  Based on the adiabatic geometric phase concerning with density matrix[1] , we
extend it to the sub-geometric phase in the non-adiabatic case. It is found
that whatever the real part or imaginary part of the sub-geometric phase can
play an important role in quantum transition. The imaginary part of
sub-geometric phase can deviate the resonance peak in the quantum transition,
which may bring modification on the level crossing, while the real part of
sub-geometric phase will determine the stability of initial state according to
the linear stability analysis theory, which can be regarded as somewhat
complement on the selection rule of quantum transition. Finally, we illustrate
them by two examples: one is the system with time-dependent perturbation, the
other is a two-level system. It indicates that both the real and imaginary
parts of sub-geometric phase have influence on quantum transition.
