# Entanglement-Enhanced Optical Atomic Clocks

Simone Colombo  Department of Physics, MIT-Harvard Center for Ultracold Atoms
and Research Laboratory of Electronics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA.  Edwin Pedrozo-Peñafiel  Department of
Physics, MIT-Harvard Center for Ultracold Atoms and Research Laboratory of
Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA.  Vladan Vuletić  Department of Physics, MIT-Harvard Center for
Ultracold Atoms and Research Laboratory of Electronics, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Recent developments in atomic physics have enabled the experimental generation
of many-body entangled states to boost the performance of quantum sensors
beyond the Standard Quantum Limit (SQL). This limit is imposed by the inherent
projection noise of a quantum measurement. In this perspective article, we
describe the commonly used experimental methods to create many-body entangled
states to operate quantum sensors beyond the SQL. In particular, we focus on
the potential of applying quantum entanglement to state-of-the-art optical
atomic clocks. In addition, we present recently developed time-reversal
protocols that make use of complex states with high quantum Fisher information
without requiring sub-SQL measurement resolution. We discuss the prospects for
reaching near-Heisenberg limited quantum metrology based on such protocols.

††preprint: AIP/123-QED

##  I introduction

Optical-transition atomic clocks [1, 2, 3, 4, 5] are the most accurate sensors
developed by humankind, reaching fractional stabilities below $10^{-18}$. This
mindboggling precision, corresponding to an uncertainty of less than one
second over the age of the universe, and the associated technological
improvements, enable a broad range of applications in the field of precision
metrology, such as the search for dark matter in the low-to intermediate-mass
sector [6, 7, 8, 9, 10, 11], the investigation of nuclear structure and matter
[12, 13, 10], the study of any time variation of fundamental constants [14,
15, 16], testing of the foundations of general relativity[17, 18, 4, 19] , the
detection of low-frequency gravitational waves [20, 21], or geodesy [22, 23].
Moreover, high timekeeping precision enables improvements in navigation
systems, both GPS-based and inertial [24, 25].

Currently, the main limitations to the clock precision are atomic collisions
leading to atomic energy shifts [26, 27, 28, 29, 30, 31], the Dick noise [32]
that is associated with the interrupted interrogation of the atomic system,
and the standard quantum limit (SQL) resulting from the quantum projection
noise of an ensemble of finite atom number. Schemes for removing the Dick
noise have been demonstrated [33, 34, 35, 2, 36, 37, 38], while collisional
line shifts are usually minimized by deploying a low-density atomic gas or
using engineered single-atom traps like 3D optical lattices [39, 40] or arrays
of optical tweezers [41, 42]. Collisional shifts impose a limit on the total
number of atoms that are used in optical clocks, which is typically between
$10^{2}$ and $10^{4}$ atoms [43]. At such relatively small atom number, the
SQL presents a significant constraint on the clock precision. The SQL can be
overcome by engineering quantum correlations (entanglement) between the atoms
[44, 45]. Appropriate collective entangled states [46, 47, 48, 49, 50, 51, 52,
53, 54] can readily boost the performance of these advanced clocks,
particularly for applications that require operation at fixed bandwidth [6,
55, 56, 21].

The generation of metrologically useful entanglement on the optical clock
transition of ytterbium-171 ($\mathrm{{}^{171}Yb}$) has been recently
demonstrated [57]. Yet, full clock operation beyond the SQL in a state-of-the-
art optical clock represents one of the most important challenges to be met.
In this article, we will give an overview and perspective on this topic that
is central to the development of future optical clocks and other sensors based
on quantum interference [45].

##  II Metrological Gain

Optical clocks measure the passing of time in terms of the frequency standard
provided by an optical transition to a long-lived atomic excited state with
frequency $f_{a}\gtrsim 100~{}$THz, corresponding to more than $10^{14}$
oscillations in a typical interrogation time of $\sim 1$ second. The
fractional stability of an optical clock is expressed as ${\sigma\equiv\delta
f/f_{a}}$ with ${\delta f}$ being the uncertainty of the frequency estimation.

To measure such high oscillation frequencies, one compares a local oscillator
(LO) to the atoms’ oscillations. The LO can be, depending on the type of
operation, either an ultrastable laser [2, 34, 35] (complete clock
implementation), or another atomic ensemble [58, 5] (differential operation).
The comparison is performed by measuring the accumulated phase difference
between the LO and the atomic reference in a given measurement time $\tau$
(called the spectroscopy or interrogation time). In a linear protocol, the
relation between phase difference $\varphi$ and frequency imbalance ${\Delta
f}=f_{a}-f_{\mathrm{LO}}$ between the LO and the atomic system is given by
[59]

${\varphi(\tau)}=2\pi\int_{0}^{\tau}w(t){\Delta f(t)}\mathrm{d}t$ (1)

where $w(t)$ is the protocol sensitivity response [59]. From the accumulated
phase, one can extract information about the average frequency difference. The
clock stability is then limited by the precision in the estimation of the
phase $\varphi$.

