Optical Response with Tunneling Coupling in a Hybrid Optomechanical System

Abstract

We theoretically analyze the phenomenon of Optomechanically Induced Transparency (OMIT) of a weak probe field in a hybrid optomechanical system, consisting of an optical cavity, a mechanical resonator and Quantum Dot molecules(QDs), in addition, in some parameter regimes, the probe transmission can exceed unity. Furthermore, the phase dispersion and group delay at the frequency of the probe field have been studied, we can control the fast light effect by tuning the driving field and tunneling coupling, attributed to quantum interference and Tunneling effect, it provides a flexible way to control optical response and propagation in such a hybrid optomechanical system.

Appendix A:: Calculation of c +

When substituting the exponential form ansatz into (15), we get eight equations as follows:

$$\big[\kappa+i {\Delta}_{1}-i \delta+i g_{1}(b_{s}+b_{s}^{*})\big]c_{+}=-i g_{1} c_{s}(b_{-}^{*}+b_{+})-i g_{2} A_{+}+ \varepsilon_{p},$$

(A1)

$$\big[\kappa+i {\Delta}_{1}+i \delta+i g_{1}(b_{s}+b_{s}^{*})\big]c_{-}=-i g_{1} c_{s}(b_{+}^{*}+b_{-})-i g_{2} A_{-},$$

(A2)

$$(\gamma_{m}+i \omega_{m}-i \delta)b_{+}=i g_{1} c_{s}^{*} c_{+}+i g_{1} c_{s} c_{-}^{*},$$

(A3)

$$(\gamma_{m}+i \omega_{m}+i \delta)b_{-}=i g_{1} c_{s}^{*} c_{-}+i g_{1} c_{s} c_{+}^{*},$$

(A4)

$$(\gamma_{1}+i {\Delta}_{2}-i \delta)A_{+}=-i g_{2} c_{+}-i T_{e} B_{+},$$

(A5)

$$(\gamma_{1}+i {\Delta}_{2}+i \delta)A_{-}=-i g_{2} c_{-}-i T_{e} B_{-},$$

(A6)

$$(\gamma_{2}+i {\Delta}_{3}-i \delta)B_{+}=-i T_{e} A_{+},$$

(A7)

$$(\gamma_{2}+i {\Delta}_{3}+i \delta)B_{-}=-i T_{e} A_{-}.$$

(A8)

One of its solutions is

$$c_{+}=\frac{1}{M_{3}}O_{2}\varepsilon_{p}\big[O_{1}O_{3}+i g_{1}c_{s}M_{1}O_{1}O_{3}+{g_{1}^{2}} \lvert{c_{s}}\rvert^{2}\big],$$

(A9)

where $O_{1}=\kappa -i{\Delta }_{1}-i\delta -i g_{1} (b_{s}+b_{s}^{*})$, $O_{2}=({\Gamma }_{1}-i\delta )({\Gamma }_{2}-i\delta )+{T_{e}^{2}}$, O₃ = γ_m + iω_m − iδ, $O_{4}=\kappa +i{\Delta }_{1}-i\delta +i g_{1} (b_{s}+b_{s}^{*})$, $M_{1}=\frac {ig_{1}c_{s}^{*}}{O_{1}O_{3}{g_{2}^{2}} \lvert {c_{s}}\rvert ^{2}}\big [O_{1}O_{3}+{g_{1}^{2}} \lvert {c_{s}}\rvert ^{2} \big ]\varepsilon _{p}$, $M_{2}=\frac {ic_{s}^{*}O_{1}O_{2}+i{g_{2}^{2}} c_{s}({\Gamma }_{2}-i\delta )}{g_{1} \lvert {c_{s}}\rvert ^{2} O_{2}}+\frac {2iM_{1}}{\varepsilon _{p}}\big [{\Delta }_{1}+g_{1}(b_{s}+b_{s}^{*})\big ]$ and $M_{3}=O_{1}O_{2}O_{3}O_{4}+i g_{1} c_{s} M_{2} O_{1}O_{2}O_{3}+2i{g_{1}^{2}} \lvert {c_{s}}\rvert ^{2} O_{2}\big [{\Delta }_{1}+g_{1}(b_{s}+b_{s}^{*})\big ]+{g_{2}^{2}} O_{1}O_{3}({\Gamma }_{2}-i\delta )$.