Two-parameter estimation in a matter-wave Sagnac interferometer
本站小编 Free考研考试/2022-01-02
Xu Yu1, Hong-Bin Liang1, Xiao-Guang Wang,1,21Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China 2Graduate School of China Academy of Engineering Physics, Beijing 100193, China
Abstract We explore the general characteristics of a matter-wave Sagnac interferometer in a two-parameter estimation scheme. We find that the measurement precisions of both parameters cannot reach the Heisenberg limit (HL) simultaneously when the input state is maximally entangled. Only one of the parameters’ uncertainties can approach the HL while the other is scaled by the standard quantum limit. We provide the conditions with which the measurement precision of the specific parameter can reach the HL. We also discuss and figure out the concrete expressions of the constraint conditions for saturating the quantum Cramér–Rao bound. To satisfy these constraint conditions, the evolution time has to be a series of discrete values. Additionally, we calculate the variances of the parameters through some examples under these constraint conditions. The results provided in our work show some intrinsic features of the matter-wave Sagnac interferometer for the two-parameter estimation, which can be valuable in actual experiments. Keywords：multi-parameter estimation;matter wave Sagnac interferometer;Heisenberg limit
PDF (650KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Xu Yu, Hong-Bin Liang, Xiao-Guang Wang. Two-parameter estimation in a matter-wave Sagnac interferometer. Communications in Theoretical Physics, 2020, 72(9): 095101- doi:10.1088/1572-9494/ab8a2e
1. Introduction
The matter-wave Sagnac interferometer, which is mainly constructed by an atomic gyroscope, has been widely used to measure parameters including rotations [1–6], accelerations [5–7], as well as gravitational fields [8–10] in the field of quantum metrology because of its easy implementation and high reliability. Compared to the general optical Sagnac interferometer, the matter-wave interferometer senses physical quantities through manipulating trapped atoms or ions, which would promote the resolution by orders of magnitude [11]. To enhance the sensitivity and stability, the way of manipulating the atoms would be crucial. Recently, researchers investigated an interferometer comprised of a single clock-type atom, which is trapped in the harmonic potential well [12]. In that scheme, the Sagnac phase is accumulated by a state-dependent manipulation of the atom. Later, the scheme was generalized to multi-particle scenario [13]. The authors calculated the rotation sensitivity with an ensemble of entangled atoms and found that the ultimate precision could reach the Heisenberg limit (HL). Based on that, in [14], the sensitivity of rotation was discussed with the Fisher information via a general multi-particle state. However, most of the works done based on the clock-type interferometer scheme are focused on the single-parameter estimation scheme. At the same time, in many practical applications, such as quantum imaging [15, 16], quantum tomography [17], microscopy [18], spectroscopy [19], as well as the application of the sensor networks [20–22], there would usually involve simultaneous estimation of multiple parameters. On the other hand, multi-parameter estimation tasks with optical Sagnac interferometer have been experimentally demonstrated recently [23, 24]. Therefore, to further explore the utilization potentiality of the matter-wave Sagnac interferometer in quantum metrology, it would be motivating to investigate its properties for multi-parameter estimation.
For quantum estimation with multiple parameters, the measurement precision is quantified by the variances of the estimators. The value scope of these variances can be obtained through the quantum Cramér–Rao bound (QCRB) [25–27], i.e. ${ \mathcal C }\geqslant {{ \mathcal F }}^{-1}$. Here ${ \mathcal C }$ is the covariance matrix of the estimators whose diagonal elements are equal to the variances of the measured parameters with unbiased estimators and ${ \mathcal F }$ is the quantum information matrix (QIFM) whose entry is defined as ${{ \mathcal F }}_{i,j}=\displaystyle \frac{1}{2}\mathrm{Tr}(\rho \{{L}_{i},{L}_{j}\})$ with ${L}_{i}({L}_{j})$ the symmetric logarithmic derivative operator [28] and $\{\cdot ,\cdot \}$ the anti-commutation [25, 28, 29]. The QCRB is defined only if ${ \mathcal F }$ is invertible. The inequality in the QCRB means for arbitrary same dimensional vector $\vec{x}$, there should have ${\vec{x}}^{\top }{ \mathcal C }\vec{x}\,\geqslant {\vec{x}}^{\top }{{ \mathcal F }}^{-1}\vec{x}$. Therefore, if we set $\vec{x}={(0,\cdots ,1,\cdots ,0)}^{\top }$ and consider ${{ \mathcal C }}_{{ii}}={\delta }^{2}{\theta }_{i}$, we would get$ \begin{eqnarray}{\delta }^{2}{\theta }_{i}={{ \mathcal C }}_{{ii}}\geqslant {{ \mathcal F }}_{{ii}}^{-1}.\end{eqnarray}$For a measurement scheme with ${\theta }_{1},{\theta }_{2},\cdots $ the measured parameters, $| {\psi }_{0}\rangle $ the initial state and U the parameterization operator, the elements of the QIFM can be rewritten as [30]$ \begin{eqnarray}\begin{array}{rcl}{{ \mathcal F }}_{{ij}} & = & 4\mathrm{Cov}({{ \mathcal H }}_{{\theta }_{i}},{{ \mathcal H }}_{{\theta }_{j}})\\ & = & 4(\langle {\psi }_{0}| ({{ \mathcal H }}_{{\theta }_{i}}{{ \mathcal H }}_{{\theta }_{j}}+{{ \mathcal H }}_{{\theta }_{j}}{{ \mathcal H }}_{{\theta }_{i}})/2| {\psi }_{0}\rangle \\ & & -\langle {\psi }_{0}| {{ \mathcal H }}_{{\theta }_{i}}| {\psi }_{0}\rangle \langle {\psi }_{0}| {{ \mathcal H }}_{{\theta }_{j}}| {\psi }_{0}\rangle ),\end{array}\end{eqnarray}$$ \begin{eqnarray}{{ \mathcal F }}_{{ii}}=4{{\rm{\Delta }}}^{2}{{ \mathcal H }}_{{\theta }_{i}}=4(\langle {\psi }_{0}| {{ \mathcal H }}_{{\theta }_{i}}^{2}| {\psi }_{0}\rangle -| \langle {\psi }_{0}| {{ \mathcal H }}_{{\theta }_{i}}| {\psi }_{0}\rangle {| }^{2}),\end{eqnarray}$with ${{ \mathcal H }}_{{\theta }_{i}}={\rm{i}}({\partial }_{{\theta }_{i}}{U}^{\dagger })U$ the characteristic operator for the parameterization of ${\theta }_{i}$. Consequently, calculating the characteristic operators is crucial to obtain the measurement precisions. On the other hand, for saturating the lower bound of the QCRB, there should be some necessary and sufficient conditions. The general forms of these conditions have been given by researchers [30–33]. Such constraint conditions may reveal intrinsic features of the measurement apparatus and provide guidelines to design and optimize a metrology process for multi-parameter estimation.
