# Compact chip-scale guided cold atom gyrometers for inertial navigation:
Enabling technologies and design study

Carlos L. Garrido Alzar<EMAIL_ADDRESS>SYRTE, Observatoire de Paris,
Université PSL, CNRS, Sorbonne Université, LNE, 61 avenue de l’Observatoire,
75014 Paris, France

###### Abstract

This work reviews the topic of rotation sensing with compact cold atom
interferometers. A representative set of compact free-falling cold atom
gyroscopes is considered because, in different respects, they establish a
rotation-measurement reference for cold guided-atom technologies. The review
first discusses enabling technologies relevant to a set of key functional
building blocks of an atom chip-based compact inertial sensor with cold guided
atoms. These functionalities concern the accurate and reproducible positioning
of atoms to initiate a measurement cycle, the coherent momentum transfer to
the atom wave packets, the suppression of propagation-induced decoherence due
to potential roughness, the on-chip detection, and the vacuum dynamics because
of its impact on the sensor stability, which is due to measurement dead time.
Based on the existing enabling technologies, the design of an atom chip
gyroscope with guided atoms is formalized using a design case that treats
design elements such as guiding, fabrication, scale factor, rotation-rate
sensitivity, spectral response, important noise sources, and sensor stability.

##  I Introduction

Sensing, metrology, and industrial applications of cold atom interferometers
drive the increasing interest in this topic by the physics and engineering
communities. Indeed, atom interferometry lays out the working principle of
cold atomic clocks, Norman ; bize  accelerometers and gravimeters, kasevich1 ;
peters ; peters1 ; abend ; kang  gradiometers, mcguirk ; rosi ; tino
gyrometers, riehle ; gustavson1 ; gustavson2 ; canuel ; zhan  and
magnetometers. hardman  The unprecedented high measurement sensitivity and
stability made possible by these instruments have naturally triggered the
development of compact and transportable atom-based quantum sensors. muquans ;
aosense  Among the most relevant applications we find high-precision
geophysics, gillot  fundamental physics bouchendira ; Weiss ; aguilera ;
fixler  measurements, and inertial navigation. savoie ; dutta ; barret1  For
inertial navigation, atom interferometers are particularly important because
they provide an absolute measurement of the physical quantity of interest, be
it acceleration or rotation. Focusing on rotation-sensing applications, this
latter fact implies that, for instance, atom gyrometers can be used to measure
and preserve the orientation of a carrier without external references, such as
a Global Positioning System (GPS) signal. In geophysics, a gyrometer can be
used for local monitoring of the variations in Earth’s rotation rate due to
seismic or tectonic-plate displacements. velikoseltsev1 ; velikoseltsev  In
the field of general relativity, for example, tests of the geodetic and Lense-
Thirring effects schiff ; schiff1  can also be foreseen given the long-term
sensitivity experimentally demonstrated in the lab. dutta1

Several research groups around the world launched the development of high
precision compact portable atom interferometers around 20 years ago. At
Stanford, Kasevich et al. built a mobile atomic gravity gradiometer prototype
instrument (MAGGPI), kasevich2  with which they measured the Newtonian
gravitational constant $G$ and mapped the gravity gradient of Earth. Yu et al.
at the Jet Propulsion Laboratory (JPL) worked to develop atom interferometer
inertial sensors for gravity mapping, geodesy, fundamental physics, and
planetary science. nanyu  In Hannover, the E. Rasel’s group developed a
compact dual-atom interferometer for testing the equivalence principle quantus
; maius  and measuring rotations berg  and the fine structure constant. A
collaboration between the groups of A. Landragin at SYRTE in Paris and of P.
Bouyer in Bordeaux developed a portable dual-species atom interferometer for
testing the equivalence principle in microgravity. ice  A portable gravimeter
has also been developed at SYRTE. It participates in international comparison
gravimetry campaigns for the SI definition of $g$ and the redefinition of the
kilogram. jiang

The rotation sensing capability of an atom interferometer has been
demonstrated by using the Sagnac effect, sagnac  in which two waves
propagating in opposite directions inside a rotating interferometer of
physical area $A$ experience a path-length difference and consequently a phase
shift $\Phi$ that depends on the rotation rate $\Omega$. With the invention of
lasers, the realization of high-sensitivity optical Sagnac gyroscopes became
possible, such as optical fiber gyroscopes vali  and gyrolasers, macek  which
are commonly used for inertial navigation. In particular, the fiber gyroscopes
used nowadays can reach sensitivities on the order of $10^{-7}$ rad s-1 over a
second of integration time and, in the case of gyrolasers, this value goes
down to $10^{-8}$ rad s-1. Following the Sagnac derivation, the minimal phase
shift that can be measured is $\Phi=2AE\Omega/(\hbar c^{2})$, which indicates
that $\Phi$ does not depend on the nature of the propagating wave. Therefore,
for the same given area $A$, the sensitivity of an interferometer using
massive particles of total energy $E$ can be several orders of magnitude
higher than that of an optical interferometer. This is the argument that
triggered the development of atom interferometry, and the technology progress
on atom optics allowed the realization of matter-wave interferometers already
in the beginning of the 1990s. mlynek ; shimizu ; keith

The first demonstrations of rotation measurements were performed, as usual,
with laboratory-scale devices. In 1991, Riehle et al. used a jet of thermal
calcium atoms in their gyrometer. riehle  In this experiment, the Sagnac
effect was observed by monitoring the displacement of interference fringes at
the output of the interferometer. In 2000 at Stanford, Gustavson et al.
gustavson1 ; gustavson2  developed an atom gyroscope capable of reaching a
short-term sensitivity of 6$\times$10-10 rad s-1 Hz-1/2, comparable to that of
the best gyrolasers at that time. They used a Ramsey–Bordé borde  symmetric
interferometer with two counterpropagating thermal jets of cesium atoms. This
configuration allowed the discrimination between rotation and acceleration
effects. The first compact cold atom gyroscope-accelerometer was developed by
the team at SYRTE. canuel  This device used two clouds of cold cesium atoms in
a configuration similar to the Stanford one. gustavson1  Table 1 presents the
state-of-the-art in rotation technologies.

Gyrometer | Sensitivity | Stability | $\tau_{I}$
---|---|---|---
technology | rad s${}^{-1}/\sqrt{{\rm Hz}}$ | rad s-1 | min
Atoms |  |  |
SYRTE 2018 | $3\times 10^{-8}$ | $3\times 10^{-10}$ | 167
SYRTE 2015 | $9\times 10^{-8}$ | $10^{-9}$ | 167
SYRTE 2013 | $2.6\times 10^{-7}$ | $2.5\times 10^{-8}$ | 3
SYRTE 2009 | $2.4\times 10^{-7}$ | $10^{-8}$ | 30
Stanford 2005 | $8\times 10^{-8}$ | $4\times 10^{-9}$ | 30
Stanford 2000 | $6\times 10^{-10}$ | $2\times 10^{-9}$ | 2
Mechanical |  |  |
Superconducting (GP-B) | $5.9\times 10^{-7}$ | $3.4\times 10^{-13}$ | 240
Optical |  |  |
Geant G-Ring laser | $1.2\times 10^{-11}$ | $1.6\times 10^{-13}$ | 300
Navigation ring laser gyro | ${\approx}10^{-8}$ | ${\approx}10^{-9}$ | –
Fiber iXBlue | $3\times 10^{-7}$ | $10^{-8}$ | –
Others |  |  |
RMN | $5\times 10^{-7}$ | $4\times 10^{-9}$ | –
He superfluid | $8\times 10^{-9}$ | – | –
NV center | $10^{-5}$ | – | –
Table 1: State-of-the-art in rotation sensing. Here, $\tau_{I}$ is the
integration time (1 rad s${}^{-1}\approx 2\times 10^{5}$ degree h-1).

To develop compact rotation sensors, area-enclosing magnetic guides realized
with macroscopic structures have been demonstrated. For instance, by using an
array of copper-tape coils in a racetrack shape, Tonyushkin and Prentiss
demonstrated a linear magnetic guide. tonyushkin  In this work, an atom
interferometer with an enclosed area was realized by splitting the initial
cloud along the guide axis and then translating the latter in the direction
perpendicular to the splitting. With this method, the authors demonstrated
smooth translations over centimeter-scale distances. The current in the coils
was as high as 50 A. Thus, this moving-guide configuration could be used for
rotation sensing. wu  In addition, by using a displaced linear magnetic guide
to enclose an area, Burke and Sackett demonstrated a scalable Sagnac atom
interferometer. burke  In this experiment, the visibility of an interferometer
realized with 3$\times 10^{4}$ 87Rb atoms was used as a measure of the
possible attainable enclosed area. It amounted to 0.05 mm2. Yan proposed using
this solution to realize a guided atom gyroscope on an atom chip. yan  To
displace the guide, state-dependent microwave potentials are generated by a
pattern of on-chip coplanar waveguides for the microwave field. Ring guiding
geometries generated with macroscopic structures have also been proposed and
realized, including the investigation of time-orbiting guiding potentials
gupta ; reeves ; arnold  and the realization of time-averaged adiabatic
potentials, gildemeister ; lesa ; pandey ; sherlock  inductively coupled ring
traps, griffin  DC storage rings, arnold1  and RF-dressed quadrupole ring
traps. morizot1 ; heathcote ; chakraborty

This paper reviews the various enabling technologies for the realization of
compact portable cold atom gyrometers based on atom chips. We mainly focus on
the research achievements relevant to inertial navigation applications.
Particular attention is devoted to atom chips because of their high potential
in the implementation of compact quantum sensors. In Sec. II we review the
general physical principles of atom interferometry. We then present the main
experimental realizations of compact cold atom gyrometers using free-falling
atoms. These are reference examples of key technological solutions to specific
inertial navigation questions to be considered when designing and implementing
guided atom interferometers. Section III is devoted to guided atom
interferometry and discusses the main inertial navigation specifications of
such a configuration. Section III.2 overviews the different enabling
technologies for atom-chip-based inertial sensors. The relevant results for a
sensor using guided atoms are presented in this section. Section IV discusses
the relevant systematic effects and noise sources based on an example of a
guided atom interferometer. Considering the expected sensor performance, a
particular application to a fundamental physics experiment is presented at the
end of Sec. IV.

