Probing of nonlinear hybrid optomechanical systems via partial accessibility

Hybrid optomechanical systems are emerging as a fruitful architecture for quantum technologies. Hence determining the relevant atom-light and light-mechanics couplings is an essential task in such systems. The fingerprint of these couplings is left in the global state of the system during nonequilibrium dynamics. However, in practice, performing measurements on the entire system is not feasible, and thus one has to rely on partial access to one of the subsystems, namely, the atom, the light, or the mechanics. Here we perform a comprehensive analysis to determine the optimal subsystem for probing the couplings. We find that if the light-mechanics coupling is known or irrelevant, depending on the range of the qubit-light coupling, then the optimal subsystem can be either the light or the qubit. In other scenarios, e.g., simultaneous estimation of the couplings, the light is usually the optimal subsystem. This can be explained as light is the mediator between the other two subsystems. Finally, we show that the widely used homodyne detection can extract a fair fraction of the information about the couplings from the light degrees of freedom.

Figure 1

Schematic of a hybrid optomechanical tripartite system. Two unknown parameters $g_1$ and $g_2$ are to be estimated in a cavity-mediated system with partial accessibility.

Figure 2

Von Neumann entropy $S\left(t\right)$ for each subsystem as a function of the scaled time ${\omega }_{m}t$ for different $g_1$ and $g_2$ coupling parameters. The dynamics evolves from an initial state as in Eq. (14)

Figure 3

Inverse of the QFI ${\left({\mathcal{Q}}_{ii}\right)}^{-1}$ for partial access and for the global state for four coupling parameters. Panels (a) to (d) show the estimation of $g_1$ by knowing $g_2$; panels (e) to (h) show the estimation of $g_2$ by knowing $g_1$.

Figure 4

Efficiency as defined as in Eq. (26) for the estimation of $g_1$ [panels (a) and (c)] and $g_2$ [panels (b) and (d)] for two time intervals. The height of each bar quantifies the performance with respect to the ultimate global bound, whereas the color for each bar represents the optimal subsystem.

Figure 5

Precision limits quantified by ${\left({\mathcal{Q}}^{-1}\right)}_{ii}$ for the estimation of $g_i$ when a nuisance parameter $g_j$ is present in the system.

Figure 6

Panels (a) and (c) show the efficiency for the estimation of $g_1$ in the presence of a nuisance unknown parameter $g_2$ for two time intervals. Similarly, panels (b) and (d) show the estimation of $g_2$ in the presence of a nuisance parameter $g_1$ for the same time windows.

Figure 7

Precision limits for the joint estimation $\mathrm{Tr}\left[{\mathcal{Q}}^{-1}\right]$ of parameters $g_1$ and $g_2$ as a function of time ${\omega }_{m}t$ for four coupling parameters. We have also included the optimized classical Fisher information scalar bound, $\overline{\mathrm{Tr}\left[{\mathcal{F}}^{-1}\right]}$, when the light field is measured using a homodyne detection scheme.

Figure 8

Panels (a) to (d) show the efficiency ratios for the single-parameter and nuisance multiparameter cases when the optimal subsystem and the global state are computed from a lossless closed dynamics. Panels (e) to (h) show the same as above when the optimal subsystem decoheres while the global state remains lossless.

Figure 9

Panels (a) and (b) show the single-parameter efficiency ratio between closed and open global states. Panels (c) and (d) show the nuisance multiparameter efficiency ratio between global states.