Localization and delocalization properties in quasi-periodically perturbed Kicked Harper and Harper models

We numerically study the single particle localization and delocalization phenomena of an initially localized wave packet in the kicked Harper model (KHM) and Harper model subjected to quasi-periodic perturbation composed of $M-$modes. Both models are localized in the monochromatically perturbed case $M=1$. KHM shows localization-delocalization transition (LDT) above $M\geq2$ as increase of the perturbation strength $\eps$. In contrast, in a time-continuous Harper model with the perturbation, it is confirmed that the localization persists for $M=2$ and the LDT occurs for $M\geq 3$. Furthermore, we investigate the diffusive property of the delocalized wave packet in the KHM and Harper model for $\eps$ above the critical strength $\eps_c$ ($\eps>\eps_c$) comparing with other type systems without localization, which takes place a ballistic to diffusive transition in the wave packet dynamics as the increase of $\eps$.