Quantum metrology in the noisy intermediate-scale quantum era

In a conventional Ramsey spectroscopy to measure the atomic frequency $\omega_{0}$, see Fig. 1(a), one chooses $N$ atoms themselves as the probe and prepares their state in $|\psi_{\text{in}}\rangle=|g\rangle^{\otimes N}$, where $|g\rangle$ is the atomic ground state. First, a $\pi/2$ microwave pulse in frequency $\omega_{L}$ and time duration $t=\pi/(2|\Delta|)$, with $\Delta=\omega_{0}-\omega_{L}$, is applied on each atom. It converts $|\psi_{\text{in}}\rangle$ into $|\psi_{1}\rangle=\big{(}{|g\rangle+|e\rangle\over\sqrt{2}}\big{)}^{\otimes N}$, where $|e\rangle$ is the excited state. Second, the microwave is switched off and the atoms experience a free evolution governed by the Hamiltonian $\hat{H}_{0}=\Delta\sum_{j}\hat{\sigma}_{j}^{\dagger}\hat{\sigma}_{j}$, with $\hat{\sigma}_{j}=|g_{j}\rangle\langle e_{j}|$. It encodes $\omega_{0}$ into the state

$$|\psi(t)\rangle=e^{-i\hat{H}_{0}t}|\psi_{1}\rangle=[(|g\rangle+e^{-i\Delta t}|e\rangle)/\sqrt{2}]^{\otimes N}.$$

(12)

Here, $\omega_{L}$ contained in $\Delta$ of $\hat{H}_{0}$ is just for the purpose of the consistency of the used rotating frame with the one in the first step. Third, a $\pi/2$ microwave pulse is switched on again, which converts $|\psi(t)\rangle$ into

$$|\psi_{\text{out}}\rangle=\big{[}\cos(\Delta t/2)|g\rangle+i\sin(\Delta t/2)|e\rangle\big{]}^{\otimes N}.$$

(13)

Finally, the excited-state population operator $\hat{O}=\hat{\sigma}_{j}^{\dagger}\hat{\sigma}_{j}$ of each atom is measured. The obtained results 1 is in a probability $P_{1}=\sin^{2}{\Delta t\over 2}$ and 0 in a probability $P_{0}=\cos^{2}{\Delta t\over 2}$. The expectation value $\langle\hat{O}\rangle=\sin^{2}{\Delta t\over 2}$, the variance $\Delta O=|\sin\Delta t|/2$, and the CFI $F_{\omega_{0}}=t^{2}$ are readily calculated. Repeating the experiment within a time duration $T$, we acquire $\upsilon=TN/t$ measurement results. Then, according to the central limit theorem, we have the uncertainty of $\hat{O}$ as

$$\delta O={\Delta O\over\sqrt{TN/t}}={|\sin\Delta t|\over 2\sqrt{TN/t}}.$$

(14)

The sensitivity of $\omega_{0}$ is evaluated via the error propagation formula as

$$\delta\omega_{0}={\delta O\over|\partial_{\omega_{0}}\langle\hat{O}\rangle|}=(NTt)^{-1/2},$$

(15)

which equals to the sensitivity evaluated from the Cramér-Rao bound (1). Since no quantum resource is used in the input state, the scaling relation of $\delta\omega_{0}$ with the atom number $N$ is just the SNL. It can be evaluated that the QFI of Eq. (13) is $\mathcal{F}_{\omega_{0}}=t^{2}$. Thus, we obtain

$$\delta\omega_{0}=(\upsilon\mathcal{F}_{\omega_{0}})^{-1/2}=(NTt)^{-1/2}.$$

(16)

Therefore, the measurement scheme used in Eq. (15) saturates the quantum Cramér-Rao bound and is optimal.

Entanglement is one of the most subtle and intriguing phenomena in the microscopic world. Its usefulness has been demonstrated in various applications such as quantum teleportation [83, 84], quantum cryptography [85, 86, 87], and quantum dense coding [85, 88, 89]. It occurs when a group of particles are generated or share spatial proximity in a way such that their state cannot be described independently by the states of the constituent particles. Measurements of physical quantities such as position, momentum, spin, and polarization performed on entangled particles can be found to be perfectly correlated. The exploitation of entanglement in quantum technologies has great potential to outperform the schemes based on classical physics. This is also true for quantum metrology.

