Entanglement of mechanical modes in a doubly resonant optomechanical cavity of a correlated emission laser

We explore the extent of the induced entanglement of the mechanical modes that can be attributed to the transfer of coherence from two-mode cavity radiation in a doubly resonant optomechanical cavity. It is expected that this scheme can support a generation of mechanical oscillations with a robust degree of entanglement combined with significant controllability. The entanglement is found to be sensitive to the specific choices of the frequencies of the bichromatic drive laser. It also turns out that the degree of entanglement would be enhanced with increasing rates of injection of the atoms but with decreasing initial lengths of a doubly resonant cavity and atomic decay rates. In addition, the entanglement is found to behave qualitatively in the same way for the measures of entanglement we have applied. Since the scheme we considered can possibly be implemented with current technology and allows the quantum features of cavity radiation to be accessible for application, we anticipate that it can be utilized in the realization of continuous-variable quantum information processing.

Figure 1

(a) Schematic representation of a two-mode laser coupled to two movable mirrors, $M_1$ and $M_2$. The doubly resonant cavity is driven simultaneously by a physically accessible bichromatic laser of frequencies ${\omega }_{L_1}$ and ${\omega }_{L_2}$. The two-mode radiations with realistic frequencies ${\omega }_1$ and ${\omega }_2$ generated by a three-level atom are filtered by a beam splitter (BS) and are coupled to their respective harmonically oscillating mirrors via radiation pressure. (b) The cascade configuration of a three-level lasing atom that emits two-mode light. An external laser of amplitude $\chi$ is applied to generate coherent superposition between the involved atomic energy levels.

Figure 2

Plots of $(\mathrm{\Delta }{u}^2+\mathrm{\Delta }{\nu }^2)$ against $\chi /\gamma$ at steady state for simultaneously applied two-input lasers driving the cavity with frequencies ${\omega }_{L_1}=741.0\pi$ THz and ${\omega }_{L_2}=564.0\pi$ THz (brown solid curve), ${\omega }_{L_1}={\omega }_{L_2}=741.0\pi$ THz (blue dash-dotted curve), ${\omega }_{L_1}={\omega }_{L_2}=564.0\pi$ THz (red dashed curve), and ${\omega }_{L_1}=564.0\pi$ THz and ${\omega }_{L_2}=741.0\pi$ THz (black dotted curve). The other parameters are as shown in Table 1.

Figure 3

Plots of $(\mathrm{\Delta }{u}^2+\mathrm{\Delta }{\nu }^2)$ against $\chi /\gamma$ at steady state for different atomic injection rates $r_a=3.12\mathrm{MHz}$ (brown dash-dotted curve), $r_a=1.30\mathrm{MHz}$ (blue dotted curve), $r_a=1.00\mathrm{MHz}$ (black dashed curve), and $r_a=0\mathrm{MHz}$ (red solid curve). The other parameters are as shown in Table 1.

Figure 4

Plots of $(\mathrm{\Delta }{u}^2+\mathrm{\Delta }{\nu }^2)$ against $\chi /\gamma$ at steady state for different initial lengths of a doubly resonant cavity, $L_1=0.426$ mm and $L_2=0.324$ mm (blue dash-dotted curve), $L_1=0.532$ mm and $L_2=0.405$ mm (red solid curve), $L_1=0.6384$ mm and $L_2=0.486$ mm (black dotted curve), and $L_1=0.7448$ mm and $L_2=0.567$ mm (brown dashed curve). The other parameters are as shown in Table 1.

Figure 5

Plots of $V_s$ against $\chi /\gamma$ at steady state for simultaneously applied two-input lasers driving the cavity with frequencies ${\omega }_{L_1}=741.0\pi$ THz and ${\omega }_{L_2}=564.0\pi$ THz (red dash-dotted curve), ${\omega }_{L_1}={\omega }_{L_2}=741.0\pi$ THz (black solid curve), ${\omega }_{L_1}={\omega }_{L_2}=564.0\pi$ THz (blue dotted curve), and ${\omega }_{L_1}=564.0\pi$ THz and ${\omega }_{L_2}=741.0\pi$ THz (brown dashed curve). The other parameters are as shown in Table 1.

Figure 6

Plots of $V_s$ against $\chi /\gamma$ at steady state for different atomic decay rates, $\gamma =14.50\mathrm{MHz}$ (black solid curve), $\gamma =13.50\mathrm{MHz}$ (red dotted curve), and $\gamma =12.50\mathrm{MHz}$ (blue dash-dotted curve). The other parameters are as shown in Table 1.

Figure 7

Plots of $(\mathrm{\Delta }{u}^2+\mathrm{\Delta }{\nu }^2)$ and $V_s$ against $\chi /\gamma$ at steady state for the parameters shown in Table 1.