The generation of extreme spin squeezed states has attracted a lot of
attention since they can approach the Heisenberg limit. However, to generate
extreme spin squeezing, more complex nonlinear interactions such as TACT
interactions (76) are required Kitagawa and Ueda (1993); Kajtoch and Witkowska
(2015). It is worth noting that the realization of TACT interactions in known
quantum systems still has not been achieved. An alternative way is using the
TNT dynamics augmenting the OAT interaction with a well-designed sequence of
transverse coherent field. Promising analytic proposals based on this idea
include transforming OAT to TACT by using a periodic train of $\pm\pi/2$ spin
rotation pulses Liu  _et al._ (2011); Zhang  _et al._ (2014), or a
periodically modulated transverse field Huang  _et al._ (2015b). However,
these scheme demand stringent experimental conditions, either requiring a
large number of high-precision pulses Liu  _et al._ (2011) or a high
modulation frequency Wu, Tey, and You (2015). One can also makes use of only a
few optimized pulses along the mean spin direction, but the required evolution
time is greatly prolonged Shen and Duan (2013). To efficiently generate an
extreme spin squeezed state, one can even seek help from machine learning such
as deep reinforcement learning Chen  _et al._ (2019) and stochastic gradient
decent Carrasco  _et al._ (2022), which can achieve an optimized policy and
providing perspectives and solutions that is difficult to find with analytic
approach.

In cavity-QED systems, it was shown that effective two-axis twisting
Hamiltonians can be realized by applying bichromatic lasers Borregaard  _et
al._ (2017). Spin squeezing of the atoms in the absence of dissipation can
reach the ideal Heisenberg limit. However, the attainable spin squeezing is
limited by collective decay and this scheme may be suited for Rydberg blockade
system without strong collective decay. By using cavity-assisted Raman
transitions, tunable two-axis spin model and spin squeezing may be realized in
two cavities with all parameters can be tuned independently Yu  _et al._
(2016). The two-axis-twisting spin squeezing may also be generated by
multipass quantum erasure, which may be extendable to an optical ring cavity
system Wang  _et al._ (2017).

Except for OAT and TACT interactions, the more general form of the quadratic
collective-spin interaction can be described by the Lipkin-Meshkov-Glick (LMG)
model Lipkin, Meshkov, and Glick (1965); Meshkov, Glick, and Lipkin (1965);
Glick, Lipkin, and Meshkov (1965); Lee  _et al._ (2012); Lee (2009). A way to
implement general LMG model with independent multi-parameter control as well
as direct spin squeezing by both TACT and OAT is proposed, with a BEC in a
ring trap with periodic angular modulation of the external potential and of
the inter-particle interactions Opatrný, Kolář, and Das (2015). It is also
found that the squeezing properties are determined by the initial states and
the anisotropic parameters. To achieve the Heisenberg-limited spin squeezing,
a pulse rotation scheme is proposed to transform the model into a two-axis
twisting model Hu  _et al._ (2023). While in generic coupled spin systems with
collective spin-spin interactions, a universal scheme by continuous drivings
is also proposed Huang  _et al._ (2023).

In a spin-1 BEC, the spin squeezed ground states is recently demonstrated
using a double-quench method with a final Hamiltonian tuned close to the
quantum-critical point Xin  _et al._ (2023). It is related to a kind of
shortcut to adiabaticity Guéry-Odelin  _et al._ (2019) for fast production of
highly entangled states. In contrast to typical nonequilibrium methods by
quenching through a QPT, these squeezed ground states are time stationary. A
squeezed ground state with $6$–$8$ dB of squeezing with a constant squeezing
angle is demonstrated. The spin squeezing can be maintained for over $2$ s at
a constant orientation angle and gradually decreases after that due to atom
loss . This double-quench method can also be easily adapted to (pseudo)
spin-$1/2$ systems such as bosonic Josephson junctions mentioned before.

###  III.3 Twin-Fock state

Twin-Fock state is another kind of metrological useful entangled state Holland
and Burnett (1993); Dunningham, Burnett, and Barnett (2002); Campos, Gerry,
and Benmoussa (2003), which belongs to a kind of Dicke states Wieczorek  _et
al._ (2009); Lücke  _et al._ (2014). A Dicke state can be described by a
specific Dicke basis $|J,m\rangle=|J+m\rangle_{a}|J-m\rangle_{b}$, which is a
two-mode Fock state with $J+m$ particles in mode $a$ and $J-m$ particles in
mode $b$, respectively. Thus, a Dicke state has precise relative particle
number between the two modes while the relative phase is completely undefined
Holland and Burnett (1993). However, it was pointed out that a Dicke state is
not spin squeezed Wang and Mølmer (2002). The Dicke states have the potential
to improve the measurement precision. One can first calculate the QFI, and
find that

$F_{Q}=\frac{N^{2}}{2}-2m^{2}+N.$ (95)

Clearly, $m=\pm N/2$ is indeed a spin coherent state with $\theta=0$ or $\pi$
and the QFI $F_{Q}=N$ reduces to the case without entanglement. While for
$-N/2<m<N/2$, the QFI $F_{Q}>N$ indicates the metrological usefulness.

In particular, when $m=0$ with $N$ an even number, it corresponds to a twin-
Fock state

$|\Psi\rangle_{TWIN}=|N/2,0\rangle=|N/2\rangle_{a}|N/2\rangle_{b},$ (96)

which is a two-mode Fock state with definite equal particle number in both
modes. According to Eq. (95), the QFI for a twin-Fock state becomes
$F_{Q}=N^{2}/2+N$. When the particle number is sufficiently large $N\gg 1$,
$F_{Q}\approx N^{2}/2$, the ultimate phase measurement precision bound can be

$\Delta\phi\geq 1/\sqrt{F_{Q}}\sim\frac{\sqrt{2}}{N}$ (97)

approaching the Heisenberg scaling with a factor of $\sqrt{2}$ above the exact
Heisenberg limit.

For a twin-Fock state, applying a $\pi/2$ pulse can convert it to a state with
well-defined relative phase,

$|\psi_{AS}\rangle=\sum_{k=0}^{N/2}(-1)^{N/2-k}\sqrt{\frac{1}{2^{N}}C_{k}^{2k}C_{N/2-k}^{N-k}}|N/2,-N/2+2k\rangle$
(98)

which is called the arcsine state Campos, Gerry, and Benmoussa (2003) and can
be visualized as a ring around the equator of the generalized Bloch sphere.
Then accumulating phase $\phi$ between two modes and applying another $\pi/2$
pulse can finalize a whole Ramsey process. In practise, one cannot directly
exploit the population difference measurement to extract the phase information
since $\langle\hat{J}_{z}\rangle$ always equals $0$. An effective way to
extract the phase is using the square of the population difference
$\hat{J}_{z}^{2}$. When $\phi\approx 0$, one can find that $\hat{J}_{z}^{2}$
can attain the ultimate phase measurement precision bound of the twin-Fock
state Lücke  _et al._ (2011). Theoretically, it was proposed that the parity
measurement $\hat{\Pi}=(-1)^{N-\hat{J}_{z}}$, which distinguishes the
population number in one mode either even or odd is also capable to estimate
$\phi$ Campos, Gerry, and Benmoussa (2003). However, in systems with large
particle number, parity measurement is not feasible to be realized in
experiments.

Figure 16:  Generation of twin-Fock state via adiabatic sweeping. (a) The
thick black solid line denotes the gap $\Delta$ between the first excited and
the ground state of Hamiltonian. The two minima at $q=\pm 2|c_{2}|$ defines
three quantum phases, illustrated by their atom distributions in the three
spin components. Here $c_{2}$ is the same as the spin-dependent collision
energy $c$. The first-order Zeeman shifts are not shown because they are
inconsequential for a system with zero magnetization. (b) Absorption images of
atoms in the three spin components after Stern-Gerlach separation, showing
efficient conversion of a condensate from a polar state into a twin-Fock state
by adiabatic sweeping. (c) Conversion efficiency as a function of time.
Adapted from Ref. Luo  _et al._ (2017a).

In spinor BECs, colliding atoms can exchange spin. The spin exchange
collisions can generate entangled atom pairs, which is suitable for generating
twin-Fock state Lücke  _et al._ (2011); Luo  _et al._ (2017a). The many-body
Hamiltonian for a spin-1 BEC under the spin-exchanging interaction can be
expressed as

$\displaystyle H_{SM}$ $\displaystyle=$ $\displaystyle
2c(\hat{a}_{0}^{\dagger}\hat{a}_{0}^{\dagger}\hat{a}_{1}\hat{a}_{-1}+\hat{a}_{0}\hat{a}_{0}\hat{a}_{1}^{\dagger}\hat{a}_{-1}^{\dagger})$
(99) $\displaystyle+$
$\displaystyle[q+c(2\hat{N}_{0}-1)](\hat{N}_{+1}+\hat{N}_{-1})$

where $\hat{a}_{m_{F}}$ ($\hat{a}_{m_{F}}^{\dagger}$) denotes annihilation
(creation) operator and
$\hat{N}_{m_{F}}=\hat{a}_{m_{F}}^{\dagger}\hat{a}_{m_{F}}$ is the atom number
in $m_{F}$ component with the total atom number $N=\sum_{m_{F}}N_{m_{F}}$ with
$m_{F}=0,\pm 1$. The Hamiltonian is obtained under the assumption of all the
three spin components being in the same spatial wavefuntion $\phi(\vec{r})$.
$c=\frac{2\pi(a_{2}-a_{0})}{3m}\int d\vec{r}|\phi(\vec{r})|^{4}$ denotes spin-
dependent collision energy where atoms in $m_{F}=0$ can be transferred to
$m_{F}=\pm 1$ in pairs and vice versa. Here, $m$ is the atomic mass, $a_{0}$
and $a_{2}$ respectively denote the s-wave scattering length for total spin
$G=0$ and $G=2$ allowed by two-particle collisions Lücke (2014). $c<0$ ($c>0$)
corresponds to ferromagnetic (antiferromagnetic) interaction respectively Wu
and You (2016). $q$ is an effective quadratic Zeeman shift, which can be tuned
by the external magnetic field (proportional to the square of magnetic field
strength) or near-resonant microwave dressing field Luo  _et al._ (2017a). In
the language of angular momentum with conserved magnetization
$\hat{N}_{+1}-\hat{N}_{-1}=0$, the Hamiltonian of Eq. (99) can be rewritten as

$H_{SM}=c\hat{L}^{2}-q\hat{N}_{0},$ (100)

where $\hat{L}=\\{\hat{L}_{x},\hat{L}_{y},\hat{L}_{z}\\}$,
$\hat{L}_{\mu}=\sum_{m,n}\hat{a}^{\dagger}_{m}[f_{\mu}]_{mn}\hat{a}_{n}$
($\mu=x,y,z$ and $m,n=0,\pm 1$) with $[f_{\mu}]_{mn}$ the matrix element for
spin-1 angular momentum. The ground state of Hamiltonian with Eq. (99) is
determined by the competition between $c$ and $q$. It is convenient to express
the system state in the three-mode Fock basis
$|N-2k\rangle_{0}|k\rangle_{-1}|k\rangle_{+1}$ with $k=0,1,...,N/2$.

Figure 17:  Matter-wave interferometry with twin-Fock state generated through
spin mixing dynamics. (a) An unentangled spin coherent state and the
corresponding state after rotation. (b) An input twin-Fock state and the
corresponding state after rotation. (c) Schematic of the sequence of the
realization of the beam splitter. Three microwave pulses are sequentially
applied to achieve the coupling of the two hyperfine states $|F=2,m_{F}=\pm
1\rangle$. The total effect is equivalent to a rotation around the $x$-axis by
an angle. Figure adapted from Ref. Lücke  _et al._ (2011).

For a ferromagnetic BEC with $c<0$, the ground state can be divided into three
distinct phases. For $q\gg 2|c|$, the ground state
$|N\rangle_{0}|0\rangle_{-1}|0\rangle_{+1}$ is in the polar phase with all
atoms condensed in the $m_{F}=0$ component. For $q\ll-2|c|$, the ground state
$|0\rangle_{0}|N/2\rangle_{-1}|N/2\rangle_{+1}$ becomes a twin-Fock state with
the atoms equally partitioned into the $m_{F}=\pm 1$ components. In the
middle, two critical points $q=\pm 2|c|$ correspond to the broken-axisymmetry
phase whose ground state acquires a transverse magnetization, spontaneously
breaking the SO(2) symmetry of the system.

To generate a twin-Fock state, one can start in the polar phase and ramp down
$q(t)$ adiabatically from $q\gg 2|c|$ to $q\ll-2|c|$ crossing the two critical
points $q=\pm 2|c|$ Zhang and Duan (2013). At the QPT points, the energy gap
$\Delta E\sim 1/N^{1/3}$. The energy gap decreases slowly with the total
particle number $N$, which suggests the possibility to maintain the
adiabaticity even for $N\sim 10^{5}$. It had been demonstrated that near-
deterministic generation of a twin-Fock state with more than 10000 atoms can
be achieved with 87Rb BEC Luo  _et al._ (2017a). Later, in a subsequent
experiment Zou  _et al._ (2018), an interferometric precision of $2.42$ dB
beyond the three-mode SQL, using balanced spin-1 (three-mode) Dicke states
containing thousands of entangled atoms are demonstrated. The input quantum
states are also deterministically generated by controlled QPT and exhibit
close to ideal quality. This work also demonstrated an interferometric
precision of $8.44$ dB beyond the two-mode SQL.

For an antiferromagnetic BEC with $c>0$, the system exhibits two phases of
ground state. In the limit of $N\rightarrow\infty$, the ground state is a
polar state $|N\rangle_{0}|0\rangle_{-1}|0\rangle_{+1}$ when $q>0$ and a twin-
Fock state $|0\rangle_{0}|N/2\rangle_{-1}|N/2\rangle_{+1}$ when $q<0$. For a
finite total particle number $N$, the minimal energy gap $\Delta E\sim 1/N$
occurs at the critical point $q/c=3.7688/N^{2}$, which is close to $q=0$ if
$N$ is large Xu, Yi, and Zhang (2019). Ideally, by adiabatically tuning the
effective quadratic Zeeman energy $q$ from positive infinity to negative
infinity, the twin-Fock state can be generated. However, since the energy gap
decreases inversely proportional to $N$, which drops much faster than the one
in a ferromagnetic spin-1 BEC, the adiabatic evolution is difficult to realize
in experiment. It was suggested that utilizing the adiabatic and multilevel-
oscillation method Xu, Yi, and Zhang (2019), the high-fidelity twin-Fock state
can be efficiently generated. By partially replacing the adiabatic evolution
near the QPT point with multilevel oscillations, this method accelerates the
state generation process and relaxes the requirement on the control accuracy
of quadratic Zeeman splitting from microgauss to milligauss.

In this many-body system with Eq. (100), excited-state quantum phase
transitions (ESQPTs) can also be appealing, which refers to quantum
criticality aroused in ground state. The existence of QPTs and ESQPTs in an
antiferromagnetic spin-1 condensate are shown and their correspondence with
dynamical phase transition is demonstrated, which can be characterized by the
QFI Zhou  _et al._ (2023). The dynamical phase transition with the condensate
initially prepared in a spin coherent state can be used to probe the quantum
criticality in excited states, which gives rise to a peak value of the QFI. It
can also be used to implement sub-SQL estimation on the effective quadratic
Zeeman energy $q$. It is interesting to note that the ground-state phase
transitions from symmetry-broken states to the symmetry-restored spin-singlet
state can also be indicated by the dynamical phase transition.

In the approximation that $N_{0}\approx N\gg 1$, the initial condensate of
$m_{F}=0$ can serve as an unlimited pump source for parametric amplification
of the output modes $m_{F}=\pm 1$. In this case, one can make use of the
parametric approximation $\hat{a}_{0}\approx\sqrt{N_{0}}$ and the Hamiltonian
(99) becomes

$\displaystyle H_{SM}\approx
A(\hat{a}_{1}\hat{a}_{-1}+\hat{a}_{1}^{\dagger}\hat{a}_{-1}^{\dagger})+B(\hat{a}_{1}^{\dagger}\hat{a}_{1}+\hat{a}_{-1}^{\dagger}\hat{a}_{-1}),$
(101)

where $A=2cN_{0}$ and $B=q+c(2N_{0}-1)$. Evolving from
$|0\rangle_{+1}|0\rangle_{-1}$ under Hamiltonian in the form of Eq. (101), a
two-mode squeezed vacuum state (a superposition state of multiple twin-Fock
state with different total particle number) can be obtained Kruse  _et al._
(2016).