First, the $N$ atoms are prepared in the ground state $|\psi_{\text{in}}\rangle=|g\rangle^{\otimes N}$. A $\pi/2$ microwave pulse on the first atom is applied, see Fig. 1(b), and converts the state into $|\psi_{1}\rangle=\frac{|g\rangle+|e\rangle}{\sqrt{2}}\otimes|g\rangle^{\otimes(N-1)}$. A CNOT gate, with the first atom acting as the control bit and other $N-1$ atoms acting as the target bits, is applied to the atoms. $|\psi_{1}\rangle$ becomes a Greenberger–Horne–Zeilinger (GHZ)-type entangled state

$$|\psi_{2}\rangle=(|g\rangle^{\otimes N}+|e\rangle^{\otimes N})/\sqrt{2}.$$

(17)

Second, the atoms experience a free evolution governed by the atomic Hamiltonian $\hat{H}_{0}=\Delta\sum_{j}\hat{\sigma}_{j}^{\dagger}\hat{\sigma}_{j}$ to encode $\omega_{0}$ into the probe state, which converts $|\psi_{2}\rangle$ into

$$|\psi(t)\rangle=(|g\rangle^{\otimes N}+e^{-iN\Delta t}|e\rangle^{\otimes N})/\sqrt{2}.$$

(18)

Third, another CNOT gate is applied again to convert $|\psi(t)\rangle$ into $|\psi^{\prime}(t)\rangle=\frac{|g\rangle+e^{-iN\Delta t}|e\rangle}{\sqrt{2}}\otimes|g\rangle^{\otimes(N-1)}$. Switching on the $\pi/2$ pulse again, $|\psi^{\prime}(t)\rangle$ becomes

$$|\psi_{\text{out}}\rangle=[\cos(N\Delta t/2)|g\rangle+i\sin(N\Delta t/2)|e\rangle]\otimes|g\rangle^{\otimes N-1}.$$

(19)

Measuring the excited-state population operator $\hat{O}=\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{1}$ of the first atom, we obtain the results 1 with probability $P_{1}=\sin^{2}{N\Delta t\over 2}$ and 0 with probability $P_{0}=\cos^{2}{N\Delta t\over 2}$. They lead to $\langle\hat{O}\rangle=\sin^{2}{N\Delta t\over 2}$, $\Delta O=|\sin(N\Delta t)|/2$, and $F_{\omega_{0}}=N^{2}t^{2}$. Repeating this experiment within a time duration $T$ yields $\upsilon=T/t$ sets of experimental results. Thus, according to the central limit theorem, the uncertainty of $\hat{O}$ is

$$\delta O=\frac{\Delta O}{\sqrt{T/t}}=\frac{|\sin(N\Delta t)|}{2}\sqrt{\frac{t}{T}}.$$

(20)

The error propagation formula results in

$$\delta\omega_{0}=(N^{2}Tt)^{-1/2},$$

(21)

which is called the HL [90, 91]. Equation (21) also equals the sensitivity evaluated from the Cramér-Rao bound (1). Obviously, by using entanglement, the HL has an $N^{1/2}$-times enhancement over the SNL in Eq. (15). The QFI of Eq. (18) is $\mathcal{F}_{\omega_{0}}(t)=N^{2}t^{2}$, from which the ultimate sensitivity of $\omega_{0}$ reads

$$\delta\omega_{0}=(\upsilon\mathcal{F}_{\omega_{0}})^{-1/2}=(N^{2}Tt)^{-1/2}.$$

(22)

It matches with Eq. (21). Therefore, the measurement scheme used above saturates the quantum Cramér-Rao bound and is optimal.

A wide class of quantum metrology using quantized lights as probes is based on the Mach-Zehnder interferometer [92, 93, 69, 94]. In order to measure a frequency parameter $\gamma$ of a system, we choose two modes optical fields with frequency $\omega_{0}$ as the probe. The encoding of $\gamma$ is realized by the time evolution $\hat{U}_{0}(\gamma,t)=\exp(-{i\hat{H}_{0}t})$ with

$$\hat{H}_{0}=\omega_{0}\sum_{m=1,2}\hat{a}_{m}^{\dagger}\hat{a}_{m}+\gamma\hat{a}_{2}^{\dagger}\hat{a}_{2},$$

