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arXiv:cond-mat/0506423 · Physical Review A

Numerical study of the stability of skyrmions in Bose-Einstein condensates

Sebastian Wüster, T. E. Argue, Craig M. Savage

Published Oct 26, 2005• 40 citations
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Abstract

We show that the stability of three-dimensional skyrmions in trapped Bose-Einstein condensates depends critically on scattering lengths, atom numbers, trap rotation, and trap anisotropy. In particular, for the $^{87}\mathrm{Rb}\ensuremath{\mid}F=1,{m}_{f}=\ensuremath{-}1⟩$, $\ensuremath{\mid}F=2,{m}_{f}=1⟩$ hyperfine states stability is sensitive to the scattering lengths at the 2% level, where the differences between them are crucial. In a cigar shaped trap, we find stable skyrmions with slightly more than $2\ifmmode\times\else\texttimes\fi{}{10}^{6}$ atoms, a number which scales with the inverse square root of the trap frequency. These can be stabilized against drift out of the trap by laser pinning.

DOI
10.1103/physreva.72.043616

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We show that the stability of three-dimensional skyrmions in trapped Bose-Einstein condensates depends critically on scattering lengths, atom numbers, trap rotation, and trap anisotropy. In particular, for the 87Rb\ensuremath∣F=1,mf=\ensuremath−1⟩^{87}\mathrm{Rb}\ensuremath{\mid}F=1,{m}_{f}=\ensuremath{-}1⟩87Rb\ensuremath∣F=1,mf​=\ensuremath−1⟩, \ensuremath∣F=2,mf=1⟩\ensuremath{\mid}F=2,{m}_{f}=1⟩\ensuremath∣F=2,mf​=1⟩ hyperfine states stability is sensitive to the scattering lengths at the 2% level, where the differences between them are crucial. In a cigar shaped trap, we find stable skyrmions with slightly more than 2\ifmmode×\else\texttimes\fi1062\ifmmode\times\else\texttimes\fi{}{10}^{6}2\ifmmode×\else\texttimes\fi106 atoms, a number which scales with the inverse square root of the trap frequency. These can be stabilized against drift out of the trap by laser pinning.
Vortex states of weakly interacting Bose-Einstein condensates confined in three-dimensional rotating harmonic traps are investigated numerically at zero temperature. The ground state in the rotating frame is obtained by propagating the Gross-Pitaevskii equation for the condensate in imaginary time. The total energies between states with and without a vortex are compared, yielding critical rotation frequencies that depend on the anisotropy of the trap and the number of atoms. Vortices displaced from the center of nonrotating traps are found to have long lifetimes for sufficiently large numbers of atoms. The relationship between vortex stability and bound core states is explored.
We study the stability of a Bose-Einstein condensate of harmonically trapped atoms with negative scattering length, specifically 7Li.{}^{7}\mathrm{Li}.7Li. Our method is to solve the time-dependent nonlinear Schr\"odinger equation numerically. For an isolated condensate, with no gain or loss, we find that the system is stable (apart from quantum tunneling) if the particle number N is less than a critical number Nc.{N}_{c}.Nc​. For N>Nc,N>{N}_{c},N>Nc​, the system collapses to high-density clumps in a region near the center of the trap. The time for the onset of collapse is on the order of one trap period. Within numerical uncertainty, the results are consistent with the formation of a ``black hole'' of infinite density fluctuations, as predicted by Ueda and Huang [Phys. Rev. A (to be published)]. We numerically obtain ${N}_{c}\ensuremath{\approx}1251.$ We then include gain-loss mechanisms, i.e., the gain of atoms from a surrounding ``thermal cloud,'' and the loss due to two- and three-body collisions. The number N now oscillates in a steady state, with a period of about 145 trap periods. We obtain ${N}_{c}\ensuremath{\approx}1260$ as the maximum value in the oscillations.
We perform accurate investigation of stability of localized vortices in an effectively two-dimensional (``pancake-shaped'') trapped Bose-Einstein condensate with negative scattering length. The analysis combines computation of the stability eigenvalues and direct simulations. The states with vorticity S=1S=1S=1 are stable in a third of their existence region, 0<N<(1∕3)Nmax(S=1)0<N<(1∕3){N}_{\mathrm{max}}^{(S=1)}0<N<(1∕3)Nmax(S=1)​, where NNN is the number of atoms, and Nmax(S=1){N}_{\mathrm{max}}^{(S=1)}Nmax(S=1)​ is the corresponding collapse threshold. Stable vortices easily self-trap from arbitrary initial configurations with embedded vorticity. In an adjacent interval, (1∕3)Nmax(S=1)<N<0.43Nmax(S=1)(1∕3){N}_{\mathrm{max}}^{(S=1)}<N<0.43{N}_{\mathrm{max}}^{(S=1)}(1∕3)Nmax(S=1)​<N<0.43Nmax(S=1)​, the unstable vortex periodically splits in twofragments and recombines. At N>0.43Nmax(S=1)N>0.43{N}_{\mathrm{max}}^{(S=1)}N>0.43Nmax(S=1)​, the fragments do not recombine, as each one collapses by itself. The results are compared with those in the full three-dimensional (3D) Gross-Pitaevskii equation. In a moderately anisotropic 3D configuration, with the aspect ratio 10\sqrt{10}10​, the stability interval of the S=1S=1S=1 vortices occupies $\ensuremath{\approx}40%$ of their existence region, hence the two-dimensional (2D) limit provides for a reasonable approximation in this case. For the isotropic 3D configuration, the stability interval expands to 65% of the existence domain. Overall, the vorticity heightens the actual collapse threshold by a factor of up to 222. All vortices with $S\ensuremath{\geqslant}2$ are unstable.
A necessary condition for the existence of a skyrmion in two-component Bose-Einstein condensates with SU(2) symmetry was recently provided by two of the authors [I. F. Herbut and M. Oshikawa, Phys. Rev. Lett. 97, 080403 (2006)], by mapping the problem to a classical particle in a potential subject to time-dependent dissipation. Here we further elaborate this approach. For two classes of models, we demonstrate the existence of the critical dissipation strength above which the skyrmion solution does not exist. Furthermore, we discuss the local stability of the skyrmion solution by considering the second-order variation. A sufficient condition for the local stability is given in terms of the ground-state energy of a one-dimensional quantum-mechanical Hamiltonian. This condition requires a minimum number of bosons, for a certain class of the trap potential. In the optimal case, the minimum number of bosons can be as small $\ensuremath{\sim}{10}^{4}$.