Confinement and precession of vortex pairs in coherently coupled Bose-Einstein condensates

The dynamic behavior of vortex pairs in two-component coherently (Rabi) coupled Bose-Einstein condensates is investigated in the presence of harmonic trapping. We discuss the role of the surface tension associated with the domain wall connecting two vortices in condensates of atoms occupying different spin states and its effect on the precession of the vortex pair. The results, based on the numerical solution of the Gross-Pitaevskii equations, are compared with the predictions of an analytical macroscopic model and are discussed as a function of the size of the pair, the Rabi coupling, and the intercomponent interaction. We show that the increase of the Rabi coupling results in the disintegration of the domain wall into smaller pieces, connecting vortices of newly created vortex pairs. The resulting scenario is the analog of quark confinement and string breaking in quantum chromodynamics.

Figure 1

(a) Relative phase ${\theta }_1-{\theta }_2$ of the two components near a vortex pair in the presence of Rabi coupling ${\mathrm{\Omega }}_{\mathrm{R}}=0.5{\omega }_{\perp }$. We can see that the phase jump between the two vortices is confined within the narrow domain wall stretching between the vortices. The dotted line shows the circle around which the phase is calculated in (b). (b) Phases ${\theta }_1$ (red dotted line) and ${\theta }_2$ (blue solid line) along a circle centered in the vortex of the first component. The phase ${\theta }_1$ makes a $2\pi$ winding around a vortex, with half of the jump concentrated in a short interval of the polar angle $\alpha$ [of the coordinate system centered in the vortex core, as shown in (a)]. The phase ${\theta }_2$ is instead single valued.

Figure 2

Dependence of ${\mathrm{\Omega }}_{\mathrm{prec}}$ on the Rabi coupling ${\mathrm{\Omega }}_{\mathrm{R}}$ for different values of the vortex separation $2d$. The numerical solution of the GP equations (component 1, circles; component 2, triangles) are in good agreement with the analytical expression, Eq. (5) (solid lines), for large values of $2d$ (in units of ${\xi }_{\text{ho}}$). The inset is a zoom of the figure for small values of ${\mathrm{\Omega }}_{\mathrm{R}}$, where ${\mathrm{\Omega }}_{\mathrm{prec}}$ changes sign.

Figure 3

Dependence of the precession frequency ${\mathrm{\Omega }}_{\mathrm{prec}}$ (component 1, circles; component 2, triangles) on $1/2d$ (in units of $\sqrt{m{\omega }_{\perp }/\hslash }$). The solid lines correspond to the prediction of Eq. (5). As in the Fig. 2, the agreement is good as long as the distance $2d$ separating the two vortices is sufficiently large.

Figure 4

Fragmentation of the domain wall after the evolution time ${\omega }_{\perp }t=1.5$. One can see fragments of different size, connecting vortices of different components. The GP simulation was carried out choosing $g_{12}=0$ and ${\mathrm{\Omega }}_{\mathrm{R}}=6{\omega }_{\perp }$. The initial condition corresponded to a domain wall between the vortex pair of length $2d\approx 5{\xi }_{\mathrm{ho}}$, symmetric with respect to the trap's center. The figure is a zoom of the central region of the cloud.

Figure 5

Relative phase distribution around a single half vortex in a two-component coherently coupled system. The vortex builds a domain wall that is attached to the nearest point of the edge of the cloud. The phase distribution corresponds to the evolution time of ${\omega }_{\perp }t=0.2$. Then, the vortex starts precessing and induces the appearance of a second vortex in component 2.