The goal is then to estimate the phase $\varphi$ with the smallest possible
uncertainty $\delta\varphi$ for a fixed set of resources, such as atom number,
spectroscopy time, and dark time. The uncertainty on the phase estimation can
have both classical (technical noise) and quantum contributions. In state-of-
the-art optical clocks, the technical noise is of the same order or smaller
than the quantum noise. For atomic sensors that employ a number $N$ of
uncorrelated atoms, the quantum noise limits the phase estimation to the SQL,
$\delta\varphi_{\mathrm{SQL}}=1/\sqrt{N}$. The SQL is not a fundamental limit
and can be overcome by proper engineering of quantum correlations
(entanglement) between the atoms prior to starting the measurement protocol
[46, 47, 48, 49].

Metrologically useful entangled states are characterized by their quantum
Fisher information $\mathcal{F}_{\mathrm{Q}}$ [60, 61, 45] which quantifies
the sensitivity of the state to a change of a parameter, here a phase angle.
Generally, the quantum limit to the phase estimation for a certain quantum
state is given by the quantum Cramér-Rao bound [45]

$\delta\varphi_{\mathrm{CR}}=\frac{1}{\sqrt{\mathcal{F}_{\mathrm{Q}}}}.$ (2)

Therefore, the quantum Fisher information is the figure of merit for the
ultimate sensitivity achievable with a given quantum state. The field of
quantum metrology investigates the class of quantum states with
$\mathcal{F}_{\mathrm{Q}}>N$, which is a necessary and sufficient condition
for a quantum-enhanced sensor (i.e. a sensor operating beyond the SQL) [60].
The fundamental limit to the phase estimation, the holy grail of quantum
metrology, is the Heisenberg limit (HL) for which
$\mathcal{F}_{\mathrm{Q}}=N^{2}$.

In this article, we will discuss collective entangled states, i.e. states with
$\mathcal{F}_{\mathrm{Q}}>N$, and how to generate them in optical-transition
clocks. We express then the performances of a quantum enhanced sensors in
terms of metrological gain beyond the SQL as

$\mathcal{G}\equiv\left(\frac{\mathrm{SNR}}{\mathrm{SNR}_{SQL}}\right)^{2},$
(3)

where $\mathrm{SNR}$ and $\mathrm{SNR}_{SQL}$ are the signal-to-noise ratios
for the actual sensor and an ideal sensor operating at the SQL, respectively.
Except for relatively simple entangled states with a Gaussian envelope, known
as spin squeezed states (SSSs), saturating the Cramér-Rao bound in phase
estimation requires probabilistic methods beyond the standard evaluation of
expectation values [45, 62, 63]. Hence the metrological gain is bounded by

$\mathcal{G}\leq\frac{\mathcal{F}_{\mathrm{Q}}}{N},$ (4)

where the equality applies when the Cramér-Rao bound is saturated.

##  III  Squeezing

Figure 1: Generalized Bloch Sphere Representation of an $N$-atom system. a.
Ramsey spectroscopy using a (not entangled) coherent spin state. The blurred
circle represents the quantum projection noise that gives rise to the Standard
Quantum Limit (SQL), limiting the precision of the determination of the phase
$\varphi$. b. Ramsey spectroscopy using a spin squeezed state, a special and
simple collective entangled state where the quantum noise is redistributed
within two orthogonal quadratures. $\xi_{-}$ and $\xi_{+}$ denote the reduced
(squeezed) and increased (anti-squeezed) variance along two orthogonal axes,
respectively. In this case, the accumulated phase $\varphi$ can be estimated
with precision beyond the SQL.

We consider each two-level atom in the clock as a spin-$\frac{1}{2}$ system,
and denote by $S=N/2$ the total spin of the ensemble. Spin squeezed states
(SSSs) represent a relatively simple class of collective entangled states. As
shown in Fig. 1b, in a SSS the noise is redistributed between two orthogonal
quadratures. The squeezed quadrature, with a variance (normalized to the SQL)
of $\xi_{-}^{2}=\frac{2}{S}(\Delta S_{min})^{2}$, is oriented along the phase
axis, providing a metrological gain over the SQL by virtue of a smaller
quantum noise along the phase direction. The gain $\mathcal{G}$ of a squeezed
state is quantified by the Wineland parameter [46]
$\xi^{-2}_{R}=\frac{C^{2}}{\xi_{-}^{2}}=\mathcal{G}$, where $C$ is the
contrast of the interferometer. As we will discuss in detail in section VI,
the maximum metrological gain achievable with squeezed states depends on
secondary detrimental effects resulting from the quadrature with increased
variance, called antisqueezing, $\xi_{+}^{2}=\frac{2}{S}(\Delta S_{max})^{2}$.

