**1. Introduction**

In recent years, some novel and fundamental experimental studies have been reported that originate from the photon localized at a nanometer scale. For example, a Si light emitter with a nanostructure of boron dopants has been demonstrated, where Si is an indirect semiconductor and such an optical transition is forbidden in the conventional optics [1,2]. For microfabrication techniques, size-selective and non-adiabatic photochemical reactions (etching [3] and deposition [4]) have been observed on rough surfaces with nanostructures and under nanometrically tapered optical fiber probes. Furthermore, a giant magnetooptical effect using a ZnO single crystal with a nanostructure of the dopant has been confirmed as a surprising experimental result [5]. To explain these experimental facts, it is necessary to step into an off-shell science [6,7], which is a concept that overcomes the conventional optics limited by energy and momentum conservation laws. The origin of the appearance of strange optical phenomena in the off-shell region is considered to be environmental effects of background materials, such as the electronic excitation field and the phonon field, on the internal photon field, which is called the dressed photon. However, it is a challenging task to build a complete theory, since this would require incorporating an unknown contribution of infinite degrees of freedom. Thus, a simple theoretical expression without losing the essence of the dressed photon is strongly desired.

In this paper, a phenomenological model of the dressed photon is proposed without touching on the specific generation process of the dressed photon. At first glance, such a model may resemble an exciton–polariton picture, but the dressed photon is considered to be a quasi-particle bounded in a finite distance with the help of the surrounding electronic and phononic excitations, and the energy transfer of the dressed photon via the off-shell region or non-resonant region is allowed. This is the difference between the dressed photon and the exciton polariton. A numerical simulation is also demonstrated for expressing the dressed photon dynamics, and discussing the extraction of energy from the internal dressed-photon system to the external field.

The logical flow in this paper is summarized as follows. The spatial distribution of the dressed photon has been decomposed into plural characteristic basis states reflecting a certain steady state. At this time, the basis states can be distinguished into strong and weak contributions to the system dynamics by referring to the formula for renormalizing the weak interaction into the strong one. The weak-interacting basis states can be regarded as quasi-particle states with a light mass that resemble a photon reservoir system. In this way, the influence of a microscopic system on a macroscopic one can be formulated, and the microscopic system can be controlled from the macroscopic one. It will be a clue to explain the emergence of optical functions via the dressed photon, such as the light emission from indirect semiconductors.

The following sections are constructed to evaluate the above concept as follows. Section 2 describes the formulation of dressed-photon dynamics. Here, in addition to providing the equation of motion in a non-equilibrium system, dressed-photon basis states characterized by the spatial distribution is introduced for the subsequent discussions. In Section 3, a method for dividing a dressed-photon system into the systems with strong and weak contributions is proposed using the renormalization technique. Section 4 gives an insight for connecting the dressed photon with the external free photon, based on the method obtained up to Section 3. In addition, we will discuss how to control the dressed photon from the external degree of freedom. Finally, Section 5 summarizes this paper.

### **2. Theoretical Model of a Dressed-Photon System**
