## 2 Methods
 \tag{9}\]  
where the parameters \(\delta=\omega_{b}-\omega_{a}\) and \(\Delta=(\mathrm{E}_{1}-\mathrm{E}_{0})/\hbar-\omega_{a}\) respectively denote the difference of the angular frequencies with reference to the pump and the probe field and the detuning from the energy gap between the energy levels \(|0\rangle\) and \(|1\rangle\). The \(|1\rangle\leftrightarrow|2\rangle\) transition corresponds to the energy difference \(\hbar\omega_{12}\). The dephasing rates are symbolized with \(\gamma_{nm}\), while the population decay rate for the transition \(|n\rangle\rightarrow|m\rangle\) is indicated with \(\Gamma_{nm}\). To calculate the first-order susceptibility of the hybrid structure, aiming at the examination of the pump-probe optical response, we apply to the first-order Taylor expansion of \(\rho_{nm}\), according to the first-order approximation, in the probe field Rabi frequency:  
\[\rho_{nm}=\rho_{nm}^{(1)}+\Omega_{b}^{*}e^{i\beta t}\rho_{nm}^{(2)}+\Omega_{b}e^ {-i\beta t}\rho_{nm}^{(3)}. \tag{10}\]  
In the expansions presented in Equation (10), the terms \(\left|\rho_{nm}^{(2)}\right|\) and \(\left|\rho_{nm}^{(3)}\right|\) are considered small, in the sense that \(\left|\rho_{nm}^{(1)}\right|>>\left|\rho_{nm}^{(2)}\right|\), \(\left|\rho_{nm}^{(3)}\right|\). Since the intensity of the pump field is assumed to be four orders of magnitude higher than the intensity of the probe field (\(\Omega_{a}\gg\Omega_{b}\)), we should maintain in our calculations all the orders of its interaction with the hybrid structure. By substituting the expanded density-matrix elements in Equations (5)-(9), the set of Equations (11)-(25) is derived. \[\dot{A}_{1}=i(\Omega_{a}^{*}+G^{*}A_{1})(E_{1}-D_{1})+i(\Delta+i\gamma_{01})A_{ 1}+iT_{e}B_{1}, \tag{11}\]  
\[\dot{A}_{2}=-i\delta A_{2}+i(\Omega_{b}^{*}+G^{*}A_{2})(E_{1}-D_{1})+i(\Omega_{ a}^{*}+G^{*}A_{1})(E_{2}-D_{2})+i(\Delta+i\gamma_{01})A_{2}+iT_{e}B_{2}, \tag{12}\]  
\[\dot{A}_{3}=i\delta A_{3}+iG^{*}(E_{1}-D_{1})A_{3}+i(\Omega_{a}^{*}+G^{*}A_{1}) (E_{3}-D_{3})+i(\Delta+i\gamma_{01})A_{3}+iT_{e}B_{3}, \tag{13}\]  
\[\dot{B}_{1}=i(\Omega_{a}^{*}+G^{*}A_{1})C_{1}+i(\Delta-\omega_{12}+i\gamma_{0 2})B_{1}+iT_{e}A_{1}, \tag{14}\]  
\[\dot{B}_{2}=-i\delta B_{2}+i(\Omega_{b}^{*}+G^{*}A_{2})C_{1}+i(\Omega_{a}^{*}+ G^{*}A_{1})C_{2}+i(\Delta-\omega_{12}+i\gamma_{02})B_{2}+iT_{e}A_{2}, \tag{15}\]  
\[\dot{B}_{3}=i\delta B_{3}+iG^{*}A_{3}C_{1}+i(\Omega_{a}^{*}+G^{*}A_{1})C_{3}+i (\Delta-\omega_{12}+i\gamma_{02})B_{3}+iT_{e}A_{3}, \tag{16}\]  
\[\dot{C}_{1}=i(\Omega_{a}+GA_{1}^{*})B_{1}-iT_{e}(1-D_{1}-2E_{1})-i(\omega_{12 }-i\gamma_{12})C_{1}, \tag{17}\]  
\[\dot{C}_{2}=-i\delta C_{2}+iGA_{3}^{*}B_{1}+i(\Omega_{a}+GA_{1}^{*})B_{2}+iT_ {e}(D_{2}+2E_{2})-i(\omega_{12}-i\gamma_{12})C_{2}, \tag{18}\]  
\[\dot{C}_{3}=i\delta C_{3}+i(\Omega_{a}+GA_{1}^{*})B_{3}+i(\Omega_{b}+GA_{2}^{* })B_{1}+iT_{e}(D_{3}+2E_{3})-i(\omega_{12}-i\gamma_{12})C_{3}, \tag{19}\]  
\[\dot{A}_{1}=i(\Omega_{a}^{*}+G^{*}A_{1})(E_{1}-D_{1})+i(\Delta+i\gamma_{01})A_{ 1}+iT_{e}B_{1}, \tag{11}\]0  
\[\dot{A}_{1}=i(\Omega_{a}^{*}+G^{*}A_{1})(E_{1}-D_{1})+i(\Delta+i\gamma_{01})A_{ 1}+iT_{e}B_{1}, \tag{11}\]1  
\[\dot{A}_{1}=i(\Omega_{a}^{*}+G^{*}A_{1})(E_{1}-D_{1})+i(\Delta+i\gamma_{01})A_{ 1}+iT_{e}B_{1}, \tag{11}\]2  
\[\dot{A}_{1}=i(\Omega_{a}^{*}+G^{*}A_{1})(E_{1}-D_{1})+i(\Delta+i\gamma_{01})A_{ 1}+iT_{e}B_{1}, \tag{11}\]3  
\[ \begin{array}{rl}\dot{E}_{2}=&-i\delta E_{2}-i(\Omega_{a}^{*}+G^{*}A_{1})A_{3} ^{*}+i\big{(}\Omega_{a}+GA_{1}^{*}\big{)}A_{2}-i(\Omega_{b}^{*}+G^{*}A_{2})A_{1 }^{*}\\ &+iGA_{3}^{*}A_{1}+iT_{e}(C_{2}-C_{3}^{*})-(\Gamma_{10}+\Gamma_{12})E_{2}, \end{array}  \tag{24}\]  
\[ \begin{array}{rl}\dot{E}_{3}=&i\delta E_{3}-i(\Omega_{a}^{*}+G^{*}A_{1})A_{2 }^{*}+i\big{(}\Omega_{a}+GA_{1}^{*}\big{)}A_{3}-iG^{*}A_{1}^{*}A_{3}\\ &+i(\Omega_{b}+GA_{2}^{*})A_{1}+iT_{e}(C_{3}-C_{2}^{*})-(\Gamma_{10}+\Gamma_{ 12})E_{3},\end{array}