**5. Conclusions**

In this paper, a phenomenological model that regards the dressed photon as a particle localized at a certain node in a collection of nodes representing nanomatter has been proposed, and its spatio-temporal dynamics has been formulated using the quantum density operator. This system includes the radiative and non-radiative dissipation processes that are given by the Lindblad-type dissipation based on the first-order Born–Markov approximation, and the external excitation, and thus, the system dynamics converges to a non-equilibrium steady state. For such a model, a mechanism for extracting a part of the dressed-photon energy to the external field, which corresponds to observation instruments, has been considered to access the localized state of the dressed photon, and to explain

interesting experimental facts mediated by the dressed photon. Specifically, the methods to describe the dressed photon by characteristic basis states inspired by a non-equilibrium steady state as well as to separate the basis states into the target and the complementary spaces have been proposed and formulated using the projection operators. Contribution of the complementary space is renormalized in the target space by means of the lowest-order perturbation approximation. These theoretical and numerical approaches are a pioneering study that elucidates the principle of continuously connecting the dressed photon to the free photon. In this research, the process in which the bound or massive dressed photon dissociates its mass and is converted to the free photon has been interpreted by considering the energy transfer among the basis states with different spatial characteristics.

In the last part of this paper, a concept for accessing the principal basis states in a nanomatter system has been discussed using the same manner of renormalization. This is a qualitative proposal, but an important finding that the external manipulation of the dressed photon associated with the concept of renormalization.

The basis transformations using a predetermined steady state are inconsistent for the purpose of simulating the dressed-photon dynamics in an unknown system. However, in the experimental systems in which the dressed photon is mediated, the structural changes of the nanomatter always appears, such as an optimal rearrangemen<sup>t</sup> of atoms. Therefore, our approach to focus on changes from the steady states seems to be effective for explaining the experimental facts. In that sense, our proposed method is worth enough aiming at solving a dressed-photon optimization problem.

In the present research stage, the theoretical formulation is somewhat insufficient to explain the experimental facts quantitatively, but this paper has provided a meaningful consideration as a challenge by stepping into the essence of the underlying physical mechanism of the dressed photon and an off-shell science. It is expected to lead to a detailed understanding of dressed-photon physics and industrial applications in the near future.

**Author Contributions:** Conceptualization, H.S. and S.S.; methodology, S.S.; validation, S.S.; investigation, S.S.; writing—original draft preparation, S.S. Both authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** The authors thank anonymous reviewers for their comments to improve the quality of this paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
