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arXiv:cond-mat/0608316 · Physical Review A

Vortex in a trapped Bose-Einstein condensate with dipole-dipole interactions

Duncan H J O’Dell, Claudia Eberlein

Published Jan 4, 2007• 57 citations
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Abstract

We calculate the critical rotation frequency at which a vortex state becomes energetically favorable over the vortex-free ground state in a harmonically trapped Bose-Einstein condensate whose atoms have dipole-dipole interactions as well as the usual $s$-wave contact interactions. In the Thomas-Fermi (hydrodynamic) regime, dipolar condensates in oblate cylindrical traps (with the dipoles aligned along the axis of symmetry of the trap) tend to have lower critical rotation frequencies than their purely $s$-wave contact interaction counterparts. The converse is true for dipolar condensates in prolate traps. Quadrupole excitations and center of mass motion are also briefly discussed as possible competing mechanisms to a vortex as means by which superfluids with partially attractive interactions might carry angular momentum.

DOI
10.1103/physreva.75.013604

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We calculate the critical rotation frequency at which a vortex state becomes energetically favorable over the vortex-free ground state in a harmonically trapped Bose-Einstein condensate whose atoms have dipole-dipole interactions as well as the usual sss-wave contact interactions. In the Thomas-Fermi (hydrodynamic) regime, dipolar condensates in oblate cylindrical traps (with the dipoles aligned along the axis of symmetry of the trap) tend to have lower critical rotation frequencies than their purely sss-wave contact interaction counterparts. The converse is true for dipolar condensates in prolate traps. Quadrupole excitations and center of mass motion are also briefly discussed as possible competing mechanisms to a vortex as means by which superfluids with partially attractive interactions might carry angular momentum.
We consider a singly quantized off-axis straight vortex in a rotating dipolar ultracold gas in the Thomas–Fermi (TF) regime. We derive analytic results for small displacements and perform numerical calculations for large displacements within the TF regime. We prove that the dipolar interaction energy increases (decreases) as the vortex moves from the trap centre to the edge in an oblate (a prolate) trap. We show that the angular velocity representing the onset of metastability is lowered (raised) in an oblate (a prolate) trap. We find that the effect of the dipole–dipole interaction is the lowered (raised) precession velocity of an off-centre straight vortex line in an oblate (a prolate) trap.
We study the stability of singly and doubly quantized vortex states of harmonically trapped dipolar Bose-Einstein condensates (BECs) by calculating the low-lying excitations of these condensates. We map the dynamical stability of these vortices as functions of the dipole-dipole interaction strength and trap geometry by finding where their excitations have purely real energy eigenvalues. In contrast to BECs with purely contact interactions, we find that dipolar BECs in singly quantized vortex states go unstable to modes with an increasing number of angular and radial nodes for more oblate trap aspect ratios, corresponding to local collapse that occurs on a characteristic length scale. Additionally, we find that dipolar BECs in doubly quantized vortex states are unstable to decay into a different topological state (with two singly quantized vortices) for all interaction strengths when the trap geometry is sufficiently prolate to make the dipoles attractive, and in windows of interaction strength when the trap geometry is sufficiently oblate to make the dipoles repulsive.
We present full three-dimensional numerical calculations of single vortex states in rotating dipolar condensates. We consider a Bose-Einstein condensate of 52Cr^{52}\text{C}\text{r}52Cr atoms with dipole-dipole and sss-wave contact interactions confined in an axially symmetric harmonic trap. We obtain the vortex states by numerically solving the Gross-Pitaevskii equation in the rotating frame with no further approximations. We investigate the properties of a single vortex and calculate the critical angular velocity for different values of the sss-wave scattering length. We show that, whereas the standard variational approach breaks down in the limit of pure dipolar interactions, exact solutions of the Gross-Pitaevskii equation can be obtained for values of the sss-wave scattering length down to zero. The energy barrier for the nucleation of a vortex is calculated as a function of the vortex displacement from the rotation axis for different values of the angular velocity of the rotating trap.
Based on the two-dimensional mean-field equations for pancake-shaped dipolar Bose-Einstein condensates in a rotating frame, for both attractive and repulsive dipole-dipole interaction (DDI) as well as arbitrary polarization direction, we study the profiles of the single vortex state and show how the critical rotational frequency changes with the sss-wave contact interaction strength, DDI strength, and polarization angle. In addition, we find numerically that at the ``magic angle'' $\ensuremath{\vartheta}=arccos(\sqrt{3}/3)$, the critical rotational frequency is almost independent of the DDI strength. By numerically solving the dipolar Gross-Pitaevskii equation at high rotation speed, we identify different patterns of vortex lattices which strongly depend on the polarization direction. As a result, we undergo a study of vortex lattice structures for the whole regime of polarization direction and find evidence that the vortex lattice orientation tends to be aligned to the dipole polarization axis for positive DDI strength and to the perpendicular direction of the dipole axis for negative DDI strength.
We study a harmonically trapped highly oblate two-component Bose-Einstein condensate, which consists of both dipolar and scalar bosonic atoms. When the dipoles are polarized along the symmetry axis of the trap, the dipolar atoms' density exhibits a flat-disk regime at the trap center. It is found that the dipolar component is easier to induce the first vortex when the trap is rotating about the symmetry axis, if the long-range dipolar interactions dominate over the contact interactions, and vice versa. Vortex lattices with the compensating triangular, square, and stripe structures can be formed respectively under rapid rotation. When the dipoles are polarized perpendicular to the rotation axis, the two components are generically phase separated into a kernel-shell structure due to the anisotropic dipole-dipole interactions. While the giant-hole states are induced under slow rotation, the compensating stripe vortex structures are stabilized under fast rotation. In particular, a stripe-splitting structure is found to be competing with the stripe ones. These features are expected to be observed in the mixture of 52Cr{}^{52}\mathrm{Cr}52Cr atoms in the two different spin states.
We present exact results in the Thomas-Fermi regime for the statics and dynamics of a harmonically trapped Bose-Einstein condensate that has dipole-dipole interactions in addition to the usual s-wave contact interactions. Remarkably, despite the nonlocal and anisotropic nature of the dipolar interactions, the density profile in a general time-dependent harmonic trap is an inverted parabola. The evolution of the condensate radii is governed by local, ordinary differential equations, and as an example we calculate the monopole and quadrupole shape oscillation frequencies.