Splitting of singly and doubly quantized composite vortices in two-component Bose-Einstein condensates

We study numerically the dynamical instabilities and splitting of singly and doubly quantized composite vortices in nonrotated 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 $(\kappa_1,\kappa_2)$ of phase-winding numbers. Our numerical 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 quasiparticle excitations of stationary axisymmetric composite vortices, we find complex-frequency excitations (dynamical instabilities) for the singly quantized $(1,1)$ and $(1,-1)$ vortices and for all variants of doubly quantized vortices, which we define by $\max_{j=1,2}|\kappa_j| = 2$. While the predictions of the linear Bogoliubov analysis are confirmed by direct time integration of the Gross--Pitaevskii equations of motion, the latter also reveals intricate long-time decay behavior not captured by the linearized dynamics. Firstly, the $(1,\pm 1)$ vortex is found to be unstable against splitting into a $(1,0)$ and a $(0,\pm 1)$ vortex. Secondly, the $(2,1)$ vortex exhibits a two-step decay process by splitting first into a $(2,0)$ and a $(0,1)$ vortex followed by the off-axis splitting of the $(2,0)$ vortex into two $(1,0)$ vortices. Thirdly, the $(2,-2)$ vortex is observed to split into a $(-1,1)$ vortex, three $(1,0)$ vortices, and three $(0,-1)$ vortices. Each of these exotic splitting modes is the dominant dynamical instability of the respective stationary vortex in a wide range of intercomponent interaction strengths and relative populations of the two condensate components and should be amenable to experimental detection.