## 4 Conclusions  
In this work, we present a detailed theoretical analysis of the linear optical response of an SQD-MNS hybrid system. More specifically, we demonstrated that the properties of the MNS core substantially affect the spectral characteristics of the linear absorption coefficient for the MNS and the SDQ, separately. In the present study, we explore how the radius and the dielectric constant of the core affect the absorption spectra of the SQD and the MNS. After deriving the non-linear density matrix equations, we implement a first-order series expansion of the density matrix elements, with respect to the incident field. Next, we obtain analytical expressions for the imaginary part of the linear optical susceptibility of the SQD and the MNS. We also derive analytical expressions for the theoretical prediction of the positions and the magnitudes of the absorption/gain resonances that arise on the absorption spectra; the analytical formula for the FWHM of the absorption peak is presented in the study, which constitutes a measure of the inverse lifetime of the hybrid excitons. More specifically, we have identified two distinct types of hybrid excitons. It is demonstrated  
Figure 7: The linear absorption spectra for the SQD (**a**) and the MNS (**b**), for different values of the center-to-center distance (green solid curve: \(\mathrm{R}=11.5\) nm, pink dashed curve: \(13.5\) nm, blue dotted curve: \(15.5\) nm and turquoise dashed-dotted curve: \(100\) nm). Captions (**a**) and (**b**) correspond to the cases in which the inner radius of the MNS \(r_{1}\) is equal to 2 nm and 7 nm, respectively. The dielectric constant of the core \(\varepsilon_{\mathrm{c}}\) is set equal to \(7.5\) and the rest physical parameters take the same values as in Figure 3.  
that both the dielectric material and the radius of the MNS core determine the type of the hybrid exciton that emerges on the optical absorption spectrum of the SQD. In particular, the first type of hybrid exciton arises on the absorption spectrum of the quantum emitter, providing that the core of the metal nanoshell is thick and the dielectric constant of its material is close to the one corresponding to vacuum. Under the strong exciton-plasmon coupling regime, the gain without population inversion is importantly enhanced; the absorption peak is also strongly quenched, and the exciton lifetime is importantly shrunk. The second type of hybrid exciton appears under the weak exciton-plasmon coupling regime. Under this regime, the value of the dielectric constant and the radius of the dielectric core are both increased. Typical features of the second type of hybrid exciton are the amplified absorption resonance, the enhanced exciton lifetime, and the suppression of the gain coefficient. The formation of this last type of the hybrid excitons is induced when the dielectric constant of the core is enhanced for any fixed value of the core's radius. When the core of the MNS has a dielectric constant that tends to unity, the increase of the radius, up until a critical value, leads to the formation of the first type of hybrid exciton. However, above this critical value, the hybrid structure starts exhibiting the characteristics associated with the second type of hybrid exciton. Lastly, the properties of the core govern the absorption response of the MNS, as well. More explicitly, the gain that emerges on the profile of the corresponding spectrum is significantly enhanced, provided that the metal shell becomes exceptionally thin and the dielectric constant of the core tends to unity. Under these conditions, the gain that emerges on the profile of the absorption spectrum of the MNS presents the highest enhancement that can be observed. Conceptualization, S.G.K. and E.P. ; methodology, S.G.K. and E.P. ; software, A.K. and S.G.K. ; validation, A.K., S.G.K. and E.P. ; investigation, S.G.K. and A.K. ; writing--original draft preparation, A.K., S.G.K. and E.P. ; writing--review and editing, S.G.K. and E.P. ; visualization, A.K., S.G.K. and E.P. ; supervision, S.G.K. and E.P.