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arXiv:cond-mat/0309511 · Physical Review A

Spin textures in slowly rotating Bose-Einstein condensates

Erich J. Mueller

Published Mar 17, 2004• 83 citations
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Abstract

Slowly rotating spin-1 Bose-Einstein condensates are studied through a variational approach based upon lowest Landau level calculus. The author finds that in a gas with ferromagnetic interactions, such as $^{87}\mathrm{Rb}$, angular momentum is predominantly carried by clusters of two different types of skyrmion textures in the spin-vector order parameter. Conversely, in a gas with antiferromagnetic interactions, such as $^{23}\mathrm{Na}$, angular momentum is carried by $\ensuremath{\pi}$ disclinations in the nematic order parameter which arises from spin fluctuations. For experimentally relevant parameters, the cores of these $\ensuremath{\pi}$ disclinations are ferromagnetic, and can be imaged with polarized light.

DOI
10.1103/physreva.69.033606

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Slowly rotating spin-1 Bose-Einstein condensates are studied through a variational approach based upon lowest Landau level calculus. The author finds that in a gas with ferromagnetic interactions, such as 87Rb^{87}\mathrm{Rb}87Rb, angular momentum is predominantly carried by clusters of two different types of skyrmion textures in the spin-vector order parameter. Conversely, in a gas with antiferromagnetic interactions, such as 23Na^{23}\mathrm{Na}23Na, angular momentum is carried by \ensuremathπ\ensuremath{\pi}\ensuremathπ disclinations in the nematic order parameter which arises from spin fluctuations. For experimentally relevant parameters, the cores of these \ensuremathπ\ensuremath{\pi}\ensuremathπ disclinations are ferromagnetic, and can be imaged with polarized light.
We theoretically investigate vortex-lattice phases of rotating spinor Bose-Einstein condensates (BEC's) with a ferromagnetic spin interaction by numerically solving the Gross-Pitaevskii equation. The spinor BEC under slow rotation can sustain a rich variety of exotic vortices due to the multicomponent order parameters, such as Mermin-Ho and Anderson-Toulouse coreless vortices (the two-dimensional Skyrmion and meron) and nonaxisymmetric vortices with shifting vortex cores. Here, we present the spin texture of various vortex-lattice states at higher rotation rates and in the presence of an external magnetic field. In addition, the vortex phase diagram is constructed in the plane of the total magnetization MMM and the external rotation frequency $\ensuremath{\Omega}$ by comparing the free energies of possible vortices. It is shown that the vortex phase diagrams in the $M\text{\ensuremath{-}}\ensuremath{\Omega}$ plane may be divided into two categories: (i) the coreless vortex lattice formed by the several types of Mermin-Ho vortices and (ii) the vortex lattice filling in the cores with the pure polar (antiferromagnetic) state. In particular, it is found that the type-(ii) state forms composite lattices of coreless and polar-core vortices. The difference between type (i) and type (ii) results from the existence of the singularity of the spin textures, which may be experimentally confirmed by the spin imaging within polarized light recently proposed by Carusotto and Mueller. We also discuss the stability of triangular and square lattice states for rapidly rotating condensates.
We investigate two kinds of coreless vortices with axisymmetric and nonaxisymmetric configurations in rotating two-component Bose-Einstein condensates. Starting from the Gross-Pitaevskii energy functional in a rotating frame, we derive a nonlinear sigma model generalized to the two-component condensates. In terms of a pseudospin representation, an axisymmetric vortex and a nonaxisymmetric one correspond to spin textures referred to as a ``skyrmion'' and a ``meron-pair,'' respectively. A variational method is used to investigate the dependence of the sizes of the stable spin textures on system parameters, and the optimized variational function is found to reproduce well the numerical solution. In the SU(2) symmetric case, the optimal skyrmion and meron-pair are degenerate and transform to each other by a rotation of the pseudospin. An external rf field that couples coherently the hyperfine states of two components breaks the degeneracy in favor of the meron-pair texture due to an effective transverse pseudomagnetic field. The difference between the intracomponent and intercomponent interactions yields a longitudinal pseudomagnetic field and a ferromagnetic or an antiferromagnetic pseudospin interaction, leading to a meron-pair texture with an anisotropic distribution of vorticity.
We explore the dielectric properties of spinor condensates. Gases with vector (nematic) order, such as spin-1 condensates with ferromagnetic (antiferromagnetic) interactions, display optical activity (birefringence). This optical activity/birefringence is respectively seen for light detuned from resonance by a frequency which is not too large compared with the fine/hyperfine structure. Careful tuning of the light frequency can isolate either of these two effects. These features can be used to image spin textures in clouds of spin-1 bosons, and to discern states showing different types of spin order.
Dynamics of skyrmionic spin texture in the spin-1 Bose-Einstein condensate is examined by analytical and numerical means. We show the skyrmion (coreless vortex) to be inherently unstable in the sense that the state initially prepared purely within the antiferromagnetic (ferromagnetic) order parameter manifold inevitably evolves into a mixture of both. The vorticity-dependent drift in the presence of the trapping potential also contributes to the disintegration of the initial spin texture manifold. We argue that the notion of a skyrmion as a topologically protected entity becomes ill defined during the dynamical evolution process.
We have numerically solved the low-energy excitation spectra of ferromagnetic Bose-Einstein condensates subject to dipolar interparticle interactions. The system is assumed to be harmonically confined by purely optical means, thereby maintaining the spin degree of freedom of the condensate order parameter. Using a zero-temperature spin-1 model, we solve the Bogoliubov excitations for different spin textures, including a spin-vortex state in the absence of external magnetic fields and a rapidly rotating polarized spin texture in a finite homogeneous field. In particular, we consider the effect of dipolar interactions on excitations characteristic of ferromagnetic condensates. The energies of spin waves and magnetic quadrupole modes are found to increase rapidly with the dipolar coupling strength, whereas the energies of density oscillations change only slightly.