*2.1. Quantum Master Equation*

From the experimental situation for a nanophotonics system, one can understand that the system always exists under an environment with the external photoexcitation and the dissipation, and the balance of the input and the output is maintained. This is a nonequilibrium open system. For describing the dressed-photon dynamics, the dressed photon is assumed to be a carrier bounded in a nanomatter system, and a part of energy dissipates to the external field as the free photon, where the optical coherence is disappeared. In other words, it is a problem to analyze the internal states of the dressed photon distributed in a non-equilibrium open system.

A nanomatter with an arbitrary shape is expressed as a collection of nodes that bind the dressed photon, and are freely arranged inside a matter. To avoid misunderstanding, note that the node does not mean the atomic site, but a center of mass for the dressed photon with spatial spreading. Therefore, the nodes are not restricted by a periodic array structure representing the translational symmetry of such an electron wave, and can set freely. This paper is not intended to describe a rigorous structure of a matter, but rather to represent adequately the essence of dressed-photon mediating phenomena. From this point of view, this model is equivalent to a quantum walk on a graph. Several studies have also been reported that sugges<sup>t</sup> that dressed-photon phenomena correspond to some stationary solution in a quantum-walk system [8,9].

The dressed photon as a carrier is assumed to be transferred among the nodes by the hopping conduction, such that the coupling strength is expressed as a function of distance between a target node and all the others. Figure 1 illustrates a dressed-photon system, where a nanoscale two-dimensional taper structure and a nanomatter are expressed as just a collection of nodes without distinguishing the two separated parts. In this system, the dressed photon is injected from the upper part of a taper structure, released to the external field radiatively, and returns to the input side non-radiatively. This model is a nonequilibrium open system. The equation of motion, that is a quantum master equation, in such a system can be described using the quantum density operator *ρ*(*t*) as follows [10–12],

$$\frac{\partial \rho^{I}(t)}{\partial t} = -\frac{i}{\hbar} \left[ H\_{\text{int}}^{I} + H\_{\text{exc}}^{I} \rho^{I}(t) \right] + \mathcal{L}^{(\text{nr})} \rho^{I}(t) + \mathcal{L}^{(\text{r})} \rho^{I}(t), \tag{1}$$

where the superscript *I* for each operator represents the interaction picture, and the hopping energy transfer and the coherent excitation are expressed as *H*int and *H*exc, respectively. The square brackets represent the commutation relation, and L(r) and L(nr) mean the Lindblad-type radiative and the non-radiative dissipations. The following devotes explanation of each component in (1), where the superscript *I* for the interaction picture is omitted to avoid the complexity of the subscript and superscript expressions. In defining the operators that are used in this research, the basis states are assumed to be a one or zero dressed photon. This means an assumption of the weak excitation limit. In the future, the many-body interaction of the dressed photon, i.e., the nonlinear problem, should be considered, and it will be reported somewhere.

**Figure 1.** Schematic illustration of a dressed-photon system that consists of plural arbitrarily arranged nodes and models a taper structure of an optical fiber probe and a nanomatter. It is not necessary to distinguish between the taper and the nanomatter, as it is regarded as just a collection of nodes. In this system, the dressed photon is coherently excited from the upper part of the taper structure, and transfers via the hopping conduction among nodes with the coupling strength according to the distance. Some dressed photon dissipates out of the system as the free photon, and some returns to the input side non-radiatively.

### 2.1.1. Dressed-Photon Excitation by External Field

The external excitation of the dressed photon is assumed to be given coherently from the upper part in a taper structure in Figure 1. When the creation and annihilation operators *a*† *i*and *ai* of the dressed photon at a node *i* are predetermined, the excitation is expressed as

$$H\_{\text{exc}} = \sum\_{i \in \text{edge}} \hbar A(a\_i + a\_i^\dagger),\tag{2}$$

where *hA*¯ denotes the strength of the excitation that is related to the amplitude of the external input field. The form of (2) is inspired by the theoretical description of the conventional electric dipole excitation.