##  II Atom interferometers for rotation sensing

###  II.1 Elements of atom interferometry

In an optical interferometer, beam splitters and mirrors are used to modify
the propagation mode of light injected at the input port of the device. In a
complementary way, laser beams modify the propagation state of atoms in an
atom interferometer (AI). At the input of the interferometer we prepare a
well-defined propagation mode in terms of momentum and internal energy of the
atomic state. Atom interferometers are typically implemented with stimulated
Raman transitions to manipulate the atomic state. They are generated by
counterpropagating laser beams, following the proposal of Bordé. borde

The Raman transition is a two-photon process in which the two laser beams
interact with the atoms described by a three-level system, as shown in Fig. 1.
Here, the laser frequencies are chosen to be significantly detuned by $\Delta$
from any possible optical transition involving an excited intermediate state
$|i\rangle$. In this way, spontaneous emission is suppressed, thereby
preserving the coherence properties of the atoms.

Figure 1: Representation of stimulated Raman transitions induced by two laser
beams of frequencies $\omega_{\rm L1}$ and $\omega_{\rm L2}$. The Raman
detuning is given by $\delta.$

Let us consider a particular realization of the process sketched in Fig. 1.
Suppose that the atom is initially prepared in an internal state of energy
$|f\rangle$ and momentum ${\bf p}$. After the absorption of a photon at
frequency $\omega_{\rm L1}$ and the subsequent stimulated emission of a photon
at $\omega_{\rm L2}$, the internal state of the atom is $|e\rangle$ and its
momentum ${\bf p}+\hbar({\bf k}_{\rm L1}-{\bf k}_{\rm L2})$. Because the laser
beams propagate in opposite directions, the atom changes its initial momentum
by $2\hbar k$, where $2k=2|{\bf k}|=|{\bf k}_{\rm L1}-{\bf k}_{\rm L2}|$. To
estimate the order of magnitude of the corresponding change in velocity, we
consider a transition on the $D_{2}$ line of 87Rb atoms. Along the axis of the
Raman beams, the change in velocity is $2\times 5.9$ mm/s, which is twice the
recoil velocity. This implies that, if the atom’s state changes from $|f,{\bf
p}\rangle$ to $|e,{\bf p}+2\hbar{\bf k}\rangle$, then after one second its
initial trajectory will be deflected by 1.2 cm.

By using light pulses (i.e., by changing the duration of the interaction
between the atoms and the laser beams), any superposition state may be
prepared between $|f,{\bf p}\rangle$ and $|e,{\bf p}+2\hbar{\bf k}\rangle$. In
particular, the pulse duration that prepares a superposition of the states
with the same weight for both states is called a “$\pi/2$” pulse and is used
to implement a matter-wave beam splitter. If the light pulse exchanges the two
states, then it is a “$\pi$” pulse and it implements a matter-wave mirror.
Typically, these pulses have a duration of a few tens of microseconds for
Raman beams with an optical power of about 400 mW and a 1/$e^{2}$ radius of 2
cm. The standard geometry of an atom interferometer is of Mach–Zehnder type
with three light pulses, as shown in Fig. 2.

Figure 2: Mach–Zehnder atom interferometer configuration. The short, black,
horizontal lines represent the equiphase planes of the laser beams in the
laboratory frame, and $T$ is the free propagation time.

To determine the atomic phase $\Phi$ at the output of the AI, we measure the
probability of detecting the atoms in states $|f,{\bf p}\rangle$ and $|e,{\bf
p}+2\hbar{\bf k}\rangle$, which can be written as $P=A+B\cos(\Phi)$. For an
apparatus subjected to an acceleration a, the phase $\Phi$ can be shown to be
given by borde

$\Phi_{\rm a}={\bf k}\cdot{\bf a}\ T^{2},$ (1)

where $T$ is free propagation time between the laser pulses. If the device is
subjected to a rotation at an angular rate ${\bf\Omega}$, then the atomic
phase is

$\Phi_{\Omega}=2{\bf k}\cdot({\bf\Omega}\times{\bf v})\ T^{2},$ (2)

where ${\bf v}$ is the initial velocity of the atoms in the state $|f,{\bf
p}\rangle$ at the AI input. The global inertial phase measured by an AI is
therefore $\Phi=\Phi_{\rm a}+\Phi_{\Omega}$, which thus contains information
about both the rotations and accelerations to which the device is subjected.
This phase is imprinted by the laser beams onto the atomic wave function
during the light-atom interactions that occur in the AI. Several techniques
are available to isolate accelerations from rotations, the most common being
to use a dual atom cloud source or a pulse configuration that renders the
interferometer sensitive to rotations but not to DC accelerations. For
instance, if a dual cloud source is used, then the two clouds are launched
following reciprocal paths at exactly the same absolute velocity. In this
situation, at the end of the interferometer sequence, the global inertial
phase measured on one cloud carries a rotation contribution with velocity
${\bf v}$, whereas the measurement of the global phase on the other cloud
gives a rotation contribution with velocity $-{\bf v}$. By adding and
subtracting these two global phases, one may distinguish accelerations from
rotations. How the global phase $\Phi$ is related to inertial forces is
presented below.

###  II.2 Free-falling atom compact rotation sensors

Compact atom interferometer inertial sensors can be divided into three large
classes: free-falling, confined, and guided atom interferometers. Out of these
three classes, free-falling atoms define the state of the art, so we consider
here a few representative implementations of this class. Almost all
realizations of inertial sensing with AIs use free-falling cold atoms. The
atoms are either dropped, launched vertically in a fountain configuration, or
follow parabolic trajectories. In this way, the atoms provide an inertial
frame with respect to which the forces exerted on the instrument can be
measured. For example, in Fig. 2 the AI is sensitive to an acceleration in the
direction of the Raman beams. On the one hand, accelerations translate into a
displacement of the lasers’ equiphase planes (short black lines) with respect
to the free-fall atomic trajectories. On the other hand, rotations of the
equiphase planes with respect to the atomic trajectory and about an axis
perpendicular to the AI’s oriented area are mapped to different light-atom
interaction strengths in the phase of the atomic wave function at every light
pulse.

One representative realization of a compact, free-falling AI was the first
cold atom gyrometer realized by Canuel et al. at SYRTE. canuel  This atom
interferometer used a dual cloud source and measured the six inertial axes
following the working principle presented in Fig. 3. It offered projection-
noise-limited performance, with a sensitivity to rotations and accelerations
of 2.4$\times$10-7 rad s-1 Hz-1/2 and 5.5$\times$10-7 m s-2 Hz-1/2,
respectively. Despite this remarkable result, the performance of this device
was limited by the initial temperature of the atoms (2 $\mu$K), the relatively
short interrogation time ($2T=60$ ms), the small area ($\approx$4 mm2) of the
interferometer, and the superposition of the atomic trajectories (Fig. 4).

Figure 3: Principle of a six-axis inertial sensor. The atomic clouds are
launched on a parabolic trajectory and interact with the Raman lasers at the
top. The four configurations (a)–(d) give access to the three rotations and
the three accelerations. In the three-pulse configuration, the Raman beam can
be horizontal or vertical. Therefore, the atom cloud can be split or deflected
in (a) the horizontal $x\text{-}y$ plane or (b), (c) in the vertical
$x\text{-}z$ plane. (d) With a butterfly four-pulse sequence of horizontal
beams, the rotation $\Omega_{x}$ can be measured. Adapted from Canuel et al.,
canuel  © 2006 by the American Physical Society. Figure 4: Picture of the
SYRTE six-axis inertial sensor. The instrument is 50 cm tall. Courtesy of A.
Landragin (Paris).

At NIST, Donley’s team recently demonstrated a multiaxis cold-atom gyrometer
by using a single source in a centimeter-scale cell. chen  Their device
simultaneously measures accelerations and rotations by using a point source
atom interferometer (PSI), as sketched in Fig. 5. After averaging for 1 s, the
measured sensitivities for the magnitude of the rotation rate and the
direction were $0.033^{\circ}$ s-1 and $0.27^{\circ}$, respectively.
Concerning acceleration measurements, the authors demonstrated a relative
precision in the gravity measurement of $\delta g/g=1.6\times 10^{-6}$ Hz-1/2.
The PSI technique consists of an ensemble of single-atom independent
interferometers.