(23)

where the first term is the Hamiltonian of the fields and the second one is the linear interaction of the second field with the system [95, 96, 97]. The evolution $\hat{U}_{0}(\gamma,t)$ accumulates a phase difference $\gamma t$ to the two fields, which is measured by the Mach-Zehnder interferometer. It has two beam splitters $\text{BS}_{i}$ ($i=1,2$) separated by the phase shifter $\hat{U}_{0}(\gamma,t)$ and two detectors $\text{D}_{i}$ (see Fig. 2). Its input-output relation reads

$$|\Psi_{\text{out}}\rangle=\hat{V}\hat{U}_{0}(\gamma,t)\hat{V}|\Psi_{\text{in}}\rangle,$$

(24)

where $\hat{V}=\exp[i\frac{\pi}{4}(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1})]$ is the action of $\text{BS}_{i}$ [94]. We are interested in the quantum superiority of $|\Psi_{\text{in}}\rangle$ in metrology subject to an explicit measurement scheme. Thus we consider Caves’s original scheme [93]; i.e., the photon difference $\hat{M}=\hat{a}^{\dagger}_{1}\hat{a}_{1}-\hat{a}^{\dagger}_{2}\hat{a}_{2}$ is measured by $\text{D}_{i}$, which is also the most general measurement in the Mach-Zehnder interferometer. We can evaluate $\langle\hat{M}\rangle=\langle\Psi_{\text{out}}|\hat{M}|\Psi_{\text{out}}\rangle$ and $\delta^{2}M=\langle\hat{M}^{2}\rangle-\langle\hat{M}\rangle^{2}$. The metrology sensitivity is obtained via

$$\delta\gamma={\delta M\over|\partial_{\gamma}\langle\hat{M}\rangle|}.$$

(25)

If the input state is a product state of a coherent state and a vacuum state, i.e., $|\Psi_{\text{in}}\rangle=\hat{D}_{\hat{a}_{1}}|0,0\rangle$, with $\hat{D}_{\hat{a}}=\exp{(\alpha\hat{a}^{\dagger}-\alpha^{\ast}\hat{a})}$, which contains a mean photon number $N=|\alpha|^{2}$, then we have $\langle\hat{M}\rangle=N\cos(\gamma t)$ and $\langle\hat{M}^{2}\rangle=N^{2}\cos^{2}(\gamma t)+N$. It can be readily evaluated that

$$\text{min}\delta\gamma=1/(t\sqrt{N})$$

(26)

when $\gamma t=m\pi/2$, with $m$ being a positive integer. In this case, the QFI reads $\mathcal{F}_{\gamma}=Nt^{2}$. The coincidence of the metrology sensitivity from $\mathcal{F}_{\gamma}$ with Eq. (26) indicates that the $\hat{M}$-measurement is optimal.

If the input state is a product state of a squeezed vacuum state and a coherent state, i.e., $|\Psi_{\text{in}}\rangle=\hat{D}_{\hat{a}_{1}}\hat{S}_{\hat{a}_{2}}|0,0\rangle$, with $\hat{S}_{\hat{a}}=\exp[(\xi^{*}\hat{a}^{2}-\xi\hat{a}^{\dagger 2}/2)]$ and $\xi=re^{i\phi}$, its total photon number is $N=|\alpha|^{2}+\sinh^{2}r$, which contains the ratio $\iota\equiv\sinh^{2}r/N$ from the squeezed mode and can be regarded as the quantum resource of the scheme. To $|\Psi_{\text{out}}\rangle$, we have $\langle\hat{M}\rangle=[\sinh^{2}r-|\alpha|^{2}]\cos(\gamma t)$ and

	$\displaystyle\delta M$	$\displaystyle=$	$\displaystyle\{\cos^{2}(\gamma t)[\left|\alpha\right|^{2}+2\sinh^{2}r\cosh^{2}r]+\sin^{2}(\gamma t)$		(27)
			$\displaystyle\times[\left|\alpha\cosh r-\alpha^{\ast}\sinh re^{i\phi}\right|^{2}+\sinh^{2}r]\}^{\frac{1}{2}}.$

Then the best precision of estimating $\gamma$ is obtained as

$$\min\delta\gamma=\frac{[(1-\iota)e^{-2r}+\iota]^{\frac{1}{2}}}{t\sqrt{N}|1-2\iota|}$$

