Excitations and number fluctuations in an elongated dipolar Bose-Einstein condensate

We study the properties of a magnetic dipolar Bose-Einstein condensate (BEC) in an elongated (cigar shaped) confining potential in the beyond quasi-one-dimensional (quasi-1D) regime. In this system the dipole-dipole interactions (DDIs) develop a momentum-dependence related to the transverse confinement and the polarization direction of the dipoles. This leads to density fluctuations being enhanced or suppressed at a length scale related to the transverse confinement length, with local atom number measurements being a practical method to observe these effects in experiments. We use meanfield theory to describe the ground state, excitations and the local number fluctuations. Quantitative predictions are presented based on full numerical solutions and a simplified variational approach that we develop. In addition to the well-known roton excitation, occurring when the dipoles are polarized along a tightly confined direction, we find an "anti-roton" effect for the case of dipoles polarized along the long axis: a nearly non-interacting ground state that experiences strongly repulsive interactions with excitations of sufficiently short wavelength.