Figure 5: Schematic of the PSI science package. The inner dimension of the
glass cell is 1 cm2. The instrument measures accelerations in the $z$
direction and the projection of the rotation vector onto the $x\text{-}y$
plane. “PBS” is a polarizing beam splitter. Adapted from Chen et al., chen  ©
2019 by the American Physical Society.

Because of the initial thermal velocity distribution of the cold atom cloud, a
strong correlation exists between position and velocity after a ballistic
expansion during the free-fall time of the cloud. This correlation means that
two different atoms with exactly opposite velocities will interact with the
Raman beams in the same way as the two clouds in the SYRTE gyrometer. It is
this correlation that is exploited to measure distinguishably accelerations
and rotations. Consequently, the PSI can be seen as a collection of dual-
source atom interferometers, but at the single-atom level. The strength of the
PSI is that it can measure rotation and acceleration in a single measurement
shot and using a single atom cloud.

Typically, the measurement rate of an atom interferometer is on the order of a
Hz. However, applications such as inertial navigation and guidance require a
high measurement bandwidth for the proper real-time integration of the vehicle
equations of motion. At Sandia National Laboratories, Rakholia et al. rakholia
tackled this critical question and realized a dual-axis device with a 60 Hz
bandwidth. The core idea of their technique is based on combining a dual
atomic source with the recapture of a fraction of the atoms after the
interferometer sequence. By simultaneously measuring accelerations and
rotations, this work presents a building block of a six-axis inertial sensor.
Although the interferometer duration is relatively short (4 ms), the loss in
sensitivity is compensated by using large atom numbers, large launching
velocities, and reducing the dead time. The measured sensitivities, which are
typical of laboratory experiments, are in the range of $\mu g$ Hz-1/2 and
$\mu$rad s-1 Hz-1/2 for acceleration and rotation, respectively. Such
performance is the result of a carefully developed device, analyzed following
a model developed by the authors. The identification and characterization of
the various noise sources also contributed to achieving the high sensitivities
shown in this work.

Another example of free-falling compact atom gyrometers was developed at
Hannover in Rasel’s group. tackmann  Their gyrometer had a Sagnac area of 19
mm2 and was 13.7 cm long. The rotation measurement was performed by using a
cold atom beam at 2.79 m s-1 and achieved a sensitivity of 6.1$\times$10-7 rad
s-1 Hz-1/2.

At Bordeaux, Barrett et al. developed a dual-species 87Rb-39K compact inertial
sensor for the study of the equivalence principle in microgravity. barret  One
of the important achievements of this experiment, from the inertial-sensing
perspective, is the operation of an atom interferometer in the strongly
vibrating environment of an aircraft. The sensor, subjected not only to
vibration levels of 10-2 $g$ Hz-1/2 but also to rotation rates as large as
$5^{\circ}$ s-1, measured the Eötvös ratio in microgravity with a systematic
uncertainty of 3$\times$10-4.

##  III Guided atom interferometry with atom chips

###  III.1 Main requirements for navigation

For an inertial navigation application, the main constraints for an atom
interferometer are small volume, low power consumption, high bandwidth, high
dynamic range, low cost, robustness, and device survivability when exposed to
or operated in a severe environment. barbour  The sensor is then suitable for
a specific inertial navigation task depending on its performance. For
instance, the angular random walk of a navigation-grade gyrometer has to be
less than $10^{-3}{}^{\circ}\,\text{h}^{-1/2}$.

To use an atom chip as a sensor platform, several technological obstacles have
to be addressed. Whether trapped or guided, coherent splitting and
recombination of a sub-Doppler–cooled thermal or degenerate atomic ensemble is
required. If we use a trapped ensemble, then symmetric splitting at the level
of 10-3 is required to have coherence times greater than 40 ms, and
acceleration must be sensed at the 10${}^{-6}~{}g$ level. In the case of a
guided atom interferometer, we need to fabricate or somehow generate
equivalent roughness-free matter-wave guides. For rotation sensing, the phase
bias stability has to be at the 10${}^{-4}{}^{\circ}\,{\rm h}^{-1}$ level.

###  III.2 Enabling technologies

Atom chip technology has been extensively covered in several reviews. keil ;
fortagh ; folman ; folman1 ; lev ; reichel  Here, we address the atom chip
enabling technologies relevant to key functional building blocks necessary for
implementing compact inertial sensors with cold guided atoms. These functions
concern

  * •

the precise and accurate positioning of the atoms to start the interferometer
sequence;

  * •

the coherent momentum transfer and splitting of the atom clouds;

  * •

the attenuation or suppression of propagation-induced decoherence;

  * •

the on-chip detection;

  * •

the vacuum dynamics because of its non-negligible contribution to the Dick
effect dick  via sensor dead time.

Addressing these points is important in order to exploit the full potential
for compactness offered by an atom chips.

Atom chips are a promising technology for manipulating cold atoms using
complicated confining geometries, which is important for developing compact
matter-wave interferometers. keil  In fact, it is possible to microfabricate
on an atom chip a complex wire pattern to create sub-micron magnetic
potentials with the shape required by the targeted application. We can, for
example, design arrays of potential wells for quantum information processing,
sinuco  traps for measuring accelerations, ammar  and toroidal waveguides for
detecting rotations. clga ; lesanovsky ; fernholz ; seungjin

The coherent beam splitting of a cold thermal cloud has already been
demonstrated for propagating atoms. cassetari  On atom chips, the coherent
manipulation of Bose–Einstein-condensed trapped and propagating atoms has been
observed in atom interferometers schumm ; maussang ; shin ; wang  and atomic
clocks. treutlin ; deutsch

Although cold atom propagation in circular macroscopic guides has already been
observed, gupta ; gildemeister  the demonstrated sensitivity to rotations was
insufficient for high-precision measurements or inertial sensing applications.
Using a linear waveguide, Wu et al. estimated the expected rotation
sensitivity of their enclosed-area interferometer. wu  Although the estimated
short-term sensitivity was 1$\times$10-9 rad s-1 Hz-1/2, the expected
stability was insufficient for measurements at a metrologically relevant
level. In other work, Qi et al. qi  built a magnetically guided atom
interferometer that uses Cs atoms and for which they used a compact
architecture to measure accelerations. The guide was generated by race-track
coils carrying a current of about 30 A. The resulting magnetic guide had a
radial frequency of 98 Hz and was loaded with 5$\times 10^{7}$ Cs atoms. The
resulting measurement uncertainty was 7$\times 10^{-5}$ m s-2 with an
interrogation time of $2T$ = 18 ms. The guide had an enclosed area of
1.8$\times 10^{-2}$ mm2 and can potentially be used to sense rotations.

Atom chips offer a convenient and flexible way for precise spatial positioning
of the atom clouds at the input port of the interferometer. Long et al. used a
magnetic conveyor to transport atoms over a total distance of 24 cm. long
Figure 6 shows the operation of this conveyor, which rotates the atom cloud
position by 90∘.

Figure 6: Illustration of positional precision that can be realized by using
an atom chip. The figure shows the rotation of a cloud of trapped 87Rb atoms.
The absorption images were acquired along two perpendicular axes parallel to
the substrate. The pictures are scaled so that the wire dimensions may serve
as a length scale for determining the cloud size. Reprinted with permission
from Long et al., long  © 2005 by Springer Eur. Phys. J. D.

Another relevant development for guided atom interferometry is their
demonstration of a magnetic guide that does not require an external bias
field. The guide is loaded by the rotating-trap device shown in Fig. 6. In
other work, Günther et al. demonstrated sub-micron accuracy in detecting the
position of a Bose–Einstein condensate, gunther  which enabled a demonstration
of magnetic-field microscope sensing at the mG level.

Different atom interferometer techniques are available to launch atoms into
the guide by subjecting them to a momentum kick. For magnetically guided
atoms, the standard solution is to use double Bragg diffraction. swu ; giese
In particular, Wu et al. developed a theory to describe a pulse sequence of
two square-shaped standing-wave light pulses. swu  By properly choosing the
strengths, durations, and separation between pulses, a Bose–Einstein
condensate at rest may be almost perfectly (99%) split into the $\pm 2\hbar k$
diffraction orders. wang ; swu  Figure 7 shows the designed pulse sequence.

Figure 7: (top) Illustration of the two square-shaped pulse sequence. (bottom)
Representation of the momentum-state vector in the Bloch sphere. Adapted from
Wu et al., swu  © 2005 by the American Physical Society.

On atom chips, coherent momentum splitting can be produced by using a
Stern–Gerlach beam splitter, as demonstrated by Machluf et al. in Ref.
machluf, . The beam splitter results from a combination of magnetic-field
gradient pulses and RF transitions between Zeeman states. Both the gradient
and the RF field are generated by feeding current to on-chip wires. Figure 8
shows absorption images of the splitting process.

