$ level during a 10 $\mu$s pulse. In order to match the
experimental result with the theoretical prediction, we introduce an empirical
multiplicative factor $\kappa=2.5$ for the laser intensity (see Supplemental
Material at […] for detailed calculations). Considering the dipole matrix
elements we can derive an estimation of the total spontaneous emission
probability during the whole interferometer both from data and theory (blue
dots and black line in Fig.5). Here again we assume $\kappa=2.5$ in the
theoretical calculations. The need of this empirical factor $\kappa$ is not
understood. It could come from experimental defaults (non perfect $\pi/2$
Raman pulses) and from our theoretical treatment of spontaneous emission that
is too simple (the resolution of Bloch optics equations could be used in a
more elaborated model Cheng  _et al._ (2016)).

The optimal detuning is given by the curve minimum: $\Delta_{2}=-100$ MHz both
theoretically and experimentally. We can conclude from Figure 5 that for
$\Delta_{2}=-100$ MHz, 70 $\%$ of the atoms undergo spontaneous emission
during the interferometer duration. Such a loss of atoms reduces the contrast
by a factor 3 and could explain our relatively low contrast value of $13\%$
(see Figure 3). As a matter of fact, we demonstrated in Perrin  _et al._
(2019a) a horizontal hybrid accelerometer on the same experimental setup with
a contrast of $40\%$. Since $40\%\times 1/3\simeq 13\%$, the visiblity loss is
probably due to spontaneous emission. Nevertheless, we investigate in the next
subsection the phase shift sensitivity to magnetic field and show how it also
affects the LPAI contrast and bias.

Figure 5: Probability of spontaneous emission (SE) as a function of the
detuning $\Delta_{2}$ from the excited state $\ket{F^{\prime}=2}$. Red dots
represent the experimental data of probability of transfer by spontaneous
emission in the $\ket{F=2}$ level during the first Raman pulse of duration 10
$\mu$s. The data was obtained from the detected background on Raman spectra
plotted for different detunings $\Delta_{2}$ at constant effective Rabi
frequency. The dashed line represents the theoretical probability of transfer
by spontaneous emission (adjusted with a parameter $\kappa=2.5$) in the
$\ket{F=2}$ level during a Raman pulse of 10 $\mu$s. Blue dots represent the
total probability of spontaneous emission in the $\ket{F=1}$ and $\ket{F=2}$
levels during the whole interferometer. It was derived from the experimental
data in red dots. The black line is the theoretical total probability of
spontaneous emission during the whole interferometer (corrected by a factor
$\kappa=2.5$).

Setting the detuning close to resonance still has an advantage: the transition
requires low Raman intensity ($\sim I_{\textrm{sat}}$) compared to the
magnetically insensitive Raman transition ($\sim 10~{}I_{\textrm{sat}}$) for
which the detuning is set far from resonance. However spontaneous emission can
be drastically reduced if the Raman transition is performed on the $D_{1}$
line instead of the $D_{2}$ line of 87Rb, since the hyperfine levels are
further apart. Theoretical calculations of spontaneous emission on the $D_{1}$
line show that only 10 $\%$ of the atoms undergo spontaneous emission during
the interferometer sequence, which is comparable to LPAIs using the
$\ket{F=1,m_{F}=0}-\ket{F=2,m_{F}=0}$ transition with a commonly used one-
photon detuning from $\ket{F^{\prime}=3}$ of $\sim-1$ GHz.

###  IV.2 Sensitivity to magnetic field

As shown in Eq.(2) the LPAI is sensitive to both inertial acceleration and
magnetic forces from field inhomogeneities. Using the
$k_{\textrm{eff}}$-reversal technique one can extract each contribution by
computing either the half sum or the half difference of the phase shifts (see
Section III). This is only valid under the assumption of a constant magnetic
field gradient from shot-to-shot. We perform in this Section the detailed
calculation of the phase shift due to magnetic field by taking into account
spatial inhomogeneities of the magnetic field up to order two. From this study
we estimate the bias and the loss of contrast induced by the magnetic field.

We have considered so far a weak magnetic field and a linear relationship
between magnetic energy levels and magnetic field, with the same shift for
$\ket{F=1,m_{F}=\mp 1}$ and $\ket{F=2,m_{F}=\pm 1}$. For the ground state
manifold of the $D$ transition, the exact calculation of the energy levels is
given by the Breit-Rabi formula Breit and Rabi (1931). In the case of 87Rb in
the $\ket{F=1,m_{F}=-1}\equiv\ket{1,-1}$ and
$\ket{F=2,m_{F}=1}\equiv\ket{2,1}$ levels, the energies are given by:

$\left\\{\begin{aligned}
E_{\ket{1,-1}}&=\hbar\left(\alpha_{m}-\Delta\alpha\right)\times
B\equiv\hbar\omega_{a}\\\
E_{\ket{2,+1}}&=\hbar\left(\alpha_{m}+\Delta\alpha\right)\times
B\equiv\hbar\omega_{b}\end{aligned}\right.$ (5)

$\mathrm{with~{}~{}}\alpha_{m}=\dfrac{g_{J}-g_{I}}{4\hbar}\mu_{B}\mathrm{~{}~{}and~{}~{}}\Delta\alpha=\dfrac{g_{I}}{\hbar}\mu_{B}.$
Here we do not take into account the hyperfine splitting, as it is a constant
which cancels out in the phase shift calculation.

