Gradient of the Adiabatic Gauge Potential in Classical Systems

TL;DR

This paper introduces a method to compute the gradient of the adiabatic gauge potential in classical systems, revealing its behavior in integrable and chaotic regimes, with implications for understanding canonical transformations and chaos.

Contribution

It presents an efficient computational approach for the classical AGP gradient and explores its divergence in chaotic systems, extending quantum concepts to classical phase space.

Findings

Canonical transformation reproduces expected results in integrable systems

Gradient diverges in chaotic systems related to Lyapunov times

Method provides insights into classical adiabatic processes

Abstract

The adiabatic gauge potential (AGP) is the generator of unitary transformations which preserve the eigenbasis of a quantum Hamiltonian under parametric variation. While its usefulness in quantum mechanics has been thoroughly demonstrated in recent years, less attention has been given to its behavior in classical systems, where the AGP is a phase space function and its gradient defines special canonical transformations. In this paper we propose an efficient method to compute the gradient of the AGP as a classical function. We demonstrate that the obtained canonical transformation reproduces expected results for simple orbits and integrable systems for which the adiabatic limit is well-defined. In chaotic systems the gradient diverges in a way that is related to Lyapunov times.