Figure 8: Time-of-flight absorption images (a) of the trapped cloud. In panel
(b), the momentum splitting is weak (less than $\hbar k$) when using 5 $\mu$s
of interaction time with the magnetic-field gradient ($\text{TOF}=14$ ms). The
splitting realized in panel (c) corresponds to $40\hbar k$ of momentum
transfer using 1 ms interaction time and 2 ms TOF. In panel (d), the
absorption images show when the internal states are separated by an additional
strong gradient pulse. Adapted from Machluf et al., machluf  © 2013 by
Springer Nature Communications.

In this work the authors demonstrate interference fringes characterized by
short-term phase fluctuations of about 1 radian and long-term drifts on the
time scale of an hour. State-dependent microwave potentials can also be used
to induce momentum transfer with on-chip microfabricated wires. bohi

Usually, the number of atoms after application of the final beam-splitter
pulse is used to determine the inertial phase at the end of a single AI cycle.
This is done for all the propagation modes in order to compute the associated
probabilities, as discussed in Sec. IV. The standard technique is based on
absorption imaging, as already presented, for instance, in Fig. 6. However,
locally detecting atoms at well-defined positions on the chip is advantageous
for three main reasons: The first reason is that individual manipulation of
the atom clouds (or single atoms) becomes possible, which can be useful for
preparing one atom cloud and, at the same time, measure the AI output on the
previously prepared cloud. This would allow, for example, an interleaved
operation configuration. savoie  The second reason is the possibility to
implement local quantum non-demolition measurements of atom number at the AI
output. Such a strategy would allow recycling of atoms in the interferometer
to increase its measurement bandwidth. Finally, the third reason is that
integrated atom detection can be implemented (for instance, in situ collection
of fluorescence by using an on-chip fiber), which is important for making
compact sensors and avoiding the use of CCD cameras.

The enabling technology in this case has already been demonstrated by using
integrated optical elements on an atom chip. wilzbach  Another solution is to
combine an atom chip with integrated photonic optical chips. In both
situations, atoms may be locally excited for position-resolved detection.
ritter ; mehta ; mehta1 ; kohnen  Figure 9 shows a schematic representation of
an optical chip used to acquire the spectrum of a hot vapor of rubidium atoms.
It is based on a Mach–Zehnder light interferometer with one arm exposed to the
atomic vapor. The evanescent field of the optical waveguide mode interacts
with the atoms, and the interaction strength is read out as a modification of
the interferometer transmission. In other words, the presence of the atoms
produces a phase shift that carries information on their number and internal
state.

Figure 9: Schematic drawing of an integrated Mach–Zehnder optical
interferometer containing an interaction or detection region for the atoms in
one of its arms. Light is coupled in by focusing a laser beam onto a grating
coupler. Another grating coupler is used to extract the light from the device.
Reprinted with permission from Ritter et al., ritter  © 2015 AIP Publishing
LLC.

In another experiment, Mehta et al. used a grating coupler to implement
position-resolved excitation of ions mehta1  and addressed a single 88Sr+
qubit, as shown in Fig. 10.

Figure 10: Fluorescence images showing single-ion addressing. The grating
coupler is aligned with the central ion, which is prepared in a dark state by
focusing a 674 nm laser onto the grating. The fluorescence is at 422 nm, and
the signal is collected for 2 s. Adapted from Mehta et al., mehta1  © 2016 by
Springer Nature Publishing AG.

To guarantee monomode propagation of guided atoms golding ; andersson ; burke1
; das ; wu1  when realizing magnetic waveguides with relatively low currents
(below hundreds of mA), the atoms must be confined close to the chip surface.
However, in this situation, the coherent properties of the atomic states can
be dramatically affected by the corrugation of the microwires. These
fabrication defects produce a magnetically rough potential that can destroy
the system coherence or, even worse, blockade the propagation of the atoms.
kruger ; Whitlock ; sinclair:031603  Fabrication defects have been a
fundamental limitation for atom chips. kraft ; Hinds ; leanhardt:100404 ;
EstevePRArug ; lukin ; sculturing ; EsteveDWell  The physical origins of this
roughness can be understood as follows: Consider a wire with a current flowing
left to right as shown by the dark blue arrow in Fig. 11.

Figure 11: Top view of current $I_{\rm wire}$ flowing in a wire with
corrugated walls.

Because of meanders in the wire’s wall, an electron flux exists in the
direction perpendicular to the wire. These electrons generate a current
$\delta I_{\rm wire}$ that in turn produces a magnetic trapping potential in
the direction parallel to the wire.

Figure 12(a) shows an in-trap absorption image of a cold thermal 87Rb cloud
that exhibits fragmentation. This image corresponds to atoms loaded in a
linear magnetic guide created with three microfabricated wires, trebbia  each
of which carried a DC current in a configuration that creates a quadrupole
trap close to the minimum of the guiding potential. When the current
directions are reversed, the extrema of the rough magnetic potential are also
reversed, as can be seen in Fig. 12(b). However, the atoms are now confined in
complementary positions with respect to Fig. 12(a), and

Figure 12: (a) Absorption images of a thermal cloud for negative DC, (b)
positive DC, and (c) AC currents. Plotted is the number of atoms per pixel
(pixel size is $6\times 6~{}\mu$m2). Panel (d) shows the potential roughness
extracted from longitudinal profiles by using the Maxwell–Boltzmann
distribution. Adapted from Trebbia et al., trebbia  © 2007 by the American
Physical Society.

it is this complementarity that inspired the modulation technique that
suppresses the roughness of the magnetic guide. trebbia  In fact, by
temporally modulating the current in the wires, Trebbia et al. demonstrated a
roughness-free guide, as can be seen in the absorption image shown in Fig.
12(c). The rough potential was extracted by fitting the linear density with a
Maxwell–Boltzmann distribution along the axis of the cloud in Figs.
12(a)–12(c). bouch  This result is presented in Fig. 12(d), which quantifies
the fragmentation complementarity observed in the previous figures and the
degree of suppression of the roughness. For this technique to work, two
criteria must be satisfied: (a) the modulation frequency must exceed the
radial guide frequency but not the Zeeman frequency; (b) the phase difference
between the currents must be constant.

One of the key components of any cold atom sensor is the vacuum system. The
simplest vacuum setup would use a single chamber incorporating the atom source
(i.e., an alkali-metal dispenser). However, this solution requires a highly
dynamic vacuum system capable of switching from high (${\approx}10^{-8}$ mbar)
to low pressure (${\approx}10^{-11}$ mbar) in a few tenths of milliseconds.
Dugrain  Such a response is needed to reduce the deleterious effect of the
dead time between measurements. Recently, a promising technique was
demonstrated kitching  that allows one to quickly and reversibly control the
Rb background pressure in a cell, with switching times close to 1 s. This
reversible operation is illustrated in Fig. 13, where the magneto-optical-trap
(MOT) loading and depletion are controlled by a voltage applied across the
electrodes.

Figure 13: (a) MOT time response is represented by experimental points (red
dots), which correspond to the measured atom number. (b) The current flowing
in this reversible source is given by the black points. Adapted from Kang et
al., kitching  © 2019 by the Optical Society of America.

Other relevant investigations on the optimized operation of compact ultrahigh
vacuum (UHV) systems have been reported in Ref. campo, . In this work, light-
induced atomic desorption is used to modulate the background pressure of 87Rb
atoms in a glass cell. For atom interferometry and atom sensors, a UHV system
was designed and tested for operation in the highly vibrating environment of a
rocket. grosse  In Ref. scherer  the authors investigated the use of passive
vacuum pumps (non-evaporable getter pumps) for the development of compact cold
atom sensors. Basu and Velásquez-Garcia basu  demonstrated microfabricated
non-magnetic ion pumps to maintain UHV conditions in miniature vacuum chambers
for atom interferometry. Finally, Martin et al. jmm  investigated the pumping
dynamics of sputtered ion pumps. This study describes the main physical
pumping mechanisms based on a nonlinear model for the ion-current pump-down
dynamics in the low-pressure regime. The results suggest that a three-
dimensional (3D) structured cathode might allow for a smaller pump, increasing
or at least holding constant the trapping cross section of this electrode,
which may have significant consequences on the design and development of
miniature ion pumps.

##  IV Case study of a sensor design

This section presents a case study of a rotation sensor design that uses an
atom chip. The working principle is based on the Sagnac effect, which
naturally suggests a circular guiding geometry for the AI.

###  IV.1 Measuring rotation with a waveguide

To establish the physics that will drive the gyrometer design, consider the
Sagnac effect as sketched in Fig. 14. Starting with the illustration on the
left side of the figure, assume that, at $t=t_{0}$, two particles $A$ and $B$
leave the AI entry port $E$ and propagate freely in the azimuthal direction of
this circular guide of radius $r$. If the guide is rotated about an axis
${\bf\Omega}$ perpendicular to the guide’s oriented area, then after a time
interval $\delta t$ particles $A$ and $B$ would travel a geometrical path
length $\delta_{A}=r(\pi-\Omega\delta t)$ and $\delta_{B}=r(\pi+\Omega\delta
t)$, respectively. Therefore, there exists a path-length difference
$\delta_{AB}=\delta_{B}-\delta_{A}$ determined by the different arrival times
of the particles at the exit port $S$. By measuring this quantity, we can
determine the magnitude and direction of the rotation of the apparatus.

Figure 14: Illustration of the Sagnac effect in a circular guide.