(28)

when $\phi=2\varphi$ and $\gamma t=(2m+1)\pi/2$ for $m\in\mathbb{Z}$. If the squeezing is absent, then $\min\delta\gamma|_{\iota=0}=(tN_{0}^{1/2})^{-1}$ is just the SNL. For $\iota\neq 0$, using $e^{-2r}\simeq 1/(4\sinh^{2}r)$ for $N\gg 1$ and optimizing $\iota$, we have

$$\min\delta\gamma|_{\iota=(2\sqrt{N})^{-1}}=(tN^{3/4})^{-1},$$

(29)

which is the Zeno limit [98, 99, 100]. It beats the SNL and manifests the superiority of squeezing in metrology. The QFI of $|\Psi_{\text{out}}\rangle$ in this case is derived to be

$$\mathcal{F}_{\gamma}=t^{2}N\iota[4N(1-\iota)+1].$$

(30)

Optimizing $\iota$, we obtain the best sensitivity $\min\delta\gamma|_{\iota=1/2}=(tN)^{-1}$, which is smaller than the Zeno limit in Eq. (29). This means that the above measurement scheme is not optimal. Reference [101] revealed that the measurement of the photon numbers of both of the two output ports can saturate the quantum Cramér-Rao bound. Benefiting from the quantum-enhanced sensitivity, the squeezing has been used in gravitational-wave observatory [16, 34, 35].

If the input state is two-mode squeezed vacuum state $|\Psi_{\text{in}}\rangle=\hat{S}|0,0\rangle$, with $\hat{S}=e^{r(\hat{a}_{1}\hat{a}_{2}-\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger})}$, whose total photon number is $N=2\sinh^{2}r$. In this case, we choose to measure the parity operator $\hat{\Pi}=e^{i\pi\hat{a}^{\dagger}_{1}\hat{a}_{1}}$. Substituting $|\Psi_{\text{in}}\rangle$ into Eq. (24), we calculate $\langle\hat{\Pi}\rangle=\langle\Psi_{\text{out}}|\hat{\Pi}|\Psi_{\text{out}}\rangle=[1+N(2+N)\cos^{2}(\gamma t)]^{-1/2}$ and $\delta\Pi=(1-\langle\hat{\Pi}\rangle^{2})^{1/2}$, where $\hat{\Pi}^{2}=1$ has been used. Then the metrology sensitivity is evaluated via the error propagation formula $\delta\gamma=\frac{\delta\Pi}{|\partial_{\gamma}\langle\hat{\Pi}\rangle|}$ as

$$\min\delta\gamma=\big{[}2t\sqrt{N(2+N)}\big{]}^{-1},$$

(31)

when $\gamma t=(2n+1)\pi/2$ with $n\in\mathbb{Z}$. It is remarkable to find that the best sensitivity is even smaller than the HL $\Delta\gamma\propto(tN)^{-1}$, which reflects the quantum superiority of the used squeezing and measured observable. The QFI of $|\Psi_{\text{out}}\rangle$ equals to

$$\mathcal{F}_{\gamma}=t^{2}N(N+2)\sin^{2}(\gamma t).$$

(32)

The sensitivity obtained from $\mathcal{F}_{\gamma}$ matches well with Eq. (31). It verifies that the parity-measurement scheme saturates the quantum Cramér-Rao bound and is optimal. We call such a sensitivity surpassing the HL the super-HL [96, 102, 103, 104].

High-performance gyroscopes for rotation sensing are of pivotal significance for navigation in many types of air, ground, marine, and space applications. Based on the Sagnac effect, i.e., two counter-propagating waves in a rotating loop accumulate a rotation-dependent phase difference, gyroscopes have been realized in optical systems [105, 106, 107, 108, 109, 110]. The records for precision and stability of commercial gyroscopes are held by optical gyroscopes [111, 112]. However, their precision, which is proportional to the surface area enclosed by the optical path [113], is still limited by the classical SNL. It dramatically constrains their practical application and further performance improvement. A quantum gyroscope based on the Sagnac interferometer can be established as follows.