$a,b$ stand for the two hyperfine states $\ket{1,-1}$, $\ket{2,+1}$ of the
LPAI. $g_{I}$ is the nuclear g-factor, $g_{J}$ is the Landé factor and
$\mu_{B}$ is the Bohr magneton. Under the assumption of spins following the
magnetic field adiabatically during free fall, $B$ represents the magnitude of
the magnetic field. We define the average energy shift $\alpha_{m}=2\pi\times
0.70$ MHz/G (see Eq.(2)) and the differential energy shift
$\Delta\alpha=2\pi\times-1.4$ kHz/G coming from the Breit-Rabi formula.
Similarly, for the $-k_{\textrm{eff}}$ transition between $\ket{1,+1}$ and
$\ket{2,-1}$, the energy levels are described by Eq.(5), only
$(\alpha_{m},\Delta\alpha)$ are replaced with $(-\alpha_{m},-\Delta\alpha)$.

We use the Feynman path integral approach to compute the magnetic phase shift
between the two arms $(u,d)$ of the interferometer. Using a perturbative
calculation for the effect of the magnetic field Storey and Cohen-Tannoudji
(1994), the phase $\varphi^{(i)}$ accumulated along each arm $i=u,d$ of the
LPAI is given by the classical action $S_{\mathrm{cl}}$ along the
unperturbated classical path divided by $\hbar$:

$\varphi^{(i)}=S^{(i)}_{\mathrm{cl}}/\hbar=\int_{0}^{2T}\mathcal{L}[\vec{r}(t)]/\hbar\;\mathrm{d}t$
(6)

The phase difference at the output of the LPAI is then:

$\displaystyle\Delta\phi_{\textrm{path}}$
$\displaystyle=\varphi^{(u)}-\varphi^{(d)}$ (7)
$\displaystyle=(S^{(u)}_{\mathrm{cl}}-S^{(d)}_{\mathrm{cl}})/\hbar$
$\displaystyle=1/\hbar\left(\int_{0}^{2T}\mathcal{L}[\vec{r}~{}^{(u)}(t)]\;\mathrm{d}t-\int_{0}^{2T}\mathcal{L}[\vec{r}~{}^{(d)}(t)]\;\mathrm{d}t\right)$
$\displaystyle=\int_{0}^{T}\left[\omega_{b}\left(\vec{r}~{}^{(u)}(t)\right)-\omega_{a}\left(\vec{r}~{}^{(d)}(t)\right)\right]\;\mathrm{d}t$
$\displaystyle+\int_{T}^{2T}\left[\omega_{a}\left(\vec{r}~{}^{(u)}(t)\right)-\omega_{b}\left(\vec{r}~{}^{(d)}(t)\right)\right]\;\mathrm{d}t$

Taking into account the Breit-Rabi correction (see Eq.5), the phase difference
at the output of the interferometer can be split into two terms:

$\displaystyle\Delta\phi_{\textrm{path}}$
$\displaystyle=\alpha_{m}\int_{0}^{2T}\left[B\left(\vec{r}~{}^{(u)}(t)\right)-B\left(\vec{r}~{}^{(d)}(t)\right)\right]\;\mathrm{d}t$
(8)
$\displaystyle+\Delta\alpha\left(\int_{0}^{T}\left[B\left(\vec{r}~{}^{(u)}(t)\right)+B\left(\vec{r}~{}^{(d)}(t)\right)\right]\right.$
$\displaystyle-\int_{T}^{2T}\left[B\left(\vec{r}~{}^{(u)}(t)\right)+B\left(\vec{r}~{}^{(d)}(t)\right)\right]\;\mathrm{d}t\Bigg{.}\Bigg{)}$

The first term (proportional to $\alpha_{m}$) arises from the magnetic field
variation between the upper and the lower arms of the interferometer, whereas
the second term (proportionnal to $\Delta\alpha$) is accounting for the
variation of the mean field $B$ between $[0-T]$ and $[T-2T]$.

The phase shift calculation is performed on the unperturbed trajectories
$\left(\vec{r}~{}^{(u)}(t),\vec{r}~{}^{(d)}(t)\right)$ whose expressions are:

$\left\\{\begin{aligned}
\vec{r}~{}^{(u)}(t)&=\vec{r}_{0}+(\vec{v}_{0}+\vec{v}_{\textrm{eff}})t-\vec{v}_{\textrm{eff}}(t-T)\mathcal{H}(t-T)+\frac{1}{2}\vec{g}t^{2}\\\
\vec{r}~{}^{(d)}(t)&=\vec{r}_{0}+\vec{v}_{0}t+\vec{v}_{\textrm{eff}}(t-T)\mathcal{H}(t-T)+\frac{1}{2}\vec{g}t^{2}\end{aligned}\right.$
(9)

where $\vec{v}_{\textrm{eff}}=\hbar\vec{k}_{\textrm{eff}}/m$,
$(\vec{r}_{0},\vec{v}_{0})$ are the position and velocity vectors at the first
Raman pulse and $\mathcal{H}(t)$ is the Heaviside function.

