Bogoliubov modes of a dipolar condensate in a cylindrical trap

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.

Figure 1

(Color online) Excitations frequencies as function of the dipole parameter $D_{*}$ for dipolar BEC in the JILA pancake trap $({\omega }_z∕{\omega }_{\rho }=\sqrt{8})$, with zero scattering length. Plotted are modes with $m=0–4$. The three lines that extend to higher $D_{*}$ are the variational results (cf. Fig. 2 of Ref. 7).

Figure 2

(Color online) Excitations frequencies as function of the dipole parameter $D_{*}$ for dipolar BEC in a spherical trap with zero scattering length. Plotted are modes with $m=0–4$. The three lines that extend to higher $D_{*}$ are the variational results.

Figure 3

(Color online) Excitations frequencies as function of the dipole parameter $D_{*}$ for dipolar BEC in a cigar trap with ${\omega }_z∕{\omega }_{\rho }=5∕6$ and with zero scattering length. Plotted are modes with $m=0–4$. The three lines that extend to higher $D_{*}$ are the variational results.

Figure 4

(Color online) Excitations frequencies as function of the dipole parameter $D_{*}$ for dipolar BEC in a JILA cigar trap $({\omega }_z∕{\omega }_{\rho }=1∕\sqrt{8})$ with zero scattering length. Plotted are modes with $m=0–4$. The three lines that extend to higher $D_{*}$ are the variational results (cf. Fig. 3 of Ref. 7).

Figure 5

(Color online) The spatial density perturbation $f=u+v$ (see text) for the lowest (non-Kohn) mode in a spherical trap. Left, for $D_{*}=1.0$; right, for $D_{*}=4.1$, close to collapse. The nodal line is the thick line.

Figure 6

Excitation spectrum in a spherical trap with $D_{*}=4$. The eigenenergies of BdG equations (26) are represented by thick solid bars. Dotted bars correspond to the single-particle spectrum of the HF Hamiltonian (31).

Figure 7

(Color online) The spectral distribution $g_{ld}\left(\omega \right)$ in the local density approximation, Eq. (41), is compared with the smoothed, numerically exact spectral distribution $g\left(\omega \right)$, Eq. (42), for BEC in a spherical trap with $D_{*}=4.0$.