Breakdown of correspondence in chaotic systems: Ehrenfest versus localization times

Breakdown of quantum-classical correspondence is studied on an experimentally realizable example of a one-dimensional periodically driven system. Two relevant time scales are identified in this system: the short Ehrenfest time $t_{ħ}\sim \ln \left({ħ}^{-1}\right)$ and the typically much longer localization time scale $T_L\sim {ħ}^{-2}.$ It is shown that surprisingly weak modification of the Hamiltonian may eliminate the more dramatic symptoms of localization without effecting the more subtle but ubiquitous and rapid loss of correspondence at $t_{ħ}.$