  We find that the quantum-classical correspondence in integrable systems is
characterized by two time scales. One is the Ehrenfest time below which the
system is classical; the other is the quantum revival time beyond which the
system is fully quantum. In between, the quantum system can be well
approximated by classical ensemble distribution in phase space. These results
can be summarized in a diagram which we call Ehrenfest diagram. We derive an
analytical expression for Ehrenfest time, which is proportional to
$\hbar^{-1/2}$. According to our formula, the Ehrenfest time for the
solar-earth system is about $10^{26}$ times of the age of the solar system.We
also find an analytical expression for the quantum revival time, which is
proportional to $\hbar^{-1}$. Both time scales involve $\omega(I)$, the
classical frequency as a function of classical action. Our results are
numerically illustrated with two simple integrable models. In addition, we show
that similar results exist for Bose gases, where $1/N$ serves as an effective
Planck constant.