Establishing any kind of quantum correlation within the atomic ensemble
requires an interaction between the atoms, since they need to know about each
other’s state. Such an interaction can be direct, driven by state-dependent
collisions or spin-spin interactions [64, 65, 62, 66], or effective [50, 67,
68, 52, 53, 69], mediated by an external field, such as a single cavity light
mode interacting with all the atoms in the ensemble. Optical techniques are
beneficial for metrological applications since they provide an effective atom-
atom interaction that can be turned off after preparation of the spin
entangled state[70, 71, 53, 52, 68, 72, 67], thus avoiding undesirable
collision-induced energy shifts during the operation of the interferometer.

Figure 2: Atom-cavity system for generation of spin squeezed states. a. An
ensemble of $N$ spin-1/2 atoms is coupled to a high-finesse optical cavity to
generate the non-linear interaction required to produce spin squeezing. b. Top
row: SSS generated by measurement-based squeezing. The squeezing is
conditioned on the detection of photons that carry information about the
collective spin state. After the optical measurement of the spin state via the
cavity, a small pulse $\beta$ can be applied to place the quantum state at the
desired position on the Bloch sphere. Bottom row: deterministic cavity
feedback squeezing. A CSS is subject to a one-axis twisting-like Hamiltonian
($H\propto S_{z}^{2}$) generated by the light-cavity interaction (see main
text).

Incoherent scattering and photon loss set a bound to the maximal achievable
entanglement with optical methods. Hence, the use of high-finesse cavities is
preferable since they enhance the interaction relative to the scattering, and
have intrinsically smaller optical losses (Fig. 2a). Moreover, schemes that
reduce dissipation [73, 74, 75, 76, 77] can increase the metrological gain.

###  III.1 Measurement-based squeezing

A highly robust and conceptually straightforward technique that is able to
produce record levels of spin squeezing is based on quantum nondemolition
(QND) measurement [49, 50, 51, 52, 53, 54], where the uncertainty in one
quadrature of the state is reduced through the measurement of photons that
have interacted with the atoms (top row of Fig. 2b). However, this method is
necessarily imperfect, due to finite total quantum efficiency of light
collection and often incomplete use of the collected information. As a result,
squeezing via QND measurement in practice has operated far from unitarity,
with antisqueezing significantly in excess of the minimum set by the
Heisenberg uncertainty principle for a given level of squeezing.

In order to avoid incoherent spontaneous emission, so far all cavity QED
experiments aimed at the generation of collective entangled atomic states have
operated far below the saturation of the atoms, i.e., in the regime of linear
atomic response. In this limit, effective photon-photon interactions are
weak[77]. We can then consider a coherent state of light that interacts with
the atom-cavity system and which remains a coherent state after that
interaction. The light state $\left|\alpha(S_{z})\right\rangle$ exiting the
cavity, where $\alpha$ is the amplitude of the coherent state, depends on the
$S_{z}$ component of the collective atomic spin. This implies that the light
field carries information about the state of the atomic ensemble. As shown by
Li et al. [77] this information, or light-atom entanglement, is characterized
by the quantum Fisher information $I$ of the light state

$\tilde{I}=4\left|\frac{\mathrm{d}\alpha(S_{z})}{\mathrm{d}S_{z}}\right|^{2}.$
(5)

In particular, the variance $\left(\Delta S_{z}\right)^{2}$ associated with
estimating the parameter $S_{z}$, using knowledge of the state
$\left|\alpha(S_{z})\right\rangle$, is given by the relative Cramér-Rao bound
$\left(\Delta S_{z}\right)^{2}=1/\tilde{I}$.

An external observer can gain knowledge about the $S_{z}$ observable so that
the spin variance is reduced below the SQL

$\left(\Delta S_{z}\right)^{2}_{\mathrm{light}}=\frac{2}{S(1+I)},$ (6)

where $I=2\tilde{I}/S$ is the normalized Quantum Fisher information of the
light field acquired by the observer ($I=1$ resolves the SQL uncertainty).
Hence, for any $I>1$ a conditionally squeezed state of the collective atomic
spin is generated.

###  III.2 Cavity feedback squeezing

As discussed above, one can generate strong conditional spin squeezing simply
by quantum non-demolition measurement [49, 50, 51, 52, 53]. The atom-cavity
interaction entangles the collective atomic spin with the light, and a high
photon detection efficiency is instrumental in obtaining large amounts of spin
squeezing. To eliminate the requirement of high detection efficiency, a
Hamiltonian method called cavity-feedback squeezing was proposed [51] and
experimentally demonstrated [68] in 2010.

