Spinor Bose-Einstein condensate flow past an obstacle

We study the flow of a spinor $\left(F=1\right)$ Bose-Einstein condensate in the presence of an obstacle. We consider the cases of ferromagnetic and polar spin-dependent interactions, and find that the system demonstrates two speeds of sound that are identified analytically. Numerical simulations reveal the nucleation of macroscopic nonlinear structures, such as dark solitons and vortex-antivortex pairs, as well as vortex rings in one- and higher-dimensional settings, respectively, when a localized defect (e.g., a blue-detuned laser beam) is dragged through the spinor condensate at a speed larger than the second critical speed.

Figure 1

Dependence of the critical velocity on the defect strength $V_0$. The result shown corresponds to the stationary solution of Eq. (4) with $A=-0.5=-C$ and $B=\sqrt{1/2}\left(\mu =1\right)$. It was confirmed that, e.g., for the solutions of Eq. (5), the analytical and numerical results were indistinguishable up to three decimal places.

Figure 2

(Color online) Time evolution of the three hyperfine components (top, middle, and bottom rows, respectively) for different defect velocities: $s=0.025<c_1$ (left column), $c_1<s=0.325\lesssim c_2$ (middle column), and $s=0.335>c_2$ (right column); here, $c_1=0.058$ and $c_2=0.329$ are the two critical velocities while the defect strength is taken to be $V_0=0.9$. The initial condition corresponds to the stationary state with wave-function amplitudes given in Eq. (4). While the results in the left and middle columns show a steady flow (apart from an oscillatory structure that is detached at $t=0$), the evolution shown in the panels of the right column is characterized by the emission of dark solitons, even from the early stages of the process. Notice that the analytically predicted first (lower) critical velocity would fall between the velocities of the results depicted in the left and middle columns but no significant change is observed in the dynamics between these two cases.

Figure 3

(Color online) Vortex and antivortex pairs nucleated by a moving defect in the 2D spinor condensate. The left column shows the density of the spinor condensate’s component ${\psi }_0$ at different times (indicated in the panels). The location and extent of the moving defect is depicted by the oval line corresponding to an isocontour of its strength at 10% of its maximum. The right column shows the corresponding vorticity (defined in the text), clearly illustrating the presence of vortex and antivortex pairs. The case depicted here corresponds to $w_y=8$, $a=2$, and $V_0=0.9$ while the defect speed is taken to be $s=0.6>c_2$.

Figure 4

(Color online) Evolution of the maximum (dark/red) and minimum (light/yellow) of the vorticity isocontours in $\left(x,y,t\right)$. One can clearly discern the emergence of vortex (in dark/red) and antivortex (in light/yellow) pairs, as the defect moves along the $x$ direction. The left and right panels correspond, respectively, to the cases depicted in Fig. 3 $\left(w_y=8\right)$ and Fig. 5 $\left(w_y=20\right)$.

Figure 5

(Color online) Same as Fig. 3 but for a wider defect with $w_y=20$. One can clearly observe the formation of numerous vortex-antivortex pairs in the wake of the defect.

Figure 6

(Color online) Vortex ring formation in the supercritical 3D setting. Panels (a)–(e) depict isodensity contours corresponding to ${\left|{\psi }_0\right|}^2=0.3$ at the indicated times. Note that the extent of the moving defect is clearly visible in these panels (it corresponds to the rightmost flat oval shape that is created by the atomic density depletion due to its presence). Panel (f) shows typical isocontours of the norm of the vorticity of ${\psi }_0$ at times $t=12,14,\dots ,30$ (left to right). In this case we use a defect with $w_y=8$, $w_z=4$, $a=2$, and speed $s=0.8>c_2$.

Figure 7

(Color online) Similar to Fig. 6 but for a slightly larger defect speed $s=1>c_2$. The top row of panels depicts the isodensity contours of ${\psi }_0$ while the bottom row depicts the respective vorticity isocontours. Note the successive nucleation of two vortex rings. The first one shrinks and collapses into itself between $t=22$ and $t=23$ while the second ring deforms and eventually splits into two separate vortex rings between $t=29$ and $t=30$.