Quantum simulation of bound-state-enhanced quantum metrology

Quantum metrology explores quantum effects to improve the measurement accuracy of some physical quantities beyond the classical limit. However, due to the interaction between the system and the environment, the decoherence can significantly reduce the accuracy of the measurement. Many methods have been proposed to restore the accuracy of the measurement in the long-time limit. Recently, it was found that the bound state can help improve measurement accuracy and recover the $t^{-1}$ scaling [Bai et al., Phys. Rev. Lett. 123, 040402 (2019)]. Here, by using $N$ qubits, we propose a method to simulate the open quantum dynamics of a hybrid system including one atom and coupled resonators. We find that the error of the measurement can decrease as the time increases due to the existence of the bound state. With both analytical and numerical simulations, we prove the $t^{-1}$ scaling of the measurement error can be recovered when there is a bound state in the hybrid system. Interestingly, we observe that there are regular oscillations which can be used for the evaluation of the atomic transition frequency. For a finite $N$, the duration of the regular oscillations doubles as one more qubit is involved.

Figure 1

Schematic illustration of a two-level atom and a coupled-cavity array. (a) Structure of coupled cavities and (b) a two-level atom.

Figure 2

The population dynamics $P_e\left(t\right)$ of an atomic excited state with small $N$, e.g., $N=3$ (blue dashed line) and $N=4$ (red solid line). Here, the parameters used in the calculation are ${\omega }_0=10\xi , \mathrm{\Omega }=11\xi$, and $J=1.3\xi$.

Figure 3

The population dynamics $P_e\left(t\right)$ of an atomic excited state with larger $N$, e.g., $N=6$ (blue dashed line) and $N=7$ (red solid line). The inset compares the numerical simulation of the regular oscillations of the population dynamics $P_e\left(t\right)$ with $N=8$ (red stars) vs the analytical solution (blue solid line) for Eq. (8) over a period of time. Here, the parameters are the same as in Fig. 2.

Figure 4

The duration of the regular oscillations as a function of the number of qubits $N$. We use the function $ln\left(\xi T\right)=0.72823N-1.0812$ (red solid line) to fit the data (blue dots) with the linear correlation coefficient $r=0.99967$. The other parameters are the same as in Fig. 2.

Figure 5

Fourier transform for the regular oscillations with different qubit numbers. The FWHMs of the main peak are, respectively, $0.288\xi ,0.135\xi$, and $0.082\xi$ for $N=6$ (blue dotted line), $N=7$ (red dashed line), and $N=8$ (green solid line). The other parameters are the same as in Fig. 2.

Figure 6

(a) The amplitude of the regular oscillations for different numbers of qubits $N$ obtained with the numerical simulation (blue dots) is compared to the one with infinite $N$ obtained with the analytical solution (blue solid line). (b) The mean of the regular oscillations for different numbers of qubits $N$ obtained with the numerical simulation (red dots) is compared to the one with infinite $N$ obtained with the analytical solution (red solid line). The other parameters are the same as in Fig. 2.

Figure 7

Comparison of analytical and numerical results for the uncertainty $\delta \mathrm{\Omega }$ vs time. The red solid line is obtained with Eq. (12), while the blue stars are obtained with the numerically exact solution with the following parameters: $N=8, {\omega }_0=20\xi ,\mathrm{\Omega }=20.5\xi$, and $J=3\xi$. With the selection of $N$, the regular oscillations for the probability of the atomic excited state stop at $\xi T=120$.

Figure 8

The scaling of the uncertainty $\delta \mathrm{\Omega }$ vs time. The blue dots are obtained with the numerical solution with the following parameters: $N=8, {\omega }_0=20\xi , \mathrm{\Omega }=20\xi$, and $J=0.3\xi$. The red solid line is obtained by fitting the data with the function $\text{ln}(\delta \mathrm{\Omega }/\xi )=0.97196\times \text{ln}(1/\xi t)+7.9211$ and the linear correlation coefficient $r=0.9995$.

Figure 9

Numerical simulation of the uncertainty $\delta \mathrm{\Omega }$ vs time with the measured atomic transition frequency $\mathrm{\Omega }$ beyond the band of the coupled resonators. The data are obtained with Eq. (9) with the following parameters: $N=8, \mathrm{\Omega }=17\xi , {\omega }_0=10\xi , J=0.3\xi$, and $\xi T=120$.