The magnetic field magnitude $B$ is supposed to be time-independent and can
therefore be expressed through its Taylor expansion in space:

$\displaystyle B(\vec{r})$
$\displaystyle=B_{0}+\vec{B}_{1}\cdot\vec{r}+\frac{1}{2}\vec{r}\cdot\bar{\bar{B}}_{2}\cdot\vec{r}$
(10)
$\displaystyle=B_{0}+\partial_{x}Bx+\partial_{y}By+\partial_{z}Bz+\frac{1}{2}(\partial_{x}^{2}Bx^{2}+\partial_{y}^{2}By^{2}$
$\displaystyle+\partial_{z}^{2}Bz^{2}+2\partial_{x}\partial_{y}Bxy+2\partial_{x}\partial_{z}Bxz+2\partial_{y}\partial_{z}Byz)$

Using Eq.(9,10) the phase shift calculation of Eq.(8) leads to two terms
proportional to $\alpha_{m}$ and $\Delta\alpha$ respectively:

$\Delta\phi_{\textrm{path}}=\Delta\phi_{1}+\Delta\phi_{2}$ (11)

where

$\left\\{\begin{aligned}
\Delta\phi_{1}=\alpha_{m}T^{2}&\Big{(}\Big{.}\partial_{x}Bv_{\textrm{eff}}\\\
&+\partial_{x}^{2}Bv_{\textrm{eff}}\left[x_{0}+(v_{0x}+v_{\textrm{eff}}/2)T\right]\\\
&+\partial_{x}\partial_{y}Bv_{\textrm{eff}}\left[y_{0}+v_{0y}T\right]\\\
&+\partial_{x}\partial_{z}Bv_{\textrm{eff}}\left[z_{0}+v_{0z}T+7/12gT^{2}\right]\Big{.}\Big{)}\\\
\Delta\phi_{2}=-\Delta\alpha
T^{2}&\Big{(}\Big{.}2\partial_{x}B(v_{0x}+v_{\textrm{eff}}/2)\\\
&+2\partial_{y}Bv_{0y}\\\
&+2\partial_{z}B(v_{0z}+gT)\Big{.}\Big{)}\end{aligned}\right.$ (12)

The first term of $\Delta\phi_{1}$ is the magnetic acceleration of Eq.(2)
since
$\alpha_{m}\partial_{x}Bv_{\textrm{eff}}T^{2}=\alpha_{m}\partial_{x}B\frac{\hbar}{m}k_{\textrm{eff}}T^{2}=k_{\textrm{eff}}a_{B}T^{2}$.
The other terms of $\Delta\phi_{1}$ are due to magnetic field curvatures
whereas the terms of $\Delta\phi_{2}$ come from the differential shift
$\Delta\alpha$ and the magnetic field gradient. From Eq.(12) one can deduce
the bias and loss of contrast induced by the $\sigma^{+}-\sigma^{-}$ method.

The $k_{\textrm{eff}}$-reversal technique enables to eliminate any systematic
effect whose sign does not change when reversing $\vec{k}_{\textrm{eff}}$. The
bias is then due to the remaining phase terms. In our case, since
$(\alpha_{m},\Delta\alpha)$ change sign when reversing
$\vec{k}_{\textrm{eff}}$, we deduce from Eq.(12) that the bias induced by
magnetic effects is:

$\frac{1}{2}\alpha_{m}\partial_{x}^{2}Bv_{\textrm{eff}}^{2}T^{3}-2\Delta\alpha\partial_{z}BgT^{3}$
(13)

The first term is an inertial phase due to the atom recoil and the magnetic
field curvature: it arises as soon as the magnetic field gradient is different
for the upper and lower arm of the interferometer. It was estimated
theoretically by computing the magnetic field produced by the horizontal field
coils ($\partial_{x}^{2}B\sim 5$ G.m-2) and leads to a bias of $2\times
10^{-6}$ m.s-2 (we recall $T=16.5$ ms). The second term is an energy dependant
phase term which comes from time variation of the differential Zeeman energy
shift induced by the magnetic field gradient $\partial_{z}B$ seen by the atoms
during free fall. In order to estimate it we measured the magnetic field
vertical gradient by Zeeman spectroscopy and found $\partial_{z}B=1.2$ G.m-1.
The resulting bias is $2\times 10^{-4}$ m.s-2. In comparison, ref.Wodey et al.
(2020) demonstrates a high-performance magnetic shield for long baseline atom
interferometry with inhomogeneities below 3 nT/m: the associated bias would be
$3\times 10^{-9}$ m.s-2. From Eq.(12) one can easily notice that this bias can
be eliminated through an atomic fountain design with a properly set vertical
velocity at the first Raman pulse $v_{0z}=-gT$. One can also set the
quantization field to its magic value Harber  _et al._ (2002) corresponding to
the magnetic field at which the derivative of the energy difference
$\hbar(\omega_{b}-\omega_{a})$ is null. But this configuration does not cancel
out the bias perfectly because it requires to change the magnetic field sign
when changing the sign of $\vec{k}_{\textrm{eff}}$, since the magic field for
the $\pm\vec{k}_{\textrm{eff}}$ transitions are respectively $\pm 3.2$ G
Harber  _et al._ (2002).

