## 2 Methods  
The hybrid nanostructure under study is comprised of an MNS and an SQD, which are coupled to each other via the Coulomb interaction, as illustrated in Figure 1. The distance between the SQD and the center of the MNS is denoted by \(\mathbf{R}\). The dielectric constants of the SQD and the environment are \(\mathbf{e_{s}}\) and \(\mathbf{e_{env}}\), respectively. The MNS has a fixed outer radius \(\mathbf{r_{2}}\) and is composed of gold, with dielectric function \(\mathbf{\eta}(\mathbf{\omega})\). The dielectric constant and the radius of the MNS core are symbolized as \(\mathbf{\varepsilon_{c}}\) and \(\mathbf{r_{1}}\); their values may vary in the present study. In the present study, we apply the semiclassical approximation, which means that the external electric field is treated classically, while the states of the quantum emitter are quantized. Thus, we assume that the hybrid nanostructure interacts with the external linearly polarized laser field \(\mathbf{E(t)=E_{0}e^{-i\omega t}+c.c. }\), with polarization direction lying along the axis defined by the centers of the nanoparticles. We treat the surface plasmons of the MNS as a quantum energy continuum. The SQD is described as a two-  
Figure 1: Schematic configuration of the hybrid system: An SQD with dielectric constant \(\varepsilon_{s}\) is coupled to an MNS with outer radius \(r_{2}\) and inner radius \(r_{1}\). The MNS is made of gold with dielectric function \(\eta(\mathbf{\omega})\). With \(\varepsilon_{c}\) and \(\varepsilon_{env}\) we represent the dielectric constants of the core and the environment, respectively. Taking into consideration that the electric field inside the SQD is expressed as a summation of the external field and the induced internal field [8], we can express the Hamiltonian that determines the equations of motion, in the dipole approximation, as follows:  
\[\mathrm{H}=\sum_{n=0}^{1}\mathrm{E_{n}}|\,n\rangle\langle n\,|-\hbar\Big{[}( \Omega+\mathrm{G}\sigma_{10})\mathrm{e}^{-\mathrm{i}\omega t}+\mathrm{c.c.} \Big{]}\ (|\,0\rangle\langle 1|+|\,1\rangle\langle 0\,|), \tag{1}\]  
with  
\[\Omega=\frac{\mathrm{E_{0}}\mu}{2\hbar\varepsilon_{\mathrm{effs}}}\Big{(}1+ \mathrm{s_{pol}}\mathrm{f_{1}}(\omega)\mathrm{r_{2}^{3}}\mathrm{R}^{-3}\Big{)}\ \mathrm{and} \tag{2}\]  
\[\mathrm{G}=\frac{\mu^{2}}{4\pi\hbar\varepsilon_{0}\varepsilon_{\mathrm{effs}}^ {2}\varepsilon_{\mathrm{env}}}\sum_{n=1}^{10}(n+1)^{2}\mathrm{f_{n}}(\omega) \mathrm{r_{2}^{2n+1}}\mathrm{R}^{-2(n+2)}, \tag{3}\]  
where \(\mathrm{E_{n}}\) represents the energy of state \(|n\rangle\) (\(\mathrm{E_{1}}=0\), \(\mathrm{E_{2}}=\hbar\omega_{01}\)) and \(\sigma_{10}\) denotes the non-diagonal density matrix element. In Equation (2), we define the Rabi frequency \(\Omega\), where \(\mu\) represents the transition dipole moment, \(\varepsilon_{\mathrm{effs}}=2/3+\varepsilon_{\mathrm{s}}/(3\varepsilon_{ \mathrm{effs}})\) and \(\mathrm{s_{pol}}=2\) (for a polarization direction that lies along the z axis). The first term of this parameter is related to the coupling of the SQD with the probe field and the second term owes its presence to the coupling of the SQD with the field sent back at its position, due to the polarization of the MNS that is purely induced by the probe field. For the derivation of the second term of Equation (2), the first order polarizability of the core-shell nanosystem has been set equal to \(4\pi r_{2}^{3}\mathrm{f_{1}}(\omega)\). The complex function \(\mathrm{f_{1}}(\omega)\) is given in Ref. [30]. The parameter \(\mathrm{G}\), the exact origin of which is discussed in Ref. [10], expresses the bidirectional interaction between the SQD and the MNS. In the multipole expansion approach, we define the higher-order polarizability as \(4\pi r_{2}^{(2n+1)}\mathrm{f_{n}}(\omega)\), where an extended mathematical formula is introduced in order to calculate the parameter \(\mathrm{f_{n}}(\omega)\), as shown in Ref. [33].