Exploring vortex dynamics in the presence of dissipation: Analytical and numerical results

In this paper, we examine the dynamical properties of vortices in atomic Bose-Einstein condensates in the presence of phenomenological dissipation, used as a basic model for the effect of finite temperatures. In the context of this so-called dissipative Gross-Pitaevskii model, we derive analytical results for the motion of single vortices and, importantly, for vortex dipoles, which have become very relevant experimentally. Our analytical results are shown to compare favorably to the full numerical solution of the dissipative Gross-Pitaevskii equation where appropriate. We also present results on the stability of vortices and vortex dipoles, revealing good agreement between numerical and analytical results for the internal excitation eigenfrequencies, which extends even beyond the regime of validity of this equation for cold atoms.

Figure 1

Isosurfaces of antidamped single-vortex motion for $\gamma =0$ (left) and $\gamma =0.0023$ (right), showing the full PDE [Eq. (1)] numerical results (green) and the analytical result obtained from the vortex ODEs (red). Here the initial position is $(x_0,y_0)=(0,1.5)$, the chemical potential $\mu =1.6$, and the trap frequency $\Omega =0.2$.

Figure 2

Motion for a symmetric (left column) and an asymmetric (right column) vortex dipole in the absence (top panels) and presence (bottom panels) of dissipation. The initial positions of the two vortices are $(x_1,y_1)=(0,1.75)$ and $(x_2,y_2)=(0,-1.75)$ (left column) and $(x_1,y_1)=(0,3.00)$ and $(x_2,y_2)=(0,-1.25)$ (right column), while the chemical potential $\mu =1.6$ and the trap frequency $\Omega =0.2$. Other parameters are as in Fig. 1.

Figure 3

The left four panels show the analytical versus numerical trajectories of the vortex dipole epicycle for the $\mu =4$, $\gamma =0$, $\Omega =0.1$, and $B=2$ case. In particular, the upper left panel is the three-dimensional comparison of the vortex instantaneous positions $(x,y)$ versus $t$, the upper right panel is the projection of the motion on the $(x,y)$ plane, the bottom left panel is the projection on the $(y,t)$ plane, and the bottom right panel is the projection on the $(x,t)$ plane. Here the thick blue and green lines correspond to the analytical predictions, while the thin red and black lines correspond to the numerical results. The right four panels are the same as the left panels, but with $\gamma =0.0023$.

Figure 4

The first panel shows the nonvanishing imaginary part of the eigenfrequency associated with the anomalous mode, as soon as the dissipative prefactor $\gamma$ is nonzero for $\mu =0.4$, $0.8$, 1, and $1.6$ (respective curves from top to bottom). The second panel is similar, but for the real part of the eigenfrequency. Here $\Omega =0.2$. The third panel illustrates the linearization (so-called BdG) spectrum of the case $\gamma =0$ (red crosses) and that of $\gamma =0.2$ (blue circles) for $\mu =1$ and $\Omega =0.2$. The fourth panel shows the spectrum for $\mu =0.4$ (blue circles), $\mu =0.8$ (red stars), $\mu =1$ (green crosses), and $\mu =1.6$ (magenta pluses) for $\gamma =0.2$.

Figure 5

Of the images on the left, the upper four panels show the comparison of the imaginary (growth) part of the vortex linearization eigenfrequency associated with the anomalous mode from the BdG numerical analysis (solid blue) with the analytical results ${\omega }_i$ [see Eq. (14)] (red dashed). The lower four panels show the comparison of the real (oscillatory) part of the eigenfrequency associated with the anomalous mode from the BdG numerical analysis (solid blue) with the analytical results ${\omega }_r$ [see Eq. (15)] (red dashed). From left to right, the cases shown are for $\mu =0.4,0.8,1,\text{and}1.6$, respectively, and $\Omega =0.2$. Notice the progressively improving agreement as $\mu$ increases and the Thomas-Fermi regime of vortex particle motion is reached. The images on the right show the internal mode eigenfrequencies of the vortex-dipole state associated with the anomalous mode from the BdG numerical analysis (dashed blue) against the dissipation term $\gamma$ for $\mu =0.71,0.80,\text{and}1,1.6$ versus the analytical prediction (solid red), for $\Omega =0.2$.