A nonlinear dynamics approach to Bogoliubov excitations of Bose-Einstein condensates

TL;DR

This paper introduces a nonlinear dynamics framework using Gaussian wave packets and variational principles to analyze Bogoliubov excitations in Bose-Einstein condensates, linking classical nonlinear systems with quantum spectra.

Contribution

It presents a novel approach that maps the Gross-Pitaevskii equation onto a nonlinear dynamical system to study excitations in condensates.

Findings

Eigenvalues of the Jacobian match the Bogoliubov spectrum

Stability analysis reveals fixed point properties

Method bridges nonlinear dynamics and quantum excitations

Abstract

We assume the macroscopic wave function of a Bose-Einstein condensate as a superposition of Gaussian wave packets, with time-dependent complex width parameters, insert it into the mean-field energy functional corresponding to the Gross-Pitaevskii equation (GPE) and apply the time-dependent variational principle. In this way the GPE is mapped onto a system of coupled equations of motion for the complex width parameters, which can be analyzed using the methods of nonlinear dynamics. We perform a stability analysis of the fixed points of the nonlinear system, and demonstrate that the eigenvalues of the Jacobian reproduce the low-lying quantum mechanical Bogoliubov excitation spectrum of a condensate in an axisymmetric trap.