In particular for the resonance case with $B=0$, the spin-mixing Hamiltonian
becomes

$H_{SM}=2cN_{0}(\hat{a}_{1}\hat{a}_{-1}+\hat{a}_{1}^{\dagger}\hat{a}_{-1}^{\dagger}),$
(102)

and the evolved state can be calculated as

$|\psi_{SM}(t)\rangle=\sum_{n=0}^{\infty}\frac{(-i\tanh r)^{n}}{\cosh
r}|n\rangle_{+1}|n\rangle_{-1},$ (103)

where $r=2cN_{0}t$. Projecting onto a certain total particle number $2n=N$,
one can immediately get the twin-Fock state
$|N/2\rangle_{+1}|N/2\rangle_{-1}$.

The quantum interferometry with twin-Fock state had been experimentally
demonstrated with spin-1 BEC via spin mixing Lücke  _et al._ (2011). In the
experiment, spin dynamics in BECs was used to create large ensembles of pair-
correlated atoms with an interferometric sensitivity beyond SQL. The
experiment starts with creating a 87Rb condensate of $2.8\times 10^{4}$ atoms
in the hyperfine state $|F=2,m_{F}=0\rangle$ in an optical dipole trap. Then
the spin-exchange collision gradually produces correlated pairs of atoms with
spins up and down. The total number of the correlated pairs of atoms in
$|F=2,m_{F}=\pm 1\rangle$ increase exponentially with time due to spin mixing.
Afterwards, the trap is switched off and the three hyperfine states are split
by a strong magnetic field gradient, and all three hyperfine states are
recorded by absorption imaging. As the hyperfine states $|F=2,m_{F}=\pm
1\rangle$ are generated in pairs, twin-Fock state in these two modes is
prepared. The twin-Fock state was effectively obtained by post-selection of a
certain particle number.

Spin-mixing dynamics in cold atoms can also be used to realize a SU$(1,1)$
interferometer Marino, Corzo Trejo, and Lett (2012); Gabbrielli, Pezzè, and
Smerzi (2015), which is analogous to parametric down conversion in optics. As
shown in Fig. 3 (b), for spin-1 BEC, the spin-exchanging collision plays the
role of Kerr nonlinearity, and the condensate atoms in $m_{F}=0$ can be
regarded as the optical pump field. The atoms that transferred to $m_{F}=\pm
1$ correspond to the signal and idler beams. Similar to the optical parametric
down conversion, the amplification can be triggered by the fluctuations of
spin orientation. Let us indicate with $N$ the total average number of atoms
transferred in pairs from a condensate prepared in $m_{F}=0$ to the initially
empty $m_{F}=\pm 1$ modes. After spin mixing, the system acquires a relative
phase $\theta=2\theta_{0}-(\theta_{+1}+\theta_{-1})$ between $m_{F}=0$ and
$m_{F}=\pm 1$ modes. A second spin-mixing dynamics ends up the interferometer.
The final populations in $m_{F}=0,\pm 1$ depend on $\theta$. A nonlinear
SU$(1,1)$ interferometry with $F=2$ manifold of 87Rb atoms was experimentally
demonstrated Linnemann  _et al._ (2016). The phase imprinting is achieved via
adding a second-order Zeeman shift and read out by counting the mean atom
number in $m_{F}=\pm 1$. In addition, if one apply a linear coupling among the
$m_{F}=0,\pm 1$ before and after phase imprinting, realizing the so-called
"pumped-up" SU$(1,1)$ interferometer Szigeti, Lewis-Swan, and Haine (2017),
the measurement precision may be further improved.

###  III.4 Spin cat states

Spin cat states are promising candidates for achieving Heisenberg-limited
quantum metrology Huang  _et al._ (2015a, 2018). A spin cat state is a typical
kind of macroscopic superposition of spin coherent states (MSSCS) Lau  _et
al._ (2014); Nolan and Haine (2017, 2018). Generally, an MSSCS is a
superposition of multiple spin coherent states Micheli  _et al._ (2003);
Ferrini  _et al._ (2010); Spehner  _et al._ (2014); Nolan and Haine (2018),
which can be written in the form of

$\displaystyle|\Psi(\theta,\varphi)\rangle_{\textrm{M}}=\mathcal{N}_{C}(|\theta,\varphi\rangle+|\pi-\theta,\varphi\rangle)$
(104)

where $\mathcal{N}_{C}$ is the normalization factor and
$\left|\theta,\varphi\right\rangle$ denotes a spin coherent state. Here,
without loss of generality, we assume $\varphi=0$ and expressing in terms of
Dicke basis,

$|\Psi(\theta)\rangle_{\textrm{M}}=\mathcal{N}_{C}\left[\sum^{J}_{m=-J}c_{m}(\theta)\left(\left|J,m\right\rangle+\left|J,-m\right\rangle\right)\right].$
(105)

Since $c_{m}(\theta)=c_{-m}(\pi-\theta)$, the coefficients are symmetric about
$m=0$.

Figure 18:  Measurement precisions offered by spin cat states. (a) The
achievable precisions versus $\theta$ for different atom loss ratios. The
triangles denote the best optimal precisions. The precision becomes worse when
the atom loss ratio becomes larger. For nonzero loss ratio, instead of the GHZ
state, the optimal state becomes a spin cat states with modest $\theta$. Here,
the initial total atomic number $N=40$. And the left side of the thick black
dashed line indicates the region of spin cat states. (b) The achievable
precisions versus the initial total atomic number $N$ for three typical input
states (GHZ state, modest spin cat state and the spin coherent state under
loss ratio $\eta=5\%$. (c) The measurement precision versus $N$ for the three
input states in the absence of loss (i.e. $\eta=0\%$). Figures adapted from
Refs. Huang  _et al._ (2015a, 2018).

The properties of the MSSCS depend on $\theta$. When $\theta=\pi/2$, it
corresponds to a spin coherent state $\left|\pi/2,0\right\rangle$. As $\theta$
decreases, the two superposition spin coherent states become separated. When
$\theta\lesssim\theta_{c}\equiv\sin^{-1}\left\\{2\left[\frac{\left((J-1)!\right)^{2}}{2(2J)!}\right]^{1/{2J}}\right\\}$
is sufficiently small Huang  _et al._ (2015a, 2018), the two spin coherent
states become quasi-orthogonal (or orthogonal), the MSSCS can be regarded as a
spin cat state. In this case, we abbreviate the spin cat states as
$\left|\Psi(\theta)\right\rangle_{\textrm{CAT}}$, and it can be approximated
as

$\displaystyle\left|\Psi(\theta)\right\rangle_{\textrm{CAT}}$
$\displaystyle\approx$
$\displaystyle\frac{1}{\sqrt{2}}\left(|\theta,\varphi\rangle+|\pi-\theta,\varphi\rangle\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\sum^{J}_{m=-J}c_{m}(\theta)\left(\left|J,m\right\rangle+\left|J,-m\right\rangle\right)\right].$

Note that spin cat states can be understood as a superposition of GHZ states
with different spin length. Particularly when $\theta=0$, it reduces to a GHZ
state, i.e.,
$\left|\Psi(0)\right\rangle_{\textrm{CAT}}=\frac{1}{\sqrt{2}}\left(|J,-J\rangle+|J,J\rangle\right)$.
The expectation of $\hat{J}_{z}$ for a spin cat state
${}_{\textrm{CAT}}\langle\Psi(\theta)|\hat{J}_{z}|\Psi(\theta)\rangle_{\textrm{CAT}}=\sum_{m}mc_{m}^{*}(\theta)c_{m}(\theta)=0$.
Hence, the variance of a spin cat state becomes
$\Delta^{2}\hat{J}_{z}=\langle\Psi(\theta)|\hat{J}_{z}^{2}|\Psi(\theta)\rangle_{\textrm{CAT}}=\sum_{m}m^{2}c_{m}^{*}(\theta)c_{m}(\theta)$.
The variance can be analytically obtained
$\Delta^{2}\hat{J}_{z}\approx\frac{1}{4}N^{2}\cos^{2}\theta$, which only
depends on $\theta$ and $N$. If spin cat state is input for quantum
interferometry, the QFI for the output state can be computed as

$F_{Q}^{CAT}=4\Delta^{2}\hat{J}_{z}\approx N^{2}\cos^{2}\theta,$ (107)

and the corresponding measurement precision can achieve the Heisenberg-limited
scaling $\propto\frac{1}{N\cos\theta}$.

OAT is one of the well-known strategies for generating entanglement in many-
body quantum systems. Under the time-evolution of OAT interaction [Eq. (63)],
an initial spin coherent state (without entanglement) can evolve to spin
squeezed states, over-squeezed states and other kinds of nonclassical states
Pezzè  _et al._ (2018). For evolution times beyond the optimal spin squeezing
time, the Husimi distribution of the evolved state in the OAT model starts to
develop into the non-Gaussian regime. Especially when $\chi t=\pi/2$, it
evolves to the GHZ state You (2003). Recently in a trapped-ion quantum
simulator, the multi-headed cat states are successfully prepared with $N=12$
ions Franke  _et al._ (2023). Yet, the generation of spin cat state with this
method Micheli  _et al._ (2003); Pezzé and Smerzi (2009); Piazza, Pezzé, and
Smerzi (2008); Pawłowski  _et al._ (2017) requires long evolution times and
the generated states are not deterministically prepared. So far, experiments
have demonstrated the generation of spin-squeezed Gross  _et al._ (2010);
Riedel  _et al._ (2010) and slightly non-Gaussian states Strobel  _et al._
(2014) in ultracold atomic ensembles. The generation of spin cat states in an
engineered Dicke model may also be achieved Groiseau, Masson, and Parkins
(2021). In an appropriate limit of the Dicke model, one can obtain OAT
dynamics that leads to the generation of spin cat states. Additionally, the
quantum jump trajectories that generate "kitten" states in a random fashion is
investigated. Besides, the effects of decoherence are also explored on spin
cat state generation and the experimental feasibility in trapped ions and atom
ensembles with cavity-mediated Raman transitions is discussed.

An effective and deterministic way to generate spin cat states is the
adiabatic evolution Lee (2006); Huang  _et al._ (2015a); Huang, Zhuang, and
Lee (2018). As shown in Fig. 19, adiabatic evolution crossing QPTs can be used
for generating non-Gaussian entangled states. The process can be described by
a time-dependent Hamiltonian Huang, Zhuang, and Lee (2018),

$H_{BS1}(t)=R_{1}(t)H_{1}+R_{2}(t)H_{2}.$ (108)

Here, the Hamiltonians $H_{1}$ and $H_{2}$ are interpolated with the time-
varying parameters $R_{1}(t)$ and $R_{2}(t)$. One can choose proper
Hamiltonians such that the groundstate of $R_{1}(0)H_{1}$ is non-degenerate
while the groundstate of $R_{2}(T)H_{2}$ at time $T$ is multi-fold degenerate.
Defining the ratio $\lambda(t)=R_{1}(t)/R_{2}(t)$, there exists a QPT at the
critical point $\lambda_{c}$, that is, $R_{1}(t)H_{1}$ dominates the system
when $|\lambda(t)|>|\lambda_{c}|$ and $R_{2}(t)H_{2}$ dominates the system
when $0\leq|\lambda(t)|<|\lambda_{c}|$. Thus, starting from the non-degenerate
groundstate of $R_{1}(0)H_{1}+R_{2}(0)H_{2}$, an entangled groundstate can be
obtained with high fidelity if $\lambda$ is adiabatically swept through
$\lambda_{c}$. For instance, preparing the ground state of a BEC trapped in a
double-well potential with attractive interaction Trenkwalder  _et al._
(2016), or in an ion crystal Safavi-Naini  _et al._ (2018). By adiabatic
sweeping the control parameter across the spontaneous symmetry-breaking
transition, spin cat states with different degree of entanglement can be
prepared. However, due to the degeneracy of the spectrum, this method may be
fragile to symmetry-breaking perturbations and finite temperature Dakić and
Radonjić (2017); Gabbrielli, Smerzi, and Pezzè (2018). It should be mentioned
that, by driving the parameter inversely across the critical point again, one
can realize recombination as a kind of nonlinear detection, which will be
illustrated in Sec. IV.3.

For the Hamiltonian of Eq. (77) with $\delta=0$, when $|\Omega/N\chi|\gg 1$,
the system ground state is an SU(2) spin coherent state. The sign of $\chi$
determines the properties of the ground state when $\Omega$ is not large
enough. When $\chi>0$ and $|\Omega/N\chi|\ll 1$, the ground state is a spin
squeezed state. While for $\chi<0$, the ground state becomes a spin cat state
when $|\Omega/N\chi|\ll 1$ Lee (2009). To generate a spin cat state, it is
natural to prepare the ground state of Hamiltonian (77) in the limit of
$|\Omega/N\chi|\ll 1$ with $\chi<0$. For atomic BEC system, the negative
twisting strength can be achieved by tuning the interspecies s-wave scattering
length via Feshbach resonance Chin  _et al._ (2010).

Under the transformation $\hat{a}(\hat{b})\rightarrow\hat{b}(\hat{a})$,
$\hat{J}_{x}\rightarrow\hat{J}_{x}$, $\hat{J}_{z}\rightarrow-\hat{J}_{z}$, the
Hamiltonian of Eq. (77) with $\delta=0$ remains unchanged. Thus, this
Hamiltonian possesses a parity symmetry and it guarantees the symmetry-
protected adiabatic evolution Xing  _et al._ (2016); Zhuang  _et al._ (2020).
Adiabatic evolution occurs due to the presence of a finite minimum energy gap
between instantaneous eigenstates of the same parity. With negative $\chi$,
one can generate spin cat states via the adiabatic evolution. Initially, we
set $\Omega(0)$ to be sufficiently large, the ground state is nearly a spin
coherent state along $x$ axis with even parity. By sweeping the Rabi frequency
across the critical point ${\Omega}/{N|\chi|}=1$, the two lowest eigenstates
change from non-degenerate to degenerate. Through adiabatically sweeping
$\Omega(t)$ to zero, the evolved state will stay in the instant ground state
and spin cat states can be prepared.

Figure 19:  Schematic of quantum phase estimation via driving through quantum
phase transitions. The two beam splitters are achieved by driving the
parameter forwardly and inversely across the critical point. In the first beam
splitter, an entangled state can be prepared while in the second beam
splitter, the state is gradually disentangled. Figure adapted from Ref. Huang,
Zhuang, and Lee (2018).

For the sweeping, one can linearly change $\Omega(t)=\Omega(0)+\upsilon t$
from the non-degenerate regime across to the degenerate regime with the fixed
sweeping rate $\upsilon$. If $\upsilon$ is sufficiently small, the adiabatic
evolution of the ground state can still be achieved with high fidelity.
However, this linear sweeping scheme is not time-saving. In order to
accelerate ground state adiabatic evolution, the sweeping rate can be modified
temporally in accordance with the instantaneous energy gaps separating the
ground state and the second excited state (both possessing the same parity),
while maintaining the adiabatic parameter $\epsilon$. It is one of the
nonlinear sweeping schemes. Since $\epsilon$ is fixed, it is called adiabatic-
parameter-fixed sweeping Zhuang  _et al._ (2020). It is shown that adiabatic-
parameter-fixed sweeping can generate spin cat states with high fidelity Huang
_et al._ (2022).

In this scheme, the time-varying Rabi frequency

$\Omega(t)=\Omega(0)+\int_{0}^{t}\upsilon(t^{\prime})dt^{\prime},$ (109)

where $\upsilon(t)=\dot{\Omega}(t)$ is the instant sweeping rate of the Rabi
frequency. Since the parity symmetry, for adiabatic-parameter-fixed sweeping,

$\upsilon(t)=\frac{\epsilon{\left[{E_{1}}(t)-E_{3}(t)\right]}^{2}}{\left|\langle{\phi_{1}(t)}|\hat{J}_{x}|{\phi_{3}(t)}\rangle\right|}.$
(110)

Here, $E_{1}(t)$ and $E_{3}(t)$ respectively represent the energy of instant
ground state $\phi_{1}(t)$ and the second excited state $\phi_{3}(t)$ of
Hamiltonian with Eq. (77). Based on the adiabatic-parameter-fixed sweeping
scheme, the total time for adiabatic evolution can be reduced compared with
the naive linear sweeping. The fidelities can exceed 0.9 with $\epsilon=0.05$
and total atom number $N=100$.

