  We theoretically investigate collective modes of a one-dimensional (1D)
interacting Bose gas in harmonic traps at finite temperatures, by using a
variational approach and local density approximation. We find that the
temperature dependence of collective mode frequencies is notably different in
the weakly and strongly interacting regimes. Therefore, the experimental
measurement of collective modes could provide a sensitive probe for different
quantum phases of a 1D trapped Bose gas, realized by tuning the interatomic
interaction strength and temperature. Our prediction on the temperature
dependence of the breathing mode frequency is in good qualitative agreement
with an earlier experimental measurement for a weakly interacting 1D Bose gas
of rubidium-87 atoms in harmonic traps [Moritz et al., Phys. Rev. Lett. 91,
250402 (2003)].
