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arXiv:0807.4905 · Physical Review B

Vortex molecules in spinor condensates

Ari M. Turner, Eugene Demler

Published Jun 22, 2009• 7 citations
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Abstract

Condensates of atoms with spins can have vortices of several types; these are related to the symmetry group of the atoms' ground state. We discuss how, when a condensate is placed in a small magnetic field that breaks the spin symmetry, these vortices may form bound states. Using symmetry classification of vortex charge and rough estimates for vortex interactions, one can show that some configurations that are stable at zero temperature can decay at finite temperatures by crossing over energy barriers. Our focus is cyclic spin-2 condensates which have tetrahedral symmetry.

DOI
10.1103/physrevb.79.214522

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Condensates of atoms with spins can have vortices of several types; these are related to the symmetry group of the atoms' ground state. We discuss how, when a condensate is placed in a small magnetic field that breaks the spin symmetry, these vortices may form bound states. Using symmetry classification of vortex charge and rough estimates for vortex interactions, one can show that some configurations that are stable at zero temperature can decay at finite temperatures by crossing over energy barriers. Our focus is cyclic spin-2 condensates which have tetrahedral symmetry.
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We develop a symmetry classification scheme to find ground states of pseudo-spin-12\frac{1}{2}21​, spin-1, and spin-2 spin-orbit-coupled spinor Bose-Einstein condensates and show that as the SO(2) symmetry of simultaneous spin and space rotations is broken into discrete cyclic groups, various types of lattice structures emerge in the absence of a lattice potential. Examples include two different kagome lattices for pseudo-spin-12\frac{1}{2}21​ condensates and a nematic vortex lattice in which uniaxial and biaxial spin textures align alternatively for spin-2 condensates. For the pseudo-spin-12\frac{1}{2}21​ system, although mean-field states always break time-reversal symmetry, there exists a time-reversal invariant many-body ground state, which is fragmented and expected to be observed in a microcondensate.
We propose a method for systematically finding ground states of spinor Bose-Einstein condensates by utilizing the symmetry properties of the system. By this method, we can find not only an inert state, whose symmetry is maximal in the manifold under consideration, but also a noninert state, which has lower symmetry and depends on the parameters in the Hamiltonian. We establish the symmetry-classification method for the spin-1, 2, and 3 cases at zero magnetic field, and find an additional phase in the last case. The properties of the vortices in the spin-3 system are also discussed.
We classify all possible fractional charge vortices of charge less than unity in spin-1 and spin-2 polar and cyclic Bose-Einstein condensates (BECs) with zero magnetization. Statics and dynamics of these vortices in quasi-two-dimensional spinor BECs are studied employing an accurate numerical solution and a Lagrange variational approximation. The results for density and collective-mode oscillation are illustrated using the fractional-charge BEC vortex of 23Na^{23}\mathrm{Na}23Na and 87Rb^{87}\mathrm{Rb}87Rb atoms with realistic interaction and trapping potential parameters.
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Extended Gross-Pitaevskii equations for the rotating F=2F=2F=2 condensate in a harmonic trap are solved both numerically and variationally using trial functions for each component of the wave function. Axially symmetric vortex solutions are analyzed and energies of polar and cyclic states are calculated. The equilibrium transitions between different phases with changing of the magnetization are studied. We show that at high magnetization the ground state of the system is determined by interaction in ``density'' channel, and at low magnetization spin interactions play a dominant role. Although there are five hyperfine states, all the particles are always condensed in one, two, or three states. Two interesting types of vortex structures are also discussed.