Exploring rigidly rotating vortex configurations and their bifurcations in atomic Bose-Einstein condensates

In the present work, we consider the problem of a system of few vortices $N\le 5$ as it emerges from its experimental realization in the field of atomic Bose-Einstein condensates. Starting from the corresponding equations of motion for an axially symmetric trapped condensate, we use a two-pronged approach in order to reveal the configuration space of the system's preferred dynamical states. We use a Monte Carlo method parametrizing the vortex particles by means of hyperspherical coordinates and identifying the minimal energy ground states thereof for $N=2,\cdots ,5$ and different vortex particle angular momenta. We then complement this picture with a dynamical system analysis of the possible rigidly rotating states. The latter reveals a supercritical and subcritical pitchfork, as well as saddle-center bifurcations that arise, exposing the full wealth of the problem even for such low-dimensional cases. By corroborating the results of the two methods, it becomes fairly transparent which branch the Monte Carlo approach selects for different values of the angular momentum that is used as a bifurcation parameter.

Figure 1

Bifurcation scenario for $N=2$ same-charge vortices and corresponding MC simulations. The bifurcation diagram, as a function of the square root of the angular momentum of the system ($\sqrt{L}$) is increased, obtained from the corresponding ODEs (1) is depicted by the solid lines with blue denoting a stable branch and red an unstable branch. The MC simulation results are depicted by the small magenta circles. The critical point beyond which (through a supercritical pitchfork bifurcation) the asymmetric configurations arise (and become a ground state of the system) is indicated by the vertical dashed line. Note how the MC simulations accurately follow the stable branches of the bifurcation diagram.

Figure 2

Bifurcation scenario for $N=3$ same-charge vortices and corresponding MC simulations. Same notation as in Fig. 1. The left panels correspond to the entire domain $0\le L\le 1$ while the right panels depict a zoomed in version near the bifurcation region. The top panels show the bifurcation diagrams in a planar view, namely, $r_i$ vs $\sqrt{L}$. The bottom panels depict the bifurcation diagram in the three-dimensional space ($\sqrt{L},r_1,r_2$), where the $C_3$ symmetry of the solution is clearly visible. The thin dashed vertical line corresponds to the symmetry-breaking bifurcation where the symmetric (equilateral triangle) solution loses its stability. The thin solid vertical line depicts the location of the saddle-center bifurcation of collisions between asymmetric (isosceles triangle) solutions. The thin dash-dotted vertical line corresponds to the location where the MC simulation switches branches (i.e., location where the energy minimum configuration switches from an equilateral to an isosceles one).

Figure 3

Bifurcation scenario for the relative angles between vortices ${\delta }_{12}$ as a function of the square root of the angular momentum for $N=3$ vortices. The data and notation are the same as in Fig. 2. The top panel is the full view while the bottom panel depicts a zoomed in version around the bifurcation region.

Figure 4

Bifurcation scenario and corresponding MC simulations showing the complete dynamical picture associated with the bifurcations in the case of $N=4$. Similarly to the $N=3$ case, the top panels show a planar representation of the solutions using only $r_1$ as a function of $\sqrt{L}$, while the bottom panels relay a three-dimensional variant thereof with $r_2$, as a function of $r_1$ and $\sqrt{L}$. The blue lines are stable branches, the red lines represent the unstable branches, and the Monte Carlo data are overlayed using small magenta circles. This conveys not only how new branches (such as the rhombus and general quadrilateral) emerge through suitable (supercritical pitchfork or saddle-center, respectively) bifurcations, but also when they become the global energy minimizers and hence are followed by the Monte Carlo simulation. Specifically, the thin dashed vertical line corresponds to the symmetry-breaking bifurcation where the symmetric (square) configuration loses its stability towards the newly created rhombus configuration, while the thin dash-dotted vertical line corresponds to the location where the MC simulation switches branches (i.e., transitions from the rhombic configuration to a more general quadrilateral without any apparent symmetry).

Figure 5

Bifurcation scenario for the relative angles between vortices ${\delta }_{12}$ as a function of the square root of the angular momentum for $N=4$ vortices. The data and notation are the same as in Fig. 4. The top panel is the full view, while the bottom panel depicts a zoomed in version around the bifurcation region. Notice that between the transition from the square to the rhombus (dashed vertical line) and that from the rhombus to the general quadrilateral (dash-dotted vertical line), no modification is noticed for the first bifurcation by looking solely at the angles, as the rhombic configuration maintains ${\delta }_{12}=\pi /2$.

Figure 6

Energy corresponding to all the bifurcation branches presented in the previous plots together with the energy computed from the MC simulations. As before, the bifurcation branches are depicted by a blue (red) line for stable (unstable) branches and the MC simulations are depicted by the magenta circles. The first, second, and third rows of panels correspond, respectively, to the $N=2$, 3, and 4 cases. The left panels present the energy as computed from Eq. (3) over the entire range $0\le L\le 1$, while the right panels depict the corresponding energy difference $\Delta H$ between each configuration and the corresponding symmetric configuration for each value of $L$ (see the text for more details).

Figure 7

Monte Carlo results for $N=5$ vortices. The top and bottom panels depict, respectively, the MC results for the radii $r_i$ and the relative angles ${\delta }_{ij}$. Two transitions are observed. The first transition at $L^{1/2}=0.9467$ (location depicted by the vertical dashed line) indicates where the configuration with a single vortex at the center turns into the ground state of the system. The second transition observed at $L^{1/2}=0.9615$ (location depicted by the vertical dash-dotted line) indicates where the asymmetric configuration (a tight cluster of four vortices plus a single vortex further away from the center) turns into the ground state of the system. The solid (blue) dots along the different branches indicate the locations of the displayed vortex configurations in the top panel.