Enhanced output entanglement with reservoir engineering

We study the output entanglement in a three-mode optomechanical system via reservoir engineering by shifting the center frequency of filter function away from resonant frequency. We find the bandwidth of the filter function can suppress the entanglement in the vicinity of resonant frequency of the system, while the entanglement will become strong if the center frequency departs from the resonant frequency. We obtain the approximate analytical expressions of the output entanglement, from which we give the optimal center frequency at which the entanglement takes the maximum. Furthermore, we study the effects of time delay between the two output fields on the output entanglement, and obtain the optimal time delay for the case of large filter bandwidth.

Figure 1

A three-mode optomechanical system with a mechanical resonator (mode $\widehat{b}$) interacted with two cavities (cavities 1 and 2). Cavity 1 is driven with a red-detuned laser, while cavity 2 is driven with a blue-detuned laser. The entanglement between the output fields of two cavities can be generated.

Figure 2

(a) The entanglement vs the normalized center frequency $\omega /\kappa$. The black solid line is the numerical results, the red dashed line is plotted according to the analytical expression, Eq. (10). (b) The squeezing parameter $R_{12}$ (red dashed line), the thermal populations ${\overline{n}}_1/{10}^3$ (blue dotted line), and ${\overline{n}}_2/{10}^3$ (black solid line) vs the normalized center frequency $\omega /\sigma$. The parameters are $\gamma =1,\sigma =10,\kappa ={10}^5,G=\kappa /10$.

Figure 3

(a) The entanglement vs the normalized center frequency $\omega /\kappa$. The black solid line is the numerical results, the red dashed line is plotted according to the analytical expression, Eq. (11). (b) The squeezing parameter $R_{12}$ (red dashed line), the thermal populations ${\overline{n}}_1/{10}^7$ (blue dotted line), and ${\overline{n}}_2/{10}^7$ (black solid line) vs the normalized center frequency $\omega /\sigma$. The parameters are $\gamma =1,\sigma =10,\kappa ={10}^5,G=10\kappa$.

Figure 4

(a) The entanglement $\text{En}$ vs the normalized center frequency $\omega /\kappa$: The red solid line is the entanglement plotted with the optimal time delay, Eq. (14); the blue dashed line is the entanglement plotted with the numerical optimal time delay making the entanglement $\text{En}$ maximum; the black solid line is the entanglement plotted according to the analytical expression, Eq. (12), without time delay; and the green dashed-dotted line is the entanglement plotted by numerical results according to the logarithmic negativity without time delay. (b) The optimal time delay ${\tau }_{\text{opt}}$ (red solid line) according to Eq. (14) and the numerical optimal time delay (blue dashed line). The parameters are $\gamma =1,\sigma =\kappa ={10}^5,G=10\kappa$.