Approximating Quantum Lyapunov Exponents in Quantum Kicked Rotor

TL;DR

This paper introduces a new measure called Quantum Lyapunov Exponent to quantify chaos in the Quantum Kicked Rotor, analyzing state fidelity and exploring diverse initial states to understand quantum chaotic dynamics.

Contribution

It proposes the Quantum Lyapunov Exponent as a novel metric for quantum chaos and develops an optimization framework for representing complex quantum states.

Findings

Quantum Lyapunov Exponent effectively measures quantum chaos.

Diverse initial states lead to varied dynamical behaviors.

Optimization framework aids in visualizing complex quantum states.

Abstract

In this work, we study quantum chaos by focusing on the evolution of initially close states in the dynamics of the Quantum Kicked Rotor (QKR). We propose a novel measure, the Quantum Lyapunov Exponent (QLE), to quantify the degree of chaos in this quantum system, analogous to its classical counterpart. We begin by modeling the momentum space and then the QLE is computed through analyzing the fidelity between evolving states, offering insights into the quantum chaotic behavior. Furthermore, we extend our investigations to various initial states: localized, uniform, spreading, contracting and oscillating in momentum space. Our results unveil a diverse range of dynamical behaviors, highlighting the complex nature of quantum chaos. Finally, we propose an innovative optimization framework to represent a complex state as a superposition of the aforementioned states, which has potential implications for visualizing and understanding the dynamics of multifaceted quantum systems.