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We study the collective modes of a one-dimensional harmonically trappedBose-Einstein condensate in the presence of the anomalous density using thetime-dependent Hartree-Fock-Bogoliubov theory. Within the hydrodynamicequations, we derive analytical expressions for the mode frequencies and thedensity fluctuations of the anomalous density which constitutes the minoritycomponent at very low temperature and feels an effective external potentialexerted by the majority component, i.e., the condensate. On the other hand, wenumerically examine the temperature dependence of the breathing modeoscillations of the condensate at finite temperature in the weak-couplingregime. At zero temperature, we compare our predictions with availableexperimental data, theoretical treatments, and Monte carlo simulations in allinteraction regimes and the remaining hindrances are emphasized. We show thatthe anomalous correlations have a non-negligible role on the collective modesat both zero and finite temperatures.

Collective modes of a one-dimensional trapped Bose gas in the presence of the anomalous density