The method can be understood in terms of a two-fold interaction process[67],
where quantum spin fluctuations of the atomic ensemble are imprinted onto the
light field that then acts back onto the atoms[51]. By placing the atoms in an
optical resonator the atom-light interaction can be significantly enhanced. In
general, such strong interaction between an ensemble of $N$ spins and the
optical field of an optical resonator can be described through the Hamiltonian
[77]:

$\displaystyle\hat{H}_{dip}=-\hbar\Omega\hat{n}_{c}\hat{S}_{z},$ (7)

where $\hat{n}_{c}$ is the intracavity photon number, and $\hbar\Omega$ is the
light shift produced by a single photon inside the cavity. In a cavity setup,
the intracavity photon number $\hat{n}_{c}=\hat{c}^{\dagger}\hat{c}$ is
$\hat{S}_{z}$-dependent, producing to lowest order the paradigmatic one-axis
twisting (OAT) Hamiltonian, initially proposed theoretically by Kitawaga and
Ueda in 1993 [48]:

$\displaystyle\hat{H}_{OAT}=\hbar\chi\hat{S}_{z}^{2}.$ (8)

This Hamiltonian produces a spin-dependent precession about the $z$-axis that
is proportional to $S_{z}$, causing an elliptical distortion of the initial
symmetric noise distribution (bottom row of Fig. 2b). $\chi$ is known as the
shearing parameter that quantifies how quickly the initial coherent spin state
distribution is “sheared”.

###  III.3  Entanglement via direct atom-atom interaction

Entanglement can also be generated by direct atom-atom interactions, e.g.,
collisions in a Bose-Einstein condensate [62, 78, 79, 80], the Coulomb
interaction between trapped ions [66, 81], or can be turned on by promoting
atoms to their Rydberg level.

Gil et al. [82] proposed and theoretically investigated an approach to
generate an OAT Hamiltonian (8) $\hat{H}\propto\hat{S}^{2}_{z}$ directly on
the optical clock transition, making use of the strong interaction between
atoms in their Rydberg states. Due to the nature of the van der Waals
interaction between Rydberg atoms decaying with the sixth power of the
distance, the interaction Hamiltonian corresponds to an effective nearest-
neighbour interaction.

Rydberg dressing methods, where atoms in their ground state are coupled to a
Rydberg excited state, are appealing for optical tweezers-array clocks, a new
platform where clock atoms are individually trapped in optical tweezers, and
arranged in one- or two-dimensional arrays [41, 36, 42]. In contrast to
optical lattice clocks, such platforms offer a high tunability and control of
interatomic distance. This allows to both control the nearest-neighbor
interaction strength, and to realize a nearly homogeneous interaction within
the whole ensemble. Recently, Schine et al. [83] have pairwise entangled
Strontium atoms on the optical clock transition via Rydberg dressing.
Generally, this technique has already been demonstrated to be efficient and
reliable for entangling atoms [84, 85] in the radiofrequency domain.

He et al. [86] noted that the weak residual spin-orbit interaction in a spin-
polarized Fermi gas can generate an effective OAT Hamiltonian that can be used
to generate collective entangled states. In particular, they consider a 3D
Fermi-degenerate optical lattice clock [40], where spin polarized atoms are
trapped in a tunable 3D optical lattice with less than one atom per lattice
site, and cooled down to the motional ground state (Fermi degeneracy). The
spin-orbit coupling strength and, consequently, the effective nonlinear
collective interaction are tuned by varying the trap depth: in a very deep
trap the interaction vanishes. It has been theoretically demonstrated that is
possible to generate deterministic entanglement via unitary dynamics in
ensembles composed of $N\approx 10^{2}-10^{4}$ atoms [86]. The potential gain
offered by these states can be as high as $\mathcal{G}\approx 14$ dB after an
evolution time $\tau\approx 1$ s. Although this timescale may look
impractically long for many clock applications, this method is particularly
interesting because it makes use of usually unwanted interactions to improve
the clock performance.

###  III.4  Optically generated Spin Squeezing in Radiofrequency Sensors

Measurement-based spin squeezing, as well as cavity feedback squeezing, have
enabled the generation of entangled states with substantial amount of
metrological gain in alkaline atoms [50, 53, 52, 68, 87]. Using both schemes,
spin squeezing has been implemented in proof-of-principles experiments to
demonstrate atomic interferometers and atomic clocks operating beyond the SQL
in the radiofrequency (RF) and microwave domain [88, 89, 52].

Recently, cavity feedback squeezing has been demonstrated in alkaline-earth
atoms [67, 72], which are of high interest for optical atomic clocks.
Braverman et al. [72] have demonstrated a substantial amount of spin squeezing
in ytterbium-171 atoms ($\mathrm{{}^{171}Yb}$), generated via cavity feedback
squeezing operating in the near-unitary regime, and offering a metrological
gain of $\mathcal{G}=6.5$ dB beyond the SQL [72], limited by the measurement
resolution of the system. The squeezing was demonstrated in the ground state
of $\mathrm{{}^{171}Yb}$ atoms, which has purely nuclear spin-$1/2$,
representing an almost ideal two-level system to coherently manipulate the
atomic state. Furthermore, using a SSS as an input state in a Ramsey protocol,
a reduction in the integration time by a factor of 3.7 over the SQL was
achieved [72].

