*3.3. Numerical Demonstration of Renormalization*

Using the above formulation, the concrete temporal evolution of the quantum density matrix is calculated numerically, and the validity of the approximation is evaluated. Figure 6 shows the steady-state solution for the three steps of the coarse graining, which are the original result without approximation, the result simply removing the basis states over *n* = 27, and the result after renormalization using (14)–(17). Prior to the renormalization, the *Q*-space components are selected as *n* = 17, 19, 20, 22, and 23, by referring the correction term of *<sup>H</sup>*int,st/*φQm*|*<sup>H</sup>*int,st|*φQm*, that weakly couple to the basis state for the excitation. Despite reducing the number of the basis states, the obtained steady-state solutions are almost

the same in all three cases, and the calculation time has been significantly reduced. In the case of Figure 6, the number of the basis states for obtaining the calculation results has been reduced by less than half against no coarse graining, and thus the number of the differential equations to be solved is 22% less. To confirm the validity of this approach, the spatial distribution of the occupancy probability of the dressed photon is reconstructed from the quantum density matrix by applying renormalization or not, that is shown in Figure 7. Both color map images of the occupancy probability before and after renormalization are in good agreemen<sup>t</sup> with each other.

**Figure 6.** Color map images of steady-state solutions for the quantum density matrix in the cases of (**a**) no eliminating the extra basis states, eliminating the states of *n* > 27, and additional renormalization of the states *n* = 17, 19, 20, 22, and 23. The number of the matrix elements decreases from (**<sup>a</sup>**–**<sup>c</sup>**) as 472, 272, and 222. In all three cases, the steady-state solutions converge to only diagonal elements, and the occupancy probability can be reproduced after coarse graining using renormalization of the quasi-static basis states.

**Figure 7.** Steady-state solutions of occupancy probability for the dressed photon are mapped in the geometrical structure of taper that are calculated using (**a**) the original basis states defined by nodes, and (**b**) reconstructed from the basis coarse-grained by eliminating extra base and renormalization. The renormalization condition is the same as that in Figure 6.

So far, a method to distinguish the heavy and the light components of the dressed photon has been proposed using the original basis transformation, and the numerical demonstration shows the potential for reducing the amount of computation. As a similar approach, a method where a macroscopic system is expressed with a small number of basis states using the basis states that are predetermined by the steady-state solutions in a small space step by step has been already published [18,19]. These papers report a large reduction in the amount of the quantum calculation. This method is very similar to our approach, in which basis transformation and renormalization are used for reducing the number of the principal basis states. Meanwhile, our main purpose in this research is to observe and control a behavior of the dressed photon localized in a nanometer space. This point will be considered in the next section.

### **4. Discussion on Control of Dressed Photon Distribution**

This section discusses the physical meaning for renormalizing particular basis states. In Section 3.3, from the characteristics of the basis states defined by a steady-state solution, the basis states with the weak contribution can be converted to dissipative component in the system by applying the renormalization method in the first-order perturbation approximation. Such a situation is equivalent to a free photon reservoir. According to the intuitive image, the dressed photon can be regarded as stripping off the mass caused by the interaction with the environment and changes into the massless free photon, where the dressed-photon basis states staying in a nanomatter system are responsible for the stripped mass via renormalization.

On the other hand, it is interesting to consider how to affect the spatial distribution characteristics of the dressed photon that stays inside a nanomatter system. As an example, let us consider extracting a certain principal basis state with the strong localization of the dressed photon into the *Q*-space. When the localization basis state is selected as *n* = 7 in Figure 3, where the dressed photon energy concentrates at a tip position, is assigned in the *Q*-space, the quantum density matrix is calculated in the same manner as explained in the previous section, where the approximation of the weak coupling has been already exceeded. Figure 8a is the numerical result of the simulation, and the quantum density matrix cannot converge on the diagonal matrix elements. If the strict quantitativeness is neglected, this corresponds to change of a steady-state solution, i.e., the spatial distribution of the dressed photon can be modified by extracting artificially the principal localization basis state. In Figure 8b, the steady-state solutions are shown as a color map image in the taper geometry reconstructed from Figure 8a. One can observe that the localization of the dressed photon at a tip position disappears after such renormalization. It should be noted that this result represents the characteristic behavior of dressed-photon mediated phenomena. Removing the dressed photon out of a nanomatter system for an experimental observation makes another new internal state of the spatial distribution of the dressed photon in the system. For controlling and optimizing dressed-photon mediated phenomena, the renormalization of the basis states of interest, that is proposed in this paper, will be an extremely important concept.

**Figure 8.** Numerical calculation result of the quantum density matrix when the basis state of *n* = 7 is additionally extracted as the *Q*-space. (**a**) Color map image of a steady-state solution for the density matrix elements, and (**b**) reconstructed color map image on the geometrical structure of taper. The extraction of the state actively contributing to the localization drastically changes convergence property of the quantum density matrix as well as spatial distribution of the dressed photon in a nanomatter.
