Ground-state and collective modes of a spin-polarized dipolar Bose-Einstein condensate in a harmonic trap

Abstract

We report results for the Thomas-Fermi ground state of a spin-polarized dipolar interacting Bose-Einstein condensate for the case when the external magnetic field B is not orientated parallel to a principal axis but is aligned parallel to a symmetry plane of a harmonic anisotropic trap. For a dipole interaction strength parameter ε D = 0 the release energy of the condensate depends on the trap orientation angle ϑ T between the principal axis e z,T of the trap and the field B. From the quasiclassical Josephson equation of macroscopic quantum physics we determine the low-lying eigenfrequencies of small-amplitude collective modes of the condensate density for various trap frequencies ω a and trap orientation angles ϑ T . For the special case of a spherical harmonic trap with trap frequency ω it is rigorously shown for − 1 2 < ε D < 1 that a pure s-wave symmetry breather excitation of the condensate density exists, that oscillates at a constant frequency s = √ 5ω around the ground-state cloud, despite the well-known fact that the shape of the ground-state cloud of a spin-polarized dipolar condensate is for ε D = 0 not isotropic. For ϑ T = 0 the small-amplitude modes of the particle density with isotropic and quadrupolar symmetry consist of two groups. There exist four modes that are combinations of basis functions, with s-wave, d x 2 -y 2 -and d z 2 -wave, and d xz -wave symmetry, and two modes that are combinations of basis functions with d yz -and d xy -wave symmetry. A characteristic difference in the dependence of the frequencies of these six collective modes on the dipole interaction strength parameter ε D for prolate and oblate harmonic triaxial traps, respectively, is suggested to be used as an experimental method to measure the s-wave scattering length a s of the atoms.

Vortices and vortex lattices in quantum ferrofluids

“…In the absence of rotation it is possible to find Thomas-Fermi solutions when the dipoles are not aligned along one trap axis [200]. Although non-trivial, it is in principle possible to generalise the work of Sapina et al [200] to include rotation, thereby providing a framework to quantify under what regimes it is favourable for a vortex to be seeded into a dipolar BEC when the dipoles are not perpendicular to the axis of rotation 11 .…”

Dynamics of a dipolar Bose-Einstein condensate in the vicinity of a superconductor

“…As expected from our previous calculations with the effective potential, we see a resonance at the position where the breather mode frequency and the double oscillation frequency 2ω ′ x coincide. If we used an symmetric trap with κ = 1, the breather mode frequency would approach the double oscillation frequency rather than cross it [40]. The strength of the resonance depends of course on the strength of the dipole-dipole interaction parameter ε x .…”

“…In the following we will focus on the so-called monopole-quadrupole modes [40]. In a harmonic trap the density distribution of a BEC within TF approximation is an ellipsoid.…”

“…The case r ∈ D TF is useful if one is interested in calculating the potential between the atoms in the BEC. For example to calculate the semi-axes of a dipolar BEC [40] or its collective modes [40,42]. Since we want to calculate the potential of the mirror cloud at the position of the actual BEC we need the case r / ∈ D TF .…”

“…They need to be determined numerically from a set of coupled self-consistency equations [40]. The central density n 0 can be determined from the requirement N =´dr ′ n (r ′ ), and is given by…”