Antiresonance and Localization in Quantum Dynamics

Abstract

The phenomenon of quantum antiresonance (QAR), i.e., exactly periodic recurrences in quantum dynamics, is studied in a large class of nonintegrable systems, the modulated kicked rotors (MKRs). It is shown that asymptotic exponential localization generally occurs for $η$ (a scaled $\hbar$) in the infinitesimal vicinity of QAR points $η_0$. The localization length $ξ_0$ is determined from the analytical properties of the kicking potential. This ``QAR-localization" is associated in some cases with an integrable limit of the corresponding classical systems. The MKR dynamical problem is mapped into pseudorandom tight-binding models, exhibiting dynamical localization (DL). By considering exactly-solvable cases, numerical evidence is given that QAR-localization is an excellent approximation to DL sufficiently close to QAR. The transition from QAR-localization to DL in a semiclassical regime, as $η$ is varied, is studied. It is shown that this transition takes place via a gradual reduction of the influence of the analyticity of the potential on the analyticity of the eigenstates, as the level of chaos is increased.