which is Eq. 1 in the Introduction. In the last line of Eq. 2, ĤM represents the molecular Hamiltonian, the second term Ĥph = â † â+ 1 2 hω c represents the Hamiltonian of the vacuum photon field inside the cavity with the frequency ω c , the third term describes the light-matter interaction in the electricdipole "d • E" form , with χ = hωc 2ε 0 V ê ≡ χê characterizing the light-matter coupling vector oriented in the direction of polarization unit vector ê, V as the quantization volume for the photon field, and ε 0 as the permittivity inside the cavity. Further, â † and â are the photon creation and annihilation operator, qc = h/2ω c (â † + â) and pc = i hω c /2(â † -â) are the photonic coordinate and momentum operator, respectively.