Dynamics of vortices followed by the collapse of ring dark solitons in a two-component Bose-Einstein condensate

We explore the effects of system parameters on the dynamics of ring dark solitons (RDSs) and vortices followed by the collapse of RDSs in a two-component Bose-Einstein condensate (BEC). The system exhibits complicated dynamical behaviors, which are quite different from those in a scalar BEC. For two shallow RDSs with equal initial depths, the dynamical trajectories of generated vortex dipoles are similar to those in a scalar BEC, but the time for vortex dipoles to perform a periodic motion is increased. In particular, there exists a critical depth, above which vortex dipoles first move along the vertical direction and then preform complicated dynamics, including their rearrangement and recombination. Finally, we consider the case of unequal initial depths and find that the number of created vortices is determined by the depth of the shallow RDS, while their initial moving direction is determined by the deeper one.

Figure 1

The initial profiles of the two ring dark solitons with trap frequency $\mathrm{\Omega }=0.028$, eccentricity $e_c=0.4$, and the initial depths $cos{\varphi }_1\left(0\right)=cos{\varphi }_2\left(0\right)=0.6$. The lengths of the semiminor axes of the elliptical configuration of two such components are taken as $R_{10}=27.9$ for panel (a) and $R_{20}=28.9$ for panel (b). Panels (c) and (d) are the corresponding total density and density difference of these two components, respectively, from which we can easily see the radius difference between two such RDSs. The field of view is $150\times 150$ in dimensionless units.

Figure 2

Evolution of two ring dark solitons of a two-component BEC for (a) $t=26$, (b) $t=60$, (c) $t=150$, (d) $t=350$, (e) $t=670$, (f) $t=783$, and (g) $t=810$. The top row is for component 1, while the bottom row is for component 2. The field of view is $150\times 150$ in dimensionless units. Other parameters are the same as those in Fig. 1.

Figure 3

Evolution of two ring dark solitons with equal initial depths of $cos{\varphi }_1\left(0\right)=cos{\varphi }_2\left(0\right)=0.75$, and the subsequent dynamics of vortices. The trap frequency is $\mathrm{\Omega }=0.028$ and the eccentricity is $e_c=0.4$. The lengths in the semiminor axis of the elliptical configuration of two such components are taken as $R_{10}=27.9$ and $R_{20}=28.9$. The top row is for component 1, while the bottom row is for component 2. The field of view is $150\times 150$ in dimensionless units. See the Supplemental Material for a movie of the dynamics [48].

Figure 4

Evolution of the vortices following Fig. 3. See the Supplemental Material for a movie of the dynamics of the vortices [48].

Figure 5

(a) Schematic illustration of the dynamics of vortices for the states in Figs. 3–3. (b) General description of the dynamics of vortex pairs, which also shows the dynamics of the vortices shown in Fig. 4. The blue circle at the origin of coordinates indicates the mini RDS formed during the dynamical process. The red and black solid circles indicate the vortices with different circulation (winding number $w=\pm 1$), while the red and black hollow circles indicate the new recombined vortex pair during the dynamical process. The moving directions are indicated by the arrows and the trajectories are indicated by the dashed lines.

Figure 6

Evolution of two RDSs for different initial depths with $cos{\varphi }_1\left(0\right)=0.6$ and $cos{\varphi }_2\left(0\right)=0.75$, and the subsequent dynamics of vortices. The trap frequency is $\mathrm{\Omega }=0.028$, the eccentricity is $e_c=0.4$, and the lengths in the semiminor axis of the elliptical configuration are taken as $R_{10}=27.9$ and $R_{20}=28.9$. The top row is for component 1, while the bottom row is for component 2, and the field of view is $150\times 150$ in dimensionless units. See the Supplemental Material for a movie of the dynamics [48].