Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system

We propose an effective approach for generating highly pure and strong cavity-mechanical entanglement (or optical-microwave entanglement) in a hybrid modulated three-mode optomechanical system. By applying two-tone driving to the cavity and modulating the coupling strength between two mechanical oscillators (or between a mechanical oscillator and a transmission line resonator), we obtain an effective Hamiltonian where an intermediate mechanical mode acting as an engineered reservoir cools the Bogoliubov modes of two target system modes via beam-splitter-like interactions. In this way, the two target modes are driven to two-mode squeezed states in the stationary limit. In particular, we discuss the effects of cavity-driving detuning on the entanglement and the purity. It is found that the cavity-driving detuning plays a critical role in the goal of acquiring highly pure and strongly entangled steady states.

Figure 1

Schematic representation of the system. An optical cavity mode $a$ driven by a two-tone laser $E_L\left(t\right)$ is coupled to an intermediate mechanical mode $b_1$ with single-photon optomechanical coupling strength $g_1$. $b_1$ is in turn coupled, with a time-dependent coupling strength $g_2\left(t\right)$, to (a) another mechanical oscillator, or alternatively (b) a transmission line resonator $b_2$.

Figure 2

Stationary cavity-mechanical entanglement $E_N$ vs the ratio of the effective couplings $G_{+}/G_{-}$ for different values of $\delta$. (a) $\delta =10{\gamma }_1$, (b) $\delta =5{\gamma }_1$, and (c) $\delta ={\gamma }_1$. The other parameters are $G_{-}=2.5{\gamma }_1$, $\kappa ={\gamma }_2$, and ${\overline{n}}_1={\overline{n}}_2={\overline{n}}_d=0$.

Figure 3

Steady-state purity of the cavity mode $d$ and the mechanical mode $b_2$ against the ratio of the effective couplings $G_{+}/G_{-}$ for different values of $\delta$. (a) $\delta =10{\gamma }_1$, (b) $\delta =5{\gamma }_1$, and (c) $\delta ={\gamma }_1$. All other parameters are the same as those in Fig. 2.

Figure 4

Stationary cavity-mechanical entanglement $E_N$ vs the effective coupling $\delta$. The sets of parameters corresponding to different lines are ${\gamma }_2={\gamma }_1/20,G_{+}/G_{-}=0.604$ (red solid line); ${\gamma }_2={\gamma }_1/200,G_{+}/G_{-}=0.786$ (blue dashed line); and ${\gamma }_2={\gamma }_1/2000,G_{+}/G_{-}=0.918$ (olive dotted line). The other parameters are $G_{-}=2.5{\gamma }_1$, $\kappa ={\gamma }_2$, and ${\overline{n}}_1={\overline{n}}_2={\overline{n}}_d=0$.

Figure 5

Steady-state purity of the cavity mode $d$ and the mechanical mode $b_2$ vs the effective coupling $\delta$. All parameters are the same as those in Fig. 4.

Figure 6

The time evolution of the entanglement [(a) and (b)] and purity [(c) and (d)] of the quantum states of the cavity mode $d$ and mechanical mode $b_2$ with [(b) and (d)] and without [(a) and (c)] the nonresonant terms. The parameters are $G_{-}=2.5{\gamma }_1$, $G_{+}=0.918G_{-}$, $\kappa ={\gamma }_2={\gamma }_1/2000$, $\delta ={\gamma }_1$, ${\overline{n}}_2={\overline{n}}_d$, ${\omega }_1=10{\gamma }_1$, and ${\omega }_2=100{\gamma }_1$.