## 2 Methods  
The hybrid system studied has two counterparts: a spherical MNP with dielectric function \(\varepsilon_{m}(\omega)\) and a molecule of two coupled SQDs with dielectric constant \(\varepsilon_{S}\), as seen in Figure 1. Moreover, we consider that the condition \(R>a\) is met, with \(R\) and \(a\) respectively denoting the interparticle distance and the radius of the MNP, and we assume that the size of the SQD is negligible as compared to the size of the MNP. The structure is surrounded by a material with dielectric constant \(\varepsilon_{env}\). The hybrid system interacts with two linearly polarized, oscillating electromagnetic fields, where \(E_{i}\) and \(\omega_{i}\) respectively represent the corresponding amplitude and the angular  
Figure 1: Schematic diagram of the MNP–double-SQ hybrid system, \(a\), \(b\), and \(R\) respectively representing the radius of MNP, the radius of the SQDs, and the center-to-center distance between an SQD and the MNP. Between the SQDs, electron tunneling is present. frequency (\(i=a\), for the pump field and \(i=b\), for the probe field), that excite plasmons on the surface of the MNP, providing a strong, continuous spectral response. An energy-level scheme is presented in Figure 2, where the possible transitions are also depicted. The coupling between the two SQDs with different band structures, which are typically placed in distances between 7 and 10 nm, owes its presence to the electron tunneling. The coupled SQDs are represented as a three-level quantum system [32; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52]. With \(|0\rangle\) we denote the ground state, and \(|1\rangle\) (direct exciton) corresponds to the state describing a single electron excitation, in which case the excited electron, as well as the hole that is created, are both confined within the same quantum dot. On the other hand, an electron can be excited in state \(|2\rangle\) (indirect exciton) when it is conveyed from the first quantum dot to the second one, due to tunneling effects following the excitation pathway. The incident field induces an oscillating dipole moment and excites electrons from the ground state \(|0\rangle\) to the (direct) exciton state, where an electron-hole pair is bound to the same SQD. The direct exciton is also coupled to the states of the surface plasmons due to the long-range Coulomb interaction that permits the exchange of energy between the counterparts [53; 54; 55; 56]. The tunneling barrier can be controlled by properly adjusting the voltage of an external gate electrode structure [39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52]. In Equation (1), we present the Hamiltonian that describes the interaction of the hybrid system with a pump and a probe field, as depicted in Figure 2, in the dipole approximation. \[H=\text{E}_{0}|0\rangle\langle 0|+\text{E}_{1}|1\rangle\langle 1|+\text{E}_{2}|2 \rangle\langle 2|+\hbar T_{e}(|1\rangle\langle 2|+|2\rangle\langle 1|)-\mu E_{SQD}(|0 \rangle\langle 1|+|1\rangle\langle 0|). \tag{1}\]  
The first three terms in Equation (1) describe the unperturbed system, with \(\text{E}_{n}\) being the energy of level \(|n\rangle\) (for the ground state, we take \(\text{E}_{0}=0\)), while the fourth term owes its presence to the tunneling effects between the SQDs, with \(T_{e}\) expressing the rate at which electrons tunnel within the SQD molecule. The last term in Equation (1) describes the interaction between the electromagnetic fields and the SQD molecule. Here, \(\mu\) and \(E_{SQD}\) respectively represent the dipole moment associated to the excitation \(|0\rangle\leftrightarrow|1\rangle\) and the total field detected at the position of the SQD molecule, which is given by the formula  
\[E_{SQD}(t)=\frac{\hbar}{\mu}\!\!\sum_{i=a,b}\!\Big{\{}[\Omega_{i}+G_{i}\rho_{ 10}(t)]e^{-i\omega_{i}t}+[\Omega_{i}^{*}+G_{i}^{*}\rho_{01}(t)]e^{i\omega_{i}t }\Big{\}}, \tag{2}\]  
Figure 2: Schematic energy-level configuration of the hybrid system.