Distributed vorticity model for vortex molecule dynamics

We analyze the effect of a hard wall trapping potential on the dynamics of a vortex molecule in a two-component Bose-Einstein condensate with linear coherent coupling. A vortex molecule consists of a vortex of the same charge in each component condensate connected by a domain wall of the relative phase. In a previous paper [S. Choudhury and J. Brand, Phys. Rev. A 106, 043319 (2022)] we described the interaction of a vortex molecule with the boundary using the method of images by separately treating each component vortex as a point vortex, in addition to a Magnus force effect from the surface tension of the domain wall. Here we extend the model by considering a continuous distribution of image vorticity reflecting the effect of the domain wall on the vortex molecule phase structure. In the case of a precessing centered vortex molecule in an isotropic trap, distributing the image vorticity weakens its contribution to the precession frequency. We test the model predictions against numerical simulations of the coupled Gross-Pitaevskii equations in a two-dimensional circular disk and find support for the improved model.

Figure 1

Vortex molecule in a disk. Numerical solution of the GPE (1) with a centered vortex molecule with molecular length $d=20\xi$ in a disk-like trap described by Eq. (5). (a) Density of condensate 1, $n_1={\left|{\psi }_1\right|}^2$. (b) Density of condensate 2, $n_2={\left|{\psi }_2\right|}^2$. (c) Relative phase $arg\left({\psi }_1{\psi }_2^{*}\right)$. (d) Total phase $arg\left({\psi }_1{\psi }_2\right)$. The black dotted circles in panels (c) and (d) denote the trap boundary at the disk diameter of $2L=70\xi$. Other parameters are $\nu =2\times {10}^{-3}\mu$, $g_{12}=0$.

Figure 2

Concept diagram of the vorticity distribution model for a centered vortex molecule in a disk trap. The image vortices are formed at locations $\tilde{\mathbf{R}}=\frac{L^2}{{\left|\mathbf{R}\right|}^2}\mathbf{R}$ where $\mathbf{R}$ is the position of the original charge. The distance of the image vortices from the center of the condensate is inversely proportional to the distance of its original vortex from the center. Hence, even though the charge distribution is evenly spaced the image charges are not and reach out to infinity as $\left|\mathbf{R}\right|\to 0$.

Figure 3

Components of the vortex molecule precession frequency according to various models as a function of the molecular size $d$. The negative-valued boundary contributions ${\mathrm{\Omega }}_{\mathrm{bound}}$ are labeled “$\mathrm{dv}$” for the distributed velocity contribution ${\mathrm{\Omega }}_{\mathrm{dv}}$ of Eq. (22) and “$\mathrm{sv}$” for the single vortex contribution ${\mathrm{\Omega }}_{\mathrm{sv}}$ of Eq. (21). The interaction components ${\mathrm{\Omega }}_{\mathrm{int}}$ follow Eq. (20) and bring positive contributions. The one labeled “$\mathrm{lin}$” follows from a purely linear interaction potential with the surface tension of the idealized Josephson vortex. The contribution from the parameterized numerically obtained interaction energy is labeled “$\mathrm{pm}.$” The frequency is expressed in units of ${\mathrm{\Omega }}_0=(\mu +\nu )/\hslash$. Parameters are the same as Fig. 1.

Figure 4

Precession frequency of a centered vortex molecule in a disk trap as a function of the molecular size $d$. The orange crosses represent real time evolution data from the GPE of Eq. (1). The curves are model predictions according to Eq. (19) with different combinations of the contributions for ${\mathrm{\Omega }}_{\mathrm{bound}}$ and ${\mathrm{\Omega }}_{\mathrm{int}}$ shown in Fig. 3. The blue solid line marked “dv $+$ pm” combines the distributed vorticity variant of the boundary contribution with the parametrized interaction energy and provides the best explanation of the numerical GPE data within the available models. The red dot-dashed line marked “sv $+$ lin” represents single vortex boundary contribution from Eq. (21) along with the linear interaction energy contribution from Eq. (24). The brown dashed line marked “sv $+$ pm” represents single vortex boundary contribution along with parametrized interaction energy contribution. The frequency is expressed in units of ${\mathrm{\Omega }}_0=(\mu +\nu )/\hslash$. Parameters are the same as Fig. 1.

Figure 5

Numerical solution of the GPE with a centered vortex molecule with molecular length $d=60\xi$. The component vortices are separated less than ${\xi }_{\mathrm{J}}$ from the trap boundaries and the assumption of a localized domain wall breaks down. (a) Density of condensate 1 $n_1={\left|{\psi }_1\right|}^2$. (b) Density of condensate 2 $n_2={\left|{\psi }_2\right|}^2$. (c) Relative phase $arg\left({\psi }_1{\psi }_2^{*}\right)$. (d) Total phase $arg\left({\psi }_1{\psi }_2\right)$. Parameters as in Fig. 1.

Figure 6

Trajectories predicted for an off-centered vortex molecule by different models and their deviations. An off-centered vortex molecule moves in a disk-shaped trap ($L=30\xi$) with initial positions ${\mathbf{R}}_1(t=0)=(3.95\xi ,7.83\xi )$ [black dot in panel (a)], and ${\mathbf{R}}_2(t=0)=(2.07\xi ,-7.71\xi )$ (not shown). (a) Trajectories. The trajectory of vortex 1 is represented by colored lines showing the predictions from different models as in Fig. 4. The dashed line “GPE” denotes the numerical solution of the full Eq. (1). The dash-dotted line “sv $+$ lin” shows the trajectory for the model with a single image vortex contribution of Eq. (21) combined with a linear interaction contribution based on Eq. (24). The full line “dv $+$ pm” is obtained from combining the distributed vorticity contribution of Eq. (21) with the parameterized interaction contribution from Appendix pp1. The colored dots mark the endpoints of the respective trajectories. (b) Deviations of the point vortex models from the GPE time evolution shown in panel (a). The instantaneous distance $|{\mathbf{R}}_1^{\mathrm{pv}}\left(t\right)-{\mathbf{R}}_1^{\mathrm{GPE}}\left(t\right)|$ between the location of vortex 1 according to the point vortex model and the GPE simulation is plotted as a function of time. Results are shown for the two point vortex models of panel (a) and labeled “sv $+$ lin” and “dv $+$ pm”, respectively. The unit of time is $1/{\mathrm{\Omega }}_0=\hslash /(\mu +\nu )$.

Figure 7

Fitting the interaction energy $V\left(d\right)$ as a function of molecular length $d$. The orange dots show data from imaginary time evolution of the GPE from Ref. [29] (only some representative data points are shown here). The inset is a closeup showing data for small values of $d$. The lines show the parametrization of $V\left(d\right)$ used for the model dynamics discussed in this paper consisting of three different segments. The blue solid line shows a third-order spline interpolation of the numerical data for $4\xi <d<55\xi$. The green dot-dashed line is a linear extrapolation for large $d>55\xi$. The red dashed line shows the quartic extrapolation of Eq. (A1) for $d<4\xi$. Parameters are $\nu =2\times {10}^{-3}\mu$, $g_{12}=0$. $W_0={\hslash }^2(\mu +\nu )/m(g+g_{12})={(\mu +\nu )}^2{\xi }^2/(g+g_{12})$ and an arbitrary offset was added to the energy.