Multiparameter estimation via an ensemble of spinor atoms

Multiparameter estimation, which aims to simultaneously determine multiple parameters in the same measurement procedure, attracts extensive interest in measurement science and technologies. Here, we propose a multimode many-body quantum interferometry for simultaneously estimating linear and quadratic Zeeman coefficients via an ensemble of spinor atoms. Different from the scheme with individual atoms, by using a $N$-atom multimode Greenberger-Horne-Zeilinger state, the measurement precisions of the two parameters can simultaneously attain the Heisenberg scaling, and they respectively depend on the hyperfine spin number $F$ in the forms of $\mathrm{\Delta }p\propto 1/\left(FN\right)$ and $\mathrm{\Delta }q\propto 1/\left(F^{2}N\right)$. Moreover, the simultaneous estimation can provide better precision than the individual estimation. Further, by taking a three-mode interferometry with Bose-condensed spin-1 atoms for an example, we show how to perform the simultaneous estimation of $p$ and $q$. Our scheme provides a novel paradigm for implementing multiparameter estimation with multimode quantum-correlated states.

Figure 1

Schematic of the multiparameter estimation with spin-$F$ atoms. The spin-$F$ atoms contain $2F+1$ Zeeman sublevels ($\left|F,m_F\right.〉$, $m_F=-F,-F+1,...,F$), and a multiparticle state involving $2F+1$ Zeeman sublevels can be prepared as the input state. Then, the input state will accumulate different phases dependent on the two unknown parameters $p$ and $q$ caused by linear and quadratic Zeeman effects, respectively. Finally, the measurement on the output state and the parameter estimation are implemented.

Figure 2

The measurement precisions $\mathrm{\Delta }{p}_{\mathrm{QCRB}}$ and $\mathrm{\Delta }{q}_{\mathrm{QCRB}}$ versus $\theta$ for different single spin-$F$ atoms in binomial distribution among Zeeman sublevels.

Figure 3

The measurement precisions $\mathrm{\Delta }{p}_{\mathrm{QCRB}}$ and $\mathrm{\Delta }{q}_{\mathrm{QCRB}}$ versus the hyperfine spin-$F$ for the three different kinds of input states of the single atom.

Figure 4

The log-log scaling of $\mathrm{\Delta }{p}_{\mathrm{QCRB}}$ and ${\mathrm{\Delta }q}_{\mathrm{QCRB}}$ versus the total atomic number $N$ under different hyperfine spin-$F$ values for the optimal GHZ state and the optimal product state.

Figure 5

The log-log scaling of $\mathrm{\Delta }{p}_{\mathrm{QCRB}}$ and $\mathrm{\Delta }{q}_{\mathrm{QCRB}}$ versus the total atomic number $N$ for $F=1$ and $F=2$ under simultaneous estimation (blue solid lines) and individual estimation (red dashed lines).

Figure 6

The log-log scaling of (a) $\mathrm{\Delta }{p}_{\mathrm{QCRB}}$ and (b) $\mathrm{\Delta }{q}_{\mathrm{QCRB}}$ versus the total atom number $N$ for the optimal prepared state via SMD (the red triangles). The blue dashed lines are the corresponding fitting results.

Figure 7

(a) The gap $\mathrm{\Delta }{E}_1$ in units of $|c_2|$ between the first excited state and the ground state of the Hamiltonian in Eq. (33) with respect to $\epsilon /|c_2|$ and $\epsilon /|c_2|=\pm 2$ defines three distinct quantum phases. The log-log scaling of (b) $\mathrm{\Delta }{p}_{\mathrm{QCRB}}$ and (c) $\mathrm{\Delta }{q}_{\mathrm{QCRB}}$ versus the total atom number $N$ for the optimal prepared state via QPT (the red triangles). The blue dashed lines are the corresponding fitting results.

Figure 8

Schematic of three-mode interferometry. First, the system state among the three Zeeman sublevels is prepared. Second, a $\frac{\pi }{2}$ pulse coupling the states of $\left|1,m_F=\pm 1\right.〉$ is applied. Third, interrogation for accumulating the two relative phases ${\phi }_1(p,q)$ and ${\phi }_{-1}(p,q)$ is implemented. Fourth, another two $\frac{\pi }{2}$ pulses coupling the three components are applied. Fifth, the square of the population difference measurement is used to extract the relative phase, and the two parameters $p$ and $q$ are derived from the relative phase.

Figure 9

Observable measurements under different values of $p$ and $q$. Panels (a) and (c): $\langle {({\widehat{N}}_1-{\widehat{N}}_0)}^2\rangle$ versus $p$ and $q$. Panels (b) and (d): $\langle {({\widehat{N}}_{-1}-{\widehat{N}}_0)}^2\rangle$ versus $p$ and $q$. In panels (a) and (b), the prepared state $\left|{\psi }_1\right.〉$ is the optimal one given in Sec. 5a. In panels (c) and (d), the prepared state $\left|{\psi }_1\right.〉$ is the optimal one given in Sec. 5b. Here the total atomic number is $N=20$.

Figure 10

FFT spectra of $\langle {({\widehat{N}}_1-{\widehat{N}}_0)}^2\rangle$ and $\langle {({\widehat{N}}_{-1}-{\widehat{N}}_0)}^2\rangle$. Panels (a) and (c): The oscillation frequency of $\langle {({\widehat{N}}_1-{\widehat{N}}_0)}^2\rangle$ is close to $\omega =\left|4p\right|$, $|-2p+2q|$, and $|2p+2q|$ (from right to left). Panels (b) and (d): The oscillation frequency of $\langle {({\widehat{N}}_{-1}-{\widehat{N}}_0)}^2\rangle$ is close to $\omega =\left|4p\right|$ and $|-2p+2q|$ (from right to left). Panels (a) and (b): The prepared state $\left|{\psi }_1\right.〉$ is the optimal one given in Sec. 5a. Panels (c) and (d): The prepared state $\left|{\psi }_1\right.〉$ is the optimal one given in Sec. 5a. Here $N=20$, $p=10$, and $q=1$.