In this paper, we explore the two-parameter measurement precision of a matter-wave Sagnac interferometer with trapped atomic systems. The rotation frequency and the trapping frequency are the parameters to be estimated simultaneously. Measuring the rotation frequency is of great value in navigation [34], seismology [35] and geodesy [36], and the measurement of the trapping frequency, or say, the trapping energy, can be used to detect electromagnetic fields. We use the maximally spin-space entangled state proposed in [14] as the input resource. The parameterization process is characterized by a state-dependent manipulation, which is provided in [12]. By calculating the minimum uncertainties of the two parameters and analyzing whether the minimum uncertainties can approach the HL simultaneously, we find the limitation of the matter-wave Sagnac interferometer in the two-parameter estimation process. We also derive the constraint conditions for saturating the QCRB. To achieve the best measurement precision, one should choose proper evolution time, input state and suitable specification of matter-wave Saganc interferometer to satisfy these constraint conditions.
2. Backgrounds
The schematic protocol of the measurement process is seen in figure 1. A couple of spin-1/2 particles that are trapped in the harmonic potential well are injected into a rotating Sagnac interferometer. Due to the interaction between the particles and the interferometer, the spin-up and down particles would be separated and driven to counter-transport along a circular path. When the particles recombine again, there would accumulate a phase difference which is called the Sagnac phase between the two types of particles. By detecting the Sagnac phase, one could obtain the information about the parameters. The interaction between the particles and the interferometer plays a core role in the measurement process, which can be described as [12]$ \begin{eqnarray}H(t)=\displaystyle \sum _{k=1}^{N}\hslash \omega {a}_{k}^{\dagger }{a}_{k}+{\rm{i}}\hslash \mu \sqrt{\omega }\left({a}_{k}-{a}_{k}^{\dagger }\right)\left({\rm{\Omega }}+{\sigma }_{z}^{(k)}{\omega }_{p}(t)\right),\end{eqnarray}$where $\mu =\sqrt{m/(2{\hslash })}R$, N is the particle number, m is the mass of a single particle, the subscript k denotes the k’th particle, ${a}_{k}^{\dagger }({a}_{k})$ is the creation (annihilation) operator for the trap mode, ${\sigma }_{z}^{(k)}=| \uparrow {\rangle }_{k}\langle \uparrow | -| \downarrow {\rangle }_{k}\langle \downarrow | $ is the spin operator with $| \uparrow (\downarrow )\rangle $ the atomic spin state, ω is the frequency of the harmonic potential, ω is the angular velocity of the Sagnac system and ${\omega }_{p}(t)$ is the relative angular velocity between the trapped particles and the rotating Sagnac interferometer. In this paper, ω and ω are the two parameters to be estimated. Suppose the true values of the parameters are ${\omega }_{0}$ and ${{\rm{\Omega }}}_{0}$. That is to say, we would calculate the variances of ω and ω around ${\omega }_{0}$ and ${{\rm{\Omega }}}_{0}$.
Figure 1.
New window|Download| PPT slide Figure 1.Schematic diagram of the parameter measurement process in a Sagnac interferometry. The Sagnac system is simplified as a clockwise rotating disc with a uniform angular velocity ω. The input state that is prepared by trapping N spin-1/2 particles in a harmonic potential with frequency ω is injected into the Sagnac system. Then the particles with different spin directions are coherently split and countertransport along the disc with sweeping angular frequency ${\omega }_{p}(t)$ in the rotating frame. The output state containing the information of ω and ω is formed when the particles recombine again. One can readout the information of these parameters by detecting the output state.
To obtain the best measurement precisions, we also suppose the particles are maximally entangled. In this paper, we use the quasi-GHZ state as the input state, which can be written as$ \begin{eqnarray}| {\psi }_{0}\rangle =\displaystyle \frac{1}{\sqrt{2}}\left(\underset{k=1}{\overset{N}{\bigotimes }}| {\psi }_{\uparrow }\rangle | \uparrow {\rangle }_{k}+\underset{k=1}{\overset{N}{\bigotimes }}| {\psi }_{\downarrow }\rangle | \downarrow {\rangle }_{k}\right),\end{eqnarray}$where $| {\psi }_{\uparrow (\downarrow )}{\rangle }_{k}$ is the k’th particle’s motional state in the harmonic potential well. Suppose the particles enter the Sagnac system and split into two parts at t=0 and recombine again at t=T. As the two opposite paths form a complete circle, T should satisfy the condition [12]$ \begin{eqnarray}{\int }_{0}^{T}{\omega }_{p}(t){\rm{d}}t=\pi .