However, the adiabatic sweeping still demands the control parameter vary
slowly to let the evolved state follow the instant ground state of the system.
Due to the limited coherence time in experiments, adiabatic process may be
time-consuming and not easy to realize. Shortcut to adiabaticity is a well-
known approach for quickly achieving the target state Ruschhaupt  _et al._
(2012); Guéry-Odelin  _et al._ (2019); Hatomura and Pawłowski (2019). However
for many-body quantum state generation, this method is not simple and may
necessitate some approximations (such as in the semiclassical limit) in order
to get the analytical best solution. To accelerate state preparation process,
TNT dynamics is proposed in which an additional rotation is introduced along
with OAT interaction as shown in Sec. III.2. One may also use TNT dynamics
with time-dependent rotation sequence for the preparation of spin cat states.
Moreover, to obtain the better performance, one can apply piecewise time-
modulation of rotations designed via machine optimization Huang  _et al._
(2022). When comparing adiabatic evolution, the fidelity to the spin cat
states can be increased while also significantly reducing the needed evolution
time. It does not require large modification to the existing experimental
setups and can be realized with state-of-the-art techniques in a BEC system or
an optical cavity system with light-mediated interactions.

Figure 20:  Stochastic generation of spin cat states under ideal conditions.
(a) The probability of measuring $N_{h}$ particles in the ground state is
represented by bars. The dashed lines represent cumulative probability
thresholds. (b) Conditional probability to find $N_{g}$ particles in the $g$
mode given a heralded measurement of $N_{h}$ particles in the $h$ mode. The
$g$ and $h$ modes are defined as the symmetric and antisymmetric combinations
of $\left|{m_{F}=+1}\right\rangle$ and $\left|{m_{F}=-1}\right\rangle$. (c)
Examples of probability (left) and Husimi distributions on the Bloch sphere
(right) of rotated heralded states for different values of $N_{h}$. (d) QFI of
the heralded states as a function of $N_{h}$. In all panels $N=100$. Figure
adapted from Ref. Pezzè  _et al._ (2019).

In a spin-1 BEC, which is described by Hamiltonian of Eq. (99), a proposal for
generating macroscopic superposition states of a large number of atoms is
presented Pezzè  _et al._ (2019). As shown in Fig. 20, measuring the number of
particles in one mode prepares with large probability highly entangled
macroscopic superposition states in the two remaining modes. The macroscopic
superposition states are heralded by the measurement outcome. This protocol is
robust under realistic conditions in current experiments, including finite
adiabaticity, particle loss, and measurement uncertainty. It is demonstrated
that MSSCS of a large number of atoms, $N>100$, can be heralded—with large
success probability—in a ferromagnetic spin-1 BEC. First, one can consider the
preparation of the ground state of the three-mode condensate by the
quasiadiabatic crossing over a QPT point, as demonstrated experimentally Hoang
_et al._ (2016); Luo  _et al._ (2017a). Once the ground state is prepared,
MSSCS can be generated stochastically, with up to 90$\%$ probability, by
measuring the number of atoms in one of the modes. The state preparation is
nondestructive — the spin cat states can be manipulated after its generation.
Remarkably, a numerical simulation of the proposed procedure reveals that the
structure of the MSSCS survives under experimentally realistic conditions.
Nonadiabatic crossing of the QPT and particle loss reduce the Fisher
information to about 50$\%$ of its ideal value, which still implies
significant entanglement and metrological sensitivity above the classical
limit.

Figure 21:  Entanglement amplification for GHZ state generation with atomic
ensemble trapped in an optical cavity. (a) A collection of $N$ atoms is
connected to a high-finesse optical cavity. The cavity is exposed to a single-
frequency light that can be switched on or off. (b) The atomic levels and the
spectra of cavity transmission. The Rabi drivings pair the states
$|\uparrow\rangle$ and $|\downarrow\rangle$. The cavity mode exhibits coupling
between the state $(\left|\uparrow\right\rangle)$ and $|\text{e}\rangle$,
characterized by a detuning $\Delta$. Since the intense connection between the
atom and the cavity, each atom in the state $|\uparrow\rangle$ causes a shift
in the resonance of the cavity. This shift is equal to
$\omega_{s}=g^{2}/\Delta$, which is more than $\kappa$. Therefore, the
incoming light with a frequency of $\omega_{1}=\omega_{c}+\omega_{s}$ only
alters the energy of the Dicke state $|-S+1\rangle$ and establishes a boundary
in the Hilbert space at this state with $S$ the total spin length. (c) The
Husimi-Q function is observed on the generalized Bloch sphere before, during,
and after the process of entanglement amplification. Figure adapted from Ref.
Zhao  _et al._ (2021).

In an optical cavity with multi-atom ensemble, an entanglement-creation scheme
named entanglement amplification is proposed, converting unentangled states
into entangled states and amplifying less-entangled ones to maximally
entangled GHZ states whose fidelity is logarithmically dependent on the atom
number and robust against common experimental noises Zhao  _et al._ (2021). As
shown in Fig. 21, the scheme starts with a multi-atom ensemble initialized in
a spin coherent state. By shifting the energy of a particular Dicke state, the
Hilbert space of the ensemble is broken into two isolated subspaces to tear
the spin coherent state into two components so that entanglement is
introduced. After that, the isolated subspaces are utilized to further enhance
the entanglement by coherently separating the two components. By single-
particle Rabi drivings on atoms in a high-finesse optical cavity illuminated
by a single-frequency light Chen  _et al._ (2015b), $2000$-atom GHZ states can
be created with a fidelity above 80$\%$ in an experimentally achievable
system. This may make resources of ensembles at Heisenberg limit practically
available for quantum metrology.

In experimental trapped-ion systems, GHZ state can also be generated and used
for precision measurement. In 2001, an experiment demonstrated that the
sensitivity of rotation angle estimation with a Ramsey spectroscopy can be
improved by using entangled trapped ions Meyer  _et al._ (2001). The
experiment used two 9Be+ ions that are confined in a linear radio-frequency
trap. Two hyperfine states
$\left|F=1,m_{F}=-1\right\rangle\equiv\left|\uparrow\right\rangle$ and
$\left|F=2,m_{F}=-2\right\rangle\equiv\left|\downarrow\right\rangle$ form the
basis of an effective spin-1/2 system. $\left|\uparrow\right\rangle$ and
$\left|\downarrow\right\rangle$ are coupled by two-photon Raman transitions.
The detection of the ions in states $\left|\uparrow\right\rangle$ and
$\left|\downarrow\right\rangle$ is done by state-sensitive fluorescence. The
use of entangled states for parity measurement and Ramsey spectroscopy has
been demonstrated with
$\left|\Psi_{P}\right\rangle=\left(e^{i\phi}\left|\uparrow\uparrow\right\rangle+\left|\downarrow\downarrow\right\rangle\right)/\sqrt{2}$
and
$\left|\Psi_{R}\right\rangle=\left(\left|\uparrow\downarrow\right\rangle+\left|\downarrow\uparrow\right\rangle\right)/\sqrt{2}$,
respectively. The experimental data show the measurement sensitivity is
improved beyond the SQL and close to the Heisenberg limit. Later the Ramsey
spectroscopy with three entangled 9Be+ ions in the GHZ state was demonstrated.
The experimental data show that the spectroscopic sensitivity is 1.45(2) times
as high as that of a perfect experiment with three independent ions, which
approaches the Heisenberg limit Leibfried  _et al._ (2004). Then the precision
spectroscopy of a pair of trapped Ca+ ions in a decoherence-free subspace of
specifically designed entangled states Roos  _et al._ (2006) was demonstrated.
In addition to the enhancement of signal-to-noise ratio in frequency
measurements, a suitably designed pair of ions enable atomic clock
measurements in the presence of magnetic field noise. Later, there have been
many other experiments Friis  _et al._ (2018); Pogorelov  _et al._ (2021);
Franke  _et al._ (2023) about the preparation of GHZ state with trapped atomic
ions for quantum metrology.

Multi-particle GHZ states have also been experimentally generated in other
quantum systems. The deterministic generation of an 18-qubit GHZ state and
multi-component atomic spin cat states of up to 20 qubits have been
demonstrated with a superconducting quantum processor interconnected by a bus
resonator Song  _et al._ (2019). By engineering an OAT Hamiltonian, the system
of qubits, once initialized, coherently evolves to multi-component atomic spin
cat states, which are superpositions of atomic coherent states including the
GHZ state at specific time intervals as expected. Moreover, a highly scalable
method for producing multipartite cat states, also known as flying cat states,
for itinerant microwave photons has also been presented Wang  _et al._
(2022b). By reflecting coherent-state photon pulses from a microwave cavity
containing a superconducting qubit, flying cat states of microwave photons
containing up to four photonic modes are successfully prepared. In addition,
the GHZ states of $20$ Rydberg atoms have been demonstrated by exploiting
optimal control methods to optimize the preparation time Omran  _et al._
(2019). The entanglement manipulation by using GHZ states to distribute
entanglement to distant sites in the array has been demonstrated, establishing
important ingredients for quantum metrology.

##  IV Interrogation and readout

Apart from state preparation, interrogation and readout are two important
processes in quantum metrology. The interrogation process is provided by a
dynamical evolution of the probe and it determines the QCRB for the parameter
$\theta$ with a probe state $\rho_{0}$. On the other hand, the readout process
determines the CRB and the measurement precision through practical observable
measurements. In this section, we will discuss the advancements made in
investigating the interrogation and readout processes. Regarding the
interrogation process, the signal can either be time-invariant or time-
varying. Therefore, we will review both the interrogation process with time-
independent Hamiltonians and the one with time-dependent Hamiltonians.
Regarding the readout process, we will cover both conventional readout methods
and the recent developments in interaction-based readout.

###  IV.1 Interrogation

The parameters to be estimated are typically embedded within the Hamiltonian
or within the dynamics of a physical system. During the interrogation process,
these parameters are encoded into quantum states by evolving the probe state
$\rho_{0}$ under the influence of the corresponding Hamiltonians or physical
dynamics, represented by $\hat{U}_{\theta}$ (as shown in Fig. 1). Here, we
illustrate the interrogation process with time-independent Hamiltonians Boixo
_et al._ (2007); Pang and Brun (2014) and time-dependent Hamiltonians Pang and
Jordan (2017); Yuan and Fung (2015); Yuan (2016); Hou  _et al._ (2019),
respectively.

A commonly studied and straightforward example in quantum metrology involves
estimating a multiplicative parameter within a Hamiltonian. For example,
estimating the parameter $\theta$ in a unitary evolution
$\hat{U}_{\theta}=e^{-i\theta\hat{H}}$, where the generator $\hat{H}$ is time-
independent. In this case, if a quantum probe with a pure state
$\rho_{0}=|\Psi\rangle\langle\Psi|$ undergoing a unitary evolution
$\hat{U}_{\theta}=e^{-i\theta\hat{H}}$ with $\theta=\omega T$, the QFI that
determines the ultimate measurement precision of $\omega$ can be given as
$F_{\omega}^{Q}=T^{2}(\Delta\hat{H})^{2}$ and its maximum is
$F_{\omega}^{Q}|_{\textrm{{max}}}=T^{2}(\lambda_{\textrm{{max}}}-\lambda_{\textrm{{min}}})^{2}$
with $\lambda_{\textrm{{max}}}$ and $\lambda_{\textrm{{min}}}$ respectively
denoting the maximal and minimal eigenvalues of $\hat{H}$. For a many-body
SU$(2)$ quantum interferometry involving $N$ two-mode particles, one can
choose a linear or nonlinear coupling of collective spin operators as the
generator $\hat{H}$.

In most cases, one may choose a linear operator such as $\hat{H}=\hat{J}_{z}$
as the generator, which is commonly used in quantum metrology Pezzè  _et al._
(2018) including magnetic field measurement Maze  _et al._ (2008); Kotler  _et
al._ (2011); Schmitt  _et al._ (2017), frequency measurement Boss  _et al._
(2017); Huelga  _et al._ (1997); Pezzè and Smerzi (2020), and more. The
maximal and minimal eigenvalues are $\lambda_{\textrm{{max}}}=N/2$ and
$\lambda_{\textrm{{min}}}=-N/2$, respectively. Thus, the maximum quantum
Fisher information is
$F_{\omega}^{Q}|_{\textrm{{max}}}=T^{2}(\lambda_{\textrm{{max}}}-\lambda_{\textrm{{min}}})^{2}=T^{2}N^{2}$,
which corresponds to the Heisenberg limit (HL). Achieving the HL with the
linear generator $H=\hat{J}_{z}$ requires the probe to be initialized in a
maximally entangled state.

In the case of a nonlinear generator $\hat{H}=\left({\hat{J}}_{z}\right)^{k}$
(where the exponent $k>1$), the measurement precision can reach a scaling of
$N^{-k}$. When $k$ is an even number, the maximal and minimal eigenvalues of
$\hat{H}$ are $\lambda_{\textrm{{max}}}=(\frac{N}{2})^{k}$ and
$\lambda_{\textrm{{min}}}=0$, respectively. The corresponding maximum of QFI
and QCRB of $\omega$ are
$F_{\omega}^{Q}|_{\textrm{{max}}}=T^{2}(\frac{N}{2})^{2k}$ and
$\omega_{\textrm{QCRB}}=\frac{2^{k}}{T}\frac{1}{N^{k}}$, respectively. When
$k$ is an odd number, the maximal and minimal eigenvalues of $\hat{H}$ are
$\lambda_{\textrm{{max}}}=(\frac{N}{2})^{k}$ and
$\lambda_{\textrm{{min}}}=-(\frac{N}{2})^{k}$, respectively. The corresponding
maximum of QFI and QCRB of $\omega$ are
$F_{\omega}^{Q}=4T^{2}(\frac{N}{2})^{2k}$ and
$\omega_{\textrm{QCRB}}=\frac{2^{k-1}}{T}\frac{1}{N^{k}}$, respectively.
Mathematically, when $k>1$, the scaling of $N^{-k}$ is better than the
Heisenberg scaling $N^{-1}$ for the case of linear generators Nie  _et al._
(2018); Boixo  _et al._ (2007); Roy and Braunstein (2008); Hall and Wiseman
(2012). However, this does not mean that the measurement precision breaks the
Heisenberg uncertainty relation. The measurement precision offered by a
nonlinear generator still obeys the Heisenberg uncertainty relation. In other
words, when we unify the definitions of HL for both linear and nonlinear
generators as the ultimate precision limit dictated by the Heisenberg
uncertainty relation, the scaling of HL depends on the form of the associated
generator. With this unified definition, it should be noted that the HL
remains unbreakable regardless of the protocol employed. A realistic
possibility for taking advantage of nonlinear generator is the case of $k=2$,
such as a quadratic OAT Hamiltonian $\hat{H}=\chi\hat{J}_{z}^{2}$.

