Quantum metrology enhanced by the $XY$ spin interaction in a generalized Tavis-Cummings model

Quantum metrology is recognized for its capability to offer high-precision estimation by utilizing quantum resources, such as quantum entanglement. Here, we propose a generalized Tavis-Cummings model by introducing the $XY$ spin interaction to explore the impact of the many-body effect on estimation precision, quantified by the quantum Fisher information (QFI). By deriving the effective description of our model, we establish a closed relationship between the QFI and the spin fluctuation induced by the $XY$ spin interaction. Based on this exact relation, we emphasize the indispensable role of the spin anisotropy in achieving Heisenberg-scaling precision for estimating a weak magnetic field. Furthermore, we observe that the estimation precision can be enhanced by increasing the strength of the spin anisotropy. We also reveal a clear scaling transition of the QFI in the Tavis-Cummings model with a reduced Ising interaction. Our results contribute to the enrichment of metrology theory by considering many-body effects, and they also present an alternative approach to improving the estimation precision by harnessing the power provided by many-body quantum phases.

Figure 1

Schematic illustration of the generalized TC model: A coherent light $|\alpha \rangle$ is injected into the cavity and coupled to $N$ trapped spins (depicted in green) with a coupling strength $g$. These $N$ spins collectively form the $XY$ spin chain and are employed for sensing a magnetic field $h$.

Figure 2

Comparison of expectation values derived from the original Hamiltonian (1), indicated by the blue circles, and those obtained from the effective Hamiltonian (3), represented by the red dashed line. We have employed the experimental parameters [76]: ${\omega }_0/\left(2\pi \right)=6.9$ GHz, ${\omega }_{\mathrm{a}}/\left(2\pi \right)=6.89$ GHz, $g/\left(2\pi \right)=1.05$ MHz. Other parameters are $N=4$, $\overline{n}=40$, $\lambda =\gamma =1$, $h={10}^{-5}$ Hz, $\theta =\pi /2$, $\varphi =0$, and $\phi =\pi /3$. The numerical computations have been conducted using the qutip software package [77].

Figure 3

Phase diagram of the $XY$ model for $\gamma \ge 0$ and $h\ge 0$ under the normalization condition $\lambda =1$. The model exhibits critical behavior along the red line. The critical magnetic field $h_{\mathrm{c}}=1$ separates the paramagnetic and the ferromagnetic phases. The blue dashed line corresponds to the Ising model. On the black dots ${\gamma }^2+h^2=1$, the ground state can be factorized as a product state.

Figure 4

Comparison of numerical and analytic results of sensitivity $1/\sqrt{{\mathcal{F}}_h}$ (in units of $1/t$) for the TC model with the Ising interaction: (a) the weak-field case $h={10}^{-5}$ Hz and (b) the strong-field case $h={10}^5$ Hz. The blue circles represent the numerical result obtained from Eq. (6), while the red dashed line denotes the analytic sensitivity given by Eqs. (16) and (17). The gray dotted-dashed line indicates SQL precision ($\propto \overline{n}$) while the black line represents HL precision ($\propto {\overline{n}}^2$). In both cases, $N=8$ and the other parameters are the same as those in Fig. 2.

Figure 5

QFI (in units of $t^2$) vs the magnetic field $h$ in the TC model with the Ising interaction. The blue circles are the numerical value of QFI obtained from Eq. (6). Here, the number of spins is $N=40$ and the average photon number is $\overline{n}=1000$. The other parameters are the same as those in Fig. 2.

Figure 6

(a) Comparison of numerical and analytic results of sensitivity $1/\sqrt{{\mathcal{F}}_h}$ (in units of $1/t$) for the TC model with the $XY$ interaction. The circles, upper triangles, lower triangles, and squares indicate the numerical results for $\gamma =0$, $\gamma =0.01$, $\gamma =1$, and $\gamma =100$, respectively. The black dashed line, black solid line, and gray solid line denote the analytical results, Eqs. (20), (16), and (21), for $\gamma =0$, $\gamma =1$, and $\gamma \gg 1$. (b) Numerical result of the rescaled variance $\mathrm{Var}\left(J_z\right)/N$ vs the anisotropy parameter $\gamma$. The number of spins is taken as $N=40$. The magnetic field is chosen as $h=0.1$ and other parameters are the same as those in Fig. 2.