where Ĥel is the electronic Hamiltonian under the Born-Oppenheimer approximation (i.e., without the nuclear kinetic energy operator), Ĥph = ω c â † â is the Hamiltonian of the cavity field, â † and â are the raising and lowering operators of the cavity field, ê is a unit vector indicating the field polarization direction, and μ is the dipole operator of the molecule. Through the light-matter coupling terms in Eq. 2, various photon-dressed electronic states will be coupled to each other. For example, |ψ g , 1⟩ ≡ |ψ g ⟩⊗|1⟩ (the ground electronic state with 1 photon) and |ψ e , 0⟩ ≡ |ψ e ⟩ ⊗ |0⟩ (an excited electronic state with 0 photons) will couple through ⟨ψ g , 1| μ • ê(â † + â)|e, 0⟩ = µ ge ⟨1|(â † + â)|0⟩, where µ ge is the transition dipole between the ground state and excited state projected along the ê direction. When the energy of these two basis states become close, the |ψ g , 1⟩ and |ψ e , 0⟩ states hybridize, leading to the formation of excited polariton states. This is the typical resonant light-matter coupling induced hybridization and generating new eigenstates, and polaritons.