Entanglement-Enhanced Optical Atomic Clocks

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.


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 lowto intermediate-mass sector [6][7][8][9][10][11] , the investigation of nuclear structure and matter 10,12,13 , the study of any time variation of fundamental constants [14][15][16] , testing of the foundations of general relativity 4,[17][18][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 2,[33][34][35][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 6,21,55,56 .
The generation of metrologically useful entanglement on the optical clock transition of ytterbium-171 ( 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 longlived atomic excited state with frequency f a 100 THz, corresponding to more than 10 14 oscillations in a typical interrogation time of ∼ 1 second.The fractional stability of an optical clock is expressed as σ ≡ δ f / f a with δ 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 5,58 (differential operation).The comparison is performed by measuring the accumulated phase difference between the LO and the atomic reference in a given measurement time τ (called the spectroscopy or interrogation time).In a linear protocol, the relation between phase difference ϕ and frequency imbalance ∆ f = f a − f LO between the LO and the atomic system is given by 59 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 ϕ.
The goal is then to estimate the phase ϕ with the smallest possible uncertainty δ ϕ 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, δ ϕ SQL = 1/ √ 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 F Q 45,60,61 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 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 F 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 In this article, we will discuss collective entangled states, i.e. states with F Q > N, and how to generate them in opticaltransition clocks.We express then the performances of a quantum enhanced sensors in terms of metrological gain beyond the SQL as where SNR and 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 where the equality applies when the Cramér-Rao bound is saturated.
III. SQUEEZING FIG. 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 ϕ. 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.ξ − and ξ + denote the reduced (squeezed) and increased (anti-squeezed) variance along two orthogonal axes, respectively.In this case, the accumulated phase ϕ can be estimated with precision beyond the SQL.
We consider each two-level atom in the clock as a spin-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 ξ 2 − = 2 S (∆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 G of a squeezed state is quantified by the Wineland parameter 46 , 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, 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 62,[64][65][66] , or effective 50,52,53,[67][68][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 52,53,67,68,[70][71][72] , thus avoiding undesirable collision-induced energy shifts during the operation of the interferometer.FIG. 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 β 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 ∝ S 2 z ) 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 metro-logical gain.

A. 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 |α(S z ) exiting the cavity, where α 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 In particular, the variance (∆S z ) 2 associated with estimating the parameter S z , using knowledge of the state |α(S z ) , is given by the relative Cramér-Rao bound (∆S z ) 2 = 1/ Ĩ.An external observer can gain knowledge about the S z observable so that the spin variance is reduced below the SQL where I = 2 Ĩ/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.

B. 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 : where nc is the intracavity photon number, and hΩ is the light shift produced by a single photon inside the cavity.In a cavity setup, the intracavity photon number nc = ĉ † ĉ is Ŝzdependent, producing to lowest order the paradigmatic oneaxis twisting (OAT) Hamiltonian, initially proposed theoretically by Kitawaga and Ueda in 1993 48 : 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).χ is known as the shearing parameter that quantifies how quickly the initial coherent spin state distribution is "sheared".

C. Entanglement via direct atom-atom interaction
Entanglement can also be generated by direct atomatom 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) Ĥ ∝ Ŝ2 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 36,41,42 .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 Fermidegenerate 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 ≈ 10 2 − 10 4 atoms 86 .The potential gain offered by these states can be as high as G ≈ 14 dB after an evolution time τ ≈ 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.

