Bifurcations, stability, and dynamics of multiple matter-wave vortex states

In the present work, we offer a unifying perspective between the dark soliton stripe and the vortex multipole (dipole, tripole, aligned quadrupole, quintopole, etc.) states that emerge in the context of quasi-two-dimensional Bose-Einstein condensates. In particular, we illustrate that the multivortex states with the vortices aligned along the (former) dark soliton stripe sequentially bifurcate from the latter state in a supercritical pitchfork manner. Each additional bifurcation adds an extra mode to the dark soliton instability and an extra vortex to the configuration; moreover, the bifurcating states inherit the stability properties of the soliton prior to the bifurcation. The critical points of this bifurcation are computed analytically via a few-mode truncation of the system, which clearly showcases the symmetry-breaking nature of the corresponding bifurcation. We complement this small(-er) amplitude, few mode bifurcation picture, with a larger amplitude, particle-based description of the ensuing vortices. The latter enables us to characterize the equilibrium position of the vortices, as well as their intrinsic dynamics and anomalous modes, thus providing a qualitative description of the nonequilibrium multivortex dynamics.

Figure 1

Dependence of the numerical factor $A$ [see Eqs. (5) and (6)] on the initial displacement $R_0$ of the vortex from the center of the trap, for $\mu =3$.

Figure 2

(Top) Number of atoms as a function of the chemical potential for the different states for $\Omega =0.2$. (Bottom) Corresponding number of atom difference between the different branches and the dark soliton stripe branch.

Figure 3

The single-vortex state for a trap of strength $\Omega =0.2$. The top panels show contour plots of the density (left) and phase (right) of the wave function for $\mu =3$, while the bottom panel shows the eigenfrequencies $\omega$ of the Bogoliubov spectrum as a function of the chemical potential $\mu$. Theoretical predictions are given by the dashed line $\omega =\mu -2\Omega$ and the dashed dotted line $\omega ={\omega }_{\mathrm{pr}}$, see Eq. (7). Note that, in the color online version, within the upper left panel, blue denotes zero and red maximal density. Similarly, in the upper right panel blue to red denotes a phase going from $0$ to $2\pi$.

Figure 4

The dark soliton state for a trap strength $\Omega =0.2$. The top panels show contour plots of the density (left) and phase (right) of the wave function for $\mu =3$, while the bottom panel shows the eigenfrequency $\omega$ of the BdG spectrum as a function of the chemical potential $\mu$.

Figure 5

The vortex dipole state for a trap strength $\Omega =0.2$. The top panels show contour plots of the density (left) and phase (right) of the wave function for $\mu =3$. The middle panel shows the dependence of the position of the vortices on the chemical potential for $\Omega =0.1$, $\Omega =0.15$, and $\Omega =0.2$ (curves from top to bottom). The lines indicate the predictions of the ODE with $A=2\sqrt{2}\pi$, $B=1.35$. The bottom panels show the real and imaginary parts of the eigenfrequencies $\omega$ of the BdG spectrum as a function of the chemical potential $\mu$. The thin solid line shows the analytical prediction ${\omega }_{\mathrm{pr}}^{\mathrm{vd}}=\sqrt{2}{\omega }_{\mathrm{pr}}$.

Figure 6

The three-vortex state for a trap strength $\Omega =0.2$. The top panels show contour plots of the density (left) and phase (right) of the wave function for $\mu =3$, while the bottom panel shows the eigenfrequencies $\omega$ (real and imaginary parts) of the BdG spectrum as a function of the chemical potential $\mu$.

Figure 7

The aligned quadrupole state for a trap strength $\Omega =0.2$. The top panels show contour plots of the density (left) and phase (right) of the wave function for $\mu =3$, while the bottom panel shows the (real and imaginary) eigenfrequencies $\omega$ of the BdG spectrum as a function of the chemical potential $\mu$.

Figure 8

The vortex quadrupole state for a trap strength $\Omega =0.2$. The top panels show contour plots of the density (left) and phase (right) of the wave function for $\mu =3$, while the bottom panel shows the (real and imaginary parts of the) eigenfrequencies $\omega$ of the BdG spectrum as a function of the chemical potential $\mu$. The dashed line shows the theoretical prediction for the mode departing from zero $\omega =\frac{4\sqrt{2}}{5}(\mu -3\Omega )$.

Figure 9

The five-vortex state for a trap strength $\Omega =0.2$. The top panels show contour plots of the density (left) and phase (right) of the wave function for $\mu =3$, while the bottom panel shows the real and imaginary parts of the eigenfrequencies $\omega$ of the BdG spectrum as a function of the chemical potential $\mu$.

Figure 10

Time evolution of the center of the vortices for the two-vortex state perturbed by the eigenvector of the anomalous mode (top panel) and the eigenvector associated to the zero eigenvalue connected to the rotational invariance (bottom panel) at $\mu =1.5$. The filled circles represent the initial positions of the vortices.

Figure 11

Time evolution of the center of the vortices for the three-vortex state perturbed by the eigenvector of the purely imaginary mode at $\mu =2.0$. The initial positions of the vortices are depicted by the filled circles.

Figure 12

Time evolution of the centers of two vortices for a vortex with positive vorticity and one with negative vorticity initially placed at $(x_0,y_0)=(3.9,0)$ (white circle) and $(x_0,y_0)=(-2.4,0)$ (black circle), respectively (left panel), and $(x_0,y_0)=(0.75,0.3)$ (white circle) and $(x_0,y_0)=(-4.5,-0.3)$ (black circle), respectively (right panel), for $\mu =3$. The thick solid lines depict the predictions of Eqs. (11) and (12) and the thin solid lines the predictions of Eq. (1).

Figure 13

Time evolution of the center of two vortices for a vortex with positive vorticity and one with negative vorticity initially placed at $(x_0,y_0)=(4.5,0)$ (white circle) and $(x_0,y_0)=(-4.5,0)$ (black circle), respectively (left panel), and $(x_0,y_0)=(0.75,0.75)$ (white circle) and $(x_0,y_0)=(-1.5,-1.5)$ (black circle), respectively (right panel), for $\mu =3$. The thick solid lines depict the predictions of Eqs. (11) and (12) and the thin solid lines the predictions of Eq. (1).