Diffusion and localization for the Chirikov typical map

TL;DR

This paper investigates the classical and quantum dynamics of the Chirikov typical map, revealing a chaos border, diffusive regimes, and a nuanced dependence of quantum localization length on classical diffusion constants.

Contribution

It introduces the Chirikov typical map with random phase shifts, analyzing its classical chaos border and quantum localization properties across different time scales.

Findings

Classical chaos border at k_c ~ T^{-3/2}

Existence of diffusive regimes on short and long time scales

Quantum localization length depends on classical diffusion constants

Abstract

We consider the classical and quantum properties of the "Chirikov typical map", proposed by Boris Chirikov in 1969. This map is obtained from the well known Chirikov standard map by introducing a finite number $T$ of random phase shift angles. These angles induce a random behavior for small time scales ($t<T$) and a $T$-periodic iterated map which is relevant for larger time scales ($t>T$). We identify the classical chaos border $k_c\sim T^{-3/2} \ll 1$ for the kick parameter $k$ and two regimes with diffusive behavior on short and long time scales. The quantum dynamics is characterized by the effect of Chirikov localization (or dynamical localization). We find that the localization length depends in a subtle way on the two classical diffusion constants in the two time-scale regime.