\end{eqnarray}$Through the Hamiltonian in equation (4) and using the Magnus expansion in [37], the evolution operator can be written as$ \begin{eqnarray}\begin{array}{rcl}U(T) & = & \underset{k=1}{\overset{N}{\bigotimes }}{U}_{k}(T)\\ & = & \underset{k=1}{\overset{N}{\bigotimes }}{{\rm{e}}}^{-{\rm{i}}\omega {a}_{k}^{\dagger }{a}_{k}T}{{\rm{e}}}^{{\rm{i}}{{\rm{\Phi }}}_{k}(\omega ,{\rm{\Omega }},T)}{D}_{k}[{\eta }_{k}(\omega ,{\rm{\Omega }},T)],\end{array}\end{eqnarray}$where$ \begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{{\rm{\Phi }}}_{k}(\omega ,{\rm{\Omega }},T) & = & {\displaystyle \int }_{0}^{T}{\displaystyle \int }_{0}^{{t}_{1}}{f}_{k}(\omega ,{\rm{\Omega }},{t}_{1}){f}_{k}(\omega ,{\rm{\Omega }},{t}_{2})\\ & & \cdot \sin (\omega ({t}_{1}-{t}_{2})){\rm{d}}{t}_{2}{\rm{d}}{t}_{1},\\ {\eta }_{k}(\omega ,{\rm{\Omega }},T) & = & -{\displaystyle \int }_{0}^{T}{f}_{k}(\omega ,{\rm{\Omega }},t){{\rm{e}}}^{{\rm{i}}\omega t}{\rm{d}}t,\\ {f}_{k}(\omega ,{\rm{\Omega }},t) & = & \sqrt{\displaystyle \frac{m\omega }{2\hslash }}R\left({\rm{\Omega }}+{\sigma }_{z}^{(k)}{\omega }_{p}(t)\right),\end{array}\end{array}\end{eqnarray}$and ${D}_{k}[\eta ]={{\rm{e}}}^{\eta {a}_{k}^{\dagger }-{\eta }^{* }{a}_{k}}$ refers to the displacement operator. Then the time-dependent characteristic operators ${{ \mathcal H }}_{\omega }$ and ${{ \mathcal H }}_{{\rm{\Omega }}}$ can be written as$ \begin{eqnarray}\begin{array}{rcl}{{ \mathcal H }}_{\omega } & = & {\rm{i}}({\partial }_{\omega }U{\left(T\right)}^{\dagger })U(T)\\ & = & \displaystyle \sum _{k=1}^{N}{\rm{i}}({\partial }_{\omega }{U}_{k}{\left(T\right)}^{\dagger }){U}_{k}(T)=\displaystyle \sum _{k=1}^{N}{{ \mathcal H }}_{\omega }^{(k)},\\ {{ \mathcal H }}_{{\rm{\Omega }}} & = & {\rm{i}}({\partial }_{{\rm{\Omega }}}U{\left(T\right)}^{\dagger })U(T)\\ & = & \displaystyle \sum _{k=1}^{N}{\rm{i}}({\partial }_{{\rm{\Omega }}}{U}_{k}{\left(T\right)}^{\dagger }){U}_{k}(T)=\displaystyle \sum _{k=1}^{N}{{ \mathcal H }}_{{\rm{\Omega }}}^{(k)}.\end{array}\end{eqnarray}$The concrete expressions of ${{ \mathcal H }}_{\omega }^{(k)}$ and ${{ \mathcal H }}_{{\rm{\Omega }}}^{(k)}$ are$ \begin{eqnarray}\begin{array}{rcl}{{ \mathcal H }}_{\omega }^{(k)} & = & ({K}_{1}{a}_{k}+{K}_{1}^{* }{a}_{k}^{\dagger })\\ & & -({K}_{2}{a}_{k}+{K}_{2}^{* }{a}_{k}^{\dagger }){\sigma }_{z}^{(k)}+\lambda {\sigma }_{z}^{(k)}-{{Ta}}_{k}^{\dagger }{a}_{k},\end{array}\end{eqnarray}$$ \begin{eqnarray}{{ \mathcal H }}_{{\rm{\Omega }}}^{(k)}=({\delta }_{1}{a}_{k}+{\delta }_{1}^{* }{a}_{k}^{\dagger })+{\delta }_{2}{\sigma }_{z}^{(k)},\end{eqnarray}$where$ \begin{eqnarray}\begin{array}{rcl}{K}_{1} & = & \mu \sqrt{\omega }{\rm{\Omega }}\left[\left(T-{\rm{i}}\displaystyle \frac{1}{2\omega }\right){q}^{* }(\omega ,T)-{\rm{i}}{\partial }_{\omega }{q}^{* }(\omega ,T)\right],\\ {K}_{2} & = & \mu \sqrt{\omega }\left[\left({\rm{i}}\displaystyle \frac{1}{2\omega }-T\right){p}^{* }(\omega ,T)+{\rm{i}}{\partial }_{\omega }{p}^{* }(\omega ,T)\right],\\ \lambda & = & {\mu }^{2}{\rm{\Omega }}\left\{\displaystyle \frac{1}{\omega }{\displaystyle \int }_{0}^{T}{\omega }_{p}(t)[\cos \omega (t-T)-\cos (\omega t)]{\rm{d}}t\right.\\ & & \left.+2{\displaystyle \int }_{0}^{T}{\omega }_{p}(t)(t-T)\sin (\omega t){\rm{d}}t\right\},\\ {\delta }_{1} & = & -{\rm{i}}\displaystyle \frac{2\mu }{\sqrt{\omega }}\sin \left(\displaystyle \frac{\omega T}{2}\right){{\rm{e}}}^{-{\rm{i}}\displaystyle \frac{\omega }{2}T},\\ {\delta }_{2} & = & 2\pi {\mu }^{2}\left(1-\displaystyle \frac{1}{\pi }{\displaystyle \int }_{0}^{T}{\omega }_{p}(t)\cos [\omega (T-t)]{\rm{d}}t\right),\end{array}\end{eqnarray}$with $p(\omega ,T)={\int }_{0}^{T}{\omega }_{p}(t){{\rm{e}}}^{{\rm{i}}\omega t}{\rm{d}}t$ and $q(\omega ,t)={\int }_{0}^{T}{{\rm{e}}}^{{\rm{i}}\omega t}{\rm{d}}t$. Now, we have obtained the characteristic operators ${{ \mathcal H }}_{\omega }$ and ${{ \mathcal H }}_{{\rm{\Omega }}}$. Next we will calculate the QFIM and then the lower bounds of the variances of the parameters ω and ω.
3. General results
The elements of the QFIM with respect to ω and ω for the input state in equation (5) are$ \begin{eqnarray}{{ \mathcal F }}_{11}=4{{\rm{\Delta }}}^{2}{{ \mathcal H }}_{\omega },\,\,{{ \mathcal F }}_{22}=4{{\rm{\Delta }}}^{2}{{ \mathcal H }}_{{\rm{\Omega }}},\,\,{{ \mathcal F }}_{12}={{ \mathcal F }}_{21}=4{\rm{Cov}}({{ \mathcal H }}_{\omega },{{ \mathcal H }}_{{\rm{\Omega }}}).\end{eqnarray}$Therefore, through equation (1), we would get$ \begin{eqnarray}\begin{array}{rcl}{\delta }^{2}\omega \geqslant {{ \mathcal F }}_{11}^{-1} & = & \displaystyle \frac{1}{4}\displaystyle \frac{{{\rm{\Delta }}}^{2}{{ \mathcal H }}_{{\rm{\Omega }}}}{{{\rm{\Delta }}}^{2}{{ \mathcal H }}_{\omega }{{\rm{\Delta }}}^{2}{{ \mathcal H }}_{{\rm{\Omega }}}-{\left|\mathrm{Cov}({{ \mathcal H }}_{\omega },{{ \mathcal H }}_{{\rm{\Omega }}})\right|}^{2}},\\ {\delta }^{2}{\rm{\Omega }}\geqslant {{ \mathcal F }}_{22}^{-1} & = & \displaystyle \frac{1}{4}\displaystyle \frac{{{\rm{\Delta }}}^{2}{{ \mathcal H }}_{\omega }}{{{\rm{\Delta }}}^{2}{{ \mathcal H }}_{\omega }{{\rm{\Delta }}}^{2}{{ \mathcal H }}_{{\rm{\Omega }}}-{\left|\mathrm{Cov}({{ \mathcal H }}_{\omega },{{ \mathcal H }}_{{\rm{\Omega }}})\right|}^{2}},\end{array}\end{eqnarray}$where the property of the second-order matrix has been used in the above formulas. Combining with equations (9)–(11), ${{\rm{\Delta }}}^{2}\omega ({\rm{\Omega }})$ and $\mathrm{Cov}({{ \mathcal H }}_{\omega },{{ \mathcal H }}_{{\rm{\Omega }}})$ can be rewritten as$ \begin{eqnarray}\begin{array}{rcl}{{\rm{\Delta }}}^{2}{{ \mathcal H }}_{\omega ({\rm{\Omega }})} & = & \displaystyle \sum _{k=1}^{N}{{\rm{\Delta }}}^{2}{{ \mathcal H }}_{\omega ({\rm{\Omega }})}^{(k)}+\displaystyle \sum _{{k}_{1}\ne {k}_{2}}^{N}\mathrm{Cov}({{ \mathcal H }}_{\omega ({\rm{\Omega }})}^{({k}_{1})},{{ \mathcal H }}_{\omega ({\rm{\Omega }})}^{({k}_{2})})\\ & = & N{{\rm{\Delta }}}^{2}{{ \mathcal H }}_{\omega ({\rm{\Omega }})}^{(1)}+({N}^{2}-N)\mathrm{Cov}({{ \mathcal H }}_{\omega ({\rm{\Omega }})}^{(1)},{{ \mathcal H }}_{\omega ({\rm{\Omega }})}^{(2)}),\end{array}\end{eqnarray}$$ \begin{eqnarray}\begin{array}{l}\mathrm{Cov}({{ \mathcal H }}_{\omega },{{ \mathcal H }}_{{\rm{\Omega }}})=\displaystyle \sum _{k=1}^{N}\mathrm{Cov}({{ \mathcal H }}_{\omega }^{(k)},{{ \mathcal H }}_{{\rm{\Omega }}}^{(k)})\\ \ \ +\,\displaystyle \sum _{{k}_{1}\ne {k}_{2}}^{N}\mathrm{Cov}({{ \mathcal H }}_{\omega }^{({k}_{1})},{{ \mathcal H }}_{{\rm{\Omega }}}^{({k}_{2})})\\ =\,N\mathrm{Cov}({{ \mathcal H }}_{\omega }^{(1)},{{ \mathcal H }}_{{\rm{\Omega }}}^{(1)})+({N}^{2}-N)\mathrm{Cov}({{ \mathcal H }}_{\omega }^{(1)},{{ \mathcal H }}_{{\rm{\Omega }}}^{(2)}),\end{array}\end{eqnarray}$where the second steps in equations (15) and (16) have considered the symmetry for the characteristic operators. The numerators and denominators in equation (14) can be expressed as$ \begin{eqnarray}\begin{array}{l}{\rm{\Delta }}{{ \mathcal H }}_{\omega }^{2}={AN}+{{BN}}^{2},\,\,\,\,\,\,\,\,{\rm{\Delta }}{{ \mathcal H }}_{{\rm{\Omega }}}^{2}={CN}+{{DN}}^{2},\\ {\rm{\Delta }}{{ \mathcal H }}_{\omega }^{2}{\rm{\Delta }}{{ \mathcal H }}_{{\rm{\Omega }}}^{2}-| \mathrm{Cov}({{ \mathcal H }}_{\omega },{{ \mathcal H }}_{{\rm{\Omega }}}){| }^{2}={{EN}}^{2}+{{FN}}^{3},\end{array}\end{eqnarray}$where $A,B,C,D,E,F$ are real coefficients that are related to the evolution time T, the particle mass m, the interferometer’s radius R and the input state. The concrete expressions of these coefficients can be seen in the supplementary material, which can be found online at stacks.iop.org/CTP/72/095101/mmedia. From equation (14), it is clearly that if we want the lower limit of the measurement precision of $\omega ({\rm{\Omega }})$ to reach the HL when $N\gg 1$, which means ${{\rm{\Delta }}}^{2}\omega ({\rm{\Omega }})\propto 1/{N}^{2}$, we should let $B(D)=0$ and $F\ne 0$. The condition which is equivalent to $B(D)=0$ can be expressed as$ \begin{eqnarray}\begin{array}{rcl}B & = & 0\,\iff \,2\mathrm{Re}\left({K}_{1}\langle {\psi }_{0}| {a}_{k}{\sigma }_{z}^{(k)}| {\psi }_{0}\rangle \right.\\ & & \left.-{K}_{2}\langle {\psi }_{0}| {a}_{k}| {\psi }_{0}\rangle \right)+\lambda =T\langle {\psi }_{0}| {a}_{k}^{\dagger }{a}_{k}{\sigma }_{z}^{(k)}| {\psi }_{0}\rangle ,\end{array}\end{eqnarray}$$ \begin{eqnarray}D=0\,\iff \,\displaystyle \frac{{\delta }_{2}}{2}+\mathrm{Re}\left({\delta }_{1}\langle {\psi }_{0}| {a}_{k}{\sigma }_{z}^{(k)}| {\psi }_{0}\rangle \right)=0,\end{eqnarray}$here ${K}_{1},{K}_{2},\lambda ,{\delta }_{1},{\delta }_{2}$ are seen in equation (12). The right-hand side of equation (18) can be denoted as the average trapping energy difference between the spin-up and down particles as $\langle {\psi }_{0}| {a}_{k}^{\dagger }{a}_{k}{\sigma }_{z}^{(k)}| {\psi }_{0}| \rangle $$=\,1/2(\langle {\psi }_{\uparrow }| {a}_{k}^{\dagger }{a}_{k}| {\psi }_{\uparrow }\rangle -\langle {\psi }_{\downarrow }| {a}_{k}^{\dagger }{a}_{k}| {\psi }_{\downarrow }\rangle )\,=$$1/(2{\hslash }{\omega }_{0})({\bar{E}}_{k}^{\uparrow }-{\bar{E}}_{k}^{\downarrow })$$=\,1/(2N{\hslash }{\omega }_{0})({\bar{E}}^{\uparrow }-{\bar{E}}^{\downarrow })$. Unfortunately, our calculation shows (seen in the supplementary material) that if conditions in equations (18) and (19) were both met, F would be equal to zero as well. Hence, the optimal measurement precisions of the two parameters cannot achieve the HL simultaneously in the current scheme. Our calculation also shows (seen in the supplementary material) if one of the last two equations was satisfied, namely, B(D) was equal to zero while D(B) was nonzero, then $A\cdot D\,=F(B\cdot C=F)$. Therefore, for large particle number cases, i.e. $N\gg 1$, only one of the parameters’ best measurement precisions can reach the HL, while the other can only approach the standard quantum limit (SQL), which signifies ${{\rm{\Delta }}}^{2}\omega ({\rm{\Omega }})\propto 1/N$. The results can be shown more clearly as$ \begin{eqnarray}\begin{array}{l}B=0\Rightarrow {AD}=F\Rightarrow {\delta }^{2}{\omega }_{\min }\sim \displaystyle \frac{1}{N}\cdot \displaystyle \frac{1}{4A},\\ \quad {\delta }^{2}{{\rm{\Omega }}}_{\min }\sim \displaystyle \frac{1}{{N}^{2}}\cdot \displaystyle \frac{1}{4D}.\end{array}\end{eqnarray}$$ \begin{eqnarray}\begin{array}{l}D=0\Rightarrow {BC}=F\Rightarrow {\delta }^{2}{\omega }_{\min }\sim \displaystyle \frac{1}{{N}^{2}}\cdot \displaystyle \frac{1}{4B},\\ \quad {\delta }^{2}{{\rm{\Omega }}}_{\min }\sim \displaystyle \frac{1}{N}\cdot \displaystyle \frac{1}{4C}.\end{array}\end{eqnarray}$In general, with the input state in equation (5) and the Sagnac System depicted in figure 1, at most one of the two measurement precisions can approach the HL while the other only be scaled by the SQL. Such as it is, one may be able to choose which one can approach the HL. This is the main result of our work in this paper.
If we consider the total variance, i.e. ${\delta }^{2}\omega +{\delta }^{2}{\rm{\Omega }}$, as a figure of merit for the two-parameter estimation, then in our scenario, it is easy to prove that the total variance is also scaled by the SQL, namely, ${({{\rm{\Delta }}}^{2}\omega +{{\rm{\Delta }}}^{2}{\rm{\Omega }})}_{\min }\propto 1/N$. On the other hand, for the optical Sagnac interferometer scenario with the mode separated parameterization process, whose unitary evolution is described as ${U}_{{\boldsymbol{\theta }}}=\exp ({\rm{i}}{\boldsymbol{\theta }}\cdot {\boldsymbol{H}})\,=\exp \left({\rm{i}}{\sum }_{m=1}^{d}{\theta }_{m}{H}_{m}\right)$ , the total variance can reach the HL with the entangled input state [28, 38–40]. The input state in our work is maximally entangled but the measurement precision cannot surpass the SQL. Such an imperfection may reflect the boundedness of the atom Sagnac interferometer for multi-parameter estimation.