In the following, we discuss the key elements of the design of an atom-chip-
based cold atom gyrometer. To be quantitative, we consider magnetic guiding of
ultracold thermal 87Rb atoms.

###  IV.2 Sensor design

#### Guiding

As we have seen, several solutions exist for realizing a matter-wave guide. To
implement an inertial sensor and exploit the potential for compactness offered
by an atom chip, we consider a circular magnetic guide produced by three on-
chip microfabricated concentric wires. Figure 15 shows the wire pattern, which
does not require an external bias field produced with coils. This is a well-
known configuration in which the external wires generate the bias magnetic
field needed to cancel the magnetic field of the central wire.

Figure 15: Microfabricated-wire configuration used to generate the circular
magnetic guide shown in Fig. 16. Here, the entry $E$ and output $S$ ports are
located at the same point in space. The laser beams implementing the matter-
wave optics are represented by the wave vectors ${{\bf k}_{1}}$ and ${{\bf
k}_{2}}$.

The resulting field is a quadrupole field and, by modulating the current fed
to the wires, the atoms can be guided at the minimum of this field in an
equivalent roughness-free stable magnetic guide. trebbia ; clgapra  Note that
the entry $E$ and output $S$ ports are located at the same point in space.
Such a geometry allows rejection of common-mode noise. Figure 16 shows the
magnetic potential generated by this configuration of microfabricated wires.
This magnetic potential is obtained when the wires are separated by 13 $\mu$m
and carry modulated currents of $-123$ mA and 121 mA in the external and
central wires, respectively. The radius of the central wire is $R=500~{}\mu$m.
The magnetic guide obtained in this way is around 13 $\mu$m from the chip
surface.

Figure 16: Magnetic guiding potential for on-chip cold atom gyrometer. It is
produced by a three-wire configuration in which the distance between the wires
is 13 $\mu$m and the central circular wire has a radius of 500 $\mu$m.

Given these parameter values, the radial guiding frequency is about 1.5 kHz,
and the potential depth is roughly 300 $\mu$K. These guide design parameters
allow the propagation of cold thermal atoms. By using this type of source we
can avoid the important contribution of atom-atom interactions to the
systematic noise and drift in the inertial phase of interest, wang ; horikoshi
; jo ; li ; stickney  and less preparation time is required than with
Bose–Einstein-condensed samples. In fact, concerning just the cooling process,
cold thermal atoms used in atom interferometry are obtained after molasses
cooling whereas degenerate samples require an additional evaporative-cooling
step. However, quantum degenerate atom sources might be relevant when
realizing, for instance, large-momentum-transfer beam splitters. mcdonald ;
debs  Actually, as discussed later, the sensitivity of the interferometer can
be enhanced by increasing the launch velocity and by letting the atoms
undertake several trips around the guide at a fixed single-loop interrogation
time $2T$.

#### Guide fabrication

The realization of the proposed configuration, shown in Fig. 15, raises
several fabrication challenges. For instance, the fabrication of the
microwires must account for the relevant practical and physics constraints of
a compact sensor for inertial navigation. From the practical point of view,
power consumption is another relevant constraint (for instance, to operate the
sensor on batteries). As already discussed, this problem can be mitigated by
guiding the atoms close to the chip surface, at a few tens of microns. In this
case, the proximity to the chip allows the production of strong magnetic-field
gradients to confine the atoms by using currents well below 1 A. note2  This
point is important for two main reasons: First, reducing the current intensity
reduces the heat dissipated by the wires. Second, developing or adding a heat-
management solution to the sensor, which can be detrimental to its
performance, is not necessary (e.g., a cooling system that introduces
parasitic vibrations). However, from the physics point of view, as already
mentioned, the proximity to the wires renders the guided atoms sensitive to
the roughness of the magnetic potential. This problem suggests a
microfabrication process based on metal evaporation, but this is a costly
solution when considering mass production of these sensors. The combined
challenge of low roughness and low cost can be addressed by using a
metallization process based on electroplating. The only requirement that
remains in this case is the realization of relatively small metal grains. lev
In addition, using an electroplating process allows the fabrication of
microwires with cross sections with large aspect ratios. In this way, their
electrical resistance and, consequently, heat dissipation can be reduced.

Figure 17 shows an example of a circular magnetic guide used to transport
atoms from the laser-cooling region to the science region of an atom chip.
huang  The picture illustrates a three-wire circular geometry of 1 cm diameter
in which the atoms are transported at 240 $\mu$m from the atom chip surface,
which demonstrates the feasibility of fabricating the wire pattern needed to
produce circular magnetic guiding potentials on an atom chip. The bottom chip
is used to realize an evaporative cooling trap and also serves as an
electrical feedthrough for the in-vacuum wires of the top chip in which the
circular trap is implemented. Both atom chips were fabricated by using
electroplating, which is a well-established technology in the microelectronics
industry.

Figure 17: Photograph of the atom chip of the LKB-SYRTE experiment Trapped
Atom Clock on a Chip. The experiment is developing an entanglement-enhanced
atomic clock with trapped atoms on a chip. More details are available in Ref.
huang, . Courtesy of J. Reichel (Paris).

#### Scale factor

When using magnetically guided atoms, it is preferable to use Bragg
transitions giltner ; ahlers  to realize the beam-splitter and mirror light
pulses. moler  In fact, in a Bragg transition, only the atomic momentum is
modified and the atoms remain in the same magnetic-trappable internal state.
In a circular waveguide, only beam-splitter pulses need be implemented because
the atomic trajectories are, by construction, deflected by the guide. For the
geometry shown in Fig. 15, we require only two Bragg transitions, as indicated
in Fig. 18. If we do not use a composite pulse sequence, swu  the initial
state $|\Psi\rangle=|{\bf p}=0\rangle$ is transformed into a three-component
superposition state $|\Psi\rangle=\alpha|{\bf p}=-2\hbar{\bf
k}\rangle+\beta|{\bf p}=+2\hbar{\bf k}\rangle+\gamma|{\bf p}=0\rangle$.

Figure 18: Beam-splitting process for the circular guiding potential
considered in Fig. 15. It represents a double Bragg diffraction, ahlers  each
realized by the pair of beams enclosed by a given red dashed ellipse.

By using the path-integral approach, pippa  one can show that the relevant
inertial (rotation) phase defining the scale factor is given by the expression

$\Phi=4kR\Omega(2T)=2\times 2\frac{M}{\hbar}A\Omega,$ (3)

where $2T=\pi R/v_{\rm r}=M\pi R/(\hbar k)$ is the interferometer
interrogation time for a round trip in a guide of area $A=\pi R^{2}$ and
$v_{\rm r}$ is the recoil velocity. In Eq. (3), we have considered an ideal
beam-splitting process to compute the probability of finding the atoms at the
output port $|{\bf p}=0\rangle$ of the AI.

#### Rotation-rate sensitivity

When the AI is operated at mid-fringe (maximum phase sensitivity), the
probability $P(\Omega)$ of finding the atoms at the output port $|{\bf
p}=0\rangle$ is

$P(\Omega)=\frac{N}{2}\big{[}1-\eta\cos\big{(}\Phi+\pi/2\big{)}\big{]},$ (4)

where $N$ and $\eta$ are the number of atoms and the contrast of the AI,
respectively. Equation (4) states that an infinitesimal variation of
$P(\Omega)$ translates into an infinitesimal variation of the rotation rate
such that

$\delta\Omega=\frac{1}{\Big{|}\frac{dP(\Omega)}{d\Omega}\big{|}_{\Omega=0}\Big{|}}\delta
P(\Omega=0).$ (5)

In Eq. (5), $\delta P(\Omega=0)$ gives the probability noise around the
working point of the AI, which can be chosen, for example, by setting a fixed
phase difference between the Bragg beams. For instance, in Eq. (4) the phase
difference is $\pi/2$, such that $P(\Omega)$ is the half-fringe value of the
interference signal in the absence of rotations. If we represent by
$\vartheta$ the latitude on Earth where the instrument is located, then Eq.
(3) becomes

$\Phi=\frac{4}{\pi}\frac{M}{\hbar}v_{\rm r}^{2}(2T)^{2}\Omega\sin(\vartheta),$
(6)

and consequently we have

$\frac{dP(\Omega)}{d\Omega}\Big{|}_{\Omega=0}=\frac{2}{\pi}N\eta\frac{M}{\hbar}v_{\rm
r}^{2}(2T)^{2}\sin(\vartheta).$ (7)

If the probability noise is defined solely by the quantum projection noise,
then

$\delta P(\Omega=0)=\sqrt{P(\Omega=0)}=\sqrt{\frac{N}{2}},$ (8)

so we finally find the following expression for the rotation-rate sensitivity:

$\delta\Omega=\frac{\pi}{2\eta\sqrt{2N}(M/\hbar)v_{\rm
r}^{2}(2T)^{2}\sin(\vartheta)}.$ (9)

Physically, Eq. (9) gives us the minimal rotation rate that can be detected if
the AI is projection-noise limited. To reach the projection-noise-limited
sensitivity, several technical problems must be overcome. In particular, for a
magnetic guide produced with modulated currents, the relative stability of the
currents supplied to the microwires and the noise level of the phase
difference between them must be considered. Both noise sources induce a
fluctuation in the separation between the guide and the chip surface. reichel
Consequently, the radial frequency of the guide (i.e., the radial confinement
energy of the guide) also fluctuates, generating a phase noise that is mapped
to the phase of the atomic wave function. As a result, the measured inertial
phase will acquire a parasitic contribution due to these physical effects. In
addition, if the guide uses a self-generated offset field, clgapra  then the
phase fluctuations will also compromise the stability of the magnetic guide.
Other relevant problems limiting the sensitivity and accuracy of this device,
and that are common to cold atom sensors, are detection noise, magnetic-field
noise, cloud temperature, and shot-to-shot atom number fluctuations (see, for
instance, Ref. szmuk, ).

