*2.2. Spatial Mode Expansion*

In this subsection, the basis states for expressing the spatial distribution of the dressed photon are discussed using the quantum master equation formulated in Section 2.1. So far, the basis states of the dressed photon are set as the dressed photon exists or not at local nodes in a nanomatter system as an implicit understanding. In Figure 2, a steady-state solution in the case of a two-dimensional taper structure is calculated and mapped with the color gradation that represents the occupancy probability of the dressed photon. Figure 2a–c denote the snapshots of the temporal evolution at the time steps, *t* = 5, 50, and 5000, respectively. The simulation parameters used in these calculations are written in the caption in Figure 2, and are commonly used in the following calculations. At each time step, the spatial distribution of the dressed photon reflects the weight coefficient *ci* of the basis states in a quantum superposition state |*φ* = *<sup>c</sup>*0|0, ··· , 0 + *<sup>c</sup>*1|1, 0, ··· , 0 + *<sup>c</sup>*2|0, 1, 0, ··· , 0 + ··· + *cN*|0, ··· , 0, <sup>1</sup>, where, for example, |0, 1, 0, ··· , 0 represents a state in which the dressed photon exists at the node labeled as the position 2. The temporal evolution can be interpreted as follows; in the early stage, the dressed photon runs down as the ballistic conduction at a part of the taper slopes, and then is reflected at a boundary of the taper tip, i.e., a spatial singular point of the system, leading to a steady state. Finally, it is found that the dressed photon makes spatial localization near around the tip, similar to a standing wave. In addition, there are locations inside the taper where the occupancy probability of the dressed photon is highly established quasi-periodically. The spatial distribution is, of course, determined depending on the shape of a matter system, such as the size of the taper structure and the steepness of the taper slopes. It also depends on the coupling strength ¯*hV*(*r*).

**Figure 2.** Temporal snapshot of the occupancy probability for the dressed photon at the time steps (**a**) *t* = 5, (**b**) 50, and (**c**) 5000. In this calculation, the parameters are set as *hA*¯ = 1, *hV*¯ 0 = 27.2, *<sup>m</sup>*eff = 0.1, *γ*(r) = 0.01, and *γ*(nr) = 10, and the matter system is assumed as a two-dimensional taper structure which is expressed by the 47 nodes. The dressed photon initially transfers on the taper slopes, and converges into a steady state with the spatial localization similar to a generation of standing wave caused by a system asymmetry.

As mentioned in the Introduction, the dressed photon should be controlled in a nanomatter system, and observed via the free photon radiated from the system. In the following, a way to extract the characteristics of the spatial distribution of the dressed photon is discussed from a viewpoint of the basis transformation. Although the Fourier transformation and/or the Bloch's theorem, which are based on translational symmetry, are used in the cases of the conventional optics and solid-state physics to catch the clear description of a wave nature, they cannot be applied for the description of the dressed photon because of the spatial singularity of the matter boundary and the impurity. Therefore, focusing on the fact that this system converges to a non-equilibrium steady state, the basis transformation which diagonalizes the steady state is proposed. According to the obtained basis states, there is no energy transfer between such basis states at a steady state, and the dressed photon dynamics can be separable depending on the spatial distribution of the basis states which strongly reflects a geometrical nature of a nanomatter.

From the steady-state solution (Figure 2c), the matrix *U* that diagonalizes the quantum density matrix can be determined numerically and uniquely. As a result of this transformation, (1) is rewritten as

$$\frac{\partial \rho\_{\rm st}(t)}{\partial t} = -\frac{i}{\hbar} [H\_{\rm int, \rm st} + H\_{\rm exc, \rm st} \rho\_{\rm st}(t)] + \mathcal{L}\_{\rm st}^{(\rm nr)} \rho\_{\rm st}(t) + \mathcal{L}\_{\rm st}^{(\rm r)} \rho\_{\rm st}(t),\tag{7}$$

$$\mathcal{O}\_{\text{st}} \equiv \mathcal{U}^{-1} \mathcal{O} \mathcal{U},\tag{8}$$

$$\mathcal{L}\_{\rm st}^{(\rm nr,r)} \rho\_{\rm st} \equiv \frac{\gamma^{(\rm nr,r)}}{2} \left( 2a\_{\rm st} \rho\_{\rm st}(t) a\_{\rm st}^{\dagger} + \left\{ a\_{\rm st}^{\dagger} a\_{\rm st} \rho\_{\rm st}(t) \right\} \right), \tag{9}$$

where the subscript "st" means the operators after the basis transformation as the quantum density matrix being diagonalized, and *O* is an arbitrary operator. To decompose the individual row of the matrix *U* is intuitive because the elements of a certain row are constructed from the weight coefficients of the linear combination of the basis states in the local-node description, and it is sorted in descending order of the occupancy probability. The basis states can be visualized as shown in Figure 3.

