Heisenberg-limited estimation of the coupling rate in an optomechanical system with a two-level system

We theoretically investigate the high-precision estimation of the coupling rate, which can reflect an actual observable, in a hybrid optomechanical system (OMS). In our scheme, apart from an optical cavity, the mechanical vibrator is also coupled to a two-level system (TLS) via a direct dispersive coupling. As for the initial state, we take into account coherent states for both optical and mechanical modes. After evolving a period of the vibrator, the final state then carries the information about the estimated coupling rate. We find that through measuring the internal state of the TLS, the precision can achieve a Heisenberg limit that indicates remarkable improvement of estimation in quantum metrology. This result is obviously shown in a simple analytical expression under the approximate condition, and further analysis demonstrates that the Heisenberg limit can be attained indeed. Moreover, we also study the dependence of the precision on the estimated parameter to reveal the appropriate measurement ranges.

Figure 1

Scheme of the tripartite hybrid optomechanical system for coupling strength estimation. The cavity confines the photons with fixed and harmonically moving mirrors, and the latter is regarded as a mechanical vibrator generating phonons. Thus, both of the resonators are coupled together with radiation pressure. Besides, a real or artificial atom, described as a two-level system (TLS), is attached to the mechanical vibrator, resulting in direct dispersive coupling.

Figure 2

Precision $\delta R$ as a function of the cavity photon number $N$ for $R={10}^{-4}$ (thin solid line), $R={10}^{-3}$ (thick solid line), and $R={10}^{-2}$ (dotted line) in both charts (a) and (b) that display different ranges of $N$. Besides, the thick and thin dashed lines display the Heisenberg limit (HL) and standard quantum limit (SQL) for comparison, respectively. Note that the overlap happens between the lines for $R={10}^{-4}$ and for HL in both charts (a) and (b), as well as between the lines for $R={10}^{-3}$ and for HL in chart (a), i.e., in the lower $N$ range.

Figure 3

The log-log plot displays the dependence of precision $\delta R$ on the estimated parameter $R$ for various cavity photon numbers $N$. The thick dashed, the thin dashed, and solid lines represent cases $N=10$, $N={10}^2$, and $N={10}^3$, respectively.

Figure 4

Precision $\delta R$ as a function of the cavity photon number $N$ for $R=0.251$ (thin solid line), $R=0.253$ (thick solid line), and $R=0.26$ (dotted line). The thick and thin dashed lines display the Heisenberg limit (HL) and standard quantum limit (SQL) for comparison, respectively. Note that the precision line for $R=0.251$ overlaps with that for HL in the considered $N$ range.