Up to now, we have not considered the constraint conditions for saturating the inequalities in equation (14) at the same time. Next we would do some further analysis of the measurement precisions under the constraint conditions. For the sake of simplicity, we set the relative rotating angular velocity ${\omega }_{p}(t)$ a constant. This is a feasible choice in actual experiments as ${\omega }_{p}(t)$ being constant means that the potential well rotates uniformly around the Sagnac apparatus, which is easier to be operated and has higher stability than the variable-rotating scheme.
4. Measurement precisions under the constraint condition
As the input state in our work is a pure state, the constraint condition for saturating the QCRB can be simplified as [30, 41]$ \begin{eqnarray}\langle {\psi }_{0}| \left[{{ \mathcal H }}_{\omega },{{ \mathcal H }}_{{\rm{\Omega }}}\right]| {\psi }_{0}\rangle =0.\end{eqnarray}$Substituting equations (9)–(11) into the last equation, and considering the commutation relation $[{{ \mathcal H }}_{\omega }^{(k)}$, ${{ \mathcal H }}_{{\rm{\Omega }}}^{(k)}]$$=({K}_{1}{\delta }_{1}^{* }-{K}_{1}^{* }{\delta }_{1})+({\delta }_{1}{K}_{2}^{* }-{\delta }_{1}^{* }{K}_{2}){\sigma }_{z}^{(k)}$$+\,({\delta }_{1}{a}_{k}-{\delta }_{1}^{* }{a}_{k}^{\dagger })\tau $ and $\langle {\psi }_{0}| {\sigma }_{z}^{(k)}| {\psi }_{0}\rangle =0$, the constraint condition of equation (22) can be further simplified into two cases. The first one is$ \begin{eqnarray}\sin \left(\displaystyle \frac{{\omega }_{0}\tau }{2}\right)=0\,\,\,\Rightarrow \,\,\,\tau =\kappa \cdot \displaystyle \frac{2\pi }{{\omega }_{0}},\,\,\,\kappa =1,2,\cdots ,\end{eqnarray}$and the second is$ \begin{eqnarray}\displaystyle \frac{\mu {{\rm{\Omega }}}_{0}}{\sqrt{{\omega }_{0}}}\sin \left(\displaystyle \frac{{\omega }_{0}\tau }{2}\right)=\mathrm{Re}\left({{\rm{e}}}^{-{\rm{i}}\displaystyle \frac{{\omega }_{0}\tau }{2}}\langle {\psi }_{0}| {a}_{k}| {\psi }_{0}\rangle \right)\ne 0.\end{eqnarray}$We call equations (23) and (24) as Condition ${\rm{I}}$ and Condition $\mathrm{II}$, respectively. Equations (23) or (24) guarantees that the lower limits in equation (14) is attainable, while equations (18) and (19) ensure that one of the lower limits can approach the HL.
4.1. Constraint condition I
In Condition equation (23) subsection, the evolution time T takes a series of discrete values, which makes ${\delta }_{1}$ equal to 0 and ${\omega }_{p}={\omega }_{0}/(2\kappa )$, with κ any positive integer. Coefficients in equation (12) are also determined as$ \begin{eqnarray}\begin{array}{rcl}{K}_{1} & = & -{\rm{i}}\displaystyle \frac{2\mu \pi \kappa {{\rm{\Omega }}}_{0}}{{\omega }_{0}^{3/2}},\,\,\,\,\,{K}_{2}={\rm{i}}\displaystyle \frac{\mu \pi }{\sqrt{{\omega }_{0}}},\\ \lambda & = & -\displaystyle \frac{2{\mu }^{2}\pi {{\rm{\Omega }}}_{0}}{{\omega }_{0}},\,\,\,\,\,{\delta }_{2}=2{\mu }^{2}\pi .\end{array}\end{eqnarray}$Such a simplification makes $C,E$ in equation (17) equal to zero and $D={\delta }_{2}^{2},A/F=1/{\delta }_{2}^{2}$. As D is not zero, the minimum uncertainty of ω cannot reach the HL. Therefore, we should let B=0 in this subsection. From equation (20), we can write the lower limits of the parameters’ variances as$ \begin{eqnarray}{\delta }^{2}{\omega }_{\min }=\displaystyle \frac{1}{N}\cdot \displaystyle \frac{1}{4A},\end{eqnarray}$$ \begin{eqnarray}{\delta }^{2}{{\rm{\Omega }}}_{min}=\displaystyle \frac{1}{{N}^{2}}\cdot \displaystyle \frac{1}{4{\delta }_{2}^{2}}=\displaystyle \frac{1}{{N}^{2}}\cdot \displaystyle \frac{1}{16{\pi }^{2}{\mu }^{4}}=\displaystyle \frac{1}{{N}^{2}}\cdot {\left(\displaystyle \frac{\hslash }{2m\pi {R}^{2}}\right)}^{2}.\end{eqnarray}$Obviously, the optimal measurement precision of ω is bounded by the SQL while the one of ω is scaled by the HL. The minimum variance of ω in equation (27) is inversely proportional to N2 with a coefficient ${({\hslash }/(2m\pi {R}^{2}))}^{2}$, which is only related to m and R and has no concern with the input state. Therefore, as long as m and R are fixed, the minimum variance of ω is constant. No matter what the motional states in equation (5) are, one would always obtain the same optimal measurement precision of ω. Such a stability may quite useful in practical experiments. The result of equation (27) is the same as the one obtained in the single-parameter measurement scheme with the GHZ state [13] or the so called partially entangled state as the input state [14], which equivalent to that the motional state is the vacuum state or the coherent state in our work. It is shown that, with the same type of input state, the measurement precision of the rotation frequency in the two-parameter estimation scenario can be the same with the one in the single-parameter estimation scenario.
