Study of optical bistability/multistability and transparency in cavity-assisted-hybrid optomechanical system embedded with quantum dot molecules

Abstract

We theoretically explore optical bistability/multistability and transmission spectrum in a hybrid optomechanical system consisting of quantum dot molecules in a semiconductor cavity which is in turn coupled to an auxiliary cavity. This auxiliary cavity exhibits both linear and quadratic couplings to a mechanical oscillator. Here, the hybrid optomechanical system is continuously driven by a strong laser from both ends and a weak probe field applied to an auxiliary cavity. The nonlinear nature of the system in the Heisenberg-Langevin equation is taken into consideration and the perturbation method is utilized to deal with problems in the continuous wave regime. The outcome reveals that optical multistability and transmission (absorption and dispersion) spectrum can be deftly manipulated by properly adjusting the parameters. Such an investigation may be useful in designing all-optical switching devices.

Appendices

Appendix A

$\alpha _{1}$ and $\beta _{1}$ are defined as follows

$$\begin{aligned} \alpha _{1}=\, & {} \frac{(\gamma _{1} \gamma _{2} \gamma _{a}- \gamma _{1} \Delta _{2} \Delta _{a} - \gamma _{2} \Delta _{1} \Delta _{a} - \gamma _{a} \Delta _{1} \Delta _{2}+\gamma _{a}T_{e}^{2}+\gamma _{2}G^{2})(\gamma _{1}\gamma _{2}-\Delta _{1}\Delta _{2}+T_{e}^{2})}{(\gamma _{1}\gamma _{2}-\Delta _{1}\Delta _{2}+T_{e}^{2})^{2}+(\gamma _{1}\Delta _{2}+\gamma _{2}\Delta _{1})^{2}}\nonumber \\+ & {} \frac{(\gamma _{1} \gamma _{2}\Delta _{a}+\gamma _{1} \gamma _{a} \Delta _{2}+\gamma _{2}\gamma _{a}\Delta _{1}-\Delta _{1}\Delta _{2}\Delta _{a}+\Delta _{a}T_{e}^{2}+\Delta _{2}G^{2})(\gamma _{1}\Delta _{2}+\gamma _{2}\Delta _{1})}{(\gamma _{1}\gamma _{2}-\Delta _{1}\Delta _{2}+T_{e}^{2})^{2}+(\gamma _{1}\Delta _{2}+\gamma _{2}\Delta _{1})^{2}} \end{aligned}$$

(38)

$$\begin{aligned} \beta _{1}= & {} \frac{(\gamma _{1} \gamma _{2}\Delta _{a}+\gamma _{1} \gamma _{a} \Delta _{2}+\gamma _{2}\gamma _{a}\Delta _{1}-\Delta _{1}\Delta {2}\Delta {a}+\Delta _{a}T_{e}^{2}+\Delta _{2}G^{2})(\gamma _{1}\gamma _{2}-\Delta _{1}\Delta _{2}+T_{e}^{2})}{(\gamma _{1}\gamma _{2}-\Delta _{1}\Delta _{2}+T_{e}^{2})^{2}+(\gamma _{1}\Delta _{2}+\gamma _{2}\Delta _{1})^{2}}\nonumber \\- & {} \frac{(\gamma _{1} \gamma _{2} \gamma _{a}- \gamma _{1} \Delta _{2} \Delta _{a} - \gamma _{2} \Delta _{1} \Delta _{a} - \gamma _{a} \Delta _{1} \Delta _{2}+\gamma _{a}T_{e}^{2}+\gamma _{2}G^{2})(\gamma _{1}\Delta _{2}+\gamma _{2}\Delta _{1})}{(\gamma _{1}\gamma _{2}-\Delta _{1}\Delta _{2}+T_{e}^{2})^{2}+(\gamma _{1}\Delta _{2}+\gamma _{2}\Delta _{1})^{2}}. \end{aligned}$$

(39)

Appendix B

The $\Lambda$, $\chi$, $\mu$ and $\tau$ are defined as follows

$$\begin{aligned} \Lambda= & {} a+\frac{ce+df}{e^{2}+f^{2}},\end{aligned}$$

(40)

$$\begin{aligned} \chi= & {} b+\frac{de-cf}{e^{2}+f^{2}} \end{aligned}$$

(41)

where,

$$\begin{aligned} a= & {} \delta \gamma _{b} \gamma _{P}+(\Delta _{b}-\delta )\delta ^{2}+(\Delta _{b}-\delta )\omega _{m}(2G_{2}|b_{2}|^{2}-\omega _{m})-\delta ^{2}(G_{1}+G_{2}x_{s})x_{s}\nonumber \\- & {} (G_{1}+G_{2}x_{s})x_{s}\omega _{m}(2G_{2}|b_{2}|^{2}-\omega _{m})+\omega _{m}(G_{1}+G_{2}x_{s})^{2} x_{s}|b_{s}|^{2},\end{aligned}$$

