Complete controllability of finite quantum systems with twofold energy level degeneracy

Complete controllability of finite-dimensional quantum systems with energy level degeneracy is investigated using two different approaches. One approach is to apply a weak constant field to eliminate the degeneracy and then control it using techniques developed for non-degenerate quantum systems. Conditions for the elimination of degeneracy are found and the issues of influence of relaxation time of a constant external field on the target state are addressed through the fidelity. Another approach is to control the degenerate system by a single control field directly. It is found that the system with twofold degenerate excited states and non-degenerate ground state is completely controllable except for the two-level system. Conditions of complete controllability are found for both systems with different energy gaps and with equal energy gaps.