2.1.4. Non-Radiative Dissipation

As shown in Figure 1, the dressed photon is simultaneously excited and dissipated from the input side of a nanomatter system because it is an open system. Since the actual system of interest should be regarded as a microscopic part of an infinite system, it is difficult to accurately model the whole picture of the matter structure that is continuously connected from microscopic to the macroscopic systems. In our formulation, the Lindbladtype non-radiative dissipation is assumed for simply realizing a non-equilibrium open system. This assumption is approximately inadequate, but deep physical consideration in this topic is beyond the scope of this paper. It is expected a theoretical model will be built that accurately incorporates the macroscopic system hidden in the background. There are several studies for connecting a microscopic system with a macroscopic one [18,19].

Here, the non-radiative dissipation, i.e., the third term in (1), is given qualitatively as a similar manner in (5) as

$$\mathcal{L}^{(\text{nr})}\rho(t) = \frac{\gamma^{(\text{nr})}}{2} \sum\_{i,j \in \text{edge}} \left( 2a\_i \rho(t) a\_j^\dagger - \left\{ a\_i^\dagger a\_j, \rho(t) \right\} \right), \tag{6}$$

where *γ*(nr) is the non-radiative relaxation constant of the local dressed photon that is obviously faster than the radiative one. This is almost the same as (5), but the region with the dissipation is limited to a part of a taper structure.