Extensive studies have been dedicated to quantum metrology utilizing time-
independent Hamiltonians. However, the investigation of time-dependent
Hamiltonians is crucial in order to broaden the applicability of quantum
metrology to more complex scenarios. Below we consider the quantum metrology
with time-dependent Hamiltonians $\hat{H}_{\theta}(t)$ Pang and Jordan (2017).
A system is initialized in a state $\left|{\Psi}\right\rangle$, and then
evolves under the time-dependent Hamiltonian $\hat{H}_{\theta}(t)$ for a time
duration of $T$, its final state reads

$\displaystyle\left|{\Psi_{\theta}(T)}\right\rangle=\hat{U}_{\theta}(T)\left|{\Psi}\right\rangle,$
(111)

where $\hat{U}_{\theta}(T)=e^{-i\int_{0}^{T}\mathcal{T}\hat{H}_{\theta}(t)dt}$
is the unitary evolution under the Hamiltonian $\hat{H}_{\theta}(t)$ with the
time-ordering operator $\mathcal{T}$. According to Eq. (II.3), the QFI of
$\left|{\Psi_{\theta}(T)}\right\rangle$ for estimating $\theta$ can be written
as

$\displaystyle
F_{\theta}^{Q}=4\left[\langle\Psi|\hat{h}^{2}_{\theta}(T)|\Psi\rangle-\langle\Psi|\hat{h}_{\theta}(T)|\Psi\rangle^{2}\right],$
(112)

where
$\hat{h}_{\theta}(T)=i\hat{U}_{\theta}(T)^{{\dagger}}\left[\partial_{\theta}\hat{U}_{\theta}(T)\right]$.
Therefore, the maximum $F_{\theta}^{Q}$ is the squared gap between the maximum
and minimum eigenvalues of $\hat{h}_{\theta}(T)$, similar to the case of a
static Hamiltonian. This means that the key to give the optimal estimation
precision is finding the maximum and minimum eigenvalues of
$\hat{h}_{\theta}(T)$. To obtain the maximum QFI, one can take an alternative
approach that breaks the unitary evolution $\hat{U}_{\theta}(T)$ into products
of small time intervals $dt$ and takes the limit $dt\rightarrow 0$. That is,

$\displaystyle\hat{h}_{\theta}(T)=\int_{0}^{T}\hat{U}_{\theta}(t)^{{\dagger}}\partial_{\theta}\hat{H}_{\theta}\hat{U}_{\theta}(t)dt.$
(113)

Thus one can decompose the global optimization of the eigenvalues of
$\hat{h}_{\theta}(T)$ into the local optimizations at each time instant $t$.
As $\hat{U}_{\theta}(t)$ is unitary, $\hat{U}_{\theta}(t)|\psi\rangle$ is a
normalized state and the upper bound of
$\langle\Psi|\hat{U}_{\theta}(t)^{{\dagger}}\partial_{\theta}\hat{H}_{\theta}(t)\hat{U}_{\theta}(t)|\Psi\rangle$
must be the maximum eigenvalue $\lambda_{\textrm{{max}}}(t)$ of
$\partial_{\theta}\hat{H}_{\theta}(t)$ at time $t$. Consequently, the maximum
eigenvalue and the minimum eigenvalue of $\hat{h}_{\theta}(T)$ are bounded by
$\int_{0}^{T}\lambda_{\textrm{{max}}}(t)dt$ and
$\int_{0}^{T}\lambda_{\textrm{{min}}}(t)dt$, respectively. Therefore the upper
bound of the QFI can be given as

$\displaystyle
F_{\theta}^{Q}(T)\leq\left[\int_{0}^{T}(\lambda_{\textrm{max}}(t)-\lambda_{\textrm{min}}(t))dt\right]^{2}.$
(114)

This indicates that the upper bound of $F_{\theta}^{Q}(T)$ is determined by
the integral of the gap between the maximum and minimum eigenvalues of
$\partial_{\theta}\hat{H}_{\theta}(t)$ over the time interval from $0$ to $T$.

From Eq. (114), in the case of a time-independent Hamiltonian
$\hat{H}_{\theta}=\theta\hat{H}$, it reduces to
$F_{\theta}^{Q}(T)|_{\textrm{{max}}}=T^{2}(\lambda_{\textrm{{max}}}-\lambda_{\textrm{{min}}})^{2}$.
For a time-dependent Hamiltonian $\hat{H}_{\theta}(t)$, a convenient way to
achieve the upper bound of QFI is to introduce a control Hamiltonian
$\hat{H}_{c}(t)$, which is independent of the parameter $\theta$,to the
original Hamiltonian. That is, the system will evolve under the total
Hamiltonian is $\hat{H}_{\text{tot}}=\hat{H}_{\theta}(t)+\hat{H}_{c}(t)$. By
applying the control Hamiltonian $\hat{H}_{c}(t)$, the state evolution under
the total Hamiltonian will always stay in the instantaneous eigenstates of
$\partial_{\theta}\hat{H}_{\theta}(t)$. To realize the above target, the
optimal control Hamiltonian $\hat{H}_{c}(t)$ can be chosen as

$\displaystyle\hat{H}_{c}(t)=\sum_{k}f_{k}(t)|\Psi_{k}(t)\rangle\langle\Psi_{k}|-\hat{H}_{\theta}(t)+i\sum_{k}|\partial_{t}\Psi_{k}(t)\rangle\langle\Psi_{k}|,$

where $f_{k}(t)$ are arbitrary real functions, and $|\Psi_{k}\rangle$ is the
$k$-th eigenstate of $\partial_{\theta}\hat{H}_{\theta}(t)$. The aim of
applying this control Hamiltonian is to guide the state evolution governed by
the total Hamiltonian $\hat{H}_{\text{tot}}$, following the paths traced by
the instantaneous eigenstates of $\partial_{\theta}\hat{H}_{\theta}(t)$. This
approach allows for acquiring the maximum amount of information about
$\theta$. Fig. 22 illustrates a schematic diagram for the state evolution
under the optimal control Hamiltonian $\hat{H}_{c}(t)$. This control approach
can also be extended to multi-particle systems. However, the optimal control
Hamiltonian may become more complex and challenging to implement.

Figure 22:  Optimal Hamiltonian control scheme. To achieve the maximal Fisher
information, the optimal control Hamiltonian needs to keep the state evolution
under the total Hamiltonian $\hat{H}_{\text{tot}}$ evolving along the tracks
of the eigenstates of $\partial_{\theta}\hat{H}_{\theta}(t)$ at all times $t$.
Figure adapted from Ref. Pang and Jordan (2017).

By employing the above method, the precision of measuring the field amplitude
and frequency of a rotating magnetic field can be significantly improved.
Similar ideas of steering quantum state evolution along specific trajectories
have also been used in other studies Garanin and Schilling (2002); Berry
(2009); Ruschhaupt  _et al._ (2012); Barnes (2013). A simple case is that the
Hamiltonian is still time-independent but the parameter to estimate is not
multiplicative, such as a spin-$1/2$ particle in a DC vector magnetic field
Yuan (2016). In this case, the Hamiltonian can be represented as
$\hat{H}_{\theta}(t)=\hat{H}_{\theta}$, and one have
$|\partial_{t}\Psi_{k}(t)\rangle=0$. Thus the optimal control Hamiltonian can
be chosen as
$\hat{H}_{c}(t)=\sum_{k}f_{k}(t)|\Psi_{k}(t)\rangle\langle\Psi_{k}|-\hat{H}_{\theta}(t)$.

In realistic entanglement-enhanced metrology, the time required to prepare
entangled states is typically not short Dooley, Munro, and Nemoto (2016);
Hayes  _et al._ (2018). Especially for highly entangled states, the
preparation time is comparable with the interrogation time. Hence, the
simultaneous state preparation and signal interrogation becomes an effective
approach to optimize time utilization in quantum metrology Hayes  _et al._
(2018). In this approach, it is more advantageous to allocate the entire
duration for the entanglement generation and signal interrogation, rather than
dividing the time between creating an entangled state and separately using the
prepared state for signal interrogation. Intuitively, this approach can
enhance sensitivity by accumulating a larger phase shift due to the extended
time duration involved. However, it generally depends on the trade-off between
entanglement generation and signal interrogation Hayes  _et al._ (2018). For
instance, in the scenario of magnetic field sensing with twisted states, it is
more advantageous to allocate the entire duration to the sensing process if
the twisting strength is relatively weak or the total time available is
limited. However, if the twisting strength is strong, it allows the twisting
dynamics and the magnetic field sensing operate concurrently will be more
effective. Moreover, chaotic dynamics can be utilized to achieve this purpose
and develop the quantum chaotic sensors Fiderer and Braun (2018); Liu  _et
al._ (2021a). Meanwhile, the machine optimized concurrent state preparation
and signal interrogation are developed Haine and Hope (2020); Schuff, Fiderer,
and Braun (2020); Huo  _et al._ (2022), which can provide better measurement
precision in a shorter time compared to traditional schemes.

###  IV.2 Conventional readout

In addition to the preparation of multiparticle entangled states, the readout
of such states poses a significant challenge in achieving entanglement-
enhanced quantum metrology. In a SU(2) quantum interferometry, it is usually
to measure the population difference $n\equiv n_{\uparrow}-n_{\downarrow}$
between the two involved modes $\left|{\uparrow}\right\rangle$ and
$\left|{\downarrow}\right\rangle$. The population uncertainty $\Delta n$
limits the attainable measurement precision $\Delta\theta=\Delta
n/(\partial\langle n\rangle/\partial\theta)$. In situations involving non-
entangled states, population measurement works well and can achieve the
optimal measurement precision provided by the input state. However, when
dealing with entangled states, approaching the optimal measurement precision
(especially the Heisenberg limit), requires single-particle-resolved state
detection. This task becomes progressively more challenging as the number of
atoms increases.

For Gaussian entangled states, population measurement can still make the
measurement precision beyond the SQL. As an example, inputting a spin squeezed
state and measuring the population difference measurement $\hat{J}_{z}$, the
measurement precision reads

$\displaystyle\Delta\theta=\frac{\Delta{{\hat{J}_{z}}}}{|\partial{\langle\hat{J}_{z}\rangle/\partial{\theta}|}}=\frac{\xi_{R}}{|\sin\theta|\sqrt{N}},$
(116)

with $\xi_{R}$ being the squeezing parameter for the input state. If
$\theta=\pi/2$, the measurement precision $\Delta\theta$ reaches its minimum
$\Delta\theta=\frac{\xi_{R}}{\sqrt{N}}$. This means that spin squeezed states
may be used to beat the SQL if $\xi_{R}<1$. Especially, if the input state is
squeezed to $\xi_{R}\propto 1/\sqrt{N}$, the measurement precision can reach
the Heisenberg scaling: $\Delta\theta\propto 1/N$.

For non-Gaussian entangled states, such as twin Fock states, spin cat states,
and over-squeezed states, the population difference measurement cannot fully
exploit their nonclassical features and achieve the optimal measurement
precisions. To resolve this issue, one can implement the parity measurement
$\hat{\Pi}=e^{i\pi\hat{b}^{{\dagger}}\hat{b}}$. According to the error
propagation formula, the measurement precision with parity measurement can be
given as Bollinger  _et al._ (1996); Gerry and Mimih (2010); Campos, Gerry,
and Benmoussa (2003)

$\displaystyle\Delta\theta=\frac{\Delta{\hat{\Pi}}}{|\partial{\langle\hat{\Pi}\rangle/\partial{\theta}|}}.$
(117)

For a twin Fock state Campos, Gerry, and Benmoussa (2003), the phase
uncertainty $\Delta\theta$ versus the particle number $N$ approaches the
Heisenberg limit $\Delta\theta=1/(2N)$ when $\theta\rightarrow 0$. For certain
values of $\theta$, the measurement precision $\Delta\theta$ may significantly
increase for specific total particle numbers, while for other total particle
numbers, it continues to approach the Heisenberg limit. For a GHZ state, the
measurement precision with parity measurement is always $\Delta\theta=1/N$,
which is independent of the values of $\theta$ Bollinger  _et al._ (1996).
However, in order to achieve entanglement-enhanced quantum metrology with
parity measurement, the requirement of single-particle resolved detection
poses a challenge, especially at larger atom numbers, and it also serves as a
bottleneck in practical experiments. Moreover, the fragility of quantum
entangled states under inevitable sources of noises Ferrini  _et al._ (2010);
Huang  _et al._ (2015a), such as phase or detection noises, presents a major
obstacle that hinders the enhancement of measurement precision.

###  IV.3 Interaction-based readout

In addition to conventional readout protocols, which involve directly
measuring the states output from an interferometer, interaction-based readout
protocols have been introduced. In interaction-based readout protocols Macrì,
Smerzi, and Pezzè (2016); Anders  _et al._ (2018); Hosten  _et al._ (2016a);
Haine (2018); Mirkhalaf, Nolan, and Haine (2018); Liu  _et al._ (2023a);
Anisimov  _et al._ (2010); Davis, Bentsen, and Schleier-Smith (2016); Fröwis,
Sekatski, and Dür (2016); Linnemann  _et al._ (2016); Szigeti, Lewis-Swan, and
Haine (2017); Nolan, Szigeti, and Haine (2017); Huang  _et al._ (2018); Burd
_et al._ (2019); Liu  _et al._ (2022); Colombo  _et al._ (2022); Mao  _et al._
(2023), the states output from an interferometer undergo a unitary operation
through specific nonlinear interactions before measurement takes place. The
introduction of nonlinear operation before implementing measurement
efficiently eliminates the need for single-particle resolved detection.

Figure 23:  Schematic of interaction-based readout. Firstly, an unentangled
initial state undergoes a unitary evolution $\hat{U}_{1}$ into a nonclassical
state. Then, the estimated physical parameter $\theta$ is encoded onto the
state in the interrogation stage via the unitary operator $\hat{U}_{\theta}$.
Finally, another unitary operator $\hat{U}_{2}$ is applied before the final
measurement.

In Fig. 23, we show how interaction-based readout works. In the preparation
stage, an unentangled initial state undergoes a unitary evolution
$\hat{U}_{1}$ into a nonclassical state. The physical parameter to be
estimated, $\theta$, is then encoded onto the state via the interrogation
stage, which is described by the unitary operator $\hat{U}_{\theta}$. The
interaction-based readout is then implemented by applying another unitary
operator $\hat{U}_{2}$ before the final measurement. Therefore the final state
reads

$\displaystyle\left|{\Psi_{\text{final}}(\theta)}\right\rangle=\hat{U}_{2}\hat{U}_{\theta}\hat{U}_{1}\left|{\Psi_{\text{in}}}\right\rangle.$
(118)

The QFI is just dependent on the unitary operator $\hat{U}_{1}$ and the
initial state. Although $\hat{U}_{2}$ does not alter the QFI, it may affect
the CFI when a measurement is made in a particular basis. Specifically,
several studies have demonstrated that the protocol with perfectly time
reverse the initial unitary operator, i.e.,
$\hat{U}_{2}=\hat{U}_{1}^{{\dagger}}$, the projection measurement onto the
initial state is optimal which can saturate the QCRB Nolan, Szigeti, and Haine
(2017); Fröwis, Sekatski, and Dür (2016); Linnemann  _et al._ (2016); Szigeti,
Lewis-Swan, and Haine (2017); Nolan, Szigeti, and Haine (2017); Haine (2018).
The corresponding interferometry also called time-reversed nonlinear
interferometry. Meanwhile, the protocols with
$\hat{U}_{2}\neq\hat{U}_{1}^{{\dagger}}$ can also improve robustness against
detection noise Liu  _et al._ (2022); Mirkhalaf, Nolan, and Haine (2018); Liu
_et al._ (2023a); Scharnagl, Kielinski, and Hammerer (2023). Below we
introduce these two interaction-based readout protocols.

Figure 24:  Interaction-based readout via one-axis twisting. (a) A spin
coherent state evolves under $\hat{H}_{\textrm{OAT}}(\chi)$ into an over-
squeezed state $\left|{\Psi_{e}}\right\rangle$. Then, rotating
$\left|{\Psi_{e}}\right\rangle$ about $\hat{\textbf{x}}$ by a small angle
$\theta$. Finally, amplifying the perturbation into a large displacement by
applying $\hat{H}_{\textrm{OAT}}(-\chi)$. (b) Metrological gain
$1/[N(\Delta\theta)^{2}]$ versus twisting strength $Q$ for the unitary
twisting echo (solid blue), compared to spin squeezing (dot-dashed green) and
the quantum Cramér-Rao bound (QCRB) on phase sensitivity (dashed orange). (c)
Metrological gain $1/[N(\Delta\theta)^{2}]$ versus measurement uncertainty
$\Delta n$ for echo with twisting strength $Q_{\textrm{{opt}}}$ (solid blue)
or $Q_{\textrm{{sq}}}$ (dotted blue), compared to direct detection of the
squeezed state at $Q_{\textrm{{sq}}}$ (dot-dashed green) and projective
measurements with the GHZ state. Figure adapted from Ref. Davis, Bentsen, and
Schleier-Smith (2016).