As an example, Fig. 19 plots the rotation sensitivity given by Eq. (9) versus
the interferometer interrogation time and for a fixed launching velocity
2$v_{\rm r}$, which means that the guide radius depends on the interferometer
duration under consideration.

Figure 19: Calculated interferometer rotation-rate sensitivity vs
interrogation time. The red (blue) line is for $N=10^{5}\ (10^{4})$ atoms and
perfect visibility ($\eta=100\%$).

The result presented in Fig. 19 indicates that, for instance, using $10^{4}$
atoms and an interferometer duration $2T=3$ s gives a short-term rotation
sensitivity of $3.4\times 10^{-8}$ rad s ${}^{-1}{\rm Hz}^{-1/2}$ or an
angular random walk, $ARW\approx 1.1\times 10^{-4}{}^{\circ}\,{{\rm
h}}^{-1/2}$. In other words, after 1 h of integration the angular standard
deviation will be $1.1\times 10^{-4}{}^{\circ}$. Such a rotation-rate
sensitivity can be realized by using a 6-mm-radius guide or after 10 round
trips in a 600-$\mu$m-radius guide. For defense applications, this ARW is
already compatible with strategic-grade inertial navigators.

#### Sensitivity function

This function is defined as

$g(t)\equiv 2\lim_{\delta\phi\to 0}\frac{\delta P}{\delta\phi},$ (10)

where $\delta P$ is the infinitesimal variation of the probability $P$ of
finding the atoms at the AI output port. This variation results from an
infinitesimal jump of the global phase $\delta\phi$ due to a perturbation
during the measurement process.

The function $g(t)$ is a measure of the impact of a given perturbation, which
happens during the interrogation time $2T$, on the determination of the atomic
phase. cheinet  In fact, inside the time window defined by $2T$, the global
phase is sensed only during the time interval defined by the beam-splitter and
mirror light pulses. Therefore, we expect the sensitivity function to vary
during the pulses and to be extremal between them. The simplest gyrometer
configuration realized with a circular magnetic guide uses two $\pi/2$ pulses
of duration $\tau$, applied at the beginning and at the end of the AI.
Therefore, the sensitivity function of this device has the form presented in
Fig 20.

Figure 20: Sensitivity function for the circular guide shown in Fig. 15.

#### Transfer function

To determine the transfer function we compute the interferometer phase noise
$\sigma_{\phi}^{2}$. We denote by $\overline{\Delta\phi}$ the difference of
the mean phases during the pulses:

$\overline{\Delta\phi}(\tau)\equiv\overline{\phi}_{2}(\tau)-\bar{\phi}_{1}(\tau),$
(11)

where

$\overline{\phi}_{1}(\tau)=\frac{1}{\tau}\int_{-T}^{-T+\tau}dt\phi(t),\quad\overline{\phi}_{2}(\tau)=\frac{1}{\tau}\int_{T-\tau}^{T}dt\phi(t).$
(12)

In this case, the phase noise measured during the pulses is

$\sigma_{\phi}^{2}=\Big{\langle}\Big{|}\frac{1}{\tau}\int_{T-\tau}^{T}dt\phi(t)-\frac{1}{\tau}\int_{-T}^{-T+\tau}dt\phi(t)\Big{|}^{2}\Big{\rangle}_{T},$
(13)

where
$\sigma_{\phi}^{2}=\big{\langle}\big{|}\overline{\Delta\phi}(\tau)\big{|}^{2}\big{\rangle}_{T}$
and $\langle\dots\rangle_{T}$ is the temporal mean evaluated over a time
interval equal to the interrogation time $2T$. Here, $\phi(t)$ is the
instantaneous phase “seen” by the interferometer (resulting from rotations,
vibrations, laser phase noise, etc.).

From Eq. (13), the transfer function of this AI in the time domain is $h(t)$,
which is shown in Fig. 21 and defined as follows:

$h(\tau-t)\equiv\left\\{\begin{array}[]{rl}0,&|t|<-T\\\ 1/\tau,&-T\leq
t\leq-T+\tau\\\ 0,&-T+\tau<t<T-\tau\\\ -1/\tau,&T-\tau\leq t\leq
T.\end{array}\right.$

Figure 21: Time-domain transfer function of a two-pulse AI rotation sensor.

The transfer function $h(t)$ expresses the fact that the AI behaves as a
bandpass filter with characteristic frequencies defined by $\tau$ and $T$.

In the frequency domain, the transfer function $H(f)$ can be easily computed
and is given by the expression

$H(f)=-\frac{2\imath}{\pi f\tau}\sin(\pi f\tau)\sin\big{[}(\pi
f(2T-\tau)\big{]}.$ (14)

The transfer function is presented in Fig. 22 for experimentally accessible
parameter values. The cutoff frequencies are given by $f_{HP}\equiv
1/(\pi\tau)$ and $f_{LP}\equiv 1/[\pi(2T-\tau)]$. As can be seen in Fig. 22,
an ensemble of frequencies exists at which the interferometer is not sensitive
to phase noise. These frequencies correspond to multiples of the inverse of
the pulse duration and the interrogation time.

Figure 22: Frequency-domain transfer function of a two-pulse AI. To obtain
this plot, we consider a pulse duration $\tau=20~{}\mu$s and an interrogation
time $2T=4$ s. The cutoff frequencies are $f_{HP}=15.9$ kHz and $f_{LP}=0.1$
Hz.

###  IV.3 Main sources of noise and systematic effects

#### Phase noise expressed via the transfer function

By using the time-domain transfer function, Eq. (13) can be written as

$\sigma_{\phi}^{2}=\Big{\langle}\Big{|}\int_{-\infty}^{+\infty}dt\phi(t)h(\tau-t)\Big{|}^{2}\Big{\rangle}_{T}=\big{\langle}\big{|}\phi(t)\otimes
h(t)\big{|}^{2}\big{\rangle}_{T},$ (15)

or, making explicit the temporal mean,

$\sigma_{\phi}^{2}=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}dt\big{|}\phi(t)\otimes
h(t)\big{|}^{2}.$ (16)

For practical purposes, we write the phase noise in terms of the
interferometer spectral properties, its frequency domain transfer function
$H(f)$, and the power spectral density (PSD) of the phase noise,
$S_{\phi}(f)$. By using Parseval’s theorem and the definition of $h(t)$, Eq.
(16) takes the form

$\sigma_{\phi}^{2}=\lim_{T\to\infty}\frac{1}{2T}\int_{-\infty}^{+\infty}dt\big{|}\phi(t)\otimes
h(t)\big{|}^{2}\operatorname*{\Longleftrightarrow}^{\rm Parseval}_{\rm
theorem}\
\lim_{T\to\infty}\frac{1}{2T}\int_{-\infty}^{+\infty}df\big{|}\phi_{T}(f)\big{|}^{2}\big{|}H(f)\big{|}^{2},$
(17)

where $\phi_{T}(f)$ denotes the Fourier transform of $\phi(t)$. Therefore, if
$S_{\phi}(f)$ is the PSD of the phase noise, then we obtain

$\sigma_{\phi}^{2}=\int_{0}^{+\infty}dfS_{\phi}(f)\big{|}H(f)\big{|}^{2}.$
(18)

#### Sensitivity of AI phase to laser phase noise

To characterize the noise in the phase accumulated during the interferometer
interrogation time $\Phi$, we use the sensitivity function. Recall that, for a
100% contrast interferometer operating at mid-fringe, the probability $P$ at
the output port is

$P=\frac{1}{2}\big{[}1-\cos(\Phi+\pi/2)\big{]}.$ (19)

Consequently, the variation of $P$ due to an infinitesimal phase jump of the
Bragg laser is

$\frac{\delta
P}{\delta\phi}=\frac{1}{2}\sin(\Phi+\pi/2)\Big{|}_{\Phi=0}\frac{\delta\Phi}{\delta\phi}=\frac{1}{2}\frac{\delta\Phi}{\delta\phi},$
(20)

from which the sensitivity function associated with a phase jump becomes

$g_{\phi}(t)=\lim_{\delta\phi\to 0}\frac{\delta\Phi}{\delta\phi}.$ (21)

Thus, the measured accumulated AI phase during the interrogation time $2T$ is

$\Phi=\int_{-T}^{T}dtg_{\phi}(t)\frac{d}{dt}(\delta\phi),$ (22)

and consequently the phase noise after one interferometer cycle is

$\sigma_{\Phi}^{2}=\Big{\langle}\Big{|}\int_{-T}^{T}dtg_{\phi}(t)\frac{d}{dt}(\delta\phi)\Big{|}^{2}\Big{\rangle}_{T}.$
(23)