**Figure 3.** Color map images of the basis states reconstructed from the transformation matrix *U* in (8). Since the state *n* = 27 corresponds to the coherent excitation, there is no meaning in the state over *n* = 27, and almost of those are excluded from visualization.

From the perspective of the spatial distribution, there are several characteristic basis states. The state of *n* = 27 in Figure 3 apparently corresponds to the excitation due to the external field at an input interface, and thus, the states in the region labeled *n* > 27 are no longer excited in this system, and most of these are excluded from the drawing. The states *n* = 14, 16, and 24 have quasi-periodic spatial structures that resemble standing waves in a waveguide. In the states of *n* = 8 and 12, the dressed photon occupies the taper slopes. Several basis states of *n* ≤ 9 show localization of the dressed photon at a taper tip; therefore, it is predicted that these states couple strongly with each other.

The dynamics of the quantum density matrix for the transformed basis states (Figure 3) can be recalculated numerically. In Figure 4, the density matrix elements are depicted as the color map images, where the time steps are similarly set as *t* = 5, 50, and 5000, and the colors represent absolute values of the density matrix elements. In an early stage of a time evolution, the occupancy probability (diagonal elements) concentrates in the basis states with a localization nature (*n* ≤ 7), and the off-diagonal elements which represent the transition probability between the different basis states also change actively. After some time, the central area of the color map becomes active, in which there are a few characteristic basis states with high occupancy probability. In the final stage, the system goes to the steady state that consists only of the diagonal matrix elements. The following two points from this basis transformation approach are noticeable. One is that the basis states labeled by *n* > 27 are not excited and negligible, and this contributes to decrease the numerical calculation volume and the calculation time. The other is that there are components growing slowly and unidirectionally without exchange of energies among the other basis states. These are reminiscent of the dissipation process for the free photon.

**Figure 4.** Color map images of the quantum density matrices at the time steps of (**a**) *t* = 5, (**b**) 50, and (**c**) 5000, respectively. The occupancy probability and the transition matrices of the dressed photon are represented as the diagonal and off-diagonal matrix elements, respectively. All simulation parameters are the same in Figure 2. Meanwhile, in the early state, the dressed photon concentrates in the states with a localization nature and goes and returns aggressively among themselves; the basis states with the intermediate spatial size show slightly calm movement, which is reminiscent of the radiative dissipation to the external field of the free photon.

### **3. Renormalization of Quasi-Static Basis States**

In the previous section, novel basis states inspired by a non-equilibrium steady state are proposed to capture the spatial property that distinguishes the dressed-photon dynamics, and the temporal evolution of the dressed photon is visualized numerically in a space of the quantum density matrix. This seems to sugges<sup>t</sup> the distinction between the matter-like and the free photon-like properties of the dressed photon. Based on this insight, this section is devoted to discussing a way to focus the principal modes of the dressed photon with a localization nature.

First, let us pay attention to the coupling strength between the unitary transformed basis states that can be observed in the interaction Hamiltonian, *H*int,st. The interaction Hamiltonian before and after the basis transformation is visualized as the color map images in Figure 5. In the case before the transformation, a quasi-periodic structure appears depending on the lattice structure of the nodes as illustrated in Figure 1. The unitary

transformation drastically changes the appearance, which is shown in Figure 5b. The effective transition among the basis states restricts in the several basis states, and many basis states stay in their own modes that are described as the diagonal matrix elements, which represent the energy shift in the system dynamics. In the following, the projection operator method is applied to extract the principal basis states with a localization nature of the dressed photon, and to eliminate the basis states with the weak contribution.

**Figure 5.** Color map images of matrix elements of the interaction Hamiltonian *H*int, which is given in (3). (**a**) The matrix before the unitary transformation has a quasi-periodic structure reflected by the range-dependent coupling strength among a certain node and nearly arranged ones, and all matrix elements in the diagonal part are zero. (**b**) The matrix after the unitary transformation shows characteristic structure. There are two distinct areas divided at *n* = 27, which corresponds to the mode of the dressed-photon excitation. In the base *n* ≤ 27, the diagonal matrix elements have large values, i.e., staying in their own modes, and it is found that several basis states dominantly contribute to the dressed-photon dynamics via off-diagonal matrix elements.
