Generalized Gaussian wave packet dynamics: Integrable and Chaotic Systems

TL;DR

This paper develops a practical method for semiclassical wave packet propagation that combines generalized Gaussian wave packet dynamics with off-center real trajectories, improving accuracy in both integrable and chaotic systems.

Contribution

It connects GGWPD with linearized wave packet methods, enabling practical saddle point trajectory calculations for complex dynamical systems.

Findings

Improved accuracy in wave packet propagation using saddle point trajectories.

Effective application to the kicked rotor model.

Demonstrates accuracy gains as Planck's constant decreases.

Abstract

The ultimate semiclassical wave packet propagation technique is a complex, time-dependent WBK method known as generalized Gaussian wave packet dynamics (GGWPD). It requires overcoming many technical difficulties in order to be carried out fully in practice. In its place roughly twenty years ago, linearized wave packet dynamics was generalized to methods that include sets of off-center, real trajectories for both classically integrable and chaotic dynamical systems that completely capture the dynamical transport. The connections between those methods and GGWPD are developed in a way that enables a far more practical implementation of GGWPD. The generally complex saddle point trajectories at its foundation are found using a multi-dimensional, Newton-Raphson root search method that begins with the set of off-center, real trajectories. This is possible because there is a one-to-one correspondence. The neighboring trajectories associated with each off-center, real trajectory form a path that crosses a unique saddle; there are exceptions which are straightforward to identify. The method is applied to the kicked rotor to demonstrate the accuracy improvement as a function of $\hbar$ that comes with using the saddle point trajectories.