$\hbar \to 0$ in Kicked Harper Model: Reassurances and Surprises

We investigate classical-quantum correspondence for kicked Harper model for extremely small values of the Planck constant $\hbar$. In the asymmetric case a pure quantum state shows clear signature of classical diffusive as well as super diffusive transitions asymptotically independent of $\hbar$. However, for the symmetric case, the $\hbar$ independent behavior occurs only for renormalized parameter $\bar{K}=K/(2\hbar)$ with intriguing features such as a sharp transition from integrable to non-integrable transport at $\bar{K}=\pi/2$, a series of transitions at multiples of $\pi$ and periodicity of the transmission probability. These results add new puzzles to the frontiers of quantum chaos.