Firstly, we introduce the interaction-based readout protocols of
$\hat{U}_{2}=\hat{U}_{1}^{{\dagger}}$ via OAT Davis, Bentsen, and Schleier-
Smith (2016). The dynamics under a OAT Hamiltonian
$\hat{H}_{\textrm{OAT}}(\chi)=\chi\hat{J}_{z}^{2}$ can generate different
entangled states. For spins initially polarized along $x$ direction, the
lowest-order effect of $\hat{H}_{\textrm{OAT}}$ can generate spin squeezed
state. For a short twisting time, the generated state is a Gaussian state.
However, for a long twisting time, the generated state becomes a non-Gaussian
state, such as spin cat state and over-squeezed state. In particular, a GHZ
state appears at $\chi t=\pi/2$. Then the generated state undergoes a
dynamical evolution to encode the information of the parameter $\theta$ to be
estimated. Before implementing measurement, a unitary operation via another
OAT Hamiltonian $\hat{H}_{\textrm{OAT}}(-\chi)=-\chi\hat{J}_{z}^{2}$ is
applied.

In Fig. 24 (a), we show the schematic of interaction-based readout with OAT.
From a spin coherent state along the x-direction
$\left|{\hat{\textbf{x}}}\right\rangle$, applying
$\hat{H}_{\textrm{OAT}}(\chi)$ for a duration $t$ yields a entangled state
$\left|{\Psi_{e}}\right\rangle=e^{-i\hat{H}_{\textrm{OAT}}t}\left|{\hat{\textbf{x}}}\right\rangle$.
Then the parameter $\theta$ is encoded into the entangled state via a unitary
operation $\hat{U}_{\theta}=e^{-i\theta\hat{J}_{y}}$. Subsequently, one can
use the interaction-based readout to undo the twisting by applying
$\hat{H}_{\textrm{OAT}}(-\chi)$. When $\theta=0$, the final state is identical
to the original spin coherent state. When $\theta\neq 0$, the
$\hat{J}_{z}$-dependent spin precession can produce a large final value of
$\langle\hat{J}_{y}\rangle$. Through measuring $\hat{J}_{y}$, the measurement
precision of $\theta$ is given by

$\displaystyle\Delta\theta=\frac{\Delta{{\hat{J}_{y}}}}{|\partial{\langle\hat{J}_{y}\rangle/\partial{\theta}|}}.$
(119)

The resulting metrological gain depends on the “shearing strength” $Q\equiv
2J\chi t$, see Fig. 24 (b). At the optimal shearing strength
$Q_{\textrm{{opt}}}=2J\textrm{arccot}\left(\sqrt{2J-1}\right)$ for $N\gg 1$,
the protocol yields the optimal measurement precision,

$\displaystyle\Delta{\theta}_{\textrm{{min}}}=\frac{\sqrt{e}}{N},$ (120)

which is very close to the Heisenberg limit. The scheme is highly robust
against detection noise, as the “untwisting" amplifies the spin rotation
signal. This implies that the measurement precision may achieve the Heisenberg
limit, even when the detection resolution of atom number is approximately
$\Delta n\approx\sqrt{N}$, see Fig. 24 (c).

Interaction-based readout with OAT interaction is also suitable for effective
detection of non-Gaussian entangled states Haine (2018); Huang  _et al._
(2018). It is shown that the interaction-based readout enables spin cat states
to saturate their ultimate precision bounds of Eq. (107). Compared with the
twisting echo scheme on spin squeezed states, the scheme with with spin cat
states via interaction-based readout is more robust against detection noise
Huang  _et al._ (2018).

Figure 25:  Twist-and-turn entanglement with interaction-based readout. (a)
Husimi $Q$ function for the state $\left|{\Psi_{\textrm{{in}}}}\right\rangle$
under TNT, where $\left|{\Psi_{\textrm{{in}}}}\right\rangle$ is chosen as the
eigenstate of $\hat{J}_{x}$ with minimum eigenvalue. Time evolution of the CFI
for a many-body entangled state produced by the TNT Hamiltonian in presence of
detection noise $\sigma$ for (b) $\chi t_{1}=0.027$ and (c) $\chi t_{1}=0.072$
with $\hat{U}_{2}=1$ (red squares), a time-reversal echo
$\hat{U}_{2}=\hat{U}_{1}^{{\dagger}}$ (dark blue diamonds), an asymmetric echo
$\hat{U}_{2}\neq\hat{U}_{1}^{{\dagger}}$ (light blue triangles), and a pseudo
echo $\hat{U}_{2}=\hat{U}_{1}$ (light turquoise pentagrams). Figure adapted
from Ref. Mirkhalaf, Nolan, and Haine (2018).

In addition to OAT, one may employ TNT to achieve interaction-based readout.
As mentioned earlier in Sec. III.2, the TNT interaction exhibits a remarkable
ability to generate entanglement at a faster rate compared to the OAT
interaction. Additionally, it enables the rapid production of both spin
squeezed states and entangled non-Gaussian spin states. Fig. 25 shows an
interaction-based readout protocol with TNT Hamiltonian Mirkhalaf, Nolan, and
Haine (2018). The TNT Hamiltonian is

$\displaystyle\hat{H}_{\textrm{{TNT}}}=\chi\hat{J}_{z}^{2}-\Omega\hat{J}_{x}$
(121)

such that $\hat{U}_{1}=e^{-i\hat{H}_{\textrm{{TNT}}}t_{1}}$ with the
adjustable parameter $t_{1}$ is the state preparation time. To align the
interferometer along the optimal Fisher information direction, one choose
$\hat{U}_{\theta}=e^{-i\theta\hat{J}_{\textbf{{n}}}}$ with
$\hat{J}_{\textbf{{n}}}=\alpha\hat{J}_{y}+\beta\hat{J}_{z}$ that normal to the
mean spin direction $\hat{J}_{x}$. Thus, the final state after the
interaction-based readout is

$\displaystyle\left|{\Psi_{\textrm{{final}}}}\right\rangle=\hat{U}_{2}e^{-i\theta\hat{J}_{\textbf{{n}}}}\hat{U}_{1}\left|{\Psi_{\textrm{{in}}}}\right\rangle$
(122)

In general, the choices of $\hat{U}_{2}$ may affect the CFI for a measurement
is made with a particular basis. For a measurement that projects into the
$\hat{J}_{x}$ eigenbasis (which can saturate the QCRB), we have
$F_{C}(\theta)=\sum_{m_{x}}\frac{1}{P_{m_{x}}(\theta)}\left(\frac{\partial
P_{m_{x}}(\theta)}{\partial{\theta}}\right)^{2}$ with
$P_{m_{x}}(\theta)=|\langle m_{x}|{\Psi_{\textrm{{final}}}}\rangle|^{2}$ being
the ideal conditional probability via projecting into the eigenbasis
$\left|{m_{x}}\right\rangle$ for a given parameter $\theta$. Further, the
robustness to detection noise with different $\hat{U}_{2}$ is shown in Fig. 25
(b, c). To model detection noise, one can take the convolution of the
probability distribution with a Gaussian distribution under detection noise
$\sigma$ Fröwis, Sekatski, and Dür (2016); Szigeti, Lewis-Swan, and Haine
(2017); Nolan, Szigeti, and Haine (2017); Haine (2018) and it is
$\tilde{P}_{m_{x}}(\theta)=\sum_{m_{x}^{{}^{\prime}}}C_{m_{x}^{{}^{\prime}}}{e^{-(m_{x}-m_{x}^{{}^{\prime}})^{2}/2{\sigma}^{2}}}{P_{m_{x}}(\theta)}$
with $C_{m_{x}^{{}^{\prime}}}$ being a normalization factor, and the
conditional probability depends on the detection noise $\sigma$. $F_{C}$ as a
function of detection noise for different choices of $\hat{U}_{2}$ are shown
in Fig. 25 (b, c). Interestingly, the optimal interaction-based readout may
not be the case of perfect time-reversal echo. In particular for weakly
entangled initial states using an asymmetric echo provides better robustness
than the symmetric case.

Figure 26:  Quantum metrology with cyclic nonlinear dynamics. (a) Husimi
representations of states with quasi-cyclic dynamics in the TNT model. Spin
coherent state (i) evolves into spin squeezed state (ii), non-Gaussian
entangled states (iii)–(v), and back to the vicinity of the initial point (vi)
in succession. (b) Probability distributions of the probe states with the
rotation $\theta$ in the $\hat{J}_{z}$ eigenbasis. Dashed lines denote the
Gaussian fitting curves. (c) Probability distributions of the output states
with the rotation $\theta$ via nonlinear detection with cyclic dynamics. (d)
Robustness to detection noise for the TNT Hamiltonian. The nonlinear cyclic
readout protocol (blue circles), the linear readout scheme (gray squares) and
exact time-reversal scheme with $t_{2}=-t_{1}$ (red dashed line). Figure
adapted from Ref. Liu  _et al._ (2023a).

Interaction-based readout can also be realized via driving through quantum
phase transitions (QPTs). As shown in Fig. 19, a nonlinear beam splitting
process, in which an initial nondegenerate ground state evolves into a highly
entangled state, can be achieved by adiabatically driving the system through a
QPT Lee (2006, 2009). Inversely, by driving the system from the entangled
state back across the QPT again to the nondegenerate regime, the state becomes
gradually disentangled Huang, Zhuang, and Lee (2018). The reversed driving
through QPTs can be regarded as a kind of nonlinear detection. Finally, the
phase shift accumulated in the interrogation process can be easily inferred
via population measurement without single-particle resolution Huang, Zhuang,
and Lee (2018); Zhuang  _et al._ (2021).

Moreover one may also use cyclic nonlinear dynamics to achieve the
interaction-based readout. Different from the time-reversed dynamics which
drives a system back to its starting point, the cyclic nonlinear
interferometer accomplish the same by forcing the system to travel along a
‘closed loop’ instead of explicitly tracing back its antecedent path. For a
TNT Hamiltonian of Eq. (121), using a spin coherent state as the input, the
first stage of the nonlinear dynamics
$\hat{U}_{1}=e^{-i\hat{H}_{\textrm{{TNT}}}t_{1}}$ serves as nonlinear
splitting leading to the non-Gaussian entangled state. After sensing a spin
rotation, the system undergoes a second stage of TNT dynamics
$\hat{U}_{2}=e^{-i\hat{H}_{\textrm{{TNT}}}t_{2}}$ for nonlinear recombination.
The non-Gaussian entangled input state evolves directly back to the vicinity
of the initial state. Compared with a direct linear readout of the non-
Gaussian entangled state, the quasi-cyclic nonlinear readout can magnify the
encoded signal and refocus the associated quantum noise, leading to a better
signal-to-noise ratio (SNR), see Fig. 26. It is found that the optimal
metrological gain with population measurement nearly saturates the QCRB and
follows the Heisenberg-limited scaling Liu  _et al._ (2023a). Meanwhile, the
interaction-based readout with cyclic nonlinear dynamics of TACT Hamiltonian
is also investigated.

At the same time of generating multiparticle entangled states in experiments,
numerous interaction-based readout protocols have been demonstrated for
achieving entanglement-enhanced quantum metrology with Bose condensed atoms
Linnemann  _et al._ (2016); Liu  _et al._ (2022), cavity-QED systems Colombo
_et al._ (2022), and ultracold trapped ions Burd  _et al._ (2019).

Figure 27:  Entanglement-enhanced sensing based on time reversal of nonlinear
dynamics in a spin-1 BEC. (a) Spin-changing collisions in a BEC are used as
the nonlinear process. Atom numbers are detected by high resolution absorption
imaging after Stern-Gerlach separation. (b) The interaction-based readout
scheme exploits a time reversal sequence. The Hamiltonian $\hat{H}$ used for
entangled state generation is inverted and reapplied for readout. (c) The
phase sensitivity is experimentally extracted by Gaussian error propagation.
The SQL (gray horizontal bar) is surpassed in close vicinity of phase
$\theta=\pi$. At phase $\pi$ the sensitivity diverges due to the vanishing
slope of the signal. Figure adapted from Ref. Linnemann  _et al._ (2016).
Figure 28:  Quantum metrology with quasi-cyclic dynamics in a spin-1 BEC. (a)
The quasi-cyclic dynamics. Temporal evolution of $\rho_{0}$ starting from the
polar state with all atoms in the $\left|{1,0}\right\rangle$ component. The
upper panel denotes the ideal case without atom loss or technical noises. The
middle and lower panels present experimentally measured data. (b) Schematic of
the three-mode cyclic dynamics nonlinear interferometry based on spin mixing
dynamics. (c) Temporal evolution of $\rho_{0}$ for different spinor phase
$\theta$ encoded at the instant marked by the vertical dashed black line. (d)
The specific sequence to implement the experiments with ‘path’ splitting
$(\hat{U}_{1})$ and recombining $(\hat{U}_{2})$ both from time-forward spin
mixing dynamics for $t_{1}=290$ ms and $t_{2}=230$ ms, respectively. (e)
Probability distributions of $\rho_{0}$ for the probe state and for the final
states (f) Metrological gain
$20\log_{10}\left[\Delta\phi/\Delta\phi_{\textrm{{SQL}}}\right]$
($\Delta\phi_{\textrm{{SQL}}}=2/\sqrt{N}$) obtained from error propagation
formula. Figure adapted from Ref. Liu  _et al._ (2022).

With a spin-$1$ BEC, its spin-exchange collision has been utilized to realize
the time-reversal interaction-based readout Linnemann  _et al._ (2016). The
spin-exchange collision is analog to the parametric amplification, in which
both the sign and the strength of the nonlinear coupling can be experimentally
adjustable, see Fig. 27 (a). As shown in Fig. 27 (b), the nonlinear detection
is experimentally demonstrated by exploiting time-reversal dynamics to
disentangle continuous variable entangled states for feasible readout. As both
preparation and readout of entangled states consisting of parametric
amplification, an active atom SU$(1,1)$ interferometer is constructed, which
is capable of utilizing the quantum entanglement by only detecting the mean
atom numbers. The phase sensitivity $(\Delta\theta)^{2}$ can surpass the SQL
in the vicinity of $\theta=\pi$, see Fig. 27 (c). Moreover, the time-reversal
readout can also be achieved by a state flip operation. Similar to the spin
echo with a $\pi$-pulse state flip that reverses linear dynamics, this process
switches the squeezing and anti-squeezing axes of the probe state, leading to
effectively time-reversed nonlinear dynamics to realize the echo-based spin-
nematic interferometry Mao  _et al._ (2023). A sensitivity of $15.6\pm 0.5$ dB
beyond the SQL for detecting small-angle Rabi rotation, as well as $16.6\pm
1.1$ dB for phase sensing in a nonlinear Ramsey-like interferometry are
observed.

Figure 29:  Time-reversal-based quantum metrology with entangled atoms in
cavity. (a) The sequence of the interaction-based readout protocol. The
protocol sequence with the quantum state evolution (top) and relevant energy
levels and cavity mode (bottom). The Wigner quasi-probability distribution
functions describing the collective quantum states. (b) 171Yb atoms are
trapped inside an optical cavity in an optical lattice. (c) The scaling of the
sensitivity with the atom number and a comparison with previous results. A
comparison with previous results including BECs experiments (blue), thermal
atoms (red), ions (dark grey) and Rydberg atoms (light grey). The green point
shows the squeezing generated in an optical lattice clock. Squares show
expected metrological gains obtained by quantum state characterization. Stars
indicate directly measured phase sensitivity gains. Solid symbols are obtained
in this work. Figure adapted from Ref. Colombo  _et al._ (2022).

