Hamiltonian reduction of quantum systems controlled by pulses

We explores Hamiltonian reduction in pulse-controlled finite-dimensional quantum systems with near-degenerate eigenstates. A quantum system with a non-degenerate ground state and several near-degenerate excited states is controlled by a short pulse, and the objective is to maximize the collective population on all excited states when we treat all of them as one level. Two cases of the systems are shown to be equivalent to effective two-level systems. When the pulse is weak, simple relations between the original systems and the reduced systems are obtained. When the pulse is strong, these relations are still available for pulses with only one frequency under the first-order approximation.