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

We study the collective modes of a one-dimensional harmonically trapped Bose-Einstein condensate in the presence of the anomalous density using the time-dependent Hartree-Fock-Bogoliubov theory. Within the hydrodynamic equations, we derive analytical expressions for the mode frequencies and the density fluctuations of the anomalous density which constitutes the minority component at very low temperature and feels an effective external potential exerted by the majority component, i.e., the condensate. On the other hand, we numerically examine the temperature dependence of the breathing mode oscillations of the condensate at finite temperature in the weak-coupling regime. At zero temperature, we compare our predictions with available experimental data, theoretical treatments, and Monte carlo simulations in all interaction regimes and the remaining hindrances are emphasized. We show that the anomalous correlations have a non-negligible role on the collective modes at both zero and finite temperatures.