Vortex precession dynamics in general radially symmetric potential traps in two-dimensional atomic Bose-Einstein condensates

We consider the motion of individual two-dimensional vortices in general radially symmetric potentials in Bose-Einstein condensates. We find that although in the special case of the parabolic trap there is a logarithmic correction in the dependence of the precession frequency $\omega$ on the chemical potential $\mu$, this is no longer true for a general potential $V\left(r\right)\propto r^p$. Our calculations suggest that for $p>2$, the precession frequency scales with $\mu$ as $\omega \sim {\mu }^{-2/p}$. This theoretical prediction is corroborated by numerical computations, not only at the level of spectral (Bogolyubov–de Gennes) stability analysis by identifying the relevant precession mode dependence on $\mu$ but also through direct numerical computations of the vortex evolution in the large-$\mu$, so-called Thomas-Fermi, limit. Additionally, the dependence of the precession frequency on the distance to the trap center of an initially displaced vortex is examined, and the corresponding predictions are tested against numerical results.

Figure 1

Radial profile $\left|\psi \right|$ as a function of $r$ for the ground state (GS, red dashed line) and the vortex state (VX, blue solid line) inside the trapping potential $V\left(r\right)=\frac{1}{p}{r}^p$ with $\mu =40$ for (a) $p=2$, (b) $p=4$, and (c) $p=6$. Note that in the Thomas-Fermi limit, the size of the vortices for the different potentials is essentially the same. This property, determined by the healing length, is controlled by the chemical potential $\mu$, which is constant ($\mu =40$) for the three cases. (d) GS profiles as a function of the rescaled distance $r/R_{\mathrm{TF}}$, where the corresponding TF radii are given by $R_{\mathrm{TF}}={\left(p\mu \right)}^{1/p}$.

Figure 2

The BdG spectrum for the vortex in the quadratic potential. The lowest mode, highlighted in orange, is the one of interest as it corresponds to the vortex precession around the trap center.

Figure 3

The vortex spectrum in the (a) quartic and (b) sextic potentials. The lowest mode, which is highlighted in orange similar to the quadratic potential, is the vortex precession mode around the trap center.

Figure 4

Scaling of the precession frequency $\omega$ on the chemical potential $\mu$. The thick lines correspond to the precession frequency extracted from the numerically obtained spectra, and the asymptotic approximations are depicted with thin lines ($C_i$ are arbitrary constants chosen for ease of exposition). The case of $p=2$ decays slower than the power scaling (see thin red dashed line) due to the logarithmic correction, while for $p=4$ and $p=6$, the predictions and the spectrum appear to compare well with each other, suggesting good agreement with the analytical prediction of Eq. (9). The black circles correspond to a correction to the asymptotic formula (6), namely, Eq. (10) with $A=8.88$ (see Ref. [34]).

Figure 5

Scaling of the precession frequency as a function of the rescaled vortex position for the different potentials $V\left(r\right)=\frac{1}{p}{r}^p$ with $\mu =40$ and $\mu =80$. A vortex is displaced from $r=0$, and its precession frequency is extracted from its dynamical evolution (see text) and depicted with orange (blue) circles (squares) for $\mu =40$ ($\mu =80$). The solid lines depict the theoretical predictions from evaluating Eq. (5) and using the limit $\xi \ll r$. The dashed curves for the $p=2$ case correspond to the corrected precession of Eq. (10), while the dashed curves for the $p=4$ and $p=6$ cases depict the rescaled theoretical predictions to match the precession frequency in the limit $r=0$. These rescaling coefficients correspond to, for $\mu =40$ and $\mu =80$, respectively, 1.7944 and 1.7804 for $p=4$ and 1.9302 and 1.9114 for $p=6$.