We choose two beams of quantized optical fields as the quantum probe. They propagating in opposite directions are input into a 50:50 beam splitter and split into clockwise and counter-clockwise propagating beams (see Fig. 3). The setup rotates with an angular velocity $\Omega$ about the axis perpendicular to its plane. Thus the two beams accumulate a phase difference $\Delta\theta={\mathcal{N}4\pi kR^{2}\Omega/c}$ when they re-encounter the beam splitter after $\mathcal{N}$ rounds of propagation in the circular path [116]. Here $k$ is the wave vector, $c$ is the speed of light, and $R$ is the radius of the quantum gyroscope. Remembering the standing-wave condition $kR=n$ ($n\in\mathbb{Z}$) of the optical fields propagating along the circular path and defining $\Delta t\equiv\mathcal{N}2\pi R/c$, we have $\Delta\omega\equiv\Delta\theta/\Delta t=2n\Omega$. Therefore, the quantum gyroscope can be equivalently treated as two counter-propagating optical fields with a frequency difference $\Delta\omega$ along the circular path. For concreteness, we choose the basic mode $n=1$. Then the optical fields in the quantum gyroscope can be quantum mechanically described by [117]

$$\hat{H}_{S}=\omega_{0}\sum_{l=1,2}\hat{a}_{l}^{\dagger}\hat{a}_{l}+\Omega(\hat{a}_{1}^{\dagger}\hat{a}_{1}-\hat{a}_{2}^{\dagger}\hat{a}_{2}),$$

(33)

where $\hat{a}_{l}$ is the annihilation operator of the $l$th field with frequency $\omega_{0}$. The optical fields couple to the beam splitter twice and output in the state $|\Psi_{\text{out}}\rangle=\hat{V}\hat{U}_{0}(\Omega,t)\hat{V}|\Psi_{\text{in}}\rangle$, where $\hat{U}_{0}(\Omega,t)=\exp(-i\hat{H}_{S}t)$ is the evolution operator of the fields and $\hat{V}=\exp[i\frac{\pi}{4}(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1})]$ describes the action of the beam splitter. Thus the angular velocity $\Omega$ is encoded into the state $|\Psi_{\text{out}}\rangle$ of the optical probe via the unitary evolution.

To exhibit the quantum superiority, we employ two-mode squeezed vacuum state as the input state $|\Psi_{\text{in}}\rangle=\hat{S}|0,0\rangle$, where $\hat{S}=\exp[r(\hat{a}_{1}\hat{a}_{2}-\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger})]$ is the squeeze operator. The total photon number of this input state is $N=2\sinh^{2}r$, which is the quantum resource of our scheme. The parity operator $\hat{\Pi}=\exp(i\pi\hat{a}_{1}^{\dagger}\hat{a}_{1})$ is measured at the output port [94]. Similar to the last example in Sec. III.2, we have

$$\min\delta\Omega=\big{[}2t\sqrt{N(2+N)}\big{]}^{-1},$$

(34)

when $\Omega t=(2n+1)\pi/4$ with $n\in\mathbb{Z}$. Super-HL sensing to the angular velocity is achieved. It can be verified that this measurement scheme saturates the quantum Cramér-Rao bound governed by the QFI.

We summarize the comparison of different scaling relations of the metrology error $\delta x$ of a physical quantity $x$ with respect to the number of quantum resources contained in different input states in the metrology schemes based on Mach-Zehnder interferometer and Ramsey spectroscopy in Table 1.

Spin-squeezed states are a class of collective-spin states having squeezed spin variance along a certain direction, at the cost of anti-squeezed variance along an orthogonal direction [8, 3, 9, 118]. Spin squeezing is one of the most successful quantum resources that can witness large-scale quantum-favored beating to the SNL in the NISQ era.

Consider an ensemble of $N$ two-level atoms or spin-$1/2$ particles in a quantum state $|\Psi\rangle$. Its collective spin operator is defined as $\hat{\bf J}=\sum_{j=1}^{N}\hat{\boldsymbol{\sigma}}_{j}/2$. The mean-spin direction is ${\bf n}_{0}=\langle\hat{\bf J}\rangle/|\langle\hat{\bf J}\rangle|$, with $\langle\hat{\bf J}\rangle=\langle\Psi|\hat{\bf J}|\Psi\rangle$. To properly use this kind of collective spin state, we design a generalized Ramsey spectroscopy as follows [118]. First, we perform a rotation with angle $\vartheta$ along the axis ${\boldsymbol{\eta}}$ such that ${\boldsymbol{\eta}}\cdot{\bf n}_{0}=0$. Then, we estimate $\vartheta$ by measuring the operator $\hat{J}_{\perp}={\boldsymbol{\alpha}}\cdot\hat{\bf J}$, where ${\boldsymbol{\alpha}}\cdot{\bf n}_{0}={\boldsymbol{\alpha}}\cdot{\boldsymbol{\eta}}={\bf n}_{0}\cdot{\boldsymbol{\eta}}=0$. In the Heisenberg picture, the rotation converts the measured operator $\hat{J}_{\perp}$ into