Regarding the contrast reduction induced by magnetic effects, it is due to the
position and velocity dependent terms in Eq.(12). Indeed any phase shift
sensitivity to position (or velocity) results in each atom (or velocity class)
providing its own fringe pattern. Since the atom detection protocol averages
over these patterns, the fringe contrast is reduced. In our case the
interrogation time $T=16.5$ ms is short enough to neglect the loss of contrast
arising from velocity dependant phase shifts in Eq.(12) (see Fig.6(a)). The
contrast reduction due to averaging over position on the other hand is much
more significant. As an example we analyse the phase shift
$\alpha_{m}\partial_{x}^{2}Bv_{\textrm{eff}}x_{0}T^{2}$: it reduces contrast
by a factor
$e^{-\frac{1}{2}(\alpha_{m}\partial_{x}^{2}Bv_{\textrm{eff}}T^{2}\sigma_{x})^{2}}$
Roura  _et al._ (2014), where $\sigma_{x}=1$ mm is the typical size of the
atomic cloud. Fig.6(b) illustrates this loss of contrast as a function of the
magnetic field curvature $\partial^{2}_{x}B$: the contrast is typically
reduced by 50 $\%$ in the presence of $\partial^{2}_{x}B\sim 50$ G.m-2. As
comparison Fig.6(a) shows how the velocity dependent phase shift
$\alpha_{m}\partial_{x}^{2}Bv_{\textrm{eff}}v_{0x}T^{3}$ does not affect
contrast. Since the magnetic field along the Raman beams cannot be measured
precisely enough on our experimental setup, we estimated theoretically the
curvature due to our magnetic field coils and found
$\partial_{x}^{2}B_{\textrm{th}}\sim 5$ G.m-2. From this we can state that
inhomogeneities of the magnetic field created by the coils do not affect the
fringe contrast. However the presence of another magnetic field source
creating a non negligible field curvature responsible of a contrast reduction
is to be considered.

Figure 6: Theoretical contrast calculated theoretically as a function of the
magnetic field curvature. Parameters: interrogation time $T=16.5$ ms, gas
temperature $\theta=2~{}\mu$K, typical atomic cloud size $\sigma_{x}=1$ mm,
velocity $\sigma_{v}=\sqrt{k_{B}\theta/m}=1.4$ cm.s-1. (a) Contrast loss due
to the velocity dependent phase shift
$\alpha_{m}\partial_{x}^{2}Bv_{\textrm{eff}}v_{0x}T^{3}$. (b) Contrast loss
due to the position dependent phase shift
$\alpha_{m}\partial_{x}^{2}Bv_{\textrm{eff}}x_{0}T^{2}$.

Loss of contrast can also be interpreted as in Roura  _et al._ (2014):
position and velocity dependent phase terms in Eq.(12) are responsible for the
opening of the interferometer in momentum and position respectively. We
introduce the notation $\Gamma=2\frac{\hbar}{m}\alpha_{m}\partial_{x}^{2}B$
since the magnetic field curvature is the exact analog of a gravity gradient
in a vertical LPAI. The forces associated with $\Gamma$ tend to open up the
trajectories of the atoms and lead to an open interferometer with nonvanishing
relative position and momentum displacements at the output of the LPAI. As
demonstrated in Roura  _et al._ (2014) a position dependent phase shift
results in a momentum displacement $\delta P$ at the output of the LPAI, and a
velocity dependent phase shift results in a position displacement $\delta X$.
Both displacements are given by the following expressions to first order in
$\Gamma T^{2}$ Roura  _et al._ (2014):

$\displaystyle\delta X$ $\displaystyle=\left(\Gamma T^{2}\right)\frac{\hbar
k_{\textrm{eff}}}{m}T$ (14) $\displaystyle\delta P$
$\displaystyle=\left(\Gamma T^{2}\right)\hbar k_{\textrm{eff}}$ (15)

In our case the change of position associated with Eq.(14) is very small
compared to the coherence length which is estimated by the thermal De Broglie
wavelength: $\delta X\sim 10^{-8}$ m $\ll
l_{c}=\lambda_{\textrm{DB}}=h/\sqrt{2\pi mk_{B}\theta}\sim 10^{-7}$ m for 87Rb
atoms at $\theta=2~{}\mu$K. On the other hand the momentum displacement given
by Eq.(15) is not as negligible as the position displacement. Even though
Roura (2017) suggests a protocol to mitigate the loss of contrast due to
gravity gradients $\Gamma$ through a suitable adjustment of the laser
wavelength at the second Raman pulse, this technique cannot be applied here.
Indeed the $\sigma^{+}-\sigma^{-}$ method requires the detuning $\Delta$ to
stay between the $\ket{F^{\prime}=1}$ and $\ket{F^{\prime}=2}$ sublevels to
avoid spontaneous emission (see Section IV.1), which means that the mitigation
technique proposed by Roura  _et al._ (2014) would inevitably reduce the
fringe contrast.

###  IV.3 Light shifts

In most cases one can eliminate the differential one-photon light shift by
adjusting the intensity ratio between the two Raman lasers. In our
$\sigma^{+}-\sigma^{-}$ Raman transition scheme, there is no intensity ratio
that cancels out the differential one-photon light shift (see Supplemental
Material at […] for complete calculation of light shifts), which means that an
intensity variation between the first and the last pulses of the LPAI leads to
a residual parasitic phase shift. For an intensity variation of 10 $\%$
between the first and the last pulses, the corresponding bias due to the one-
photon light shift is $-1.5\times 10^{-5}$ m.s-2. However this light shift is
nearly rejected through the $k_{\textrm{eff}}$-reversal technique, since
$\delta_{\textrm{LS1}}^{\textrm{diff}}(+k_{\textrm{eff}})\simeq\delta_{\textrm{LS1}}^{\textrm{diff}}(-k_{\textrm{eff}})$.
More generally it is essential to note that the one-photon light shift doesn’t
represent a limit to our $\sigma^{+}-\sigma^{-}$ technique, since one can find
an intensity ratio canceling it out on the $D_{1}$ line of 87Rb.

