Theory of localization and resonance phenomena in the quantum kicked rotor

TL;DR

This paper develops an analytic theory connecting quantum interference and Anderson localization in the quantum kicked rotor, revealing how localization depends on the effective Planck's constant and rational approximations, with implications for energy dynamics.

Contribution

The paper introduces a novel analytic framework linking the quantum kicked rotor's localization phenomena to a disordered ring analogy, providing quantitative predictions for energy behavior based on system parameters.

Findings

For rational $ heta$ values, the system resembles a disordered ring with flux.

Energy scales as $ ilde t^2$ for small $q$, indicating non-localized, non-diffusive dynamics.

Energy saturates at $ o \xi^2$ for large $q$, indicating localization.

Abstract

We present an analytic theory of quantum interference and Anderson localization in the quantum kicked rotor (QKR). The behavior of the system is known to depend sensitively on the value of its effective Planck's constant $\he$. We here show that for rational values of $\he/(4\pi)=p/q$, it bears similarity to a disordered metallic ring of circumference $q$ and threaded by an Aharonov-Bohm flux. Building on that correspondence, we obtain quantitative results for the time--dependent behavior of the QKR kinetic energy, $E(\tilde t)$ (this is an observable which sensitively probes the system's localization properties). For values of $q$ smaller than the localization length $\xi$, we obtain scaling $E(\tilde t) \sim \Delta \tilde t^2$, where $\Delta=2\pi/q$ is the quasi--energy level spacing on the ring. This scaling is indicative of a long time dynamics that is neither localized nor diffusive. For larger values $q\gg \xi$, the functions $E(\tilde t)\to \xi^2$ saturates (up to exponentially small corrections $\sim\exp(-q/\xi)$), thus reflecting essentially localized behavior.