  Quantum adiabatic evolution is a dynamical evolution of a quantum system
under slow external driving. According to the quantum adiabatic theorem, no
transitions occur between non-degenerate instantaneous eigen-energy levels in
such a dynamical evolution. However, this is true only when the driving rate is
infinitesimally small. For a small nonzero driving rate, there are generally
small transition probabilities between the energy levels. We develop a
classical mechanics framework to address the small deviations from the quantum
adiabatic theorem order by order. A hierarchy of Hamiltonians are constructed
iteratively with the zeroth-order Hamiltonian being determined by the original
system Hamiltonian. The $k$th-order deviations are governed by a $k$th-order
Hamiltonian, which depends on the time derivatives of the adiabatic parameters
up to the $k$th-order. Two simple examples, the Landau-Zener model and a
spin-1/2 particle in a rotating magnetic field, are used to illustrate our
hierarchical theory. Our analysis also exposes a deep, previously unknown
connection between classical adiabatic theory and quantum adiabatic theory.
