High-order geometric integrators for the variational Gaussian approximation

TL;DR

This paper introduces high-order geometric integrators for the variational Gaussian approximation of the Schrödinger equation, significantly improving efficiency and accuracy while preserving key physical properties like symplectic structure and norm conservation.

Contribution

The authors develop and demonstrate high-order symplectic integrators that enhance the efficiency of the variational Gaussian method for quantum dynamics, maintaining essential geometric properties.

Findings

High-order integrators drastically speed up convergence.

They are time-reversible and conserve norm and symplectic structure.

Effective in high-dimensional systems like twenty-dimensional Morse oscillators.

Abstract

Among the single-trajectory Gaussian-based methods for solving the time-dependent Schr\"{o}dinger equation, the variational Gaussian approximation is the most accurate one. In contrast to Heller's original thawed Gaussian approximation, it is symplectic, conserves energy exactly, and may partially account for tunneling. However, the variational method is also much more expensive. To improve its efficiency, we symmetrically compose the second-order symplectic integrator of Faou and Lubich and obtain geometric integrators that can achieve an arbitrary even order of convergence in the time step. We demonstrate that the high-order integrators can speed up convergence drastically compared to the second-order algorithm and, in contrast to the popular fourth-order Runge-Kutta method, are time-reversible and conserve the norm and the symplectic structure exactly, regardless of the time step. To show that the method is not restricted to low-dimensional systems, we perform most of the analysis on a non-separable twenty-dimensional model of coupled Morse oscillators. We also show that the variational method may capture tunneling and, in general, improves accuracy over the non-variational thawed Gaussian approximation.