General Theory of the Quantum Kicked Rotator. I

TL;DR

This paper reviews fundamental issues of the quantum kicked rotator, critiques existing proof methods, and analytically and numerically demonstrates anomalous localization phenomena near rational kick periods.

Contribution

It provides a critical review of the quantum kicked rotator, identifies flaws in the inverse Cayley transform method, and analytically proves and numerically confirms anomalous localization near rational periods.

Findings

Flaws in the inverse Cayley transform method for localization

Localization length increases near rational period ratios

Analytical proof and numerical confirmation of anomalous localization

Abstract

This is the first of a series of two papers. We discuss some basic problems of the quantum kicked rotator (QKR) and review some important results in the literature. We point out the flaws in the inverse Cayley transform method to prove dynamic localization. When $\tau/2\pi$, where $\tau$ is the kick period, is very close to a rational number, the localization length is larger than the typical localization length. We analytically prove anomalous localization and confirm it by numerical calculations. We point out open problems that need further work.