Since ĤM is a pure matter operator, it is invariant upon mode truncation and therefore commutes with P, P(m) ĤM P(m) = ĤM P(m) . In the case of a single mode m = 1, Eq. 44 reduces to the well-known single-mode minimal coupling Hamiltonian (see Eq. 4). Interestingly, if we apply a simple mode truncation, Ĥ′(m) p•A = P(m) Ĥp•A P(m) has the same form of Eq. 44 up to a constant that represents the zero-point energy of all modes.