By using the definition of the sensitivity function (Fig. 20) and considering
that this function vanishes outside the interval $[-T;T]$, we find

$\displaystyle\sigma_{\Phi}^{2}$ $\displaystyle=$
$\displaystyle\lim_{T\to\infty}\frac{1}{2T}\int_{-\infty}^{+\infty}dt\big{|}g_{\phi}(t)\otimes\frac{d}{dt}(\delta\phi)\big{|}^{2}$
(24) $\displaystyle=$
$\displaystyle\lim_{T\to\infty}\frac{1}{2T}\int_{-\infty}^{+\infty}d\omega\big{|}G(\omega)\big{|}^{2}\big{|}\omega\
\delta\phi_{T}(\omega)\big{|}^{2},$

and finally

$\sigma_{\Phi}^{2}=\int_{0}^{+\infty}d\omega
S_{\phi}(\omega)\omega^{2}\big{|}G(\omega)\big{|}^{2}.$ (25)

Comparing Eq. (18) with Eq. (25) shows that the frequency-domain transfer
function of the AI and the Fourier transform of the sensitivity function
$G(\omega)$ are linked by

$\big{|}H(\omega)\big{|}^{2}=\omega^{2}\big{|}G(\omega)\big{|}^{2}.$ (26)

In addition, considering Eqs. (6) and (25), the rotation rate measured by the
AI may be characterized by the standard deviation

$\sigma_{\Omega}=\frac{\pi}{4}\frac{\hbar}{M}\frac{1}{v_{\rm
r}^{2}(2T)^{2}\sin(\vartheta)}\sigma_{\Phi}.$ (27)

#### Sensitivity to vibrations or acceleration noise

We have previously seen that the AI accumulated phase is [since
$g_{\phi}(t)=0\ \forall\ t\notin[-T;T]$ ]

$\Phi=\int_{-\infty}^{+\infty}dtg_{\phi}(t)\frac{d}{dt}(\delta\phi).$ (28)

In the case of Bragg transitions with counterpropagating laser beams in the
$z$ direction, the phase (wavefront or equiphase plane) at the location of the
atoms and at time $t$ is $\phi(t)=2kz(t)$. Thus,

$\Phi=2k\int_{-\infty}^{+\infty}dtg_{\phi}(t)\frac{d}{dt}(\delta z).$ (29)

Computing the integral in Eq. (29) gives

$\Phi=\int_{-\infty}^{+\infty}dt\Bigg{[}-2k\int_{-\infty}^{t}dt^{\prime}g_{\phi}(t^{\prime})\Bigg{]}\delta\Big{(}\frac{d^{2}z}{dt^{2}}\Big{)},$
(30)

which leads to the following definition of the sensitivity function of the AI
to accelerations:

$g_{a}(t)\equiv-2k\int_{-\infty}^{t}dt^{\prime}g_{\phi}(t^{\prime}),$ (31)

so that the interferometer phase due to an acceleration can be written in the
compact form

$\Phi_{a}=\int_{-\infty}^{+\infty}dtg_{a}(t)\delta a(t).$ (32)

From the physics point of view, Eq. (32) is the sum of all the infinitesimal
acceleration variations $\delta a(t)$ experienced by the device. These
contributions are taken at time $t$ and weighted by the sensitivity function
to accelerations; the latter is also evaluated at the same time instant. The
noise in the measured phase at the output of an AI experiencing vibrations is
therefore

$\sigma_{\Phi_{a}}^{2}=\Big{\langle}\Big{|}\int_{-T}^{T}dtg_{a}(t)\delta
a(t)\Big{|}^{2}\Big{\rangle}_{T}.$ (33)

Considering the relationship between the acceleration and the phase $\phi(t)$,
we can rewrite Eq. (33) in the form

$\sigma_{\Phi_{a}}^{2}=\Big{\langle}\int_{-\infty}^{+\infty}dt\Big{|}\frac{1}{2k}g_{a}(\tau-t)\frac{d^{2}}{dt^{2}}(\delta\phi)\Big{|}^{2}\Big{\rangle}_{T}.$
(34)

Next, by using the definition for the temporal mean and for the convolution
product, we get

$\displaystyle\sigma_{\Phi_{a}}^{2}$ $\displaystyle=$
$\displaystyle\lim_{T\to\infty}\frac{1}{2T}\int_{-\infty}^{+\infty}dt\Big{|}\frac{1}{2k}g_{a}(t)\otimes\frac{d^{2}}{dt^{2}}(\delta\phi)\Big{|}^{2}.$
(35)

As before, we can use Parseval’s theorem to find the final expression for the
vibration phase noise. It is given by

$\sigma_{\Phi_{a}}^{2}=\int_{0}^{+\infty}d\omega\frac{\omega^{4}}{(2k)^{2}}S_{\phi}(\omega)\big{|}G_{a}(\omega)\big{|}^{2}.$
(36)

Equation (36) has a simple physical meaning: ${(\sigma_{\Phi_{a}}^{2})^{1/2}}$
is the rms radians added by vibrations or acceleration noise to the atomic
phase one would like to measure with the AI.

Again, by comparing Eq. (36) with Eq. (25), we can derive the following
practical expressions linking the spectral properties of the acceleration
noise with those of the phase noise. In fact, isolating
$\big{|}G_{a}(\omega)\big{|}^{2}$ in Eq. (36), we find

$\displaystyle\big{|}G_{a}(\omega)\big{|}^{2}$ $\displaystyle=$
$\displaystyle\frac{(2k)^{2}}{\omega^{2}}\big{|}G_{(}\omega)\big{|}^{2},$ (37)
$\displaystyle S_{a}(\omega)$ $\displaystyle=$
$\displaystyle\frac{\omega^{4}}{(2k)^{2}}S_{\phi}(\omega),$ (38)
$\displaystyle\big{|}H_{a}(\omega)\big{|}^{2}$ $\displaystyle=$
$\displaystyle\frac{(2k)^{2}}{\omega^{4}}\big{|}H(\omega)\big{|}^{2}.$ (39)

Thus, if we independently measure the power spectral density of the
interferometer vibrations $S_{a}(\omega)$, then we get [because $H(\omega)$
can always be computed]

$\sigma_{\Phi_{a}}^{2}=\int_{0}^{+\infty}d\omega\frac{(2k)^{2}}{\omega^{4}}S_{a}(\omega)\big{|}H(\omega)\big{|}^{2}$
(40)

for the rms vibration phase in radians.

#### Sensitivity to rotation noise

Here we refer to the rotation noise caused by the fluctuations of the
interferometer rotation sensing axis i.e., the axis perpendicular to the
oriented area of the interferometer. The starting point in this calculation is
the scale factor (3). Writing this equation as

$\phi(t)=4kR\theta(t)$ (41)

with $\theta(t)=\Omega t$, we find the following expression for the
accumulated phase at the end of the interferometer cycle:

$\Phi=4kR\int_{-\infty}^{+\infty}dtg_{\phi}(t)\frac{d}{dt}(\delta\theta).$
(42)

After defining the function of sensitivity to rotations as
$g_{\Omega}(t)\equiv 4kRg_{\phi}(t)$, Eq. (42) becomes

$\Phi=\int_{-\infty}^{+\infty}dtg_{\Omega}(t)\delta\Omega(t).$ (43)

In analogy with the derivation of Eq. (36), we show that the measured rotation
noise is given by

$\sigma_{\Phi_{\Omega}}^{2}=\int_{0}^{+\infty}d\omega\frac{\omega^{2}}{(4kR)^{2}}S_{\phi}(\omega)\big{|}G_{\Omega}(\omega)\big{|}^{2},$
(44)

where $G_{\Omega}(\omega)$ is the Fourier transform of the rotation-
sensitivity function. In the present case, the useful relationships between
the spectral properties of the phase noise and the rotation noise are

$\displaystyle\big{|}G_{\Omega}(\omega)\big{|}^{2}$ $\displaystyle=$
$\displaystyle(4kR)^{2}\big{|}G_{(}\omega)\big{|}^{2},$ (45) $\displaystyle
S_{\Omega}(\omega)$ $\displaystyle=$
$\displaystyle\frac{\omega^{2}}{(4kR)^{2}}S_{\phi}(\omega),$ (46)
$\displaystyle\big{|}H_{\Omega}(\omega)\big{|}^{2}$ $\displaystyle=$
$\displaystyle\frac{(4kR)^{2}}{\omega^{2}}\big{|}H(\omega)\big{|}^{2}.$ (47)

Once again, if one determines the PSD of the rotation noise
$S_{\Omega}(\omega)$, then, from the equation

$\sigma_{\Phi_{\Omega}}^{2}=\int_{0}^{+\infty}d\omega\frac{(4kR)^{2}}{\omega^{2}}S_{\Omega}(\omega)\big{|}H(\omega)\big{|}^{2},$
(48)

we find ${(\sigma_{\Phi_{\Omega}}^{2})^{1/2}}$ rms radians of rotation noise
contributing to the measured phase signal of the interferometer.