Besides the time-reversal dynamics, interaction-based readout can be achieved
by the quasi-cyclic dynamics resulted from spin-exchange collision. It has
demonstrated that a three-mode nonlinear interferometer with quasi-cyclic
dynamics can attain a metrological gain of $5.01$ dB over the classical limit
for a total of $26,500$ atoms can be achieved Liu  _et al._ (2022). As shown
in Fig. 28, utilizing the spin mixing dynamics $\hat{U}_{1}$ at the beginning,
paired atoms are created in $\left|{1,\pm 1}\right\rangle$ components from the
$\left|{1,0}\right\rangle$. Then the dressing microwave is switches on and
encodes a spinor phase $\theta=\theta_{1}+\theta_{-1}-2\theta_{0}$ via
$\hat{U}_{\theta}=e^{i\theta\hat{N}_{0}/2}$. At last, the microwave field is
turned off and the spin mixing dynamics $\hat{U}_{2}$ is used again to
annihilates paired atoms in $\left|{1,\pm 1}\right\rangle$ components into the
$\left|{1,0}\right\rangle$. The encoded spinor phase $\theta$ can be extracted
from the measured fractional population $\rho_{0}$ in the end. The time
evolution of the measured $\rho_{0}$ for different phase $\theta$ is shown in
Fig. 28 (c). Without an encoded phase, the system nearly returns to its
initial state at the end of the interferometry. However, a nonzero phase shift
of $2/\sqrt{N}\approx 0.012$ rad causes an evident decrease in the mean value
of $\rho_{0}$ as well as broadening of its distribution, as shown in Fig. 28
(e). The metrological gain for different phase $\theta$ is shown in Fig. 28
(f). Especially, a maximal gain of $4.28^{+0.56}_{0.58}$ dB beyond the SQL at
$=0.011$ rad is observed.

With a cavity-QED system of 171Yb atoms, it has been experimentally
demonstrated that the phase sensitivity can achieve a Heisenberg scaling of
$(b/N)$ at a fixed distance $b=12.6$ dB from the Heisenberg limit by employing
interaction-based readout with the OAT interaction Colombo  _et al._ (2022).
Especially, when used in a Ramsey sequence via an atomic interferometer, the
highest metrological gain over the SQL with $11.8\pm 0.5$dB can be achieved.
As shown in Fig. 29, an entangling light pulse is passed through the cavity to
generate the nonlinear OAT Hamiltonian $\hat{H}=\chi\hat{J}_{z}^{2}$ which
shears the initial spin coherent state. A rotation $\theta$ about the
${\hat{J}_{y}}$ axis displaces the state such that
${\langle\hat{J}_{z}\rangle=\theta S_{0}}$ with $S_{0}={N/2}$. Subsequently, a
(dis)entangling light pulse is sent through the cavity, which generates the
negative OAT Hamiltonian $-\hat{H}$ causing the quantum state to evolve
effectively ‘backwards in time’. For a small angle ($\theta\neq 0$) the final
state is displaced by an angle $m\theta$ from the original spin coherent
state, where $m$ is the signal amplification. The scaling of the sensitivity
with the atom number $N$ is shown in Fig. 29 (c), when the particle number $N$
varies between $50$ and $370$, a gain over the SQL is measured, which can
achieve the Heisenberg scaling. More Recently, a time-reversal protocol is
used to observe a simultaneous exponential growth of both the metrological
gain and the out-of-time-order correlation, thereby experimentally verifying
the relation between quantum metrology and quantum information scrambling Li
_et al._ (2023a).

Figure 30:  Quantum amplification in a trapped-ion mechanical oscillator. (a)
Conceptual illustration of the amplification protocol. Each panel shows a
Wigner function phase space distribution. A displacement $\alpha_{i}$ of an
initially squeezed ground state is amplified by subsequent anti-squeezing,
resulting in a final coherent state with amplitude $G\alpha_{i}$ with no added
noise. (b) Pulse sequence for displacement sensing protocol with phase-
sensitive red sideband (PSRSB) detection. (c) Population $P_{\downarrow}$ as a
function of the carrier $\pi/2$ pulse phase with no squeezing (blue inverted
triangles) and amplification (red circles). (d) Measurement sensitivity
enhancement in the linear small-displacement regime as a function of the ideal
gain $G=\exp(gt)$. Figure adapted from Ref. Burd  _et al._ (2019).

With a trapped-ion mechanical oscillator, interaction-based readout has been
employed to amplify and detect coherent motional displacements that are
considerably smaller than the quantum zero-point fluctuations Burd  _et al._
(2019). As shown in Fig. 30 (a), by squeezing the motional state firstly, one
can suppress the quantum fluctuations along a particular phase space
quadrature. Then, a small displacement $\alpha_{i}$ is applied along the
squeezed axis. Finally, by reversing the squeezing interaction, the oscillator
returns to a minimum-uncertainty coherent state with a larger amplitude
$\alpha_{f}=G\alpha_{i}$ with $G$ is the gain. The maximum amplification is
achieved when the displacement is along the squeezed axis. To demonstrate
experimentally, a single trapped 25Mg+ ion is used. In each experiment, the
ion is initialized in the electronic and motional ground state
$\left|{\downarrow}\right\rangle\left|{0}\right\rangle$ via optical pumping,
sideband laser cooling and microwave pulses. The pulse sequence for
displacement sensing protocol is shown in Fig. 30 (b). The displacement
$\alpha_{f}$ is obtained by measuring probability into state
$\left|{\downarrow}\right\rangle$ with
$P_{\downarrow}=\frac{1}{2}[1-C(|\alpha_{f}|)\cos\phi]$, where
$C(|\alpha_{f}|)$ is the signal contrast and $\phi$ is the phase of a carrier
$\pi/2$ pulse. Without amplification, $\alpha_{i}=\alpha_{f}$ and the contrast
is $C(|\alpha_{i}|)$. However, the initial displacement amplitude $\alpha_{i}$
is increased by utilizing the anti-squeezing and the contrast become
$C(|G\alpha_{i}|)$. This increase in contrast is shown in Fig. 30 (c).
Especially, if the displacement is along the squeezed axis, the measurement
sensitivity enhancement as a function of the ideal gain $G=\exp(gt)$ in the
linear small-displacement regime, as shown in Fig. 30 (d). More recently, by
modulating the trapping potential at close to twice of the center-of-mass mode
frequency, a motional squeezing of $5.4\pm 0.9$ dB below the ground-state
motion has been achieved via a stroboscopic protocol Affolter  _et al._
(2023).

##  V Promising applications in quantum sensors

Recent technological advancements have paved the way for the development of
highly stable and precise instruments such as lasers, optical frequency combs,
and microwave electronics. These advancements have enabled the creation of
quantum interferometers with unprecedented sensitivity and the world’s most
accurate and precise clocks. In these devices, the stable quantum states of
atoms play a crucial role. By harnessing controlled atomic entanglement, it is
possible to achieve entanglement-enhanced atom interferometry, surpassing the
precision limits of conventional methods. Entanglement-enhanced atom
interferometry allows for the precise measurement of various physical
quantities, including frequency, magnetic field, gravity, acceleration, and
rotation. This breakthrough has significant implications for the development
of atomic clocks, quantum magnetometers, quantum gravimeters, and quantum
gyroscopes, offering unprecedented levels of precision in these applications.
Quantum sensors, empowered by the remarkable precision offered by quantum
entanglement, have emerged as highly promising tools in various domains. They
hold great potential in fields such as inertial navigation, mineral
exploration, groundwater monitoring, dark matter detection, and space-based
experiments aimed at testing theories of general relativity or quantum
gravity. Below we will introduce the exciting and promising applications of
entanglement in the realm of quantum sensors.

###  V.1 Atomic clocks

Atomic clocks are the most accurate and precise clocks that have been
developed to date. Atomic clocks utilize the reliable transition frequency of
stable long-lived energy levels found in atoms. They work by comparing the
frequency of a local oscillator (LO) to the reference atomic transition
frequency. This comparison is used to ensure that the ticking period of the LO
remains stable, as shown in Fig. 31 (a). For atomic clocks the selected states
are often magnetically insensitive internal states. It makes them excellent
frequency references as they do not experience the aging process observed in
mechanical clocks. The atomic transition frequency is universally present and
highly suitable for usage as a standard frequency. For example, the current
major frequency standard relies on the hyperfine transition in 133Cs, which is
utilized to precisely define the SI second Markowitz  _et al._ (1958). The
systematic error achieved by a solitary optical clock based on an aluminum ion
Brewer  _et al._ (2019) can be $9.4\times 10^{-19}$. Nevertheless, as
systematic effects are fully understood and accounted for, atomic clocks that
make use of a large number of atoms have the capacity to far exceed the
performance of clocks that rely on a single ion. Atomic clocks with a
performance beating the SQL have been demonstrated in several experiments
Gross  _et al._ (2010); Nichol  _et al._ (2022); Schulte  _et al._ (2020);
Robinson  _et al._ (2024).

Entanglement-enhanced microwave-frequency clocks have been achieved using
neutral atoms with spin-squeezed states as input Gross  _et al._ (2010);
Leroux, Schleier-Smith, and Vuletić (2010b); Ockeloen  _et al._ (2013); Kruse
_et al._ (2016); Hosten  _et al._ (2016b). Although these experiments serve as
spectacular examples of entanglement-enhanced atomic clocks, they have not yet
achieved the same level of frequency stability as state-of-the-art fountain
clocks. This is mostly due to their Ramsey interrogation time (limited by the
noise of the microwave LO) is much shorter than the ones typically operating
in fountain clocks using uncorrelated individual atoms Weyers  _et al._
(2018).

Figure 31:  Measurement noises of a squeezing-enhanced atomic clock. (a)
Measurement and feedback loop to stabilize the laser frequency to an atomic
transition. (b) Periodic measurements with Ramsey time $T_{R}$ and dead time
$T_{D}$ in each cycle of total time $T_{C}$ lead to increased instability from
the Dick effect. (c) Quantum projection noise for $N$ particles limits the
clock stability for short interrogation times but can be decreased with
squeezed states thus reducing the inferred phase uncertainty where $\zeta$ is
the Wineland spin squeezing parameter of Eq. (56). (d) For longer $T_{R}$ the
distribution of phases broadens substantially due the laser’s decoherence.
Inefficient feedback for phases outside the interval gives the coherence time
limit. Figure adapted from Ref. Schulte  _et al._ (2020).

The performance of atomic clocks could be greatly improved if the reference
transition occurred in the optical range instead of the microwave range Szmuk
_et al._ (2015). Typically, optical frequencies are approximately $10^{4}$
times higher than microwave frequencies, resulting in a corresponding
improvement of the fractional stability by a factor of $10^{4}$. With the
advent of optical frequency combs spanning a wide range of octaves Eckstein,
Ferguson, and Hänsch (1978), there has been notable progress in optical clock
technology Ludlow  _et al._ (2015). The lasers consists of a series of closely
spaced spectral lines in frequency space. The separation between these lines
is a few hundred MHz, and it is dictated by the repetition rate of the
femtosecond laser. Therefore, by utilizing a frequency comb as an intermediate
stage, two lasers operating at distinct wavelengths might effectively compete.
The usage of optical frequency comb in an optical atomic clock enables an
experimentally feasible way to measuring optical transition frequencies and
comparing optical clocks.

Optical transition-based atomic clocks have gained significant attention in
recent years due to their remarkable precision and stability Katori (2011);
Colombo, Pedrozo-Peñafiel, and Vuletić (2022). Currently, optical atomic
clocks are the most precise and accurate clocks. These clocks generally
interrogate the optical transitions with groups of atoms that are confined in
an optical lattice. So far, the optical-transition atomic clocks can reach
fractional stability below $10^{-18}$ Bloom  _et al._ (2014); Schioppo  _et
al._ (2017); Oelker  _et al._ (2019), and approaching $10^{-19}$ Zheng  _et
al._ (2022); Li  _et al._ (2024). Technical noises from imperfect readout,
intrinsic atom–atom interactions, or aliased frequency noise of the
interrogating clock laser pose challenges for observing clock performance at
the quantum projection noise (QPN) limit. Since the atomic collisions would
disturb the transition frequency and the resulted collision shift would
decrease the accuracy of the atomic clocks. Thus the collision shifts impose a
limit on the total number of atoms used in atomic clocks. Typically, the
atomic number is about $10^{2}\sim 10^{4}$. At such relatively small atom
number, the SQL places a significant constraint on the clock frequency
measurement.

By using suitable entangled states such as spin squeezed states, the
performance of optical atomic clocks can be improved beyond the SQL. The spin
squeezing has been used to improve an optical lattice clock of 171Yb atoms
Pedrozo-Peñafiel  _et al._ (2020). The strategy used is to generate a spin
squeezed state in the ground state of 171Yb atoms and then transfer it to the
ultra-narrow optical clock transition by applying an optical $\pi$ pulse, see
Fig. 32. Using the spin squeezed state on the optical transition as the input
state, a Ramsey interferometry is perform in the optical domain with a five-
order improved Q-factor compared to the microwave transitions. After the
Ramsey interference, the spin squeezed state is mapped back onto the ground
state and read out via the cavity. As a result, a measurement precision of 4.4
dB below the SQL is demonstrated, corresponding to a 2.8-fold reduction of the
averaging time. More recently, a protocol based on time-reversal dynamics
called Signal Amplification through Time-Reversal Interaction was demonstrated
for optical atomic clocks approaching the Heisenberg limit. As shown in Sec.
IV.2, a non-Gaussian over-squeezed state is generated under a prolonged OAT
dynamics and with a time-reversal process for readout, it enables one to reach
near-Heisenberg sensitivity.

Figure 32:  Optical atomic clock operated with spin squeezing. (a) 171Yb atoms
are confined within an optical cavity in a two-dimensional magic-wavelength
optical lattice, namely in the $x$ and $z$ directions (shown by the color
red). The optical pumping and spin squeezing light (green) is directed
parallel to the cavity axis $z$, while the clock laser (yellow) travels
parallel to the $x$-axis. (b) Levels of energy and changes between them. The
ground-state RF transition
$\left|{\downarrow}\right\rangle\rightarrow\left|{\uparrow}\right\rangle$ is
shown by purple pulses, the squeezing transition at $\omega_{at}$ is
represented by green pulses, and the optical-clock transition
$\left|{\uparrow}\right\rangle\rightarrow\left|{e}\right\rangle$ is
represented by yellow pulses. The system undergoes evolution inside either the
ground-state manifold (the purple Bloch sphere) or the clock-state manifold
(the yellow Bloch sphere). The resonance of the squeezing transition is
achieved by adjusting the cavity frequency $\omega_{c}$. (c) The atoms’ strong
interaction with the cavity leads to the division of the cavity resonance into
two distinct peaks, known as vacuum Rabi splitting (red peaks). A laser is
used at a frequency $\omega_{sq}$ (green peak) that is detuned from the Rabi
peak. This laser is employed to generate a spin squeezed state by cavity
feedback. (d) Squeezed-clock sequence. A spin squeezed state is initially
created in the lowest energy level and then rotated by an angle $\alpha$. It
is then moved to the clock energy level and undergoes a Ramsey sequence for a
duration of $\tau_{R}$. Finally, it is mapped back to the states
$\left|{\uparrow}\right\rangle,\left|{\downarrow}\right\rangle$ and a
measurement of the state is made. The quantum state’s development is
illustrated on the Bloch spheres, representing the RF (purple) and optical
(yellow) transitions. Figure adapted from Ref. Pedrozo-Peñafiel  _et al._
(2020).

The clock stability can be assessed by its Allan deviation, which depends on
several factors such as the transition frequency, interrogation time, dead
time, average time, and input state. For optical atomic clocks, the laser is
stabilized by a control loop to an optical atomic transition of frequency
$\nu_{0}$, as shown in Fig. 31 (a). The Ramsey interrogation $T_{R}$ and the
dead time $T_{D}$ comprise of the duration $T_{C}=T_{R}+T_{D}$ for each cycle.
By measuring the spin component $\langle\hat{S}_{y}\rangle$, one can get the
information of $\nu_{0}$ from the laser frequency. The measurement result is
converted into an error signal that is used to correct the laser frequency.
Averaging over a long time $\tau\gg T_{C}$, the clock stability can be
measured in terms of Allan deviation $\sigma_{y}(\tau)$. For an atomic clock
with spin squeezing, the stability limited only by the QPN of the spin
measurement results in the Allan deviation

$\sigma_{QPN}(\tau)=\frac{1}{2\pi\nu_{0}T_{R}}\sqrt{\frac{T_{C}}{\tau}}\frac{\xi_{R}}{\sqrt{N}},$
(123)

where $N$ is the number of atoms participating in the clock, $\xi_{R}$ is the
Wineland spin squeezing parameter. It is important to note that under
practical conditions, optical atomic clocks are not exclusively limited by
QPN.

