Bogoliubov spectrum and Bragg spectroscopy of elongated Bose-Einstein condensates

The behavior of the momentum transferred to a trapped Bose-Einstein condensate by a two-photon Bragg pulse reflects the structure of the underlying Bogoliubov spectrum. In elongated condensates, axial phonons with different number of radial nodes give rise to a multibranch spectrum which can be resolved in Bragg spectroscopy, as shown by Steinhauer {\it et al.} [Phys. Rev. Lett. {\bf 90}, 060404 (2003)]. Here we present a detailed theoretical analysis of this process. We calculate the momentum transferred by numerically solving the time dependent Gross-Pitaevskii equation. In the case of a cylindrical condensate, we compare the results with those obtained by linearizing the Gross-Pitaevskii equation and using a quasiparticle projection method. This analysis shows how the axial-phonon branches affect the momentum transfer, in agreement with our previous interpretation of the observed data. We also discuss the applicability of this type of spectroscopy to typical available condensates, as well as the role of nonlinear effects.