$$\hat{J}^{\text{out}}_{\perp}=e^{i\vartheta\hat{J}_{\eta}}\hat{J}_{\perp}e^{-i\vartheta\hat{J}_{\eta}}=\cos\vartheta\hat{J}_{\perp}-\sin\vartheta\hat{J}_{n_{0}}.$$

(35)

It follows that $\langle\hat{J}^{\text{out}}_{\perp}\rangle=-\sin\vartheta\langle\hat{J}_{n_{0}}\rangle$ and

	$\displaystyle\delta^{2}J^{\text{out}}_{\perp}$	$\displaystyle=$	$\displaystyle\cos^{2}\vartheta\delta^{2}J_{\perp}+\sin^{2}\vartheta\delta^{2}\hat{J}_{n_{0}}$		(36)
			$\displaystyle-{\sin(2\vartheta)\over 2}\langle[\hat{J}_{\perp},\hat{J}_{n_{0}}]_{+}\rangle.$

The best phase sensitivity is calculated via the error propagation formula as

$$\delta\vartheta=\min_{\vartheta}{\delta J^{\text{out}}_{\perp}\over|\cos\vartheta\langle\hat{J}_{n_{0}}\rangle|}={\delta J_{\perp}\over|\langle\hat{\bf J}\rangle|},$$

(37)

where $|\langle\hat{\bf J}\rangle|=|\langle\hat{J}_{n_{0}}\rangle{\bf n}_{0}|=|\langle\hat{J}_{n_{0}}\rangle|$ has been used.

Consider the phase sensitivity obtained in the generalized Ramsey spectroscopy by using the so-called spin-coherent state defined as $|\theta,\phi\rangle={e^{\zeta\hat{J}_{-}}\over(1+|\zeta|^{2})^{j}}|j,j\rangle$, where $\zeta=e^{i\phi}\tan{\theta\over 2}$ and $|j,j\rangle$ is the common eigen state of $|\hat{J}^{2},\hat{J}_{z}\rangle$ with $j=N/2$. For this state, the mean-spin direction is ${\bf n}_{0}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ and the expectation $|\langle\hat{\bf J}\rangle|=N/2$. After some algebra, the variance $\delta J_{\perp}=\sqrt{N}/2$ is evaluated. Therefore, we have

$$\delta\vartheta_{\text{scs}}=N^{-1/2},$$

(38)

which is just the SNL. The squeezing parameter of a spin-squeezed state is defined as the ratio of the phase sensitivities obtained via the spin-squeezed state and the spin-coherent state [8], i.e.,

$$\xi_{R}\equiv{\delta\vartheta\over\delta\vartheta_{\text{scs}}}=\sqrt{N}{(\delta J_{\perp})_{\text{min}}\over|\langle\hat{\bf J}\rangle|}.$$

(39)

When $\xi_{R}<1$, the state is said to be spin squeezed, by which the obtained metrology sensitivity of measuring $\vartheta$ in the generalized Ramsey spectroscopy beats the SNL $\xi_{R}$ times. This requires that the state has a minimal spin fluctuation in the plane orthogonal to the mean-spin direction smaller than the one of the spin coherent state, i.e., $\min\delta\hat{J}_{\perp}<\sqrt{N}/2$. $\xi_{R}$ gives the quantum gain of the frequency sensitivity in the generalized Ramsey spectroscopy obtained by the spin-squeezed state over the SNL obtained by the spin coherent state. It has been revealed that a collective-spin state with spin squeezing contains genuine many-body entanglement [119], which is the intrinsic reason for the superiority of the spin squeezed state in enhancing metrology sensitivity.

