We study the nonlinear dynamics of the splitting of a doubly quantized vortex in a trapped Bose-Einstein condensate. The dynamics is studied in detail by solving the Gross-Pitaevskii equation. The main dynamical features are explained in terms of a nonlinear three-level system, to which we find an analytical solution. This result is compared to linear Bogoliubov analysis and to the full numerical time evolution. It is concluded that the time scale for the splitting is mainly determined by the instability of the linearized system, and nonlinear effects contribute logarithmically.
The recent experiment by Shin et al. [Phys. Rev. Lett. 93, 160406 (2004)] on the decay of a doubly quantized vortex is analyzed by numerically solving the Gross-Pitaevskii equation. Our results demonstrate that the vortex decay is mainly a consequence of dynamical instability. The monotonic increase observed in the vortex lifetimes is a consequence of the fact that the measured lifetimes incorporate the time it takes for the initial perturbation to reach the central slice. When considered locally, the splitting occurs approximately at the same time in every condensate.
The stability of doubly quantized vortices in dilute Bose-Einstein condensates of
23Na is examined at zero temperature. The eigenmode spectrum of the Bogoliubov equations for a harmonically trapped cigar-shaped condensate is computed and it is found that the doubly quantized vortex is spectrally unstable towards division into two singly quantized vortices. By numerically solving the full three-dimensional time-dependent Gross-Pitaevskii equation, it is found that the two singly quantized vortices intertwine before decaying. This work provides an interpretation of recent experiments [A. E. Leanhardt et al., Phys. Rev. Lett. 89, 190403 (2002)].
The Bogoliubov equations are solved for a three-dimensional Bose-Einstein condensate containing a doubly quantized vortex, trapped in a harmonic potential. Complex frequencies, signifying dynamical instability, are found for certain ranges of parameter values. The existence of alternating windows of stability and instability, respectively, is explained qualitatively and quantitatively using variational calculus and direct numerical solutions. It is seen that the windows of stability disappear in the limit of a cigar-shaped condensate, which is consistent with recent experimental results on the lifetime of a doubly quantized vortex in that regime.
We study numerically the dynamical instabilities and splitting of singly and doubly quantized composite vortices in two-component Bose-Einstein condensates harmonically confined to quasi two dimensions. In this system, the vortices become pointlike composite defects that can be classified in terms of an integer pair $({\ensuremath{\kappa}}_{1},\phantom{\rule{0.16em}{0ex}}{\ensuremath{\kappa}}_{2})$ of phase winding numbers. Our simulations based on zero-temperature mean-field theory reveal several vortex splitting behaviors that stem from the multicomponent nature of the system and do not have direct counterparts in single-component condensates. By calculating the Bogoliubov excitations of stationary axisymmetric composite vortices, we find nonreal excitation frequencies (i.e., dynamical instabilities) for the singly quantized
(1,1) and $(1,\phantom{\rule{0.16em}{0ex}}\ensuremath{-}1)$ vortices and for all variants of doubly quantized vortices, which we define by the condition ${max}_{j=1,2}|{\ensuremath{\kappa}}_{j}|=2$. While the short-time predictions of the linear Bogoliubov analysis are confirmed by direct time integration of the Gross-Pitaevskii equations of motion, the time integration also reveals intricate long-time decay behavior not captured by the linearized dynamics. First, the $(1,\phantom{\rule{0.16em}{0ex}}\ifmmode\pm\else\textpm\fi{}1)$ vortex is found to be unstable against splitting into a
(1,0) vortex and a $(0,\phantom{\rule{0.16em}{0ex}}\ifmmode\pm\else\textpm\fi{}1)$ vortex. Second, the
(2,1) vortex exhibits a two-step decay process in which its initial splitting into a
(2,0) vortex and a
(0,1) vortex is followed by the off-axis splitting of the
(2,0) vortex into two
(1,0) vortices. Third, the $(2,\phantom{\rule{0.16em}{0ex}}\ensuremath{-}2)$ vortex is observed to split into a $(\ensuremath{-}1,\phantom{\rule{0.16em}{0ex}}1)$ vortex, three
(1,0) vortices, and three $(0,\phantom{\rule{0.16em}{0ex}}\ensuremath{-}1)$ vortices. Each of these splitting processes is the dominant decay mechanism of the respective stationary composite vortex for a wide range of intercomponent interaction strengths and relative populations of the two condensate components and should be amenable to experimental detection. Our results contribute to a better understanding of vortex physics, hydrodynamic instabilities, and two-dimensional quantum turbulence in multicomponent superfluids.
The dynamical instabilities and ensuing dynamics of singly and doubly quantized vortex states of Bose-Einstein condensates with attractive interactions are investigated using full three-dimensional numerical simulations of the Gross-Pitaevskii equation. With increasing the strength of attractive interactions, a series of dynamical instabilities such as quadrupole, dipole, octupole, and monopole instabilities emerge. The most prominent instability depends on the strength of interactions, the geometry of the trapping potential, and deviations from the axisymmetry due to external perturbations. Singly quantized vortices split into two clusters and subsequently undergo split-merge cycles in a pancake-shaped trap, whereas the split fragments immediately collapse in a spherical trap. Doubly quantized vortices are always unstable to disintegration of the vortex core. If we suddenly change the strength of interaction to within a certain range, the vortex splits into three clusters, and one of the clusters collapses after a few split-merge cycles. The vortex split can be observed using a current experimental setup of Leanhardt et al. [Phys. Rev. Lett. 89, 190403 (2002)].