Figure 3: Scheme for generating entanglement on the optical clock transition.
The entanglement is first generated via cavity QED techniques within two
atomic levels in the RF-domain, here the nuclear sublevels of the 171Yb
electronic ground state. Collective quantum states manipulations are easier
and have very high fidelity in this domain. The entangled state is then mapped
with a coherent optical $\pi$-pulse onto the optical clock transition.

##  IV Mapping of squeezed states to the Optical Transition

In 2020, the first demonstration of spin squeezing on an optical clock
transition was achieved [57]. The strategy used in that work was to generate a
SSS in the ground state of $\mathrm{{}^{171}Yb}$ atoms, and transfer it onto
the ultra-narrow optical clock transition by applying an optical $\pi$-pulse
(see Fig. 3). Then using the SSS on the optical transition as the input state,
a Ramsey protocol was demonstrated in the optical domain, where the quality
factor of the transition ($Q=\Delta f/f$) was improved by five orders of
magnitude compared to microwaves transitions [43]. After the Ramsey protocol
was implemented in the optical domain, the squeezed state was mapped back onto
the ground state, and read out via the cavity [72, 90, 91, 52, 57, 92].
Comparing the clock operations using a SSS and a coherent spin state state as
input states, a precision of $\mathcal{G}=4.4$ dB below the SQL was
demonstrated, corresponding to a 2.8-fold reduction of the averaging time [57]
(Fig. 4 ).

Figure 4: Entanglement on ytterbium-171 optical clock transition from Pedrozo-
Peñafiel et al. [57] a.Tomography of the generated entangled state, in this
case a squeezed state, on the optical clock transition of ytterbium-171. The
squeezed $\xi_{-}^{2}$ and the antisqueezed $\xi_{+}^{2}$ axes are indicated
in the tomography data. b. Allan deviation plot (fractional frequency
instability) for a Ramsey sequence of the 171Yb clock. Blue symbols represents
the data obtained without entanglement, while red symbols correspond to a
squeezed input state. The blue area is only accessible with entangled states.

In this particular experiment, the LO stability was the limiting factor and
was independently characterized and subtracted from the Allan deviation to
quantify the amount of improvement with respect to the noise imposed by the
atomic spins alone. In this respect, improvement of the LO coherence is
required to achieve performances at the level of state-of-the-art optical
lattice clocks. In particular, optical lattice clocks use highly stable lasers
that offer coherence times on the order of $1$ second, corresponding to mHz
linewidths [93].

##  V Beyond Spin Squeezing Via Time-Reversal Protocols

SSSs are very simple entangled states, and can provide only a limited
improvement over the SQL. To approach the fundamental HL where the
metrological gain is $\mathcal{G}=1/N$, it is necessary to generate more
complex entangled states [45]. For example, a Greenberger-Horne-Zeilinger
(GHZ) state, also called "cat-state", is an extreme example of a collective
entangled state, and can attain the HL. Other highly-entangled states, with
non-Gaussian envelopes, can carry very large quantum Fisher information
$\mathcal{F}_{Q}$, and hence permit metrology close to the HL [94, 95, 96, 97,
45, 98]. Moreover, the metrological gain of some highly entangled states
follows the Heisenberg Scaling (HS) [73, 94, 99, 98], i.e. $\mathcal{G}=b/N$
with $b\geq 1$; this class of states is a constant factor away from the away
from the HL, independent of atom number.

Generating and utilizing highly-entangled states is a difficult task due to
their fragility, and remains an important challenge in quantum metrology. In
practice, the metrological gain is limited by the detector resolution, the
curvature of the Bloch sphere, and the impossibility to access all the
statistical information carried by entangled states. In particular, harnessing
the resources provided by such states requires advanced statistical methods
well beyond the usual evaluation of averages.

To relax those constraints, interaction-based readout methods have been
proposed [94, 99, 100, 101, 102, 103]. A particularly interesting class of
protocols, that allow one to approach the HL in realistic system, is based on
Loschmidt-echo-like time-reversal of the many-body Hamiltonian [94, 99, 100,
101, 102, 103, 104, 105, 106, 107]. Such protocols are based on the
application of a many-body Hamiltonian with both positive and negative sign,
corresponding to an effective evolution forward and backward in time.

