Floquet Engineering to Overcome No-Go Theorem of Noisy Quantum Metrology

Permitting a more precise measurement to physical quantities than the classical limit by using quantum resources, quantum metrology holds a promise in developing many revolutionary technologies. However, the noise-induced decoherence forces its superiority to disappear, which is called no-go theorem of noisy quantum metrology and constrains its application. We propose a scheme to overcome the no-go theorem by Floquet engineering. It is found that, by applying a periodic driving on the atoms of the Ramsey spectroscopy, the ultimate sensitivity to measure their frequency characterized by quantum Fisher information returns to the ideal $t^2$ scaling with the encoding time whenever a Floquet bound state is formed by the system consisting of each driven atom and its local noise. Combining with the optimal control, this mechanism also allows us to retrieve the ideal Heisenberg-limit scaling with the atom number $N$. Our result gives an efficient way to avoid the no-go theorem of noisy quantum metrology and to realize high-precision measurements.

Figure 1

Schematic illustration of noisy quantum metrology under periodic control. ${\widehat{\mathcal{U}}}_j(t)$ is the evolution operator of the $j$th periodically driven atom interacting with its environment.

Figure 2

(a) Quasienergy spectrum in different driving amplitude $A$. The shaded backgrounds mark the regions with the FBS. (b) Evolution of ${\mathcal{F}}_{{\omega }_0}/t^2$ in different $A$ via numerically solving Eq. (5). The red line shows the result evaluated from Eq. (7) at $t=31T$. (c) Evolution of ${\mathcal{F}}_{{\omega }_0}$ in the absence of the FBS when $A=16h$ (magenta dot-dashed line), $20h$ (green dashed line), and 0 (black line). (d) Evolution of ${\mathcal{F}}_{{\omega }_0}/t$ in the presence of the FBS when $A=11h$ (light-red line), whose stroboscopic values are given by the red line marked by the red squares. The lines marked by the orange triangle and the cyan diamond are the stroboscopic values of ${\mathcal{F}}_{{\omega }_0}/t$ when $A=8h$ and $32h$, respectively. The blue dashed lines are the QFI evaluated from Eq. (6), which match with the stroboscopic values in the long-time limit. The insert is the comparison of the numerical result (light-red line) with the one evaluated from the FBS (black dot-dashed line). We use ${\omega }_T=12h$, $g={\omega }_0=h$, ${\omega }_c=0$, and $N=20$.

Figure 3

Stroboscopic values of ${\mathcal{F}}_{{\omega }_0}/N$ at $t=50T$ as a function of the atom number $N$ in (a) for $A=8h$ (the orange triangle line), $11h$ (the red square line), and $32h$ (the cyan diamond line), and in (b) for all $A\in [0,100]h$ supporting the FBS (the gray lines). The inset of (a) shows the large-$N$ behavior for $A=32h$, where the blue dashed and green dot-dashed lines are from the first and second terms of Eq. (7), respectively. The cyan in (b) is the ideal HL. The red line is the numerical result under the optimized $A$ such that ${\mathrm{opt}}_A{y}_N(Z^2)=0.5$. The black dashed line is the analytic result evaluated from Eq. (7). (c) $f_N(Z^2)$ as a function of $Z^2$ for $N$ ranging from 1 to 19 with a fixed interval 2 by the lines from left to the right. The dependence of $Z^2$ on $N$ to make $y_N(Z^2)=0.5$ is denoted by the blue dots in the inset, which are numerically fitted as $Z^2=e^{-1.1/N}$. (d) Optimized $A$ in different $N$ such that $y_N(Z^2)=0.5$, which causes ${\lim }_{t\to \infty }{\mathcal{F}}_{{\omega }_0}\propto N^2$ shown by the red line in (b). The inset shows $Z^2$ in different $A$. The shaded backgrounds mark the regions where the FBS is formed. Other parameters are the same as Fig. 2.