Transition from vortices to solitonic vortices in trapped atomic Bose-Einstein condensates

Motivated by recent experiments, we study theoretically the dynamics of vortices in the crossover from two dimensions to one in atomic condensates in elongated traps. We explore the transition from the dynamics of a vortex to that of a dark soliton as the one-dimensional limit is approached, mapping this transition out as a function of the key system parameters. Moreover, we probe this transition dynamically through the hysteresis under time-dependent deformation of the trap at the dimensionality crossover. When the solitonic regime is probed during the hysteresis, significant angular momentum is lost from the system, but remarkably, the vortex can reemerge.

Figure 1

Evolution of the condensate (i) density and (ii) phase profiles for trap ratios (a) ${\omega }_y/{\omega }_x=1$, (b) 4, and (c) 15. We take $g=400$ and the vortex initial position $(x_{V,0},y_{V,0})=(1.5l_x,0)$. From left to right the columns represent $t=0, T_V/4, T_V/2$, and $3T_V/4$, where $T_V=2\pi /{\omega }_V$ is the vortex precession period [calculated from Eq. (4)]. All units are dimensionless.

Figure 2

(a) Vortex and solitonic vortex oscillation frequency ${\omega }_V$ versus trap ratio ${\omega }_y/{\omega }_x$ for interaction strengths $g=100$ (blue triangles), 200 (red crosses) and 400 (magenta circles), according to simulations (markers) and Eq. (4) (lines). The inset shows the relationship between $l_y/\xi$ and the trap ratio. The vortex is started at $(x_{V,0},y_{V,0})=(1.5l_x,0)$. Shading indicates the solitonic regime. (b) As in (a) but with ${\omega }_V$ plotted versus $l_y/\xi$ and also versus $R_y/\xi$ (inset). (c) Oscillation frequency versus trap ratio for different initial positions $x_{V,0}$ as per the legend (and $g=400$). The black dashed lines indicate the predicted soliton frequency ${\omega }_S={\omega }_x/\sqrt{2}$. All units are dimensionless.

Figure 3

Dynamics under the trap deformation. (a) The imposed time-dependent deformation of the trap ratio. (b) Evolution of the condensate density during a hysteresis protocol from axisymmetric to elongated and back again. Here $g=100, (x_{V,0},y_{V,0})=(1.5,0)l_x, \varepsilon =8, t_{\mathrm{ramp}}=70{\omega }_x^{-1}$, and $t_{\mathrm{hold}}=40{\omega }_x^{-1}$. All units are dimensionless.

Figure 4

Angular momentum per particle $L_z$ versus time (left) and trap ratio ${\omega }_y/{\omega }_x$ (right). From (a) to (e) we show the cases for different initial vortex positions, hold times, ramp times, interaction strengths, and maximum trap anisotropies. For the $|L_z({\omega }_y/{\omega }_x)|$ hysteresis plots the data are smoothed with Bézier curves. Unless varied, the parameter values are $g=100, t_{\mathrm{ramp}}=70{\omega }_x^{-1}, t_{\mathrm{hold}}=40{\omega }_x^{-1}{x}_{V,0}=1.5l_x$, and $\varepsilon =8$. The gray triangles (here out of scale) in the top left panel show the relative increase of the total energy of the system $E/E(t=0)$ that grows as large as 3. See text for details. All units are dimensionless.