A Loschmidt echo-like protocol, called Signal Amplification through Time-
Reversed Interaction (SATIN) [105], is described in Fig. 5. Here, an initial
coherent spin state pointing along $\hat{x}$ evolves under the action of the
OAT Hamiltonian, Eq. (8), for a prolonged time to generate an over-squeezed
state with large quantum Fisher information. If the process is sufficiently
unitary, time-reversing the Hamiltonian would produce the initial coherent
spin state. However, if this entangled state is perturbed, or subjected to a
rotation that makes it orthogonal to the unperturbed entangled state, then the
subsequent application of the negative Hamiltonian will produce a near-
coherent spin state which is now also orthogonal to the initial state. As a
major advantage over direct spin squeezing, detecting the distance between
these two states requires detection resolution on the order of the SQL only.

Figure 5: Signal enhancement by SATIN protocol. The output of the measurement
sequence is represented by the final state projection along the $z$ axis. Blue
histograms are the results in the absence of a signal ($\varphi=0$). a. Ramsey
protocol with a coherent input state: the imprinted signal $\varphi$ is
directly reflected in the readout. b. SATIN protocol: the signal $\varphi$ is
enhanced in the readout through the time-reversed application of the many-body
Hamiltonian. The Fisher information contained in the small features of the
highly entangled state is mapped onto a large signal after the time-reversal
($-H$) operation.

This is the feature that makes SATIN a powerful method for using highly
entangled states in atom interferometry, and that enables one to reach near-
Heisenberg sensitivity with Heisenberg scaling even for a many-particle
system, provided the evolution of the system can be kept nearly unitary, as
recently demonstrated by Colombo et al. [105] . Specifically, highly non-
Gaussian states were generated and used for atom interferometry. The SATIN
protocol allowed to utilize most of the quantum Fisher information carried by
these highly entangled states, reaching an improvement of $11.8$ dB below the
SQL (a factor of 15 in reduction of averaging time) in phase sensitivity. In
addition, that protocol demonstrated HS in metrological gain $\mathcal{G}$,
with a linear improvement in $\mathcal{G}$ with respect to the atom number, at
a fixed distance to the HL of 12.6 dB. In the future, this protocol can be
used to perform metrology in the optical domain, by mapping the entanglement
onto the optical transition, as was similarly done for a SSS [57].

Other time-reversal-like protocols have also been demonstrated in Bose-
Einstein condensates through phases shifts in a three-level system for a few
neutral atoms [108], and in cold trapped ion systems consisting of up to
${\sim}150$ ions, where the coupling with a motional mode plus spin rotations
are used for such purpose [81]. Other approaches that mimic time-reversal-type
protocols have been demonstrated, alternating spin squeezing with state
rotations [109]. In particular, Hosten et al.[110] demonstrated a gain of
$\mathcal{G}=8$ dB beyond the SQL without sub-SQL measurement resolution.

The experimental demonstration of the SATIN protocol opens the door for
quantum metrology with highly-entangled many-body systems, paving the way to
achieve nearly Heisenberg-limited operation of quantum sensors, significantly
enhancing the bandwidth at fixed precision, or the precision at fixed
bandwidth.

##  VI Usefulness of Entanglement

In this article, we discuss mostly clocks operating with Ramsey spectroscopy
[111]. In this case, the spectroscopy time is also called Ramsey time. For the
Ramsey protocol we have $w(t)=1$, and equation (1) becomes
${{\Delta\varphi(\tau)}=2\pi\,\tau\,\langle{\Delta f}\rangle}$, with
$\langle\Delta f\rangle$ being the average frequency difference in the Ramsey
time interval $\tau$. The resulting fractional stability of a clock operated
with Ramsey spectroscopy is then

$\sigma(\tau,T,T_{c})=\frac{1}{2\pi
f_{a}}\sqrt{\frac{1}{\tau\,D\,T}}\frac{1}{\sqrt{N\mathcal{G}}},$ (9)

with $T$ denoting the total measurement time, and $T_{c}$ the individual cycle
time. When the clock is operated with a duty cycle $D$ smaller than $100\%$,
one should add the Dick noise term to equation (9). The latter arises from
aliased noise of the LO [32]. In the absence of dark time the Dick noise
vanishes[32, 43], while it is irrelevant in applications where two or more
ensembles of atoms are simultaneously probed [2, 34, 35]. Schulte et al. [112]
have rigorously theoretically analyzed under which conditions entanglement can
provide a precision gain in the presence of Dick noise. They found that
entanglement is useful for atom numbers below a certain threshold that depends
on the experimental conditions, such as the LO noise, the dark time, and the
Ramsey time $\tau$.