In our setup, the two-photon light shift arises from the off-resonant
copropagating Raman transitions detuned by $\pm\dfrac{2\mu_{B}g_{F}}{h}B$ from
the considered $\pm k_{\textrm{eff}}$ counterpropagating transition. Its
effect decreases with the magnetic field. When $B=400$ mG, it is calculated to
be negligible compared to the one-photon light shift and it cancels out as
well when the $k_{\textrm{eff}}$-reversal technique is applied.

##  V Applications

In this section we propose some possible applications of our technique in both
metrology and for development of new designs of atom interferometers dedicated
to field applications.

###  V.1  $h/m_{X}$ measurement

Atom interferometers allow to determine the fine-structure constant $\alpha$
based on measuring the recoil velocity $v_{r}=\hbar k/m_{X}$ of an atom $X$ of
mass $m_{X}$ absorbing a photon of momentum $\hbar k$, where $\hbar=h/2\pi$,
and $k$ is the photon wave number. With an accurate measurement of $k$,
$h/m_{X}$ can me measured and $\alpha$ can be determined allowing to test the
standard model and beyond Parker  _et al._ (2018); Morel  _et al._ (2020).

Here, using our interferometric design we propose to measure $h/m_{X}$ by
combining the interferometric measurement of the magnetic acceleration $a_{B}$
and an independent measurement of the magnetic field gradient $\partial_{x}B$
thanks to a micro-wave spectroscopy. In Eq.(2), using the
$k_{\textrm{eff}}$-reversal technique, one can isolate the magnetic
acceleration, leading to:

$\frac{h}{m_{X}}=2\pi\frac{a_{B}}{\alpha_{m}\partial_{x}B}$ (16)

Considering state-of-the-art atom accelerometers at their best level of
accuracy ($\sim 10^{-8}$ m.s-2) Karcher  _et al._ (2018), combined with a
micro-wave Ramsey interferometer with a free-evolution time $T$=10 ms and a
signal-to-noise ratio SNR $=10^{3}$, one could measure magnetic fields at the
level of $\sim 100$ nG. Thus, under an acceleration $a_{\mathrm{B}}=1$ m.s-2
one could obtain a relative uncertainty at the level of $10^{-8}$ on $h/m_{X}$
measurement.

###  V.2 Force-balanced atom accelerometer

The supplementary internal degree of freedom provided by the magnetically-
sensitive states used in the interferometer allows to exploit the sensitivity
of the atoms to magnetic field gradients and transfer a magnetic acceleration
onto them. The magnetic acceleration of a Rb atom in a $m_{F}=\pm 1$ sublevel
due to a magnetic field gradient $\partial_{z}B$ (T.m-1) is $a_{B}\sim
32.1\times\partial_{z}B$ m.s-2. For example, applying a magnetic field
gradient of magnitude $\partial_{z}B\sim 30$ G.cm-1 could compensate for
gravity acceleration on Earth. Thus, we propose to use our interferometric
scheme to create a force-balanced atom accelerometer where the inertial
acceleration undergone by the atoms, and measured by an auxiliary classical
accelerometer, could be compensated by applying a magnetic acceleration. The
basic principle of the technique is depicted in Fig.(7).

Figure 7: Scheme of the force-balanced accelerometer. AHC: Anti-Helmholtz Coil
; HC: Helmholtz Coil.

The atom accelerometer is hybridized with a conventional classical
accelerometer in order to track the bias of the classical accelerometer such
as in Bidel  _et al._ (2020). Our coil configuration consists of two vertical
coils above and below the atoms. A pair of Helmholtz coils generates a bias
field $\vec{B}_{q}$ along the $z$-axis and defines the quantization axis. A
second pair of coils with counterpropagating currents (anti-Helmholtz
configuration) in the same housing, is set to create the magnetic field
gradient along the same direction. The current output of the classical
accelerometer, proportional to the inertial acceleration, is then used as an
input signal to counter-balance the inertial force undergone by the atoms. In
pratice, depending on the electric current fed into the anti-Helmholtz coils,
one can adjust the magnetic acceleration to $a_{B}=\gamma i$ where $i$ is the
electric current and $\gamma$ a scale factor which can be precisely determined
through calibration of the magnetic acceleration in a lab-based environment.
Thus, it is for example possible to levitate the atoms against gravitational
acceleration and therefore extend the evolution time in earthbound
laboratories. Additionnally, using three pairs of Helmholtz coils in
3-orthogonal directions, this scheme could be further extended to compensate
for any inertial acceleration in the 3 dimensions, where 3 classical
accelerometers are fixed to 3 atom accelerometers to form a 3-axis hybrid
accelerometer. With this scheme, one could simultaneously apply a magnetic
force on the atoms in the 3 dimensions. This could benefit onboard atom
accelerometers submitted to spurious accelerations which limit the dynamic
range because the atoms drop out from the laser beams and the detection zone.

##  VI Conclusion

We reported on the experimental demonstration of a horizontal cold-atom
interferometer using counterpropagating Raman transitions between
$\ket{F=1,m_{F}=\mp 1}$ and $\ket{F=2,m_{F}=\pm 1}$ of 87Rb. Using the same
$\sigma^{+}-\sigma^{-}$ polarized light arrangement as the MOT, we generated
Raman coupling between the two states of the interferometer and showed that
this technique allows to perform single-diffraction Raman process in a close-
to-zero velocity regime without the need for alternative techniques Perrin
_et al._ (2019a). We demonstrated that this technique presents both advantages
and disadvantages compared to the $\ket{F=1,m_{F}=0}-\ket{F=2,m_{F}=0}$
transition usually used in atom interferometry (see TABLE 1).

