The derivation of the SBEs starts from a system Hamiltonian that fully includes the free-particles, Coulomb interaction, dipole interaction between classical light and electronic states, as well as the phonon contributions. Like almost always in many-body physics, it is most convenient to apply the second-quantization formalism after the appropriate system Hamiltonian H ^ ^ S y s t e m is identified. One can then derive the quantum dynamics of relevant observables O ^ ^ by using the Heisenberg equation of motion