(42)

$$\begin{aligned} b= & {} (\Delta _{b}-\delta )\delta \gamma _{P}-(G_{1}+G_{2}x_{s})\delta \gamma _{P} x_{s}-\gamma {b}\delta ^{2}-\gamma _{b}\omega _{m}(2G_{2}|b_{2}|^{2}-\omega _{m}), \end{aligned}$$

(43)

$$\begin{aligned} c=\, & {} J^{2}(T_{e}^{2}\gamma _{P}\delta +\gamma _{1}\gamma _{2}\gamma _{P}\delta -(\Delta _{1}-\delta )(\Delta _{2}-\delta )\gamma _{P}\delta +\delta ^{2}\gamma _{2}(\Delta _{1}-\delta )+\delta ^{2}\gamma _{1}(\Delta _{2}-\delta )\nonumber \\+ & {} \omega _{m}\gamma _{2}(\Delta _{1}-\delta )(2G_{2}|b_{2}|^{2}-\omega _{m})+\omega _{m}\gamma _{1}(\Delta _{2}-\delta )(2G_{2}|b_{2}|^{2}-\omega _{m})),\end{aligned}$$

(44)

$$\begin{aligned} d= \,& {} J^{2}(-T_{e}^{2}\delta ^{2}-\omega _{m}T_{e}^{2}2G_{2}|b_{s}|^{2}+\omega _{m}^{2}T_{e}^{2}-\gamma _{1}\gamma _{2}\delta ^{2}-\gamma _{1}\gamma _{2}\omega _{m}2G_{2}|b_{s}|^{2}+\omega _{m}^{2}\gamma _{1}\gamma _{2}\nonumber \\{} & {} \quad +(\Delta _{1}-\delta )(\Delta _{2}-\delta )\delta ^{2}\nonumber \\+ & {} (\Delta _{1}-\delta )(\Delta _{2}-\delta )\omega _{m}2G_{2}|b_{s}|^{2}-(\Delta _{1}-\delta )(\Delta _{2}-\delta )\omega _{m}^{2}+\gamma _{P}\gamma {2}\delta (\Delta _{1}-\delta )+\gamma _{1}\gamma _{P}\delta (\Delta _{2}-\delta )),\end{aligned}$$

(45)

$$\begin{aligned} e= \,& {} G^{2}\gamma _{2}+T_{e}^{2}\gamma _{a}+\gamma _{1}\gamma _{2}\gamma _{a}-\gamma _{a}(\Delta _{1}-\delta )(\Delta _{2}-\delta )-\gamma _{1}(\Delta _{a}-\delta )(\Delta _{2}-\delta )-(\Delta _{a}-\delta )(\Delta _{1}-\delta )\gamma _{2},\end{aligned}$$

(46)

$$\begin{aligned} f= \,& {} G^{2}(\Delta _{2}-\delta )+T_{e}^{2}(\Delta _{a}-\delta )+\gamma _{1}\gamma _{a}(\Delta _{2}-\delta )+\gamma _{2}\gamma _{a}(\Delta _{1}-\delta )\nonumber \\+ & {} (\Delta _{a}-\delta )\gamma _{1}\gamma _{2}-(\Delta _{1}-\delta )(\Delta _{2}-\delta )(\Delta _{a}-\delta ). \end{aligned}$$

(47)

$$\begin{aligned} \mu= \,& {} \gamma _{b}\gamma _{P}\delta -\Delta _{b}\delta ^{2}-\Delta _{b}\omega _{m}2G_{2}|b_{s}|^{2}+\Delta _{b}\omega _{m}^{2}-\delta ^{3}-\delta \omega _{m}2G_{2}|b_{s}|^{2}+\delta \omega _{m}^{2}+\delta ^{2}G_{1}x_{s}\nonumber \\{} & {} \quad +G_{1}x_{s}\omega _{m}2G_{2}|b_{s}|^{2}\nonumber \\- & {} G_{1}x_{s}\omega _{m}^{2}+G_{2}x_{s}^{2}\delta ^{2}+G_{2}x_{s}^{2}\omega _{m}2G_{2}|b_{s}|^{2}-G_{2}x_{s}^{2}\omega _{m}^{2}-\omega _{m}(G_{1}+2G_{2}x_{s})^{2}|b_{s}|^{2},\end{aligned}$$

(48)

$$\begin{aligned} \tau= \,& {} \gamma _{b}\delta ^{2}+\gamma _{b}\omega {m}2G_{2}|b_{s}|^{2}-\gamma _{b}\omega _{m}^{2}+\Delta _{b}\delta \gamma _{P}+\delta ^{2}\gamma _{P}-G_{1}x_{s}\delta \gamma _{P}-G_{2}\delta \gamma _{P}x_{s}^{2}. \end{aligned}$$

(49)