Absolute horizontal acceleration measurement with a short-term sensitivity of
$25\times 10^{-5}$ m.s-2.Hz-1/2 was achieved. In our setup, limitations of the
sensitivity arise from spontaneous emission, leading to a reduction of the
interferometer contrast. The accuracy of the atom accelerometer is mainly
limited by the bias caused by the magnetic field gradient at the level of
$\sim 2\times 10^{-4}$ m.s-2.

Although the short-term sensitivity is bigger by almost one order of magnitude
in comparison with state-of-the-art horizontal atom accelerometers Xu  _et
al._ (2017); Perrin  _et al._ (2019a), it could be improved by changing the
Raman excitation scheme to the D1 line, thus reducing spontaneous emission.
Additionally, one could reduce this acceleration bias at the level of $\sim
3\times 10^{-9}$ m.s-2 (with $T=16.5$ ms) considering a high-performance
magnetic shielding leading to a magnetic-field inhomogeneity of 3 nT.m-1 Wodey
et al. (2020). Moreover, we showed that acceleration bias could be suppressed
by performing the atom interferometer using a fountain geometry. Finally, we
believe that using the supplementary internal degree of freedom provided by
atoms manipulated in magnetically sensitive levels provides interesting
features such as levitation schemes for inertial applications requiring
compact setups.

Advantages |  Disadvantages
---|---
Same polarization arrangement MOT/Raman |  Spontaneous emission
(reduced on the D1 line of 87Rb)
Single counterpropagating transition in retroreflective geometry despite zero Doppler shift |  Remaining copropagating Raman transition
Supplementary internal degree of freedom provided by adressing $m_{F}\neq 0$
Zeeman sublevels
(force-balanced accelerometer) |  Require precise control and mapping of the magnetic field
Table 1: Summarized advantages and disadvantages of a Mach-Zehnder cold-atom
interferometer using counterpropagative Raman transitions between the state
$\ket{F=1,m_{F}=\mp 1}$ and $\ket{F=2,m_{F}=\pm 1}$ of 87Rb.

###### Acknowledgements.

M. C acknowledges funding from ONERA through research project CHAMOIS
(Centrale Hybride AtoMique de l’Onera et Inertie Strapdown).

Supplemental Material

In this Supplemental Material we present the theoretical calculation of the
probability of transfer by spontaneous emission and the light shifts featured
in the $\sigma^{+}-\sigma^{-}$ Raman transition technique.

##  I Probability of transfer by spontaneous emission

We start by calculating the effective two-photon Rabi frequency
$\Omega_{\textrm{eff}}$ describing the Raman coupling between the hyperfine
states $\ket{F=1,m_{F}=-1}$ and $\ket{F=2,m_{F}=1}$ of the interferometer. We
compute the single-photon scattering rate $R_{\textrm{sc}}^{F}$ from atoms
starting in $\ket{F=1}$ or $\ket{F=2}$. From these the probability of transfer
by spontaneous emission for an entire interferometer can be calculated.

We describe the intensity $I_{n}$ of each EOM sideband $n$ at the output of
the phase modulator as $I_{n}=I\cdot\mathcal{J}_{n}(\beta)^{2}$ where $I$ is
the total laser intensity, $\mathcal{J}_{n}$ is the Bessel function of the
first kind of order $n$, and $\beta$ is the modulation index of the EOM. Here
we only take into account two sidebands ($n=0$ and $n=1$), the others being
detuned enough to be neglected. $I_{\textrm{sat}}$ stands for the saturation
intensity and $\Gamma$ is the natural line width. We calculate the effective
two-photon Rabi frequency:

$\Omega_{\textrm{eff}}=\dfrac{\Gamma^{2}}{2}\dfrac{I}{I_{\textrm{sat}}}\mathcal{J}_{0}(\beta)\mathcal{J}_{1}(\beta)\sum_{F^{\prime}}\dfrac{M_{1,-1}^{F^{\prime},+}\cdot
M_{2,1}^{F^{\prime},-}}{\Delta_{F^{\prime}}}$ (17)

where
$M_{F,m_{F}}^{F^{\prime},\pm}=\dfrac{\bra{F,m_{F}}er\ket{F^{\prime},m_{F}\pm
1}}{\bra{J=1/2||er|}\ket{J^{\prime}=3/2}}$ are the rubidium $D_{2}$ dipole
matrix elements for $\sigma^{\pm}$ transitions, expressed as multiples of
$\bra{J=1/2||er|}\ket{J^{\prime}=3/2}$, as given in Steck (data).
$\Delta_{F^{\prime}}=\omega_{L}-\omega_{2}^{F^{\prime}}$ is the detuning of
the carrier (of frequency $\omega_{L}$) relative to the transition
$F=2\rightarrow F^{\prime}$ (of energy $\hbar\omega_{2}^{F^{\prime}}$).

Assuming a $\pi/2$ Raman pulse of duration $\tau$ with Rabi frequency
$\Omega_{\textrm{eff}}\cdot\tau=\pi/2$, we can deduce from Eq.(17) the laser
intensity $I(\Delta_{F^{\prime}})$ as a function of the detuning
$\Delta_{F^{\prime}}$.