###  VI.1 Laser as Local Oscillator

Figure 6: Stability of optical clock in a given averaging time $T$ as a
function of Ramsey time $\tau$. a. Fractional stability as a function of
$\tau$ in the presence of decoherence at rate $\Gamma_{LO}$. The blue, red,
and purple lines indicate the operation with an unentangled coherent spin
state, a unitary SSS with $11$ dB of metrological gain), and a non-unitary SSS
with $11$ dB squeezing and $20$ dB of antisqueezing, respectively. For short
Ramsey time, the SSS can always outperform the coherent spin state,
corresponding to an effective increase of the sensor bandwidth by
$\mathcal{G}$ at fixed precision. b. A QND measurement can be is used to
rotate the SSS closer to the equator, thus avoiding the leakage of the
antisqueezed quadrature noise onto the measurement quadrature. c. Maximal
metrological gain achievable in a state-of-the-art system at fixed bandwidths
as a function of the atom number. We considered here decoherence rates
$\Gamma_{nat}=0.01~{}\mathrm{s}^{-1}$ (natural linewidth),
$\Gamma_{deph}=0.025~{}\mathrm{s}^{-1}$ (dephasing rate), and
$\Gamma_{loss}=0.01~{}\mathrm{s}^{-1}$ (atom loss rate) from the recent clock
realization of Young et al. [42].

In full clock operation, the frequency of the laser LO is locked to the
optical atomic transition, and it can be used as a time standard by means of
an optical frequency comb. In a Ramsey sequence, the phase difference
$\varphi$ between the atoms and the LO is mapped onto the final population
difference between the ground state and the clock state, i.e., $2S_{z}$, by
means of a $\pi/2$ pulse. Since $S_{z}\propto\sin(\varphi)$, the protocol
works correctly only if the phase accumulation does not exceed $\pm\pi/2$.
Thus, as shown by Braverman et al. [113], the LO noise limits the clock
performance when the standard deviation of the total accumulated phase noise
exceeds $\tau\Gamma_{LO}\approx 0.3$ (see Fig. 6a), where $\Gamma_{LO}$ is the
dephasing rate of the LO. At this level of noise, the probability of having a
total Ramsey phase exceeding $\pm\pi/2$, and thus a wrong reading, is
significant, and induces an overall increase of error in the feedback.

LO noise does not induce spin-spin decoherence; however, in the case of a SSS
input state, the approximation that the antisqueezing noise is orthogonal to
the measurement fails. Due to the limited number of atoms utilized in optical
clocks ($N\lesssim 10^{4}$) and the associated non-zero curvature of the
generalized Bloch sphere, the anti-squeezing couples into the measurement
projection, introducing extra noise and also effectively shortening the
average spin vector (i.e., reducing the contrast). Larger noise and smaller
contrast reduce the Wineland [46] parameter and the metrological gain [114,
115, 112].

To avoid the effect of this leakage, squeezed optical clocks need to operate
in a regime with a small accumulated phase $\varphi$. This implies a Ramsey
time $\tau$ limited to a value smaller than that necessary to avoid phase
errors exceeding $\pm\pi/2$ in an SQL clock operation. On the other hand, as
can be seen from equation (9), the clock stability $\sigma$ in a time $T$
improves with $\sqrt{\tau}$. Therefore, to maximize clocks performance, one
needs to choose a compromise between a short Ramsey time for greatest phase
noise suppression and a long Ramsey time for increased clock frequency
stability (see Fig. 6a).

Schemes that allow to profit from the full metrological gain offered by
squeezed states in the presence of local oscillator noise have been proposed
[97, 116]. Borregaard and Sørensen [97] argue that a large accumulated phase
can be strongly reduced by performing a series of QND measurements and
feedback before the final strong projective measurement. Compared to the case
of the measurement based squeezing, here the QND measurement does not need to
resolve the $S_{z}$ below the SQL. It only needs to preserve the atomic
coherence, i.e., contrast. The reading obtained from this weak QND measurement
will be used to rotate the squeezed state closer to the equator, where the
anitsqueezing noise does not leak into the measurement quadrature (see Fig.
6b). In a similar spirit, the large accumulated phase can be first estimated
by using multiple ensembles [116].

###  VI.2 Second atomic ensemble as Local Oscillator

When two or more ensembles are compared, i.e., for differential operation of
the sensor, the LO decoherence is a common-mode noise source and is cancelled.
The coherence time is then ultimately limited by the spontaneous emission rate
of the atomic excited state $\Gamma_{nat}$. However, in state-of-the-art
atomic clocks other processes, such as atomic collisions and environmental
inhomogeneities, can induce faster atom-atom decoherence.

Escher et al. [117] and Demkowicz-Dobrzański et al. [118] have thoroughly
investigated the limitations imposed by atomic decoherence on the metrological
gain. They theoretically demonstrated that, for a given uncorrelated noise
there is a maximal enhancement that can be achieved, and that such a limit
depends on the Ramsey time of the sensor, i.e., on the bandwidth of the
measurement.

In particular, they show that the precision in a total time $T$ as a function
of the Ramsey time $\tau$ is limited not only by the LO noise (as in Fig: 6a),
but also by atom-atom dephasing ($\Gamma_{deph}$), and by $\Gamma_{nat}$. As
illustrated in Fig. 6a, for sufficiently short Ramsey times the full
metrological gain is recovered.