#### Stability

As is well known, the most informative quantity about the stability of a
sensor (or instrument in general) is the Allan variance. allandev  By using
equations (27) and (48), the Allan variance for the measured rotation rate
with this AI may be shown to be

$\displaystyle\sigma_{\Omega}^{2}(\tau_{I})$ $\displaystyle=$
$\displaystyle\Big{[}\frac{\pi}{4}\frac{\hbar}{M}\frac{1}{v_{\rm
r}^{2}(2T)^{2}\sin(\vartheta)}\Big{]}^{2}$
$\displaystyle\times\frac{4\pi}{\tau_{I}}\sum_{m=0}^{\infty}\frac{(4kR)^{2}}{[2\pi
m/(2T)]^{2}}\big{|}H(m/T)\big{|}^{2}S_{\Omega}(m/T),$

where $\tau_{I}$ is the integration time. To obtain an order of magnitude of
this quantity, let us consider a projection-noise-limited AI with $2T=10$ s of
interrogation time. note0  If $10^{5}$ atoms are launched at $2v_{\rm r}$ by a
$\pi/2$ pulse with a duration of $\tau=20~{}\mu$s, then the Allan standard
deviation is $1.9\times 10^{-9}~{}{\rm rad\,s^{-1}}/\sqrt{\tau_{I}({\rm s})}$.
Figure 23 shows the computed Allan standard deviation for this particular
example. note1

Note that, after 12 months of integration, the interferometer reaches a
stability of $3.5\times 10^{-13}~{}{\rm rad\,s^{-1}}$, which is theoretically
compatible with applications in geophysics and the realization of tests in
fundamental physics, such as the observation of the geodetic effect.

Figure 23: Computed rotation rate in the form of the Allan standard deviation
for a projection-noise-limited AI using a circular waveguide.

In fact, in the 1960s, using general relativity, Leonard Schiff predicted that
a free-falling inertial frame in a polar orbit around a rotating gravitational
source experiences two orthogonal rotations with respect to the fixed inertial
frame of the Universe. schiff ; schiff1  These two phenomena, called the
geodetic and Lense-Thirring effects, are characterized by rotation rates of
6.6 arcseconds/year and 33 milliarcseconds/year, respectively (1
milliarcsecond = $4.848\times 10^{-9}$ rad s-1), for an orbit located 642 km
from the Earth.

Recently, in 2011, the Gravity Probe B (GP-B) experiment developed by Stanford
University and NASA confirmed these predictions with a precision of 1% by
using a satellite. The science (data recording) phase of this mission lasted
353 days. GPB ; GPB1  If, in an analogous way, we would like to measure, for
instance, the geodetic effect by using cold guided atoms on an atom chip with
a precision of 5% in 1 year, then we would need a gyrometer with a stability
of about $5.2\times 10^{-14}$ rad s-1. From the result presented in Fig. 23
and using the parameter values stated above, such a measurement would require
47 years!

However, this problem should be tackled in a different way. The meaningful
question is whether we can design a compact atom-chip-based gyrometer by using
cold guided atoms to reliably measure the geodetic effect from a satellite
orbiting at 642 km above Earth. Part of the answer to this question is
presented in Fig. 24, which shows the $v\text{-}2T$ diagram (i.e.,
interrogation time versus launching speed) with a 5% precision for 12 months
of integration time.

Figure 24: Experimental limits of the launching speed and the AI interrogation
time compatible with the observation of the geodetic effect with 5% precision
in 12 months of integration.

We learn from Fig. 24 that, if the atoms are launched with, say, an initial
velocity of 4$v_{\rm r}$, then we need a minimum interrogation time $2T=9$ s
to achieve 5% precision in the measurement. For this particular case, we would
need a 37-mm-radius circular magnetic guide.

Suppose now that we can develop an atom chip gyrometer compatible with the
CubeSat technology. cubesat  In such a scenario, we could foresee, for
example, the simultaneous measurement of the geodetic effect using two
cubesats orbiting at different distances from Earth. The more distant AI could
provide a reference measurement in the common frame of the two satellites, and
a differential signal could provide better precision to demonstrate this
effect of general relativity. Moreover, if a multi-axis atom-chip-based
inertial sensor is developed, then it could be used to realize a drag-free
satellite configuration for this scenario. Finally, to have an idea of the
potential applications accessible with this AI configuration, Fig. 25 shows
the order of magnitude of the rotation rate associated with different
fundamental physics phenomena.

Figure 25: Order of magnitude of rotation rates characteristic of different
phenomena in geophysics and fundamental physics. These rotation rates are
scaled to the rotation rate of Earth: $\Omega_{\rm E}=7.29\times 10^{-5}$ rad
s-1.

An important point to have in mind in the design of a cold atom sensor is the
scale of the instrument in terms of volume, power consumption, and weight. For
an atom-chip-based sensor, the core of the physics package (i.e., the chip) is
a few square centimeters, its power consumption can be kept below 1 W, and,
considering the holder, it can weigh up to a few hundred grams. The next
important element in the physics package is the vacuum system. By using a
glass cell, its volume can be as small as 10 cm3. However, a physics package
with such a small vacuum chamber would certainly require a second chamber to
realize a double-MOT configuration to avoid reducing the lifetime of the cold
samples. In fact, the vacuum system plays a key role in determining the dead
time between measurements. Dugrain ; jmm  For instance, assume that the
interferometer uses molasses cooled atoms. If we use a single vacuum chamber,
then the expected dead time is on the order of 5 s. If we use a double-chamber
vacuum system (2DMOT + 3DMOT), then the dead time can be reduced to less than
1 s. In both cases, if we use evaporatively cooled atoms, then we need to add,
typically, at least 3 s to the previous values. From the reported results
(see, for instance, Ref. grosse, ), one can expect that a physics package of 1
L and 1 kg is feasible for this type of sensor.

Another key element of the sensor is the laser source. Today, low-phase-noise
transportable laser sources are commercially available. muquans  To best
exploit the compactness of atom chips, an important technological effort is
required to integrate such laser sources onto an atom chip substrate.

Two other important sensor requirements for inertial navigation and guidance
are the bandwidth and the dynamic range. High-data-rate sensors are
demonstrated in Refs. rakholia, ; savoie, , and a sensor with relatively large
dynamic range was demonstrated in Ref. barret, . These experimental
realizations are very encouraging because, given the time required to cool the
atoms, inertial sensors using atom interferometry technology have in general a
very small bandwidth, typically below 1 Hz. This is a clear drawback if the
sensor is expected, for instance, to provide guidance to a carrier. In
addition, the performance (sensitivity and stability) of a cold atom inertial
sensor is dramatically affected by vibrations. Therefore, one might not expect
cold atom sensors to supplant the conventional inertial sensors that are
commercially available today (i.e., ring laser gyros, fiber-optic gyros, MEMS
accelerometers, and gyroscopes). However, contrary to cold atom inertial
sensors, conventional sensors lack the requisite stability for long-term
navigation, are not absolute, require calibration, and usually need an
external reference signal that is susceptible to jamming. Therefore, as an
optimal navigation solution one can foresee the hybridization of cold atom and
conventional inertial sensors. lautier

##  V Outlook and conclusion

This review presents the state-of-the-art results in atom interferometry that
are relevant for inertial navigation applications. These results were obtained
by using laboratory instruments that, given their volume, are not yet
compatible with mobile applications. However, they are undoubtedly relevant to
fundamental studies and to define the ultimate performance that can be
realized with the compact portable sensors. We also discuss a representative
set of portable and compact atom interferometers that have demonstrated an
inertial sensitivity. In particular, for rotation sensing, examples are given
for two classes of interferometers: free-falling atoms and guided atom
interferometers. The latter is considered in detail when using an atom chip as
the sensor platform. The enabling technologies and a case study of a sensor
design for inertial navigation applications are also presented. Notably, from
the computed sensitivity in Fig. 19, the expected angular random walk
[$1.1\times 10^{-4}$ ${}^{\circ}{\rm h}^{-1/2}$] after 1 h of integration
suggests that a sensor based on the design under consideration would be
compliant with strategic-grade inertial navigators.

Projection-noise-limited atom interferometric inertial sensors have now been
demonstrated in the free-falling configuration. When dealing with atom chips
with relatively low atom number, this result suggests that a possible solution
to reach the desired performance is to implement quantum metrology protocols
by using squeezed and entangled states. eckert ; caves ; riedel ; reinhard ;
leo ; koschorreck  In this case, the device sensitivity would scale as
$\Delta\Phi_{\rm min}\approx\xi(N)/\sqrt{N}$, where $\xi(N)<1$ defines the
degree of atomic noise squeezing. In the context of atom interferometry, these
states have been generated in optical dipole traps, appel  optical cavities in
the QED regime, smith  and in optical lattices. gross

##  VI Acknowledgments

I would like to thank the members of the atom interferometry and inertial
sensors team at SYRTE. This work was funded by the Délégation Générale de
l’Armement (DGA) through the ANR ASTRIDE program (Contracts
ANR-13-ASTR-0031-01, ANR-18-ASMA-0007-01), the Institut Francilien de
Recherche sur les Atomes Froids (IFRAF), and the Emergence-UPMC program
(Contract A1-MC-JC-2011/220).

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