## 2 Methods
\[\chi_{\mathrm{SQD}}=\frac{2\Gamma\mu}{\varepsilon_{0}\mathrm{E_{0}}\mathrm{ V}}\,\sigma_{10}(\mathrm{t}). \tag{4}\]  
The susceptibility of the SQD is proportional to the non-diagonal density-matrix element calculated in the steady state, as in Ref. [10], with \(\mathrm{V}/\Gamma\) denoting the ratio of the SQD volume to the optical confinement factor. Next, we expand the density matrix elements in a first-order Taylor series about the weak probe field and substitute these expressions  
Figure 2: Energy-level configuration of the hybrid system. The two-level diagram with the ground state \(|0\rangle\) and the direct exciton state \(|1\rangle\) describes the quantum emitter (SQD) and the electronic continuum states are associated with the MNS. into the equations of motion. After solving the derived equations, in the steady state, we calculate the linear part of the \(\chi_{\text{SQD}}\) coefficient, based on the following analytical formula:  
\[\chi_{\text{SQD}}=\text{H}\frac{\text{Z}_{\text{I}}\text{T}_{2}(\delta+\text{G} _{\text{R}})-\text{Z}_{\text{R}}(1+\text{T}_{2}\text{G}_{\text{I}})}{\text{T}_{2 }^{2}(\delta+\text{G}_{\text{R}})^{2}+\left(1+\text{T}_{2}\text{G}_{\text{I}} \right)^{2}}, \tag{5}\]  
where \(\text{G}_{\text{R}}\) and \(\text{G}_{\text{I}}\) express the real and the imaginary parts of the self-interaction coefficient \(\text{G}\), respectively, \(\delta=\omega-\omega_{01}\) is the detuning of the applied field from resonance and \(\text{T}_{2}\) represents the dephasing time. Moreover, the parameter \(\text{H}\) is equal to \(-\Gamma\mu^{2}\text{T}_{2}/(\varepsilon_{0}\text{V}\hbar)\). In Equation (5), \(\text{Z}_{\text{R}}\) and \(\text{Z}_{\text{I}}\) denote the real and the imaginary parts of \(Z=\left(1+\text{s}_{\text{pol}}\text{f}_{1(\omega)}r_{2}^{3}\text{R}^{-3} \right)/\varepsilon_{\text{effs}}\), respectively, which can be considered as frequency-independent. The critical points of the imaginary part of the \(\chi_{\text{SQD}}\) function are calculated by the analytical expression:  
\[\text{Im}\left[\chi_{\text{SQD}}\right]_{\frac{\text{max}}{\text{min}}}=\mp \text{H}\frac{\text{Z}_{\text{I}}^{2}}{2(1+\text{T}_{2}\text{G}_{\text{I}}) \left(\sqrt{\text{Z}_{\text{I}}^{2}+\text{Z}_{\text{R}}^{2}}\mp\text{Z}_{ \text{R}}\right)}. \tag{6}\]  
The minimum value of Equation (6) corresponds to the emergence of gain without inversion. If the interparticle distance between the components of the hybrid nanosystem is high enough so that we can practically consider that the interaction between them is infinitesimal, the minimum and the maximum values of \(\text{Im}\left[\chi_{\text{SQD}}\right]\) converge to zero and \(-\text{H}/\varepsilon_{\text{effs}}\), respectively. The absorption resonance arises at  
\[\delta_{\text{res}}=-\text{G}_{\text{R}}-\frac{\text{Z}_{\text{R}}}{\text{Z}_ {\text{I}}}\left(\text{G}_{\text{I}}+\frac{1}{\text{T}_{2}}\right) \tag{7}\]  
and the corresponding full width at half maximum is calculated by Equation (8):  
Figure 1:0  
Next, we determine the first-order optical susceptibility of the core-shell nanoparticle [10]  
Figure 1:1  
and we derive an analytical expression for the linear absorption coefficient:  
Figure 1:2  
In Equation (10), the parameter \(\text{P}=-3\text{s}_{\text{pol}}\mu^{2}\text{T}_{2}/\left(4\hbar\pi\varepsilon _{0}\varepsilon_{\text{effs}}\text{R}^{3}\right)\) is R-dependent, \(\widetilde{\text{Z}}=\text{f}_{1}\cdot\text{Z}=\widetilde{\text{Z}}_{\text{R }}+\text{i}\widetilde{\text{Z}}_{\text{I}}\) is a complex quantity that can be considered as frequency-independent, within the short range of frequencies of interest and \(\text{f}_{1\text{I}}\) symbolizes the imaginary part of the parameter \(\text{f}_{1}\). Based on Equation (10), we can determine the magnitude of the gain peak which arises on the MNS absorption spectrum:  
Figure 1:3