Figure 1

Schematic of one realization of a three-mode optomechanical system, where two cavity modes are coupled to a single mode of a mechanical resonator. By driving cavity 1 (2) at the red (blue) detuned mechanical sideband, a dissipative entanglement mechanism is realized. To enhance the scheme, the mechanical resonator is cavity cooled and optically damped via a coupling to a third driven cavity mode, so that its total damping rate $\gamma$ is greater than the damping rate $\kappa$ of each cavity.Reuse & Permissions

Figure 2

Stationary intracavity entanglement (as quantified by log negativity $E_N$, left scale) as a function of the entangling interaction $G_2$. We take ${\overline{n}}_{\mathrm{th},1}={\overline{n}}_{\mathrm{th},2}=0$, a mechanical frequency of ${\omega }_M=2\pi \times 10\mathrm{MHz}$, damping $\gamma =2\pi \times 0.8\mathrm{MHz}$, and do not make the rotating wave approximation. The solid red thin curve corresponds to a fixed value of $G_1=2\pi \times 2\mathrm{MHz}$, and ${\kappa }_1={\kappa }_2=2\pi \times 50\mathrm{kHz}$. One clearly sees a nonmonotonic dependence on $G_2$. The solid blue thick curve and short-dashed blue curve instead correspond to tuning $G_1$ to the value $G_1^{\mathrm{opt}}$ for each $G_2$, such that the dissipative entangling mechanism can be optimized. The value of $G_1^{\mathrm{opt}}-G_2$ [cf. Eq. (10)] is indicated by the long-dashed brown curve (right scale). The solid blue (dashed blue) curve corresponds to ${\kappa }_1={\kappa }_2=50\mathrm{kHz}$ (${\kappa }_{1/2}/2\pi =45\mathrm{kHz}$, 55 kHz) and ${\overline{n}}_{\mathrm{th},M}=0$ (${\overline{n}}_{\mathrm{th},M}=0.3$). The green dash-dotted line represents the maximum stationary entanglement achievable with a coherent two-mode squeezing interaction $E_N=\ln 2$.Reuse & Permissions

Figure 3

Intracavity stationary entanglement (quantified by $E_N$) versus cooperativity $C_2$, where we use an optimized choice for $G_1$ [as given by Eq. (10)], and have taken ${\kappa }_1={\kappa }_2$ and zero temperature. The solid lines correspond to different choices of the damping ratio $\gamma /\kappa$ as indicated; increasing $\gamma /\kappa$ increases the amount that one can cool the delocalized ${\widehat{\beta }}_A$ mode, and hence enhances entanglement. For large $C_2$, these curves asymptote to the value in Eq. (12). The dashed blue line is the asymptotic expression of Eq. (11). The green dashed-dotted line indicates $E_N=\ln 2$ (the maximum $E_N$ possible with a two-mode squeezing interaction).Reuse & Permissions