  We study the electron dynamics in a 2D waveguide bounded by a periodically
rippled surface in the presence of the time-periodic electric field. The main
attention is paid to a possibility of a weak quantum diffusion along the
coupling resonance, that can be associated with the classical Arnol'd
diffusion. It was found that quantum diffusion is possible only when the
perturbation is large enough in order to mix many near-separatrix levels. The
rate of the quantum diffusion turns out to be less than the corresponding
classical one, thus indicating the influence of quantum coherent effects.
Another important effect is the dynamical localization of the quantum
diffusion, that may be compared with the famous Anderson localization occurring
in 1D random potentials. Our estimates show that the quantum Arnol'd diffusion
can be observed in semi-metal rippled channels, for which the scattering and
decoherence times are larger than the saturation time due to the dynamical
localization.