First, we consider the superposition of two Dicke states [118]

$$|\Psi_{\text{SDS}}\rangle=(|j,{1/2}\rangle+|j,-{1/2}\rangle)/\sqrt{2},$$

(40)

where $j=N/2$ with $N$ being odd. One can evaluated $\langle\hat{\bf J}\rangle=({j+1/2\over 2},0,0)$. Thus $\hat{J}_{\perp}=\cos x\hat{J}_{y}+\sin x\hat{J}_{z}$, with $x\in[0,2\pi]$. It is straightforward to derive that

	$\displaystyle\min_{x}\delta^{2}J_{\perp}$	$\displaystyle=$	$\displaystyle\{\langle(\hat{J}_{y}^{2}+\hat{J}_{z}^{2})\rangle-[\langle(\hat{J}_{y}^{2}-\hat{J}_{z}^{2})\rangle^{2}$		(41)
			$\displaystyle+\langle[\hat{J}_{y},\hat{J}_{z}]_{+}\rangle^{2}]^{1/2}\}/2.$

According to $\langle[\hat{J}_{y},\hat{J}_{z}]_{+}\rangle=0$ for $|\Psi_{\text{SDS}}\rangle$, we have $(\delta^{2}J_{\perp})_{\text{min}}=\langle\hat{J}_{z}^{2}\rangle=1/4$. Substituting it into Eq. (39), we obtain

$$(\xi_{R})_{\text{SDS}}=2\sqrt{N}/(N+1)\simeq 2N^{-1/2}.$$

(42)

It indicates that the HL of the frequency sensitivity could be obtained if $|\Psi_{\text{SDS}}\rangle$ is used in the generalized Ramsey spectroscopy.

Second, we consider the one-axis twisted state defined as [9]

$$|\Psi_{\text{OAT}}\rangle=e^{-i\mu\hat{J}_{z}^{2}/2}|\pi/2,0\rangle.$$

(43)

The expectation value of $\hat{\bf J}$ is $\langle\hat{\bf J}\rangle=(j\cos^{2j-1}(\mu/2),0,0)$. Thus $(\delta{J}_{\perp})_{\text{min}}$ has the same form as Eq. (41). Using $\langle\hat{J}_{z}^{2}\rangle=j/2$ and

	$\displaystyle\langle\hat{J}_{y}^{2}\rangle={j\over 2}[j+{1\over 2}-(j-{1\over 2})\cos^{2j-2}\mu],$		(44)
	$\displaystyle\langle[\hat{J}_{y},\hat{J}_{z}]_{+}\rangle=j(2j-1)\cos^{2j-2}(\mu/2)\sin(\mu/2),$		(45)

we obtain

$$\min_{x}\delta^{2}J_{\perp}={j\over 2}[1+{j-1/2\over 2}(A-\sqrt{A^{2}+B^{2}})],$$

(46)

where $A=1-\cos^{2j-2}\mu$ and $B=4\cos^{2j-2}(\mu/2)\sin(\mu/2)$. Under the condition $j\gg 1$ and $\mu\ll 1$, Eq. (46) is approximated as $\min_{x}\delta^{2}J_{\perp}={j\over 2}({1\over 4\alpha^{2}}+{2\beta^{2}\over 3})$, where $\alpha=j\mu/2$ and $\beta=j\mu^{2}/4$. It reaches its minimum $(\delta J_{\perp})_{\text{min}}={1\over\sqrt{2}}({j\over 3})^{1/6}$ when $\mu=24^{1/6}j^{-2/3}$. Substituting it into Eq. (39), we obtain

$$(\xi_{R})_{\text{OAT}}\propto j^{-1/3}\propto N^{-1/3}.$$

(47)

Thus a metrology sensitivity $N^{-5/6}$ would be achieved if $|\Psi_{\text{OAT}}\rangle$ is used in the generalized Ramsey spectroscopy.

Third, we consider the two-axis twisting state defined as

$$|\Psi_{\text{TAT}}\rangle=e^{-\theta(\hat{J}_{+}^{2}-\hat{J}_{-}^{2})/2}|j,-j\rangle^{\otimes N},$$

(48)

where the mean-spin direction is along the $z$-axis. Unfortunately, the two-axis twisting model cannot be solved analytically for arbitrary $N$. The numerical calculations reveal that the spin-squeezing parameter $\xi_{R}$ scales with the atom number $N$ as $(\xi_{R})_{\text{TAT}}\sim N^{-1/2}$, which leads to a sensitivity scaling as the HL $\delta\vartheta\sim N^{-1}$.

