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

摘要

We assume the macroscopic wave function of a Bose-Einstein condensate as asuperposition of Gaussian wave packets, with time-dependent complex widthparameters, insert it into the mean-field energy functional corresponding tothe Gross-Pitaevskii equation (GPE) and apply the time-dependent variationalprinciple. In this way the GPE is mapped onto a system of coupled equations ofmotion for the complex width parameters, which can be analyzed using themethods of nonlinear dynamics. We perform a stability analysis of the fixedpoints of the nonlinear system, and demonstrate that the eigenvalues of theJacobian reproduce the low-lying quantum mechanical Bogoliubov excitationspectrum of a condensate in an axisymmetric trap.