Bogoliubov modes of a dipolar condensate in a cylindrical trap

Abstract

The calculation of properties of Bose-Einstein condensates with dipolar interactions has proven a computationally intensive problem due to the long range nature of the interactions, limiting the scope of applications. In particular, the lowest lying Bogoliubov excitations in three dimensional harmonic trap with cylindrical symmetry were so far computed in an indirect way, by Fourier analysis of time dependent perturbations, or by approximate variational methods. We have developed a very fast and accurate numerical algorithm based on the Hankel transform for calculating properties of dipolar Bose-Einstein condensates in cylindrically symmetric traps. As an application, we are able to compute many excitation modes by directly solving the Bogoliubov-De Gennes equations. We explore the behavior of the excited modes in different trap geometries. We use these results to calculate the quantum depletion of the condensate by a combination of a computation of the exact modes and the use of a local density approximation

Cold Dipolar Gases in Quasi-One-Dimensional Geometries

“…To a very good approximation the combined effects of the short-range interactions and the DDI can be understood by means of a pseudopotential theory, which includes a contact interaction, characterized by a scattering length a, and the DDI itself [13,20]. However, the correct value of a is in general not that in absence of DDI, but the result of the scattering problem including both the short-range and the DDI potentials.…”

“…Interestingly, the scattering properties in quasi-1D (also in 2D [7]) may be crucially affected by the contrained geometry. In particular the transversal confinement can induce a novel resonance known as confinement-induced resonance (CIR) [8] To a very good approximation the combined effects of the short-range interactions and the DDI can be understood by means of a pseudopotential theory, which includes a contact interaction, characterized by a scattering length a, and the DDI itself [13,20]. However, the correct value of a is in general not that in absence of DDI, but the result of the scattering problem including both the short-range and the DDI potentials.…”

“…However, the correct value of a is in general not that in absence of DDI, but the result of the scattering problem including both the short-range and the DDI potentials. Hence the DDI may affect the value of a, even very severely in the vicinity of the so-called shape resonances [13,20,21]. In addition, the combination of the DDI and the dressing of rotational excitations with static and microwave fields may allow for the engineering of novel types of interaction potentials for polar molecules in 2D geometries [22].…”

Thomas-Fermi versus one- and two-dimensional regimes of a trapped dipolar Bose-Einstein condensate

“…For a pancake dipolar BEC, radial density wave structures have been predicted for ε dd → ∞ [23], and so the assumption of homogeneous radial density leading to Eq. (20) would not hold.…”

“…Furthermore, a stable ground state may not exist; the Bogoliubov spectrum for a uniform dipolar BEC [10] predicts an instability to density fluctuations when ε dd is outside of the range −1/2 < ε dd < 1 for a > 0. Although trapping can significantly extend this range of stability, it must be mapped out by solving the Bogoliubov de Gennes equations numerically [23]. However, within this range of stability and away from density wave structures, the TF regime is both stable and has the inverted parabola profile, and so the criteria should hold.…”