*3.1. Projection Operator Method*

The projection operator method is a mathematical technique that divides the entire system into a target space (*P*) and a complementary space (*Q*), and inserts the influence of the complementary space into the target space [14,20]. A state vector of the entire system |*ψ*st is divided into the two sub-spaces using the projection operators,

$$|\psi\_{\rm st}^{P}\rangle = P|\psi\_{\rm st}\rangle,\tag{10a}$$

$$|\psi\_{\rm st}^{Q}\rangle = \mathbb{Q}|\psi\_{\rm st}\rangle,\tag{10b}$$

where the projection operators *P* and *Q* satisfy the following relations,

$$P + Q = 1,\tag{11a}$$

$$P^2 = P\_\prime \tag{11b}$$

$$
\mathbb{Q}^2 = \mathbb{Q}.\tag{11c}
$$

Using the Schrödinger equation,

$$H\_{\rm exc,st} \vert \psi\_{\rm st} \rangle = \Delta E \vert \psi\_{\rm st} \rangle,\tag{12}$$

the state vector in the *Q*-space can be expressed as the sum of the contributions from the state vector in the *P*-space, i.e.,

$$
\langle \psi\_{\rm st}^{Q} \rangle = \sum\_{n=1}^{\infty} \left( \Delta E^{-1} Q H\_{\rm int, \rm st} \right)^{n} |\psi\_{\rm st}^{P}\rangle \approx \Delta E^{-1} Q H\_{\rm int, \rm st} |\psi\_{\rm st}^{P}\rangle. \tag{13}
$$

In (12), Δ*E* corresponds to the energy shift of the basis states unitary transformed from the basis states expressed by the local nodes. It should be noted that the contribution of the radiative and non-radiative dissipations, and the excitation of the dressed photon are ignored in (12) and (13) because the dissipation and the excitation originate from the interaction with the external field, but it is qualitatively negligible by assuming that only the hopping conduction of the dressed photon contributes to the transition between the *P* and the *Q*-spaces. In the last part of (13), the first-order perturbation is applied by assuming that the basis states in the *Q*-space weakly affect the *P*-space dynamics.
