## 2 Methods
 \tag{25}\]  
where we have introduced the simplified notation  
\[\dot{\rho}_{01}^{(n)}=A_{n},\ \rho_{02}^{(n)}=B_{n},\ \rho_{12}^{(n)}=C_{n},\ \rho_{00}^{(n)}=D_{n},\ \rho_{11}^{(n)}=E_{n}, \tag{26}\]  
for the coefficients defined in Equations (10) and \(G=G_{a}+G_{b}\). For a hybrid system that is strongly pumped, the probe-field first-order absorption and the dispersion are respectively equal to \(-\text{Im}[\rho_{01}^{(2)}]\) and \(\text{Re}[\rho_{01}^{(2)}]\). More specifically, the first-order probe-field optical susceptibility, associated with the double SQD molecule, under the interaction with a pump field, is described by the following analytical expression:  
\[\chi_{\text{SQD}}^{(1)}=2\bigg{(}\frac{\Gamma}{V}\bigg{)}\frac{\mu\rho_{01}^{(2 )*}}{\varepsilon_{0}E_{0}}, \tag{27}\]  
where the coefficient \(\rho_{01}^{(2)}(T)\) is calculated at time \(T\), which is a sufficient time interval for our system to reach its steady state. In Equation (27), \(\varepsilon_{0}\) denotes the dielectric constant on vacuum, and with \(V\) and \(\Gamma\) we respectively indicate the SQD volume and the optical confinement factor [7]. To perform this calculation, we numerically solve Equations (11)-(26). The expression describing the MNP optical susceptibility at the first order [6] is given by the following expression:  
\[ \begin{array}{ll}\chi^{(1)}_{MNP}&=\frac{3\gamma_{1}\epsilon_{\rm{em}}}{ \epsilon_{0}}+\frac{3\gamma_{1}\epsilon_{\rm{em}}R^{3}}{2\pi E_{\rm{eff}} \epsilon_{\rm{eff}}R^{3}}\rho^{(2)*}_{01}\\ &=\frac{3\gamma_{1}\epsilon_{\rm{em}}}{\epsilon_{0}}+\frac{3\gamma_{1} \epsilon_{\rm{em}}V}{4\pi\epsilon_{\rm{eff}}R^{3}}\chi^{(1)}_{SQD}.\end{array}  \tag{28}\]  
At this point, if we sum the first-order susceptibilities of the individual components, we derive an expression for the optical susceptibility for the total hybrid structure at the first order:  
\[\chi^{(1)}_{tot}=\chi^{(1)}_{SQD}+\chi^{(1)}_{MNP}. \tag{29}\]  
The dressed-states picture provides us with an illuminating physical insight for the results that are presented in the next section, in the specific case of a double SQD molecule non-interacting with the MNP (practically, for an interparticle distance above 50 nm). Under this condition, the expression of the Hamiltonian, given by Equation (1), is simplified. Under the assumption of interaction only with a strong pump field, the population amplitudes of the energy levels satisfy the expression  
\[i\hbar\frac{d}{dt}\left[ \begin{array}{c}a_{0}(t)\\ a_{1}(t)\\ a_{2}(t)\end{array} \right]=H_{a}\left[ \begin{array}{c}a_{0}(t)\\ a_{1}(t)\\ a_{2}(t)\end{array} \right],\;\text{with}\;H_{a}=\left[ \begin{array}{ccc} \text{E}_{0}&-\hbar\Omega_{a}^{*}\epsilon^{i\omega_{a}t}&0\\ -\hbar\Omega_{a}\epsilon^{-i\omega_{a}t}&\text{E}_{1}&\hbar T_{e}\\ 0&\hbar T_{e}&E_{2}\end{array} \right]. \tag{30}\]  
If we apply the transformations \(a_{0}(t)=c_{0}(t)\), \(a_{1}(t)=c_{1}(t)e^{i(\Delta-iE_{1}/\hbar)t}\), and \(a_{2}(t)=c_{2}(t)e^{i(\Delta-\omega_{12}-iE_{2}/\hbar)t}\) in Equation (28), the Hamiltonian is written as follows:  
\[i\hbar\frac{d}{dt}\left[ \begin{array}{c}c_{0}(t)\\ c_{1}(t)\\ c_{2}(t)\end{array} \right]=H_{b}\left[ \begin{array}{c}c_{0}(t)\\ c_{1}(t)\\ c_{2}(t)\end{array} \right],\;\text{with}\;H_{b}=\left[ \begin{array}{ccc}0&- \hbar\Omega_{a}^{*}&0\\ -\hbar\Omega_{a}&\hbar\Delta&\hbar T_{e}\\ 0&\hbar T_{e}&\hbar(\Delta-\omega_{12})\end{array} \right].