Figure 33:  Generation of spin squeezing in a Rydberg-dressed array of 88Sr
atoms. (a) Atomic states (black lines) and transitions (colored arrows) of
88Sr relevant for clock interrogation and Rydberg dressing. (b) Diagram
illustrating the configuration of the experimental arrangement. The spacing of
$a_{\textrm{lat}}$ is approximately $575$ nm, and it is organized into several
subarrays. The upper plot illustrates the relationship between atoms at a
distance $r_{ij}$, which is defined by the potential energy $V_{ij}$. The
black circles represent the data points. The solid black line represents a
numerical fit. The laser beams that connect the states in (a) are shown by
colored arrows and correspond to the Hamiltonians $H_{c}$ and
$H_{\textrm{Ryd}}$. Optical tweezers, which have a green double cone shape and
operate at a wavelength of $515$ nm, allow for the application of the operator
$\hat{\sigma}_{z}^{(i)}$ to a specific atom $i$. (c) Diagram depicting the
procedural method for generating spin squeezed states in an experimental
setting. The pulse sequences for (II) and (III) display clock and Rydberg
laser pulses in the colors red and purple, respectively. In this context, the
symbol $\alpha$ represents the phase of the clock laser relative to a
reference. Individual photographs are captured after the process of preparing
the state (located at the bottom left) and for the purpose of determining the
atomic populations (located at the bottom right). In this context, the red
circles represent atoms in the state $\left|{g}\right\rangle$ that have been
deliberately eliminated prior to the final imaging. The generalized Bloch
spheres, located in the bottom center, depict the progression of the atomic
state for a subarray consisting of 16 atoms, where $N=4\times 4$. Figure
adapted from Ref. Eckner  _et al._ (2023).

In practice, the overall stability of a clock is influenced by various sources
of noise, including laser phase noise and dead time. Among these factors, the
Dick effect and the laser coherence time limit play significant roles in
compromising the stability. The Dick effect arises from the interrupted
interrogation of the atomic system while the finite laser coherence time
contributes to the stability with an additional diffusion process, which
refers to the laser coherence time limit. Incorporating these effects, the
optimal operating point of the control loop is determined by a trade-off
between QPN, Dick effect, and coherence time limit. This trade-off is achieved
by minimizing the combined stability Schulte  _et al._ (2020),

$\sigma_{y}(\tau)=\sqrt{\sigma_{QPN}^{2}(\tau)+\sigma_{Dick}^{2}(\tau)+\sigma_{CTL}^{2}(\tau)}.$
(124)

Considering all these critical factors of instability, it has been revealed
that a typical single-ensemble Ramsey clock holds the potential for improving
performance through spin squeezing, especially in systems with relatively low
atom numbers Schulte  _et al._ (2020). The primary limitations arise from the
instability of the lasers in the system. However, these limitations can be
addressed by employing an adaptive measurement scheme Borregaard and Sørensen
(2013b). This scheme involves performing a weak measurement of the collective
spin state, followed by a feedback process that rotates the state by an amount
determined by the measurement outcome. By doing so, the measurement backaction
is minimized, resulting in a final state that remains unaffected by undesired
noises.

Using two clock ensembles for comparison, the common-mode noises such as the
LO decoherence can be cancelled and so that the Dick effect can be completely
eliminated Schioppo  _et al._ (2017). Likewise, employing cascaded systems
with multiple ensembles can effectively mitigate the impact of laser coherence
limit Borregaard and Sørensen (2013a); Kessler  _et al._ (2014a). An
experimental implementation of a "multiplexed" one-dimensional optical lattice
clock has been realized, where spatially resolved ensembles of strontium atoms
are trapped in the same optical lattice, allowing for simultaneous
interrogation by a shared clock laser and parallel read-out processes Zheng
_et al._ (2022). The results of these experiments showcase that applications
involving optical clock comparisons can overcome the limitations imposed by LO
instability, making it a viable option for implementing a quantum network of
clocks as well K m r  _et al._ (2014).

To improve spin-squeezing-enhanced atomic clocks, it is crucial to consider
the coupling of anti-squeezing, as it couple with the measurement projection,
and then introduce extra noise and reduce the contrast. To overcome this
problem, one may utilize adaptive measurement for a single clock ensemble
Borregaard and Sørensen (2013b); Vanderbruggen  _et al._ (2013); Cox  _et al._
(2016), or combination of measurement and feed forward for multiple clock
ensembles Borregaard and Sørensen (2013a); Hume and Leibrandt (2016); Pezzè
and Smerzi (2020). Recently, the protocol of multiple clock ensembles with
feedback was analyzed Pezzè and Smerzi (2020); Li  _et al._ (2022b), where one
squeezed clock was considered together with another unentangled clock. It is
shown that by feeding forward the measurement result of the unentangled
ensemble to the squeezed ensemble, it is possible to sustain the entanglement
enhancement for extended periods of time, even when there is LO noise present.
The two methods can thereby increase the interrogation times that still allow
an enhancement due to spin squeezing.

Through integrating an optical clock platform with collective strong-coupling
cavity QED for QND measurements, the direct comparison of two spin-squeezed
optical clock ensembles has been achieved Robinson  _et al._ (2024). The QND
measurements of the clock state Bowden  _et al._ (2020) can be implemented for
spin squeezing and clock readout. Optimizing the competition between spin
measurement precision and loss of coherence, a metrological enhancement for a
large ensemble of atoms beyond the initial spin coherent state can be
measured. Using a movable lattice, the cavity can individually address two
independent sub-ensembles, enabling one to spin squeeze two clock ensembles
successively and compare their performance without the influence of clock
laser noise. Although the clock comparison remains above the effective SQL,
the performance directly verifies $1.9(2)$ dB clock stability enhancement at
the $10^{-17}$ level without subtracting any technical noise contributions.
Moreover, resonant electric dipole-dipole interactions in a cubic array of
atoms in the many-excitation limit has been observed directly Hutson  _et al._
(2024). With the 87Sr atoms loaded in a cubic optical lattice, the effects of
resonant dipole-dipole interactions is measured. The ensemble-averaged shifts
can be suppressed below the level of evaluated systematic uncertainties for
optical atomic clocks. It is found that the interactions caused a tiny clock
shift, the magnitude of which could be controlled by varying the relative
orientation of the probe light and the atomic dipoles.

In comparison to an optical lattice clock, the trapped ion clock offers the
advantage of less dead time. This is because that the same ion can be trapped
and interrogated for extended durations due to the presence of a strong
confinement potential. Despite their lower signal-to-noise ratios when
compared to optical lattice clocks, ion clocks remain among the most stable
clocks now accessible. The physics component of trapped ion clock systems is
much more amenable to miniaturization, and ongoing attempts are being made to
enable the transmission and reception of light from the ions through phonics
integrated into the trap structure itself Schulte  _et al._ (2020). In recent,
several promising trapped ion-based clocks such as highly charged ion clocks
King  _et al._ (2022) and nuclear clocks Peik  _et al._ (2021) are being
pursued and they offer further improvement in the performance of these types
of clocks. In particular, the enhanced frequency comparisons using an
elementary quantum network of two entangled trapped-ion atomic clocks has been
demonstrated Nichol  _et al._ (2022). The network’s ability to generate
entanglement with high fidelity and speed, resulting in a large signal and an
efficient duty cycle, demonstrates the potential for practical enhancements in
atomic clocks. Furthermore, as shown in Fig. 14, a potential avenue for
leveraging large-scale entanglement is demonstrated on an optical transition
by utilizing one-dimensional chains of ions. The observation of emergent
collective behaviors related to OAT with finite-range interactions represents
a significant milestone in the integration of entanglement into high-
performance ion clocks that operate with a large number of ions.

Another intriguing avenue for clock applications involves utilizing long-range
Rydberg-dressing interactions to generate spin-squeezed and non-Gaussian
entangled states. With the versatile capabilities of optical tweezer arrays in
manipulating atomic ensembles, optical clocks have been developed using
current breakthroughs in optical tweezers Young  _et al._ (2020); Schine  _et
al._ (2022); Eckner  _et al._ (2023). The tweezer array optical clocks combine
the advantages of both single-ion-based optical clocks and optical lattice
clocks. They incorporate the superb isolation and high duty cycles
demonstrated by single-ion-based optical clocks, while also benefiting from
the large ensembles and consequent low QPN characteristic of optical lattice
clocks. The demonstration of a half-minute-scale atomic coherence and high
relative stability in a tweezer clock, as reported in a recent experiment
Young  _et al._ (2020), lays a solid foundation for the engineering of
entanglement on an optical-clock transition. The utilization of expansive 2D
arrays and close proximities in this context is crucial for utilizing limited-
range Rydberg interactions. This approach enables for the study of larger
samples with enhanced interconnections, more potent interactions, and
therefore, increased entanglement. Despite the exponential decline in many-
body entanglement due to single-particle decoherence, the coherence times
demonstrated in the experiment show remarkable potential for achieving a
highly precise entangled optical clock. This clock may operate with tens of
atoms and interrogation durations lasting several seconds, making it suitable
for metrological applications. Furthermore, by incorporating precise control
and detection of individual particles along with a diverse set of tunable
entangling interactions, the experimental generation of spin squeezing was
achieved in an optical atomic clock. It has been demonstrated that such a
clock can be built upon a programmable array of interacting optical qubits
Eckner  _et al._ (2023), see Fig. 33. In subarrays of $16$ atoms and $70$
atoms, spin squeezed states with $\xi_{R}^{2}=-3.8$ dB and $\xi_{R}^{2}=-3.4$
dB have been prepared, respectively. This enables the performance of a
synchronous clock comparison with a stability that surpasses the SQL by up to
$3.5$ dB, which paves an avenue to entanglement-enhanced optical clocks.

###  V.2 Magnetometers

The precise measurement of magnetic fields with high sensitivity is a critical
requirement in various domains, including physics Berglund  _et al._ (1995,
1995); Mamin  _et al._ (2007); Díaz-Michelena (2009), biology Taylor and Bowen
(2016); Thiel  _et al._ (2016), and medical applications Romalis and Dang
(2011); Carlton  _et al._ (2010); Yen  _et al._ (2004), among others. For
instance, it is essential for detecting geomagnetic anomalies, studying
magnetic fields in space, and mapping electric and magnetic fields generated
in the brain for bio-magnetic measurements. Quantum magnetometry uses the
principles of quantum mechanics to measure magnetic fields with high precision
and sensitivity. Based upon the interaction between quantum systems and
external magnetic fields, various quantum systems including atoms,
superconducting quantum interference devices, and nitrogen-vacancy centers
have been employed to detect magnetic fields. The atomic magnetometers mainly
include optical Faraday magnetometers Budker and Romalis (2007); Davis,
Bentsen, and Schleier-Smith (2016); Koschorreck  _et al._ (2010); Wasilewski
_et al._ (2010); Wilson  _et al._ (2019); Troullinou, Lucivero, and Mitchell
(2023), coherent-population-trapping magnetometers Turner  _et al._ (2022);
Wu, Turner, and Wang (2021); Bevilacqua  _et al._ (2005); Hu  _et al._ (2014);
Andryushkov, Radnatarov, and Kobtsev (2022); Fang  _et al._ (2021), spin-
exchange relaxation-free magnetometers Shah and Romalis (2009); Savukov
(2017); Ledbetter  _et al._ (2008); Karaulanov, Savukov, and Kim (2016); Liang
_et al._ (2022); Zhang  _et al._ (2019a); Hong  _et al._ (2021), and cold-atom
magnetometers Koschorreck  _et al._ (2010); Wasilewski  _et al._ (2010);
Muessel  _et al._ (2014); Ockeloen  _et al._ (2013). Below, we will highlight
the progresses in entanglement-enhanced magnetometry.

Optical Faraday magnetometers operate by measuring the polarization rotation
of light as it traverses a group of atoms exposed to a magnetic field Budker
and Romalis (2007). Generally, the device consists of $N$ spin-$F$ atoms
initially optically pumped into a fully polarized state (pointing along the
$y$ axis, for instance), such that the collective spin of the ensemble has
length $F_{N}$. A weak magnetic field along the $x$ axis causes the precession
of the collective spin in the $y-z$ plane at a rate $g\gamma B$, where $g$ is
the gyromagnetic ratio, $\gamma$ is the Bohr magneton, and $B$ is the magnetic
field strength. Polarization of light propagating along $z$ will be rotated
proportional to the collective atomic spin $J_{z}$ due to the Faraday effect.
Thus, the atomic spin precession is detected by measuring transmission or
rotation of the polarized probe light, as show in Fig. 34 (a). The rotation of
the polarized probe light are described via the Stokes parameters $S_{x}$,
$S_{y}$, $S_{z}$, and the detected signal is Wolfgramm  _et al._ (2010)

$\displaystyle S_{y}^{\text{out}}=S_{y}^{\text{in}}+S_{x}(\nu B_{z}+\alpha
J_{z})l$ (125)

where $\nu$ is the Verdet constant of the vapor, $B$ is the magnetic field,
$\alpha$ is proportional to the vector component of the atomic polarizability,
and $l$ is the length of the medium. Quantum mechanics sets fundamental limits
on the best sensitivity that can be achieved in a magnetic-field measurement.
According to Eq. (125), the sensitivity of this magnetometers is fundamentally
limited by quantum noise main in the form of photon shot noise presented in
$S_{y}^{\text{in}}$ and atomic projection noise presented in $J_{z}$. It is
obvious that one can realize the entanglement-enhanced magnetic-field
measurement via using a squeezed state in $S_{y}$ as an input.

Figure 34:  Optical Faraday magnetometers. (a) The schematic of an all-optical
atomic magnetometer. Pump light polarizes the atoms, atomic evolves in the
magnetic field, and the resultant state of the atoms is detected by the probe
light. Figure adapted from Ref. Budker and Romalis (2007). (b) Experimental
setup of optical atomic magnetometer with optical parametric oscillator. Rb
cell, rubidium vapor cell with magnetic coil and magnetic shielding; OPO,
optical parametric oscillator; PPKTP, phase-matched nonlinear crystal; LO,
local oscillator beam; PBS, polarizing beam splitter; HWP, half-wave plate;
SMF, single-mode fiber; PD, photodiode. Figure adapted from Ref. Davis,
Bentsen, and Schleier-Smith (2016). (c) Faraday rotation measurement. The
black curve shows the sensitivity with a polarized (but not squeezed) probe.
The green line depicts the sensitivity with a polarization-squeezing probe. A
zoomed view around the calibration peak at $120$ kHz is shown in the inset.
Figure adapted from Ref. Davis, Bentsen, and Schleier-Smith (2016). (d)
Experimental setup of optical atomic magnetometer with atomic squeezer. The
squeezer prepares an optical field with reduced noise properties which is used
as a probe for the magnetometer. SMPM fiber: single-mode polarizationb
maintaining fiber; $\lambda/2$: half-wave plate; PhR: phase-retarding wave
plate; PBS: polarizing beam splitter; GP: Glan-laser polarizer; BPD: balanced
photodetector. Insets show the polarization of the squeezed-vacuum field and
the laser field before the magnetometer cell (a) and right before the last PBS
(b). Figure adapted from Ref. Horrom  _et al._ (2012).

In experiments, the improvement in magnetic field sensing via using
polarization-squeezed probe lights has been demonstrated with Rb atoms Davis,
Bentsen, and Schleier-Smith (2016); Horrom  _et al._ (2012). By using off-
resonant, polarization-squeezed probe light, the sensitivity of the
magnetometer is improved by $3.2$ dB. The magnetometer consists of a source of
polarization squeezed light, a rubidium vapor cell at room temperature, and a
shot-noise-limited polarimeter. The experimental setup is shown schematically
in Fig 34 (b). Firstly, the squeezed vacuum state is produced by pumping a
sub-threshold optical parametric oscillator (OPO). The nonlinear medium in the
OPO is used to polarize the squeezed vacuum. Then, the polarization-squeezed
light is sent through a atomic cell at room temperature. At last, the optical
rotation is detected by a shot-noise-limited polarimeter. The sensitivity of
the Faraday rotation measurement is shown in Fig. 34 (c). Moreover, one can
also use an atomic squeezer to generate the squeezed probe light, which is
based on the polarization self-rotation (PSR) effect Horrom  _et al._ (2012).
The setup is depicted in Fig. 34 (d), the atomic squeezer can generate about
$2$ dB of noise reduction, and the magnetometer with sensitivities close to
$1\text{pT}/\sqrt{\text{Hz}}$. More recently, by using squeezed probe light
and evasion of measurement backaction, one can enhance the sensitivity and
measurement bandwidth of the magnetometer at sensitivity-optimal atom number
density Troullinou, Lucivero, and Mitchell (2023).

