Non-linearity and chaos in the kicked top

TL;DR

This paper investigates how varying degrees of non-linearity in the quantum kicked top model influence the emergence and suppression of chaos, revealing distinct behaviors depending on the non-linearity parameter.

Contribution

It introduces a parametrized non-linearity in the kicked top model and analyzes how chaos intensity changes with this parameter, providing new insights into quantum-classical chaos relationships.

Findings

Chaos increases with non-linearity parameter p in [1,2]

Chaos diminishes for p > 2, leading to regular oscillations as p approaches infinity

The phase space structure becomes more complex in non-chaotic regimes

Abstract

Classical chaos arises from the inherent non-linearity of dynamical systems. However, quantum maps are linear; therefore, the definition of chaos is not straightforward. To address this, we study a quantum system that exhibits chaotic behavior in its classical limit: the kicked top model, whose classical dynamics are governed by Hamilton's equations on phase space, whereas its quantum dynamics are described by the Schr\"odinger equation in Hilbert space. We explore the critical degree of non-linearity signifying the onset of chaos in the kicked top by modifying the original Hamiltonian so that the non-linearity is parametrized by a quantity $p$. We find two distinct behaviors of the modified kicked top depending on the value of $p$. Chaos intensifies as $p$ varies within the range of $1\leq p \leq 2$, whereas it diminishes for $p > 2$, eventually transitioning to a purely regular oscillating system as $p$ tends to infinity. We also comment on the complicated phase space structure for non-chaotic dynamics. Our investigation sheds light on the relationship between non-linearity and chaos in classical systems, offering insights into their dynamic behavior.