## 3 Results  
Here, we examine the forms of the absorption spectra, with regard to the SQD and the MNS. The MNS has an outer radius \(\mathrm{r}_{2}=7.5\) nm and it consists of gold. The value of the corresponding dielectric function \(\mathrm{\eta}(\omega)\), within a short range of frequencies around 2.5 eV, can be approximately set equal to \((-2.28+3.81\mathrm{i})\), according to Ref. [34]. The hybrid nanosystem is surrounded by vacuum \((\varepsilon_{\mathrm{env}}=1)\). The dielectric constant of the SQD is \(\varepsilon_{\mathrm{s}}=6\), the dephasing time is \(\mathrm{T}_{2}=0.3\) ns and \(\Gamma/\mathrm{V}=5\times 10^{23}\) m\({}^{-3}\)[35]. The dipole moment for the \(|\,0\rangle\rightarrow\!\!|1\rangle\) transition is \(\mathrm{\mu}=0.65\) q\({}_{\mathrm{e}}\). We also assume that the corresponding energy gap is approximately equal to the excitation energy of the localized MNS surface plasmon. In Figure 3a,b, we present the spectra of the \(\mathrm{Im}\!\left[\chi_{\mathrm{SQD}}\right]\) and \(\mathrm{Im}\!\left[\chi_{\mathrm{MNS}}\right]\) coefficients, respectively, as a function of the detuning of the weak probe field from resonance \(\hbar\delta\), for different values of the center-to-center distance (R = 11.5 nm: green solid curve, 13.5 nm: pink dashed curve and 15.5 nm: blue dotted curved). The radius and the dielectric constant of the core are \(r_{1}=5\) nm and \(\varepsilon_{\mathrm{c}}=1\), respectively. We note that, when the interparticle distance is reduced, the absorption peak becomes highly asymmetric and is transposed to negative values of the detuning parameter, Equation (7); additionally, its magnitude decreases, Equation (6), and the corresponding FWHM expands, Equation (8). Hence, the lifetime of the hybrid exciton is suppressed, and the excitonic energy exhibits a red shift. The previously reported modifications of the spectral characteristics are caused by the enhancement of the coefficients \(\mathrm{Z}_{1}^{2}/\mathrm{Z}_{\mathrm{R}}^{2}\) and \(\mathrm{G}_{\mathrm{I}}\). In the limiting case where the SQD does not practically interact with the MNS, the profile of the absorption resonance obtains a Lorentzian-like line shape and the gain feature extinguishes, because \(\mathrm{Z}_{\mathrm{I}}\to 0\). The absorption peak has a magnitude and an FWHM equal to \(-\mathrm{H}/\varepsilon_{\mathrm{effs}}\) and \(2/\mathrm{T}_{2}\), respectively. This indicates that symmetry is a characteristic of the \(\mathrm{Im}\!\left[\chi_{\mathrm{SQD}}\right]\) spectrum, provided that the interaction between both nanoparticles is significantly weakened. The emergence of gain without population inversion on the same spectrum, within a specific range of frequencies, can also be predicted mathematically, based on Equation (6). According to this equation, for any set of the physical parameters, the coefficient \(\mathrm{Im}\!\left[\chi_{\mathrm{SQD}}\right]\) is negative, due to the negative sign of the constant H. The profile of the \(\mathrm{Im}\!\left[\chi_{\mathrm{SQD}}\right]\) spectrum manifests a strong dependence on both the material and the radius of the dielectric core. It is found that the magnitude of the gain dip is maximized for intermediate values of R, as long as the dielectric core has a dielectric constant close to the one corresponding to vacuum and the radius of the core is about 5 nm, as in the case illustrated in Figure 3a. Specifically, for \(r_{1}\leq 4.5\) nm, the increase of \(r_{1}\) leads to the enhancement of the gain dip. However, the characteristics of the absorption resonance are not modified. On the other hand, for \(r_{1}\geq 5\)nm, the increase of \(r_{1}\) is responsible for the suppression of the gain dip, as well as for the enhancement and the narrowing of the absorption peak.