Recall that a position shift operator, Ûq = e -i ℏ q0 p displaces q by the amount q 0 , such that Ûq Ô(q) Û † q = Ô(q + q 0 ). With this in mind, a shift operator in both photonic and matter coordinates is introduced, which transforms Eq. 18 into the AD representation. where π = i( b † -b) is the photonic momentum in the Bogoliubov transformed space and ξ g = gx Ω /Ω is an effective coupling parameter in the AD representation. This leads to the introduction of the AD Hamiltonian,