The coefficient $1/(4A)$ in equation (26) is related to the input state, which means the best measurement precision of ω varies with the input state. For example, if the motional state in equation (5) is Fock state, i.e. $| {\psi }_{\uparrow }\rangle =| {n}_{1}\rangle $, $| {\psi }_{\downarrow }\rangle =| {n}_{2}\rangle $, then to satisfy B=0, there should have$ \begin{eqnarray}{n}_{2}-{n}_{1}=2{{\rm{\Omega }}}_{0}{\mu }^{2}/\kappa .\end{eqnarray}$And the minimum variance of ω yields$ \begin{eqnarray}\begin{array}{l}{\left({\delta }^{2}\omega \right)}_{\min }\\ =\,\displaystyle \frac{1}{N}\displaystyle \frac{{\omega }_{0}^{2}}{4{\mu }^{2}{\pi }^{2}[({n}_{1}+{n}_{2}+1)({\omega }_{0}+4{\kappa }^{2}{{\rm{\Omega }}}_{0}^{2}/{\omega }_{0})-8{\mu }^{2}{{\rm{\Omega }}}_{0}^{2}]},\end{array}\end{eqnarray}$where $n={n}_{1}+{n}_{2}$ is proportional to the trapping energy of a single particle. If the motional state is coherent state, i.e. $| {\psi }_{\uparrow }\rangle =| {\alpha }_{1}\rangle $, $| {\psi }_{\downarrow }\rangle =| {\alpha }_{2}\rangle $, with ${\alpha }_{1}={r}_{1}{{\rm{e}}}^{{\rm{i}}{\theta }_{1}},{\alpha }_{2}={r}_{2}{{\rm{e}}}^{{\rm{i}}{\theta }_{2}}$, and $\{{\theta }_{1},{\theta }_{2}\}\in [-\pi ,\pi ]$, then to obtain the minimum variance of ω and satisfy B=0, there should have$ \begin{eqnarray}\begin{array}{c}{\theta }_{1}=-\displaystyle \frac{\pi }{2},\,\left\{\begin{array}{ll}{\theta }_{2}=\tfrac{\pi }{2},\,{r}_{2}={r}_{1}+\tfrac{2\mu {{\rm{\Omega }}}_{0}}{\sqrt{{\omega }_{0}}}, & {\rm{if}}\,{\omega }_{0}\geqslant 2\kappa {{\rm{\Omega }}}_{0}\\ {\theta }_{2}=-\tfrac{\pi }{2},\,{r}_{2}={r}_{1}+\tfrac{\mu \sqrt{{\omega }_{0}}}{\kappa }, & {\rm{if}}\,{\omega }_{0}\lt 2\kappa {{\rm{\Omega }}}_{0}\end{array}\right.,\end{array}\end{eqnarray}$And the minimum variance of ω would be$ \begin{eqnarray}\begin{array}{l}{\left({\delta }^{2}\omega \right)}_{\min }=\displaystyle \frac{1}{N}\cdot \displaystyle \frac{{\omega }_{0}^{2}}{16{\pi }^{2}{\kappa }^{2}}\\ \cdot \,\left\{\begin{array}{ll}\tfrac{1}{{\left(\sqrt{(n-2{\mu }^{2}{{\rm{\Omega }}}_{0}^{2}/{\omega }_{0})/2}+\mu \sqrt{{\omega }_{0}}/(2\kappa \right)}^{2}}, & \mathrm{if}\ {\omega }_{0}\gt 2\kappa {{\rm{\Omega }}}_{0}\\ \tfrac{1}{{\left(\sqrt{(n-{\mu }^{2}{\omega }_{0}/(2{\kappa }^{2}))/2}+\mu {{\rm{\Omega }}}_{0}/\sqrt{{\omega }_{0}}\right)}^{2}}, & \mathrm{if}\ {\omega }_{0}\lt 2\kappa {{\rm{\Omega }}}_{0}\end{array}\right.,\end{array}\end{eqnarray}$where $n={r}_{1}^{2}+{r}_{2}^{2}$ can be seen as the trapping energy of a single particle.
As the input state only influences the measurement precision of ω with fixed m and R in this subsection, one could judge which of any two input states is better for measurement by comparing the variances of ω with these states. For example, if we want to do a comparison about the measurement precisions between two input states with coherent state and Fock state as the motional state, respectively, we could just compare the values in equations (31) and (29) with the same n. The result is seen in figure 2. We can see the comparative result varies with the evolutionary time, the true values ${\omega }_{0}$ and ${{\rm{\Omega }}}_{0}$. When ${\omega }_{0}$ is large and κ is small or ${{\rm{\Omega }}}_{0}$ is large and ${\omega }_{0}$ is small, which corresponds to the green–yellow–red areas in the pictures, the measurement precision of ω with the Fock state always precede the one with the coherent state. On the contrary, for the light blue–dark blue areas of the pictures, the input state with coherent state as the motional state performs better than the Fock state case.
Figure 2.
New window|Download| PPT slide Figure 2.The distribution of the $\mathrm{log}$ scaling ratio between the minimum variances of ω in the coherent state and Fock state. We set ${\mu }^{-2}$ as the unit of the frequencies and let the total energy levels, i.e. ${n}_{1}+{n}_{2}$ or $| {\alpha }_{1}{| }^{2}+| {\alpha }_{2}{| }^{2}$ is 100. κ represents the evolutionary time. Points with value larger than zero means fork state is better, otherwise the coherent state case gets more precise measurement result.
4.2. Constraint condition II
Assuming $\langle {\psi }_{0}| {a}_{k}| {\psi }_{0}\rangle =x+{\rm{i}}{y}$ with $x,y$ real numbers. Then the constraint condition of equation (24) yields$ \begin{eqnarray}\displaystyle \frac{\mu {{\rm{\Omega }}}_{0}}{\sqrt{{\omega }_{0}}}\sin \left(\displaystyle \frac{{\omega }_{0}T}{2}\right)=x\cos \left(\displaystyle \frac{{\omega }_{0}T}{2}\right)+y\sin \left(\displaystyle \frac{{\omega }_{0}T}{2}\right)\ne 0.\end{eqnarray}$It is hard to calculate the measurement precisions with the general solution of T in equation (32). In this subsection, we only discuss a relatively simple case with $\cos ({\omega }_{0}T/2)=0$. Then the constraint condition of equation (32) becomes$ \begin{eqnarray}T=\displaystyle \frac{\pi }{{\omega }_{0}}(2\kappa +1),\,\,\,\,\,y=\displaystyle \frac{\mu {{\rm{\Omega }}}_{0}}{\sqrt{{\omega }_{0}}},\end{eqnarray}$where κ is an arbitrary natural number. So ${\omega }_{p}={\omega }_{0}/(2\kappa +1)$ and the factors in equation (12) are$ \begin{eqnarray}\begin{array}{rcl}{K}_{1} & = & \displaystyle \frac{{{\rm{\Omega }}}_{0}\mu }{{\omega }_{0}^{3/2}}\left[1-{\rm{i}}(2\kappa +1)\pi \right],\,{K}_{2}=\displaystyle \frac{\mu }{\sqrt{{\omega }_{0}}}\left(-\displaystyle \frac{1}{2\kappa +1}+{\rm{i}}\pi \right),\\ \lambda & = & -\displaystyle \frac{2{\mu }^{2}\pi {{\rm{\Omega }}}_{0}}{{\omega }_{0}},\,\,\,{\delta }_{1}=-\displaystyle \frac{2\mu }{\sqrt{{\omega }_{0}}},\,\,\,{\delta }_{2}\,=\,2{\mu }^{2}\pi .\end{array}\end{eqnarray}$Unlike the previous subsection, here the coefficient B or D is not necessarily equal to zero. If $B=0 \& D \ne 0$, the measurement precisions would have the form as equation (20) while if $D=0 \& B\ne 0$, the precisions would have the form as equation (21). So either of the two parameters’ minimum uncertainties can reach the HL. The general expressions of the coefficients $A \sim D$ are still too complex, for the sake of simplicity, we would like to calculate the measurement precisions just through some examples. It is clearly that the Fock state cannot be the motional state in this subsection. We choose the coherent state, i.e. $| {\psi }_{\uparrow }\rangle =| {\alpha }_{1}\rangle $, $| {\psi }_{\downarrow }\rangle =| {\alpha }_{2}\rangle $ as the motional state. Then the coefficients of $A\sim D$ are simplified as$ \begin{eqnarray}\begin{array}{rcl}A & = & \displaystyle \frac{1}{2}\left(| {\alpha }_{1}T+{K}_{2}^{* }-{K}_{1}^{* }{| }^{2}+| {\alpha }_{2}T-{K}_{1}^{* }-{K}_{2}^{* }{| }^{2}\right),\\ B & = & \left[\lambda +\mathrm{Re}({K}_{1}({\alpha }_{1}-{\alpha }_{2}))\right.\\ & & {\left.-\mathrm{Re}({K}_{2}({\alpha }_{1}+{\alpha }_{2}))-\displaystyle \frac{1}{2}T(| {\alpha }_{1}{| }^{2}-| {\alpha }_{2}{| }^{2})\right]}^{2},\\ C & = & | {\delta }_{1}{| }^{2},\\ D & = & {\left({\delta }_{2}+\mathrm{Re}({\delta }_{1}({\alpha }_{1}-{\alpha }_{2})\right)}^{2}.\end{array}\end{eqnarray}$We also set ${\alpha }_{1}={x}_{1}+{{\rm{i}}{y}}_{1},{\alpha }_{2}={x}_{2}+{{\rm{i}}{y}}_{2}$. Consider the fact that $\langle {\psi }_{0}| {a}_{k}| {\psi }_{0}\rangle =x+{\rm{i}}{y}=1/2[{x}_{1}+{x}_{2}+{\rm{i}}({y}_{1}+{y}_{2})]$, then$ \begin{eqnarray}{y}_{1}+{y}_{2}=2y=\displaystyle \frac{2\mu {{\rm{\Omega }}}_{0}}{\sqrt{{\omega }_{0}}}.\end{eqnarray}$Next we would analyze the ultimate measurement precisions with the condition B=0 or D=0.
