Consider an atom with two states, ground | g ⟩ ⟩ and excited | e ⟩ ⟩ . The atom has a Hamiltonian H = ℏ ℏ ω ω | e ⟩ ⟩ ⟨ ⟨ e | , where ω ω is the frequency of light associated with the g-e transition. Now suppose we illuminate the atom with a drive at frequency ω ω d which couples the two states, and that the time-dependent driven Hamiltonian is