Figure 7: Network of clocks. Several individually trapped ensembles are
prepared in two optical-lattices, and can be shuttled in and out of a high-
finesse optical cavity. The light-mediated interaction through the cavity
field can be used to engineer entanglement between the ensembles.

#### Quantum Network of Clocks

The comparison of multiple clock ensembles is the core of the idea of a
quantum network of clocks, proposed by Kómár et al.[119]. It consists of
multiple, coherently interconnected clocks in which the individual devices
(nodes) will benefit from the larger resources of the composed system. In
particular, this approach may serve both as a way to distribute time at an
international scale, as well as a new platform for testing fundamental
physics. Yet, to achieve this goal, entanglement between the different nodes
is necessary to improve the performance beyond the SQL. In the best-case
scenario, a Greenberger-Horne-Zeilinger (GHZ) state consisting of the product
of individual GHZ states of the nodes would allow the network to operate at or
near its collective HL.

A possible implementation is illustrated in Fig. 7. Two or more spatially
separated ensembles can be entangled via their interaction with the same
cavity mode. Each of these ensembles is effectively an individual clock and,
as recently demonstrated by Zheng et al. [58], they can be easily moved in and
out of the cavity. In this way the atomic ensembles can be coupled to the
cavity field on demand, and entangled individually, as well as with each
other.

Recently, Nichol et al. [120] have demonstrated the first quantum network of
optical atomic clocks by comparing the frequencies of two entangled
${}^{88}\mathrm{Sr}^{+}$ ions separated by 2 m. Nichol et al. generated
heralded entanglement between the two ions (i.e., the two clocks) via a
photonic link [121].

###  VI.3 Fixed-bandwidth applications

Many proposed applications of optical clocks, including gravitational-wave
detection [20, 21] and the search for axions[6, 122, 7, 11], require a large
clock bandwidth and short Ramsey time, since the clock needs to track time-
varying signals. Thus, for operations where the Ramsey time $\tau$ is much
less than the decoherence time, $\tau\Gamma_{tot}\ll 1$, entanglement remains
an important resource. In this situation it is possible to profit from the
full metrological gain offered by the entangled input state.

Figure 6c. shows the maximum gain of an optical clock as a function of the
total atom number deployed. For bandwidths set in the range 1 Hz to 1 kHz, the
transition from Heisenberg-limited performance to decoherence-limited
performance occurs at atom numbers ranging from $\sim 50$ to $\sim 5\times
10^{4}$. It worth noting that for typical atom numbers used in optical atomic
clocks ($10^{2}$ to $10^{4}$), a bandwidth of $\approx 10$ Hz or larger
implies that the maximum metrological gain is close to the Heisenberg Limit.

##  VII Summary and perspective

In this article, we have described recent advances and offered a perspective
on strategies to improve atomic sensors via engineered quantum correlations
(entanglement). The generation and application of spin squeezed states to atom
interferometers and atomic clocks have enabled sub-SQL performance of these
devices. While in most cases these have been proof-of-principle experiments,
some more recent implementations have shown practical applications for sensing
beyond the SQL [81] in state-of-the-art sensors.

We have highlighted the generation of entanglement in optical clocks,
presenting operating regimes where collective entanglement can readily bring a
sensing advantage. We have also discussed a novel technique, based on
effective time-reversal through switching the sign of a many-body Hamiltonian,
for generating and harnessing highly entangled states (non-Gaussian states) in
atomic sensors [105]. Such time-reversal protocols allow one to reach near-
Heisenberg-limited quantum metrology in many-atoms sensors. This sensing
paradigm strongly relaxes the most stringent limitations present in standard
entangled-enhanced sensing protocols, namely, the detection of states with a
large Fisher information, and the curvature of the generalized Bloch sphere.

We expect that the time-reversal technique will become a major paradigm in
quantum metrology in the years to come. Thanks to its applicability to non-
Gaussian entangled states [94, 105], such a paradigm is ideal for the
application to optical clocks and other state-of-the-art atomic sensors.
Further investigations to scale up the size of entangled quantum systems, as
well as reduce and tame the different decoherence mechanisms, will enable
near-Heisenberg limited resolution for large systems with many atoms[117,
118]. Entangled optical clocks and atomic sensors have the potential to become
a leading platform in the search for new physics, ranging from probing
different types of physics beyond the Standard Model [123, 124, 125, 55, 126]
to testing the fundamentals of gravity [21, 127, 128].

###### Acknowledgements.

We would like to thank Emily Davis, Adam Kaufman, Shimon Kolkowitz, Zeyang Li,
Mikhail Lukin, Monika Schleier-Smith, and Jun Ye for inspiring discussions.
This work was supported by NSF, NSF CUA, ONR, and DARPA.

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