Cold-atom magnetometers generally operate based on the measurement of the
collective spin of an ensemble of atoms after a Ramsey interferometry. The
device consists of $N$ two-mode particles (single-particle states) and the two
modes are labelled as $\left|{a}\right\rangle$ and $\left|{b}\right\rangle$.
According to the description of the Ramsey interferometry in Sec. II, the
magnetic field to be estimated is encoded in the quantum system during the
interrogation process. After the beam splitter, the information of the
magnetic field can be obtained via measuring the collective spin. For atomic
ensembles, trapped BECs are particularly well suited for sensing applications
requiring both high sensitivity and spatial resolution, taking advantage of
their small size, high degree of coherence, and the availability of
sophisticated techniques for precise controlling of motional degrees of
freedom Muessel  _et al._ (2014). Recently, Ramsey interferometry using spin-
squeezed states, which generated via OAT dynamics with BEC Muessel  _et al._
(2014); Gross  _et al._ (2010), has applied to sense magnetic fields.

Figure 35:  Entanglement-enhanced magnetic field sensing with an atom chip.
(a) Experimental setup. Central region of the atom chip is the atomic probe
and the scanning trajectory. The probe is used to measure the magnetic near-
field potential generated by an on-chip microwave guide. (b) Experimental
sequence of the scanning probe interferometer, showing Rabi (red) and on-chip
microwave (blue) pulses and the trap position (purple). Spheres show the
Wigner function of the collective spin state at different stages of the Ramsey
experiment. (c) Measured phase shift $\langle\varphi\rangle$ induced by a
microwave near-field pulse as a function of the atom-surface distance. Figure
adapted from Ref. Ockeloen  _et al._ (2013).

With small BECs on an atom chip, one can overcome the SQL by $4$ dB and yield
a microwave magnetic field sensitivity of $77$ pT via using the spin-squeezed
state as input Ockeloen  _et al._ (2013). The experimental schematic of the
scanning probe interferometer is shown in Fig. 35. Firstly, a spin coherent
state on the equator of the generalized Bloch sphere is created via a $\pi/2$
pulse. Then, applying OAT Hamiltonian
$\hat{H}_{\text{OAT}}=\chi\hat{J}_{z}^{2}$ to generate spin squeezed state.
Next, applying a suitable rotation around the $J_{x}$ axis to align the anti-
squeezed quadrature with the equator. A full Ramsey interferometer sequence
consisting of a $\pi/2$ pulse to make it maximally phase sensitive, a free
evolution with time $T_{R}$ for phase accumulation $\phi\propto BT_{R}$, and a
final $\pi/2$ pulse mapping $\phi$ onto the population difference. The
measurements as a function of the atom-surface distance is shown in Fig. 35
(c). For all positions, the interferometer performs well below the SQL, and
the microwave field sensitivity approach to $\Delta B=2.4\mu T$.

Figure 36:  Quantum-enhanced magnetometry with BECs in an optical lattice. (a)
Independent squeezing of $25$ binary BECs in a 1D lattice. (b) The implemented
experimental sequence for entanglement-enhanced magnetometer, which are
displayed on generalized Bloch sphere. (c) Ramsey fringes for left and right
parts of the full sample. (d) Magnetic field sensitivity of the Ramsey
magnetometer versus interrogation time. Figure adapted from Ref. Muessel  _et
al._ (2014).

With BECs in an optical lattice, one can achieve a quantum-enhanced single-
shot sensitivity of $310(47)$ pT for a static magnetic field in a probe volume
as small as $90~{}\mu m^{3}$ Muessel  _et al._ (2014). The experimental
schematic and results of the entanglement-enhanced magnetometer are shown in
Fig. 36. Firstly, generating the spin squeezed state in the levels
$\left|{a}\right\rangle$ and $\left|{b}\right\rangle$ under OAT Hamiltonian
$\hat{H}_{\text{OAT}}=\chi\hat{J}_{z}^{2}$, the state is rotated to achieve
the maximum phase sensitivity. Subsequently, a microwave $\pi$ pulse transfers
the squeezing to a magnetic-field-sensitive hyperfine transition. After
magnetic field-dependent phase evolution with a hold time, swapping the state
back to the original level to readout. During this sequence, the state
acquires a phase $\phi\propto B$. At last, a Ramsey fringe is obtained by a
final $\pi/2$ pulse, as shown in Fig. 36 (d). Magnetic field sensitivity of
the Ramsey magnetometer versus interrogation time is shown in Fig. 36 (c), the
entanglement-enhanced sensitivity can be maintained up to interrogation time
of $342$ ms. The array configuration is particularly well suited for
differential measurements and magnetic-field gradiometry.

Figure 37:  Squeezing-enhanced magnetic field sensing with cold atomic
ensembles. (a) Experimental geometry. PD: photodiode; L: lens; WP: wave plate;
BS: beam splitter; PBS: polarizing beam splitter. (b) Measurement sequence for
generating spin squeezed state. (c) Entanglement-enhanced field measurement.
(d) Log-log plot of measurement sensitivity with an input spin coherent state
(blue circles) and spin squeezed state (red diamonds). Figure adapted from
Ref. Sewell  _et al._ (2012).

Using synthesized QND measurements in an ensemble of laser-cooled spin-$1$ Rb
atoms, the spin squeezed state has been generated and employed to perform
high-precision magnetic measurement Sewell  _et al._ (2012). Based upon a
Ramsey sequence, the squeezing-enhanced field measurement in a volume of
$V=3.7\times 10^{-6}$ $\textrm{cm}^{3}$ can achieve a sensitivity of $\Delta
B=105fT\sqrt{\textrm{cm}^{3}\textrm{Hz}}$ in single-shot measurements. The
experimental geometry is shown in Fig. 37 (a), an ensemble of laser cooled
87Rb atoms in the ground state. Then the spin squeezed state is generated via
utilizing the pulses sequence, as shown in Fig. 37 (b). These pulses are used
in pairs to synthesise the QND measurement, or singly for the alignment-to-
orientation conversion measurement. The Ramsey sequence for field measurement
is shown in Fig. 37 (c). Firstly, using synthesized QND measurements to
prepares a spin squeezed state. Then, the state is allowed to evolve for a
time due to Zeeman shifts. Finally, a pulse is used for measurement, giving an
integrated signal proportional to the Zeeman shift. The measurement
sensitivity with an input spin coherent state and spin squeezed state is shown
in Fig. 37 (d).

In recent, through combining the techniques of optical pumping and continuous
QND measurement, a sustained spin squeezed state with $4\times 10^{10}$ hot
atoms has been achieved Duan  _et al._ (2024). The steady spin squeezing with
$\xi^{2}_{R}={-3.23\pm 0.24}$ dB is prepared by applying the prediction and
retrodiction QND protocol and can be maintained for about one day. The
entanglement enhancement due to the steady spin squeezing is ultilized by the
atomic magnetometer. The system can be employed to track different types of
continuous time-fluctuating magnetic fields, in which deep learning models are
constructed to decode the measurement records from the optical signals. This
work represents an important progress towards using long-lived quantum
entanglement resources in realistic sensors.

In principle, utilizing well-developed Ramsey techniques, one can detect DC
magnetic fields with ultrahigh sensitivity. While for measurement of AC
magnetic fields, just using Ramsey techniques is insufficient, and various
modulations need to be employed. It has been demonstrated that dynamical
decoupling method, one of the well-known quantum control techniques, provides
an excellent tool for measuring AC magnetic fields de Lange  _et al._ (2010);
Biercuk  _et al._ (2009); Choi, Yao, and Lukin (2017); Maurer  _et al._
(2012); Jiang and Imambekov (2011); Kuo and Lidar (2011); Schmitt  _et al._
(2017); Kotler  _et al._ (2011); Shaniv and Ozeri (2017); Maze  _et al._
(2008); Boss  _et al._ (2017). In particular, by combining many-body Ramsey
interferometry and rapid periodic $\pi$ pulses, one can estimate DC and AC
magnetic fields simultaneously. Meanwhile, the measurement precisions of DC
and AC components can both exhibit Heisenberg scaling simultaneously for
entangled particles in GHZ state Zhuang, Huang, and Lee (2020). Moreover, the
dynamical decoupling method can be used to realize quantum lock-in measurement
for high-precision magnetometers Boss  _et al._ (2017); Kotler  _et al._
(2011); Boss  _et al._ (2017) and force detector Shaniv and Ozeri (2017).
Utilizing many-body entanglement, one can realize many-body quantum lock-in
amplifiers, measuring the AC magnetic field with Heisenberg-limited scaling
even under a strongly noisy background Zhuang, Huang, and Lee (2021).

###  V.3 Gravimeters

Quantum gravimeters make use of quantum systems, such as ultracold atoms or
molecules, to probe the gravity and achieve high sensitivity via using quantum
resources. The first-generation of quantum atomic gravimeter Kasevich and Chu
(1991); Peters, Chung, and Chu (1999) was developed in 1990s. With the
unprecedented precision offered by quantum resources, atomic gravimeters are
capable of detecting extremely small changes in the gravitational field and
provide broad applications in geophysics, geodesy, metrology, and exploration
of fundamental physics. Below we will introduce the progresses in
entanglement-enhanced atomic gravimeters.

The basic principle of a quantum gravimeter involves the interferometric
measurement of the phase shift induced by gravity on a matter wave Asenbaum
_et al._ (2017); Hardman  _et al._ (2016). As an example, Fig. 38 shows a
schematic of quantum gravimetry on an atom chip Abend  _et al._ (2016).
Different from atomic clocks and magnetometers, quantum gravimeters are a type
of inertial sensors and they generally require distinct spatial paths.
Generally, quantum gravimeters are implemented via standard Mach-Zehnder
interferometers Peters, Chung, and Chu (2001); Kritsotakis  _et al._ (2018),
in which beam splitters ($\pi/2$ pulses) and mirrors ($\pi$ pulses) are
achieved via Raman transitions. The Raman transitions are realized with two
counterpropagating lasers of wave vector $k_{L}$, which coherently couple two
internal states. As a result of momentum conservation, a transition between
the two internal states will impart a momentum of $2\hbar k_{L}$ to an atom,
giving the momentum separation needed for gravity measurement. The phase of an
interferometer is related the gravity acceleration $g$ as $\Phi=k_{eff}\cdot
gT^{2}$ with the effective wave vector $k_{eff}$ and the evolution time $T$.

Figure 38:  Quantum gravimetry on an atom chip. (a) Schematic diagram of atom
interferometry for gravity measurement. (b) Pulse sequence of atom
interferometry. (c) Relaunch mechanism using Bloch oscillation and double
Bragg mechanism to increase the sensitivity. Figure adapted from Ref. Abend
_et al._ (2016).

To improve the precision of quantum gravimeters, in addition to large momentum
transfer Asenbaum  _et al._ (2017) via Bloch oscillations or double Bragg
diffraction, multiparticle entangled momentum states Shankar  _et al._ (2019);
Anders  _et al._ (2021); Greve  _et al._ (2022); Ding  _et al._ (2015) have
been introduced. It has been demonstrated that one can increase the
interferometric area via using large momentum transfer Lévèque  _et al._
(2009); Clade  _et al._ (2009); Malossi  _et al._ (2010); wey Chiow  _et al._
(2011); Altin  _et al._ (2013); Ahlers  _et al._ (2016). This allows to
improve the sensitivity by increasing the effective separation between the two
interfering atomic samples. However, there may appear extra phase noises
induced by the effective multi-photon process for achieving large momentum
transfer. Quantum entanglement offers another promising route to improve their
measurement precision. Using spin squeezed states, the sensitivity can be
given as

$\Delta g=\frac{\xi_{R}}{\sqrt{N}k_{eff}T^{2}}.$ (126)

where $N$ is the number of atoms, $\xi_{R}$ is the squeezing parameter. In an
atom interferometer with entangled momentum states Anders  _et al._ (2021), a
spin squeezing of –3.1 dB has been achieved in experiments. Thus, quantum-
enhanced atomic inertial sensors in a more matured form should be promising in
the near future Greve  _et al._ (2022).

With currently available experimental techniques, a theoretical study Szigeti
_et al._ (2020) shows that spin squeezed ultracold atoms can improve the
sensitivity up to 14 dB beyond the SQL. As shown in Fig. 39 (b), before the
standard Mach-Zehnder interference process, an initial state preparation
generating spin squeezing via the interatomic interactions during the
expansion period is introduced. Unlike the trapped schemes, where interatomic
collisions cause unwanted multimode dynamics that make it difficult to match
the two modes upon recombination Haine  _et al._ (2014), a BEC’s spatial mode
is almost perfectly preserved under free expansion, even for large atom
numbers and collisional energies. Furthermore, since the collision energy is
converted to kinetic energy during expansion, the interatomic interactions
effectively “switch off” after $\sim 10$ ms, minimizing their effect during
most of the interferometer sequence. This scheme allows high-precision
gravimetry up to a factor of five below the SQL and is robust to a range of
experimental imperfections.

Figure 39:  Entanglement-enhanced gravity measurement with BEC. (a) Space-time
diagram illustrating gravimetry with a BEC attaining the SQL. Unwanted
interatomic interactions are reduced by freely expanding the BEC for duration
$T_{\textrm{exp}}$. A $\pi/2-\pi-\pi/2$ Raman pulse sequence then creates a
Mach-Zehnder interferometer of interrogation time $T$. The two interferometer
modes correspond to internal states $\left|{1}\right\rangle$ (red) and
$\left|{2}\right\rangle$ (blue) with momentum separation. (b) Quantum-enhanced
ultracold-atom gravimetry. During initial expansion duration, the BEC’s
interatomic interactions generate spin squeezing via OAT. (c) Bloch sphere
representation of state during entanglement-enhanced gravimetry. Figure
adapted from Ref. Szigeti  _et al._ (2020).

Due to its compact size, the atomic interferometer via Bloch oscillations
Ferrari  _et al._ (2006); Ivanov  _et al._ (2008); Poli  _et al._ (2011)
provides another excellent candidate for building quantum gravimeters. In such
a setup, cold atoms are held in a vertical optical lattice tilted by the
gravity. Therefore the cold atoms will undergo Bloch oscillations whose
frequency is determined by the gravity and the lattice constant. To harness
entanglement, a scheme has been developed utilizing an ensemble of ultracold
atoms in tilted spin-dependent optical lattices along the direction of gravity
Ke  _et al._ (2018). The fast coherent separation and recombination of atoms
can be realized via polarization-synthesized optical lattices. The input
atomic wave packet is coherently split into two parts by a spin-dependent
shift and a subsequent $\pi/2$ pulse. Then the two parts are held for
accumulating a relative phase related to the gravity. Lastly the two parts are
recombined for interference by a $\pi/2$ pulse and a subsequent spin-dependent
shift. The pulses not only preclude the spin-dependent energies in the
accumulated phase, but also avoid the error sources, such as dislocation of
optical lattices in the holding process. In particular, by using entangled
states as initial states, the measurement precision of such an interferometer
can beat the SQL.

However, even in laboratory-based proof-of-principle apparatus, there is still
a considerable gap to overcome in demonstrating entanglement-enhanced
gravimetry. The main challenge stems from the fact that most methods of
generating entangled atomic states do not meet the stringent requirements of
precision gravimetry. Most cold-atom gravimeters necessitate the creation and
precise control of distinct and well-isolated atomic matter-wave momentum
modes. To achieve a significant fringe contrast, it is crucial to maintain
coherence between these modes for extended periods during the interrogation
process, while also ensuring proper alignment of the modes at the output of
the interferometer.

###  V.4 Gyroscopes

High-performance gyroscopes for rotation sensing are crucial for navigation in
a wide range of air, ground, marine, and space applications. Rotation
frequency is commonly measured by using the Sagnac effect, which is the
accumulation of a rotation-dependent phase difference by two counter-
propagating waves in a revolving loop. Gyroscopes operate with either lasers