## 3 Results
Figure 3: The imaginary part (absorption coefficient) of the linear susceptibility for the SQD (**a**) and the MNS (**b**), as a function of the detuning energy of the probe field \(\hbar\delta\), for various values of the interparticle distance: \(\mathrm{R}=11.5\) nm (green solid curve), 13.5 nm (pink dashed curve), 15.5 nm (blue dotted curve) and 100 nm (turquoise dashed-dotted curve). We also set \(r_{1}=5\) nm, \(\varepsilon_{\mathrm{c}}=1\), \(\varepsilon_{\mathrm{env}}=1\), \(r_{2}=7.5\) nm, \(\mathrm{\mu}=0.65\) q\({}_{\mathrm{e}}\), \(\varepsilon_{\mathrm{s}}=6\) and \(\mathrm{T}_{2}=0.3\) nm. In Figure 4a,b, we present the absorption spectra for the SQD and the MNS, respectively. Here, we assume that \(\mathrm{r_{1}}=7\) nm. For the rest physical parameters, we use the same values as in Figure 3. As long as the thickness of the MNS is low enough, the increase of the inner radius of the MNS leads to the transposition of the absorption resonance towards the spectral center. This pattern indicates an important attenuation of the exciton-plasmon interaction and was not observed in the case of Figure 3, where the radius of the core \(\mathrm{r_{1}}\) was set equal to 5 nm; additionally, the gain region that emerges on the \(\mathrm{Im}\left[\chi_{\mathrm{SQD}}\right]\) spectrum is substantially shrunk. Moreover, the rise of the gain peak on the \(\mathrm{Im}\left[\chi_{\mathrm{MNS}}\right]\) spectrum, for intermediate values of the interparticle distance, exhibits a strong dependence on the radius and the dielectric constant of the core. Practically, for \(\mathrm{\varepsilon_{c}}=1\), a prominent gain dip appears on the spectral form of the MNS absorption coefficient, provided that \(\mathrm{r_{1}}>6.5\) nm. Under the same conditions, the \(\mathrm{Im}\left[\chi_{\mathrm{MNS}}\right]\) spectrum becomes highly asymmetric and manifests a dispersion-like line shape. For an extremely thin metal shell of thickness about 0.01 nm, it was found that gain is detected on the MNS absorption spectrum within a much broader range of values of the interparticle distance. This suggests that, when the radius of the core is modified by 5 to 7 nm, the direction of the energy transfer process, due to the coherent interaction between the SQD and MNS, is reversed. This effect can be explained as follows. As we increase the inner radius \(\mathrm{r_{1}}\), the positive factor \(\mathrm{f_{II}}(\omega)\) becomes more and more insignificant and, thus, the R-dependent term is the one that practically determines the value of the \(\mathrm{Im}\left[\chi_{\mathrm{MNS}}\right]_{\mathrm{min}}\) coefficient, according to Equation (11). At very high interparticle distances, as in the case with \(\mathrm{R}=100\) nm, the second term of Equation (10) vanishes. This happens because the variable \(\mathrm{Z_{I}}\), which is inversely proportional to the third order of R, converges to zero. The spectral profile of \(\mathrm{Im}\left[\chi_{\mathrm{MNS}}\right]\) is a horizontal line (turquoise dashed-dotted curve), as the first term of Equation (10), which is constant, determines the value of the absorption coefficient. In Figures 5 and 6, we present the absorption spectra for the SQD, for the same values of the distance between the centers of the coupled nanoparticles as those considered in Figure 2. The radius of the core \(r_{1}\) is set equal to 5, and 7 nm, in Figures 5 and 6, respectively, while, in both figures, the dielectric constant of the core is \(\mathrm{\varepsilon_{c}}=7.5\). After comparing Figure 5 to Figures 3a and 6 to Figure 4a, we deduce that the increase of \(\mathrm{\varepsilon_{c}}\) is responsible for the enhancement and the narrowing of the absorption resonance and the suppression of the gain dip. This is expected, based on a brief mathematical analysis based on Equation (6). More specifically, the enhancement of \(\mathrm{\varepsilon_{c}}\) leads to the decrease of \(\mathrm{G_{I}}\). As a result, the denominator of Equation (6) decreases, thus leading to the enhancement of the absorption peak.