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

摘要

We study the relation between the eigenfrequencies of the Bogoliubov excitations of Bose-Einstein condensatesnand the eigenvalues of the Jacobian stability matrix in a variational approach that maps the Gross-Pitaevskiinequation to a system of equations of motion for the variational parameters. We do this for Bose-Einsteinncondensates with attractive contact interaction in an external trap and for a simple model of a self-trappednBose-Einstein condensatewith attractive 1/r interaction. The stationary solutions of theGross-Pitaevskii equationnand Bogoliubov excitations are calculated using a finite-difference scheme. The Bogoliubov spectra of thenground and excited state of the self-trapped monopolar condensate exhibit a Rydberg-like structure, which can benexplained by means of a quantum-defect theory. On the variational side, we treat the problem using an ansatz ofntime-dependent coupled Gaussian functions combined with spherical harmonics. We first apply this ansatz to ancondensate in an external trap without long-range interaction and calculate the excitation spectrum with the helpnof the time-dependent variational principle. Comparing with the full-numerical results, we find good agreementnfor the eigenfrequencies of the lowest excitation modes with arbitrary angular momenta. The variational methodnis then applied to calculate the excitations of the self-trapped monopolar condensates and the eigenfrequenciesnof the excitation modes are compared.