4.2.1.$B=0$ and $D\ne 0$
First we consider the case of $B=0 \& D\ne 0$. In this part, the ultimate measurement precisions of ω and ω can approach the SQL and HL, respectively, and their variances are inversely proportional to the coefficients A and D, respectively. B=0 leads to a constraint relation between x1 and x2, which can be written as$ \begin{eqnarray}\begin{array}{rcl}{x}_{2} & = & \displaystyle \frac{1}{\pi {\left(2\kappa +1\right)}^{2}{\sqrt{\omega }}_{0}}\left\{\mu (-{\omega }_{0}+{{\rm{\Omega }}}_{0}+2\kappa {{\rm{\Omega }}}_{0})\right.\\ & & +\left[{\pi }^{2}{x}_{1}^{2}{\left(2\kappa +1\right)}^{4}{\omega }_{0}+{\mu }^{2}{\left(-{\omega }_{0}+{{\rm{\Omega }}}_{0}+2\kappa {{\rm{\Omega }}}_{0}\right)}^{2}\right.\\ & & \left.{\left.-2\pi {x}_{1}{\left(2\kappa +1\right)}^{2}\mu {\sqrt{\omega }}_{0}({\omega }_{0}+{{\rm{\Omega }}}_{0}+2\kappa {{\rm{\Omega }}}_{0})\right]}^{\tfrac{1}{2}}\right\}.\end{array}\end{eqnarray}$From the expressions of A and D in equation (35) and consider equations (36) and (37), it is easy to find that D can be seen as a function of x1 and A is a function of x1 and y1. Hence, the ultimate measurement precisions of the two parameters would be determined with fixed x1 and y1 when ${\omega }_{0}$, ${{\rm{\Omega }}}_{0}$, R and m are given. In the previous subsection we have calculated the measurement precisions with the coherent state as the input motional state, just as seen in equations (27) and (31). Here we also use the coherent state as the input motional state. Suppose the two types of coherent states have the same trapping energy, i.e. ${x}_{1}^{2}+{x}_{2}^{2}+{y}_{1}^{2}+{y}_{2}^{2}={r}_{1}^{2}+{r}_{2}^{2}$, we would like to inquiry whether the measurement precisions in this part would surpass that in the previous subsection.
Note that ${\delta }_{2}$ here is the same as the one in equation (27) and ${\delta }_{1}$ is a negative number, so the coefficient D here may be larger than the one in Condition ${\rm{I}}$ subsection and hence we would get a more precise measurement result of ω if ${x}_{1}-{x}_{2}$ is also negative, which can be testified always true when x1 is less than zero. figure 3(a) intuitively shows that the ultimate measurement precision of ω in this part is always better than the result in equation (27) when ${x}_{1}\lt 0$, no matter what the true value of ${{\rm{\Omega }}}_{0}$ is. Besides, the comparative advantage becomes more obvious when the absolute value of x1 is larger. The comparison of the measurement precision of ω is seen in figure 3(b). The ratio varies with the true value of ${\omega }_{0}$. If ${\omega }_{0}$ is close to ${\mu }^{-2}$, then the measurement precision in equation (31) is better. If ${\omega }_{0}$ is far away from ${\mu }^{-2}$, the measurement result in this part would be more precise than the one in equation (31) with large absolute value of x1.
Figure 3.
New window|Download| PPT slide Figure 3.The ratio of ${\delta }^{2}{\rm{\Omega }}(\omega )$ with coherent motional input state in Condition $\mathrm{II}$ to that in Condition ${\rm{I}}$ versus the true value of ${{\rm{\Omega }}}_{0}({\omega }_{0})$ with different negative x1. We set the coherent states that under both conditions have the same trapping energy, i.e. ${x}_{1}^{2}+{x}_{2}^{2}+{y}_{1}^{2}+{y}_{2}^{2}={r}_{1}^{2}\,+{r}_{2}^{2}$ and let ${y}_{1}=10$, $\kappa =10$.
Combining these two pictures, we can say with the same trapping energy coherent motional input state, in some special cases, such as ${\omega }_{0}\approx {\mu }^{-2}$, ${{\rm{\Omega }}}_{0}\approx 0.1{\mu }^{-2}$ and short evolutionary time, such as $T\sim 21\pi /{\omega }_{0}$, the measurement precisions of both parameters under Condition $\mathrm{II}$ with negative x1 would be more precise than the one under Condition ${\rm{I}}$.
