There are several interesting limits of Ĥp•A (Eq. 76). Under the limiting case when A 0 = 0 or μ • Â = 0, both the -∇ j μ Â and Pj terms become 0, and Û † = Û → P ⊗ 1R ⊗ 1ph . Thus, under a such limit, Ĥp•A → ĤM + Ĥph ; hence, the matter and the cavity becomes decoupled. When using adiabatic states for the truncation, one can show that