The rate of spontaneous emission for atoms starting in $\ket{F=1,m_{F}=-1}$ or
$\ket{F=2,m_{F}=1}$ is:

$R_{\textrm{sc}}^{F}=\Gamma\dfrac{I}{I_{\textrm{sat}}}\sum_{n=\left\\{0,1\right\\},F^{\prime},\sigma=\pm}\dfrac{\left(M_{F,m_{F}}^{F^{\prime},\sigma}\right)^{2}}{1+4\dfrac{\left(\Delta_{F,F^{\prime}}^{n}\right)^{2}}{\Gamma^{2}}+\dfrac{I}{I_{\textrm{sat}}}}\mathcal{J}_{n}(\beta)^{2}$
(18)

where
$\Delta_{F,F^{\prime}}^{n}=\omega_{L}+n\omega_{\textrm{hfs}}-\omega_{F}^{F^{\prime}}$
is the detuning of the sideband $n$ (of frequency
$\omega_{L}+n\omega_{\textrm{hfs}}$) relative to the transition $F\rightarrow
F^{\prime}$ (of energy $\hbar\omega_{F}^{F^{\prime}}$). The ground state
hyperfine splitting $\omega_{\textrm{hfs}}$ is also the EOM driving frequency.

From these, the probability of spontaneous emission for an entire
interferometer can be calculated:

$P_{\textrm{SE}}=1-\exp\left(-R_{\textrm{sc}}^{1}\cdot 2\tau-
R_{\textrm{sc}}^{2}\cdot 2\tau\right)$ (19)

where we consider that the atoms spend as much time ($2\tau$) in state
$\ket{F=1}$ and in state $\ket{F=2}$, the total Raman interaction duration
being $4\tau$.

We measure experimentally the number of atoms that are transferred from
$\ket{F=1}$ to $\ket{F=2}$ when the Raman detuning is off the two-photon
resonance. In order to compare these measurements with our theoretical model,
we calculate the scattering rate $R_{\textrm{sc}}^{1\rightarrow 2}$ for atoms
initially in the state $\ket{F=1,m_{F}=-1}$, undergoing a single-photon
transition to an excited state $\ket{F^{\prime},m_{F^{\prime}}}$, and
transferred to $\ket{F=2}$ by spontaneous emission. This corresponds to the
scattering rate $R_{\textrm{sc}}^{1}$ presented in Eq.(18) with the difference
that one needs to take into account the spontaneous emission rates
$\Gamma_{F^{\prime}\rightarrow 2}$ of each de-excitation
$\ket{F^{\prime},m_{F^{\prime}}}\rightarrow\ket{F=2,m_{F}}$. Therefore we
have:

$\displaystyle R_{\textrm{sc}}^{1\rightarrow
2}=\dfrac{I}{I_{\textrm{sat}}}\sum_{n=\left\\{0,1\right\\},F^{\prime},\sigma=\pm}$
$\displaystyle\Gamma_{F^{\prime}\rightarrow 2}$ (20)
$\displaystyle\times\dfrac{(M_{1,-1}^{F^{\prime},\sigma})^{2}}{1+4\dfrac{(\Delta_{1,F^{\prime}}^{n})^{2}}{\Gamma^{2}}+\dfrac{I}{I_{\textrm{sat}}}}\mathcal{J}_{n}(\beta)^{2}$

where the spontaneous emission rates $\Gamma_{F^{\prime}\rightarrow 2}$ are
the following:

$\left\\{\begin{aligned} \Gamma_{0\rightarrow 2}&=0\\\ \Gamma_{1\rightarrow
2}&=\frac{\Gamma}{6}\\\ \Gamma_{2\rightarrow 2}&=\frac{\Gamma}{2}\\\
\Gamma_{3\rightarrow 2}&=0\\\ \end{aligned}\right.$ (21)

As presented in the associated article (see Figure 5), we introduce an
empirical parameter $\kappa$ in the theoretical formulas to account for the
difference between the effective Raman intensity and the theoretical
prediction. Thus we write the total loss of atoms (i.e. in the $\ket{F=1}$ and
$\ket{F=2}$ levels) by spontaneous emission during the whole interferometer as
:

$P_{\textrm{SE}}=1-\exp\left(-\kappa\left[R_{\textrm{sc}}^{1}\cdot
2\tau+R_{\textrm{sc}}^{2}\cdot 2\tau\right]\right)$ (22)

Likewise, we calculate the probability of transfer by spontaneous emission in
$\ket{F=2}$ during a pulse of duration $\tau$ as follows :

$P_{\textrm{SE~{}F=2}}=1-\exp\left(-\kappa R_{\textrm{sc}}^{1\rightarrow
2}\cdot\tau\right)$ (23)

##  II One-photon and two-photon light shifts

In a Mach-Zehnder type atom interferometer, uncompensated differential light
shifts $\delta_{\textrm{LS}}^{\textrm{diff}}$ from the Raman lasers result in
an additional phase contribution given by Gillot  _et al._ (2016):

$\displaystyle\Delta\phi_{\textrm{LS}}=$
$\displaystyle-\arctan\left[\tan\left(\Omega_{\textrm{eff,1}}\dfrac{\tau}{2}\right)\dfrac{\delta_{\textrm{LS,1}}^{\textrm{diff}}}{\Omega_{\textrm{eff,1}}}\right]$
(24)
$\displaystyle+\arctan\left[\tan\left(\Omega_{\textrm{eff,3}}\dfrac{\tau}{2}\right)\dfrac{\delta_{\textrm{LS,3}}^{\textrm{diff}}}{\Omega_{\textrm{eff,3}}}\right]$

where the effective Rabi frequencies and the differential light shifts for
each pulse $i$ are given by $\Omega_{\textrm{eff,i}}$ and
$\delta_{\textrm{LS,i}}^{\textrm{diff}}$ respectively. $\tau$ is the duration
of the first and last pulses.