4.2.2.$D=0$ and $B\ne 0$
In this part, the ultimate measurement precision of ω can approach the HL while the one of ω only be scaled by the SQL. The variances of the two parameters are inversely proportional to the coefficients B and C, respectively. When $B\ne 0 \& D=0$, we would have$ \begin{eqnarray}{x}_{2}={x}_{1}-\mu \pi \sqrt{{\omega }_{0}}.\end{eqnarray}$And the coefficients of $B,C$ can be written as$ \begin{eqnarray}B=\displaystyle \frac{{\mu }^{2}}{4{\omega }_{0}^{2}}{\left[2\pi \mu {{\rm{\Omega }}}_{0}-2\sqrt{{\omega }_{0}}{x}_{0}\left(\displaystyle \frac{-2}{2\kappa +1}+{\pi }^{2}(2\kappa +1\right)\right]}^{2},\end{eqnarray}$$ \begin{eqnarray}C=\displaystyle \frac{4{\mu }^{2}}{{\omega }_{0}},\end{eqnarray}$where ${x}_{0}=({x}_{1}-\mu \pi \sqrt{{\omega }_{0}}/2)$. On the other hand, the trapping energy of a single particle in the potential well has the property that $2{\bar{E}}_{k}=| {\alpha }_{1}{| }^{2}+| {\alpha }_{2}{| }^{2}$$=\,{x}_{1}^{2}+{x}_{2}^{2}+{y}_{1}^{2}+{y}_{2}^{2}$$=2{x}_{0}^{2}+{\mu }^{2}{\pi }^{2}{\omega }_{0}/2+{y}_{1}^{2}+{y}_{2}^{2}$$\geqslant \,2{x}_{0}^{2}+{\mu }^{2}{\pi }^{2}{\omega }_{0}/2+2{\mu }^{2}{{\rm{\Omega }}}_{0}^{2}/{\omega }_{0}$$\geqslant \,{\mu }^{2}{\pi }^{2}{\omega }_{0}/2+2{\mu }^{2}{{\rm{\Omega }}}_{0}^{2}/{\omega }_{0}$, with the second last equality sign holds only if ${y}_{1}={y}_{2}=\mu {{\rm{\Omega }}}_{0}/\sqrt{{\omega }_{0}}$. Such property shows at least two facts: first, the trapping energy must be larger than ${\mu }^{2}{\pi }^{2}{\omega }_{0}/2+2{\mu }^{2}{{\rm{\Omega }}}_{0}^{2}/{\omega }_{0}$, or the measurement procedure cannot proceed. Second, the sign of x0 makes no difference when calculating the trapping energy. So when the trapping energy is fixed, we can always choose a negative x0 in equation (39) and then B would have a positive correlation with $| {x}_{0}| $. As ${y}_{1},{y}_{2}$ do not affect the results, for a fixed trapping energy input state, we’d better let ${y}_{1}={y}_{2}\,=\mu {{\rm{\Omega }}}_{0}/\sqrt{{\omega }_{0}}$ that makes $| {x}_{0}| $ and as well B largest. On the other hand, as the value of C in equation (40) has no concern with the input state, the optimal uncertainty of ω would always reach the same SQL in this part no matter what motional coherent state we choose, which also embodies a sense of stability. The best measurement result of $\omega ({\rm{\Omega }})$ is expressed as$ \begin{eqnarray}{\left({\delta }^{2}\omega \right)}_{\min }\sim \displaystyle \frac{1}{{N}^{2}}\cdot \displaystyle \frac{{\omega }_{0}^{2}}{4{\mu }^{2}{\left[(-2/{\kappa }_{0}+{\pi }^{2}{\kappa }_{0})\sqrt{{\omega }_{0}}{r}_{0}+\mu \pi {{\rm{\Omega }}}_{0}\right]}^{2}},\end{eqnarray}$$ \begin{eqnarray}{\left({\delta }^{2}{\rm{\Omega }}\right)}_{\min }\sim \displaystyle \frac{1}{N}\cdot \displaystyle \frac{{\omega }_{0}}{16{\mu }^{2}},\end{eqnarray}$where ${r}_{0}=\sqrt{({r}^{2}-{\mu }^{2}{\pi }^{2}{\omega }_{0}/2-2{\mu }^{2}{{\rm{\Omega }}}_{0}^{2}/{\omega }_{0})/2}$ is the maximum $| {x}_{0}| $, ${r}^{2}=| {\alpha }_{1}{| }^{2}+| {\alpha }_{2}{| }^{2}$ indicates the trapping energy and ${\kappa }_{0}=2\kappa +1$ refers to any positive odd number. It is clearly that larger r, and ${\kappa }_{0}$, namely, higher trapping energy and longer evolution time would lead to a more precise measurement result. For fixed μ and r, to obtain the minimum value of equation (41), the true values of ${\omega }_{0}$ and ${{\rm{\Omega }}}_{0}$ should be selected as$ \begin{eqnarray}{\omega }_{0}\sim \displaystyle \frac{{r}^{2}}{{\pi }^{2}{\mu }^{2}},\,\,{{\rm{\Omega }}}_{0}\sim \displaystyle \frac{{\kappa }_{0}{r}^{2}}{2{\mu }^{2}\sqrt{{\pi }^{4}{\kappa }_{0}^{4}-3{\pi }^{2}{\kappa }_{0}^{2}+4}},\end{eqnarray}$and the minimum variance of ω would be written as$ \begin{eqnarray}{\left({\delta }^{2}\omega \right)}_{{M}}\sim \displaystyle \frac{1}{{N}^{2}}\cdot \displaystyle \frac{{\kappa }_{0}^{2}}{{\pi }^{2}{\mu }^{4}({\pi }^{4}{\kappa }_{0}^{4}-3{\pi }^{2}{\kappa }_{0}^{2}+4)}.\end{eqnarray}$To make the value of the last equation as small as possible, ${\kappa }_{0}$ should be as large as possible. In other words, we could enhance the measurement precision by prolonging the evolution time.
We analyze the ultimate measurement precisions of ω and ω under the constraint conditions of equation (23) and equation (24) in this section. In the Condition ${\rm{I}}$ subsection, only the measurement precision of ω can approach the HL and the result has no concern with the input state. Additionally, the HL is the same as the one obtained in the single-parameter estimation scenario. In the Condition $\mathrm{II}$ subsection, either of the two parameters’ measurement precisions can approach the HL. We analyze the measurement precisions with the coherent state as the input motional state in that subsection, which can be further distributed into two parts. In the first part that the variance of $\omega ({\rm{\Omega }})$ is scaled by the SQL(HL), we find both of the two measurement precisions vary with the input state. With some special input states, the measurement precisions can surpass the ones in the Condition ${\rm{I}}$ subsection. In the second part that the variances of $\omega ({\rm{\Omega }})$ is scaled by the SQL(HL). We provide the concrete expressions of the ultimate measurement precisions of these two parameters.
5. Conclusion
Our work in this paper reveals some inherent features of the Sagnac interferometer in the two-parameter estimation process. We choose the multi-particle globally entangled state as the input state. The measured parameters are the rotation frequency of the Sagnac system and the trapping frequency. We provide the general expressions of the measurement precisions and find that no matter what the motional state of the particle is, the ultimate measurement precisions of these two parameters cannot reach the HL simultaneously, only one can approach the HL while the other only be scaled by the SQL. The explanation of such a phenomenon remains for further study. We also provide the constraint conditions for saturating the QCRB. We analyze the measurement precisions under these constraint conditions. Through these conditions, one can design and optimize a proper Sagnac interferometer and measurement scheme aiming at multi-parameter estimation.
Acknowledgments
The authors thanks Yanming Che, Fei Yao and Jie Chen for useful discussions.
Funding
National Key Research and Development Program of China (Grants No. 2017YFA0304202 and No. 2017YFA0205700), the NSFC (Grants No. 11 875 231 and No. 11 935 012), and the Fundamental Research Funds for the Central Universities through Grant No. 2018FZA3005.
Appendix. Derivation of of the generator operators of the measured parameters
,1,21Zhejiang Institute of Modern Physics, Department of Physics, 
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