2.1.2. Inter-Node Energy Transfer

The hopping energy transfer means mathematically an exchange of the dressed photon between two different nodes, which is given by the following equation,

$$H\_{\rm int} = \sum\_{i \neq j} \hbar V(|\mathbf{r}\_i - \mathbf{r}\_j|) (a\_i^\dagger a\_j + a\_i a\_j^\dagger),\tag{3}$$

where the coupling strength *hV*¯ (*r*) is assumed to have a finite interaction range for expressing the localization nature of the dressed photon. Readers with knowledge of quantum theory may pay attention to a positive sign of the interaction Hamiltonian in comparison with some known models of material systems, such as the Bose–Hubbard model and the tight-binding model [13]. The interaction Hamiltonian is based on the theoretical derivation of the transition probability of the electronic excitation between two nanomatters in our published reports [14–16], where the transition probability was obtained by assuming that the constraint of the energy and momentum conservation lows can be overcome. According to the detailed explanation in [17], the coupling strength *hV*¯ (*r*) with a finite interaction distance and a positive sign is derived as the form so-called Yukawa potential,

$$V(r) = \frac{V\_0 e^{-m\_{\rm eff}r}}{r},\tag{4}$$

where *V*0 and *<sup>m</sup>*eff are an appropriate constant and an effective mass which determines the interaction range, respectively. The Yukawa function often appears to give a screening effect in a many-body interaction system. In the case of the dressed-photon energy transfer, degrees of freedom of an environment leads to the equivalent effect to the many-body interaction.
