  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 issue of influence of
relaxation time of constant external field to 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 two-fold
degenerate excited states and non-degenerate ground state are 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.