Spin-squeezed states reflect the absolute superiority of the entanglement-enhanced sensitivity in quantum metrology. The distinctive role of atomic spin squeezing in improving the sensitivity makes it valuable for the applications in quantum gyroscope [120, 121], atomic clocks [122, 123, 124, 21, 125], magnetometers [126, 127, 26, 128], and gravimetry [5].

Quantum criticality can also act as a resource to enhance the sensitivity in metrology [129, 130]. It takes advantage of criticality associated with continuous quantum phase transitions, in which the energy gap above the ground state closes in the thermodynamic limit, to realize high-precision measurements to physical quantities. Consider a Hamiltonian $\hat{H}(g)$ with the decomposition $\hat{H}(g)=\sum_{n}E_{n}(g)|\psi_{n}(g)\rangle\langle\psi_{n}(g)|$, the QFI with respect to the parameter $g$ that drives the quantum phase transition for the ground state $|\psi_{0}(g)\rangle$ reads [131, 132]

$$\mathcal{F}_{g}=4\sum_{n\neq 0}\frac{|\langle\psi_{n}(g)|\partial_{g}\hat{H}(g)|\psi_{0}(g)\rangle|^{2}}{[E_{n}(g)-E_{0}(g)]^{2}}.$$

(49)

It is clear that, if the energy gap above the ground state closes, the QFI diverges due to the vanishing denominator. This property leads to an arbitrarily high estimation precision in the thermodynamic limit. In general, the lowest excitation energy tends to zero as $(E_{1}-E_{0})\sim\xi^{-z}$ near the critical point $g_{c}$, where $z$ is the dynamical critical exponent and $\xi\sim\Lambda^{-1}|g-g_{c}|^{-\nu}$ is the correlation length. Here $\Lambda$ is the momentum cutoff determined by the inverse lattice spacing and $\nu$ is the critical exponent [133]. Combined with Eq. (49), the QFI diverges as $\mathcal{F}_{g}\sim|g-g_{c}|^{-2z\nu}$ near the critical point. A commonly used quantum critical metrology protocol is to adiabatically prepare a ground state with the parameters being near the critical point, then measure the observable to estimate the quantity that drives the occurrence of the quantum phase transition.

First, we consider applying the one-dimensional quantum transverse-field Ising model [134, 135] in quantum metrology. Its Hamiltonian reads

$$\hat{H}=-J\sum_{i=1}^{L-1}\hat{\sigma}_{i}^{x}\hat{\sigma}_{i+1}^{x}-h\sum_{i=1}^{L}\hat{\sigma}_{i}^{z},$$

(50)

where $\hat{\sigma}_{i}^{x/z}$ are Pauli operators and $L$ is the size of spin chain. Using the standard Jordan-Wigner, Fourier, and Bogoliubov transformations, Eq. (50) is rewritten as $\hat{H}=\sum_{k>0}\Lambda_{k}(\hat{\eta}_{k}^{\dagger}\hat{\eta}_{k}-1)$, where $k=(2n+1)\pi/L$, with $n=0,\cdots,L/2-1$, $\hat{\eta}_{k}$ is the fermion annihilation operator, and $\Lambda_{k}=\sqrt{(J\cos k+h)^{2}+(J\sin k)^{2}}$ is the excitation energy. It has a quantum phase transition from an ordered phase to a paramagnetic phase at the critical point $h=J$. The QFI for the parameter $J$ in the ground state is

$$\mathcal{F}_{J}=\sum_{k}\frac{(J+\frac{z_{c}}{L})^{2}\sin^{2}k}{[\frac{z_{c}^{2}}{L^{2}}+4J(J+\frac{z_{c}}{L})\cos^{2}{k\over 2}]^{2}},$$

(51)

where $z_{c}\equiv L(h-J)$ is the scaling variable. Since $\partial_{z_{c}}\mathcal{F}_{J}|_{z_{c}\rightarrow 0}=0$, $\mathcal{F}_{J}$ has a maximum at $z_{c}=0$ for all values of $L$. It scales with $L$ as the HL, i.e., $\mathcal{F}_{J}\approx\frac{L^{2}}{8J^{2}}$ in the critical region, while it scales as $L^{1}$ in the off-critical region [135, 136].