  We decompose the quantum adiabatic evolution as the products of gauge
invariant unitary operators and obtain the exact nonadiabatic correction in the
adiabatic approximation. A necessary and sufficient condition that leads to
adiabatic evolution with geometric phases is provided and we determine that in
the adiabatic evolution, while the eigenstates are slowly varying, the
eigenenergies and degeneracy of the Hamiltonian can change rapidly. We
exemplify this result by the example of the adiabatic evolution driven by
parametrized pulse sequences. For driving fields that are rotating slowly with
the same average energy and evolution path, fast modulation fields can have
smaller nonadiabatic errors than obtained under the traditional approach with a
constant amplitude.
