Relation between the eigenfrequencies of Bogoliubov excitations of Bose-Einstein condensates and the eigenvalues of the Jacobian in a time-dependent variational approach

Abstract

We study the relation between the eigenfrequencies of the Bogoliubov excitations of Bose-Einstein condensates, and the eigenvalues of the Jacobian stability matrix in a variational approach which maps the Gross-Pitaevskii equation to a system of equations of motion for the variational parameters. We do this for Bose-Einstein condensates with attractive contact interaction in an external trap, and for a simple model of a self-trapped Bose-Einstein condensate with attractive 1/r interaction. The stationary solutions of the Gross-Pitaevskii equation and Bogoliubov excitations are calculated using a finite-difference scheme. The Bogoliubov spectra of the ground and excited state of the self-trapped monopolar condensate exhibits a Rydberg-like structure, which can be explained by means of a quantum defect theory. On the variational side, we treat the problem using an ansatz of time-dependent coupled Gaussians combined with spherical harmonics. We first apply this ansatz to a condensate in an external trap without long-range interaction, and calculate the excitation spectrum with the help of the time-dependent variational principle. Comparing with the full-numerical results, we find a good agreement for the eigenfrequencies of the lowest excitation modes with arbitrary angular momenta. The variational method is then applied to calculate the excitations of the self-trapped monopolar condensates, and the eigenfrequencies of the excitation modes are compared.

15 pages, 12 figures

References in corpus (8)

• Bose-Einstein condensation of chromium
• Observation of dipole-dipole interaction in a degenerate quantum gas
• Radial and angular rotons in trapped dipolar gases
• Ground-state structure and stability of dipolar condensates in anisotropic traps
• Bose-Einstein condensates with attractive 1/r interaction: The case of self-trapping
• Variational methods with coupled Gaussian functions for Bose-Einstein condensates with long-range interactions. II. Applications
• Variational methods with coupled Gaussian functions for Bose-Einstein condensates with long-range interactions. I. General concept
• Dynamics and stability of Bose-Einstein condensates with attractive 1/r interaction

Cited by in corpus (5)

• A variational approach to Bogoliubov excitations and dynamics of dipolar Bose-Einstein condensates
• Macroscopic quantum tunneling of Bose-Einstein condensates with long-range interaction
• Verification of exceptional points in the collapse dynamics of Bose-Einstein condensates
• Normal form expansions and thermal decay rates of Bose-Einstein condensates with short- and long-range interaction
• A nonlinear dynamics approach to Bogoliubov excitations of Bose-Einstein condensates