Fourier methods for the perturbed harmonic oscillator in linear and nonlinear Schr\"odinger equations

TL;DR

This paper develops advanced Fourier-based numerical methods for solving the Gross-Pitaevskii equation with time-dependent harmonic potentials, improving accuracy and efficiency especially for complex, nonlinear, and time-varying systems.

Contribution

It introduces new methods that combine Fourier techniques with exact or highly accurate solutions of the time-dependent harmonic oscillator, overcoming limitations of traditional splitting methods.

Findings

Enhanced accuracy in numerical integration of the Gross-Pitaevskii equation.

Efficient handling of time-dependent harmonic potentials.

Improved performance for nonlinear and strongly varying systems.

Abstract

We consider the numerical integration of the Gross-Pitaevskii equation with a potential trap given by a time-dependent harmonic potential or a small perturbation thereof. Splitting methods are frequently used with Fourier techniques since the system can be split into the kinetic and remaining part, and each part can be solved efficiently using Fast Fourier Transforms. To split the system into the quantum harmonic oscillator problem and the remaining part allows to get higher accuracies in many cases, but it requires to change between Hermite basis functions and the coordinate space, and this is not efficient for time-dependent frequencies or strong nonlinearities. We show how to build new methods which combine the advantages of using Fourier methods while solving the timedependent harmonic oscillator exactly (or with a high accuracy by using a Magnus integrator and an appropriate decomposition).