Antiresonance and Localization in Quantum Dynamics

TL;DR

This paper investigates quantum antiresonance in modulated kicked rotors, revealing how it causes exponential localization near QAR points and transitions to dynamical localization as chaos increases.

Contribution

It demonstrates the connection between quantum antiresonance and dynamical localization, providing analytical and numerical evidence for QAR-localization in nonintegrable systems.

Findings

QAR induces exponential localization near specific parameter points.

QAR-localization closely approximates dynamical localization near QAR points.

Transition from QAR-localization to dynamical localization involves reduced potential analyticity influence.

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 $\eta$ (a scaled $\hbar$) in the infinitesimal vicinity of QAR points $\eta_0$. The localization length $\xi_0$ 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 $\eta$ 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.