Dissipation-induced optomechanical entanglement with the assistance of Coulomb interaction

We consider a hybrid three-mode optomechanical system where one mechanical oscillator couples to a second one via a Coulomb force and to a cavity mode via an optomechanical interaction. It is shown that stationary cavity-mechanical entanglement can be achieved by effectively cooling a Bogoliubov mode via the intermediate mode acting as an engineered reservoir. The entanglement can be maximized by carefully balancing the two confronting effects of the ratio of the effective couplings. Moreover, we analyze in detail the effects of the nonresonant terms. It is found that while the stability zone of the system shrinks in the parameter plane for large cooperativity, the maximal entanglement is not significantly reduced by the nonresonant terms. We numerically optimize the ratio in the stable region and obtain remarkably strong entanglement in the large cooperativity regime.

Figure 1

Schematic representation of our system. The cavity optomechanical system consists of a fixed mirror $A$ and a mechanical oscillator $B_1$ which is coupled to the other mechanical oscillator $B_2$ by the Coulomb interaction. The cavity is driven by the pumping field $\varepsilon$ with frequency ${\omega }_d$. The electrode carrying charge $Q_1$ $\left(Q_2\right)$ on $B_1$ $\left(B_2\right)$ is charged by the bias gate voltage $V_1$ ($V_2$). $d_0$ is equilibrium separation of $B_1$ and $B_2$. $x_1$ and $x_2$ are, respectively, the small deviations of $B_1$ and $B_2$ from their equilibrium positions due to the optomechanical and Coulomb interactions.

Figure 2

(a) Stationary cavity-mechanical entanglement $E_N$ and (b) steady-state occupancies of the the Bogoluibov modes $\langle {\widehat{\beta }}_j^{†}{\widehat{\beta }}_j\rangle$ versus the ratio of the effective couplings $G_{+}/G_{-}$. The solid blue and dashed red curves in (a) correspond, respectively, to the entanglement $E_N$ generated with and without the effects of the nonresonant terms ${\widehat{H}}_{\mathrm{cr}}$ in Eq. (3). All plots are obtained with fixed $G_{-}=2.5{\gamma }_1$ and a zero cavity bath thermal occupation ${\overline{n}}_d=0$. The other parameters are ${\omega }_1={\omega }_2=10{\gamma }_1$, ${\overline{n}}_1={\overline{n}}_2=0.2$, and $\kappa ={\gamma }_2={\gamma }_1/20$.

Figure 3

(a) The maximized entanglement $E_N$ and (b) the optimal ratio $G_{+}/G_{-}$ as functions of the effective optomechanical coupling $G_{-}$ for different mechanical bath thermal occupations. Here $E_N$ in (a) is obtained with the corresponding optimal $G_{+}/G_{-}$ in (b) for every $G_{-}$ under the rotating-wave approximation. Remaining parameters are the same as Fig. 2.

Figure 4

(a) The entanglement $E_N$ and (b) the indicator of the stability $\tilde{\lambda }$ as functions of the ratio $G_{+}/G_{-}$ for a large cooperativity $C_{-}=4G_{-}^2/\left({\gamma }_2{\gamma }_1\right)=5000$. The solid blue and dashed green lines in (a) correspond, respectively, to the results with and without the contributions of the nonresonant terms ${\widehat{H}}_{\mathrm{cr}}$. The plots in (b) are obtained including ${\widehat{H}}_{\mathrm{cr}}$. Here $\tilde{\lambda }$ is defined as the maximum of real parts of the eigenvalues of the full coefficient matrix $M$ in Eq. (9). The system is stable when $\tilde{\lambda }\le 0$; otherwise the system is unstable. Remaining parameters are the same as Fig. 2.

Figure 5

The entanglement $E_N$ versus the cooperativity $C_{-}$ for increasing decay ratio ${\gamma }_1/{\gamma }_2$. $E_N$ is evaluated with full Hamiltonian using the numerically optimized $G_{+}/G_{-}$. We take ${\omega }_1={\omega }_2=20{\gamma }_1$ and zero bath temperatures for all modes, i.e., ${\overline{n}}_d={\overline{n}}_1={\overline{n}}_2=0$, for the purposes of comparing our results with related ones.

Figure 6

Phase diagram of system stability with full Hamiltonian ${\widehat{H}}_{\mathrm{lin}}$ [Eq. (3)] in the parameter plane. The color and white regions represent the stable and unstable zones, respectively. The dashed blue curve is the numerically optimized $G_{+}/G_{-}$ in the stable region that is used to calculated $E_N$ (the line with ${\gamma }_1/{\gamma }_2=20$) in Fig. 5. The solid red curve corresponds to the approximate analytical optimal $G_{+}/G_{-}$ [Eq. (22)], which is derived under the rotating-wave approximation and is exploited for evaluating the entanglement in the large cooperativity limit in Ref. [19]. We see from the inset that the solid red curve is no longer confined to the stable (color) zone for sufficiently large cooperativity. The other parameters are the same as Fig. 5.