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 $\mathbf{B}$ 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 ${\ensuremath{\varepsilon}}_{D}\ensuremath{\ne}0$ the release energy of the condensate depends on the trap orientation angle ${\ensuremath{\vartheta}}_{T}$ between the principal axis ${\mathbf{e}}_{z,T}$ of the trap and the field $\mathbf{B}$. 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 ${\ensuremath{\omega}}_{a}$ and trap orientation angles ${\ensuremath{\vartheta}}_{T}$. For the special case of a spherical harmonic trap with trap frequency $\ensuremath{\omega}$ it is rigorously shown for $\ensuremath{-}\frac{1}{2}<{\ensuremath{\varepsilon}}_{D}<1$ that a pure $s$-wave symmetry breather excitation of the condensate density exists, that oscillates at a constant frequency ${\ensuremath{\Omega}}_{s}=\sqrt{5}\ensuremath{\omega}$ 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 ${\ensuremath{\varepsilon}}_{D}\ensuremath{\ne}0$ not isotropic. For ${\ensuremath{\vartheta}}_{T}\ensuremath{\ne}0$ 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, ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$- and ${d}_{{z}^{2}}$-wave, and ${d}_{\mathit{xz}}$-wave symmetry, and two modes that are combinations of basis functions with ${d}_{\mathit{yz}}$- and ${d}_{\mathit{xy}}$-wave symmetry. A characteristic difference in the dependence of the frequencies of these six collective modes on the dipole interaction strength parameter ${\ensuremath{\varepsilon}}_{D}$ 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.