This partially diagonalized expectation value in Eq. 119 can be fully diagonalized in the coherent state basis 148 defined by the unitary transformation ÛZ = e Z â † -Z * â, (120) which will shift the photonic creation and annihilation operators for an arbitrary complex Z, such that â † → â † + Z * and â → â + Z. This is in the same spirit of the polarized Fock state idea in the previous section (Sec 3.1), which is a polaron-like transform on the photonic DOF. Choosing the particular Z as follows