ϕ p/ℏ |n⟩, and n ϕ = 0, 1, 2..., ∞ is the quantum number for the PFS. Further, b † ϕ = (q ′ ϕ + ip)/ √ 2 and bϕ = (q ′ ϕ -ip)/ √ 2 are the creation and annihilation operators of the PFS |n ϕ ⟩, with the photon field momentum operator p and polarized photon field coordinate operator q′ ϕ = q + q 0 ϕ (R). Compared to the vacuum's Fock state |n⟩, these PFS depends on the diabatic state (or more generally, the eigenstate of μ) of the molecule, and the position of the nuclei (through the R dependence in µ ϕ (R)). Due to the electronic state-dependent nature of the polarization, the PFS associated with different electronic diabatic states become non-orthogonal, i.e., ⟨n ϕ |m φ ⟩ ̸ = δ ϕφ . Under the special case of the atomic cavity QED, the PFS representation reduces to the qubitshifted Fock basis {|n + ⟩, |m -⟩}, which has been used to solve the polariton eigen-spectrum for the quantum Rabi model throughout the entire range of light-matter coupling and derive the generalized rotating-wave approximation . These non-orthogonal Fock states and their overlap ⟨m -|n + ⟩ have shown to effectively capture the light-matter interactions in a quantum Rabi model. To express the Hamiltonian in the PFS representation, the PF Hamiltonian (Eq. 29) undergoes a unitary shift operator (similar to a polaron transformation ), Ûpol of the form,