Localization and delocalization in the quantum kicked prime number rotator

TL;DR

This paper investigates the localization and delocalization phenomena in the quantum kicked prime number rotator, revealing how the system's behavior depends on the kick period and strength, with implications for quantum chaos and number theory.

Contribution

It introduces the quantum kicked prime number rotator model and analyzes its localization-delocalization transition based on the parameters.

Findings

QKPR is localized when τ/2π is irrational due to equidistribution.

QKPR is localized for small k when τ/2π is rational, resembling a generalized kicked dimer.

QKPR delocalizes for large k at rational τ/2π, indicating a transition in behavior.

Abstract

The quantum kicked prime number rotator (QKPR) is defined as the rotator whose energy levels are prime numbers. The long time behavior is decided by the kick period $\tau$ and kick strength $k$. When $\frac{\tau}{2\pi}$ is irrational, QKPR is localized because of the equidistribution theorem. When $\frac{\tau}{2\pi}$ is rational, QKPR is localized for small $k$, because the system seems like a generalized kicked dimer model. We argue for rational $\frac{\tau}{2\pi}$ QKPR delocalizes for large k.