Splitting dynamics of doubly quantized vortices in Bose-Einstein condensates

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.

Figure 1

Energy levels in two dimensions for a condensate with an $m=2$ vortex. The panel (a) shows the real parts and panel (b) the imaginary parts. All imaginary parts except at most two are zero at any point in this phase space.

Figure 2

(a) Time development of the vortex distance $d$, and (b) time development of the populations $P_{0,m}\left(t\right)$ of the harmonic-oscillator eigenstates, for a two-dimensional condensate with coupling parameter $C=20$. In (b), the curves represent from the bottom up, $m=6$, $m=4$, $m=0$, $m=2$, and the sum of all four. The initial state was perturbed by means of a seeding of the harmonic oscillator ground state with an amplitude $P_{00}=0.001$.

Figure 3

(a) Square of the vortex distance $d$, and (b) total population in the $m=0$ (upper curve) and $m=4$ (lower curve) states, for a condensate when the coupling parameter $C=380$. The initial seeding of the harmonic-oscillator ground state is $P_{00}=1\times {10}^{-5}$.

Figure 4

Time for splitting of a doubly quantized vortex. Full lines represent the imaginary part of the complex Bogoliubov eigenvalue, asterisks represent the inverse of the time $T_{\max }$ taken for the vortex distance to achieve its first maximum according to the full GPE solution, and the dashed line is the same time scale in the three-state model, Eq. (C16).

Figure 5

(a) Maximum radius of the motion of the vortices and (b) the maximum occupation of the $m=0$ state, as a function of coupling parameter $C$ with initial seeding $P_0\left(0\right)=0.01$. The chosen range of $C$ values lies in the first instability region.

Figure 6

(a) Maximum radius of the motion of the vortices and (b) the maximum probability of the $m=0$ state, $P_0$, as a function of coupling parameter $C$ with initial seeding $P_0\left(0\right)=0.001$. The chosen range of $C$ values lies in the second instability region.

Figure 7

Time dependence of the amplitude of the lowest $m$ states in the truncated three-state model in Eq. (23), compared with the full GPE simulation. The interaction strength is $C=5$ and the initial seeding $P_0=0.01$. From top to bottom, the full lines represent the amplitudes $P_2$, $P_0$, and $P_4$ in the truncated model and the dotted lines represent the corresponding amplitudes for the full GPE simulation.

Figure 8

Contour plots of the Hamiltonian (B13) for (a) $\Delta =-1.5$, (b) $\Delta =0$, and (c) $\Delta =1.5$. The other parameters are chosen as $I_0=10$ and $K=1$. The thick line is the separatrix, dividing phase space into running and oscillating solutions.

Figure 9

Time development of the population $p$ of the core mode. Full line represents the full numerical time integration, while the dashed line is obtained from the three-state approximation. The coupling strength is chosen as $C=380$.

Figure 10

Maximum amplitude $p$ of the core mode as a function of energy difference between the two mixing Bogoliubov modes, $\Delta$, divided by the mode coupling parameter $K$. The full line is the analytical solution of the three-state model, and the symbols represent data from the GPE solution in successive instability regions; circles for the second instability region, crosses for the third, and dots for the fourth.