The one-photon Raman light shift is imprinted onto both hyperfine states
$\ket{F=1,m_{F}=-1}$ and $\ket{F=2,m_{F}=1}$ by out-of-resonance Raman lasers.
Each light shift $\delta_{\textrm{LS1}}^{F,m_{F}}$ has the following
expression:

$\delta_{\textrm{LS1}}^{F,m_{F}}=\dfrac{\Gamma^{2}}{4}\dfrac{I}{I_{\textrm{sat}}}\sum_{n=\left\\{0,1\right\\},F^{\prime},\sigma=\pm}\dfrac{\left(M_{F,m_{F}}^{F^{\prime},\sigma}\right)^{2}}{\Delta_{F,F^{\prime}}^{n}}\mathcal{J}_{n}(\beta)^{2}$
(25)

The differential one-photon light shift is then:

$\delta_{\textrm{LS1}}^{\textrm{diff}}=\delta_{\textrm{LS1}}^{2,1}-\delta_{\textrm{LS1}}^{1,-1}$
(26)

There is no intensity ratio that cancels out the differential light shift,
contrary to LPAIs using the $\ket{F=1,m_{F}=0}-\ket{F=2,m_{F}=0}$ transition.
This means that a difference of intensity between the first and the third
pulses leads to an additional phase shift Gillot  _et al._ (2016). However
this light shift is nearly rejected through the $k$-reversal technique, since
$\delta_{\textrm{LS1}}^{\textrm{diff}}(+k_{\textrm{eff}})\simeq\delta_{\textrm{LS1}}^{\textrm{diff}}(-k_{\textrm{eff}})$.
We recall that reversing the sign of $k_{\textrm{eff}}$ means preparing the
atoms alternatively in the $\ket{F=1,m_{F}=\mp 1}$ states.

In our setup, the two-photon light shift arises from the off-resonant
copropagating Raman transitions detuned by $\pm\Delta_{B}=\pm
2\times\frac{\mu_{B}g_{F}}{h}B$ from the considered $\pm k_{\textrm{eff}}$
counterpropagating transition. The level $\ket{F=1,m_{F}=-1}$ is perturbed by
the coupling $\Omega_{\textrm{eff,co}}^{-1}$ of the copropagating transition
$\ket{F=1,m_{F}=-1}\leftrightarrow\ket{F=2,m_{F}=-1}$. Likewise, the level
$\ket{F=2,m_{F}=1}$ is perturbed by the coupling
$\Omega_{\textrm{eff,co}}^{+1}$ of the copropagating transition
$\ket{F=1,m_{F}=1}\leftrightarrow\ket{F=2,m_{F}=1}$. The corresponding light
shifts are:

$\left\\{\begin{aligned}
\delta_{\textrm{LS2}}^{1,-1}&=\dfrac{\left(\Omega_{\textrm{eff,co}}^{-1}\right)^{2}}{4\Delta_{B}}\\\
\delta_{\textrm{LS2}}^{2,1}&=\dfrac{\left(\Omega_{\textrm{eff,co}}^{+1}\right)^{2}}{-4\Delta_{B}}\\\
\end{aligned}\right.$ (27)

where $\Delta_{B}=2\times\frac{\mu_{B}g_{F}}{h}B$ is the first order Zeeman
splitting between the magnetic sublevels $m_{F}=0$ and $m_{F}=2$. The
effective Rabi frequency of the copropagating transitions are:

$\left\\{\begin{aligned}
\Omega_{\textrm{eff,co}}^{-1}&=\dfrac{\Gamma^{2}}{2}\dfrac{I}{I_{\textrm{sat}}}\mathcal{J}_{0}(\beta)\mathcal{J}_{1}(\beta)\sum_{F^{\prime},\sigma=\pm}\dfrac{M_{1,-1}^{F^{\prime},\sigma}\cdot
M_{2,-1}^{F^{\prime},\sigma}}{\Delta_{F^{\prime}}}\\\
\Omega_{\textrm{eff,co}}^{+1}&=\dfrac{\Gamma^{2}}{2}\dfrac{I}{I_{\textrm{sat}}}\mathcal{J}_{0}(\beta)\mathcal{J}_{1}(\beta)\sum_{F^{\prime},\sigma=\pm}\dfrac{M_{1,1}^{F^{\prime},\sigma}\cdot
M_{2,1}^{F^{\prime},\sigma}}{\Delta_{F^{\prime}}}\end{aligned}\right.$ (28)

Finally, the differential two-photon light shift is:

$\delta_{\textrm{LS2}}^{\textrm{diff}}=\delta_{\textrm{LS2}}^{2,1}-\delta_{\textrm{LS2}}^{1,-1}$
(29)

Just as for the one-photon light shift, the differential two-photon light
shift is almost completely rejected through the $k$-reversal technique,
because
$\delta_{\textrm{LS2}}^{\textrm{diff}}(+k_{\textrm{eff}})\simeq\delta_{\textrm{LS2}}^{\textrm{diff}}(-k_{\textrm{eff}})$.

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