D. 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 50,52,53,68,87 .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 52,88,89 .
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 ( 171 Yb), generated via cavity feedback squeezing operating in the near-unitary regime, and offering a metrological gain of 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 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 .FIG. 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 171 Yb 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 π-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 171 Yb atoms, and transfer it onto the ultra-narrow optical clock transition by applying an optical π-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 = ∆ 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 52,57,72,90-92 .Comparing the clock operations using a SSS and a coherent spin state state as input states, a precision of G = 4.4 dB below the SQL was demonstrated, corresponding to a 2.8-fold reduction of the averaging time 57 (Fig. 4 ).
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 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 F Q , and hence permit metrology close to the HL 45,[94][95][96][97][98] .Moreover, the metrological gain of some highly entangled states follows the Heisenberg Scaling (HS) 73,94,98,99 , i.e.G = b/N with b ≥ 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.
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 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 subse-quent 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.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 G , with a linear improvement in 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 ∼150 ions, where the coupling with a motional mode plus spin rotations are used for such purpose 81 .Other approaches that mimic timereversal-type protocols have been demonstrated, alternating spin squeezing with state rotations 109 .In particular, Hosten et al. 110 demonstrated a gain of 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 highlyentangled 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 ∆ϕ(τ) = 2π τ ∆ f , with ∆ f being the average frequency difference in the Ramsey time interval τ.The resulting fractional stability of a clock operated with Ramsey spectroscopy is then 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 , 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 τ.

A. Laser as Local Oscillator
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 ϕ 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 π/2 pulse.Since S z ∝ sin(ϕ), the protocol works correctly only if the phase accumulation does not exceed ±π/2.Thus, as shown by Braverman et al. 113 , the LO noise limits 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 Γ nat = 0.01 s −1 (natural linewidth), Γ deph = 0.025 s −1 (dephasing rate), and Γ loss = 0.01 s −1 (atom loss rate) from the recent clock realization of Young et al. 42 .
the clock performance when the standard deviation of the total accumulated phase noise exceeds τΓ LO ≈ 0.3 (see Fig. 6a), where Γ LO is the dephasing rate of the LO.At this level of noise, the probability of having a total Ramsey phase exceeding ±π/2, and thus a wrong reading, is significant, and in-duces 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 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 112,114,115 .
To avoid the effect of this leakage, squeezed optical clocks need to operate in a regime with a small accumulated phase ϕ.This implies a Ramsey time τ limited to a value smaller than that necessary to avoid phase errors exceeding ±π/2 in an SQL clock operation.On the other hand, as can be seen from equation ( 9), the clock stability σ in a time T improves with √ τ.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 .

B. 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 Γ nat .However, in state-of-theart atomic clocks other processes, such as atomic collisions and environmental inhomogeneities, can induce faster atomatom 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 τ is limited not only by the LO noise (as in Fig: 6a), but also by atom-atom dephasing (Γ deph ), and by Γ nat .As illustrated in Fig. 6a, for sufficiently short Ramsey times the full metrological gain is recovered.FIG. 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 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 .

C. Fixed-bandwidth applications
Many proposed applications of optical clocks, including gravitational-wave detection 20,21 and the search for axions 6,7,11,122 , 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 τ is much less than the decoherence time, τΓ tot 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 Heisenberglimited performance to decoherence-limited performance occurs at atom numbers ranging from ∼ 50 to ∼ 5 × 10 4 .It worth noting that for typical atom numbers used in optical atomic clocks (10 2 to 10 4 ), a bandwidth of ≈ 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 op-tical 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-Heisenberglimited 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 55,[123][124][125][126] to testing the fundamentals of gravity 21,127,128 .

FIG. 4 . 2 +
FIG. 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 ξ 2 − and the antisqueezed ξ 2 + axes are indicated in the tomography data.b.Allan deviation plot (fractional frequency instability) for a Ramsey sequence of the 171 Yb 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.

FIG. 5 .
FIG.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 (ϕ = 0).a. Ramsey protocol with a coherent input state: the imprinted signal ϕ is directly reflected in the readout.b.SATIN protocol: the signal ϕ is enhanced in the readout through the timereversed 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.

FIG. 6 .
FIG. 6. Stability of optical clock in a given averaging time T as a function of Ramsey time τ.a. Fractional stability as a function of τ in the presence of decoherence at rate Γ 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 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 Γ nat = 0.01 s −1 (natural linewidth), Γ deph = 0.025 s −1 (dephasing rate), and Γ loss = 0.01 s −1 (atom loss rate) from the recent clock realization of Young et al.42 .