Analytic Phase Solution and Point Vortex Model for Dipolar Quantum Vortices

Abstract

We derive an analytic expression for the phase of a quantum vortex in a dipolar Bose-Einstein condensate, capturing anisotropic effects from long-range dipole-dipole interactions. This solution provides a foundation for a dipolar point vortex model (DPVM), incorporating both phase-driven flow and dipolar forces. The DPVM reproduces key features of vortex pair dynamics seen in full simulations, including anisotropic velocities, deformed orbits, and directional motion, offering a minimal and accurate model for dipolar vortex dynamics. Our results open the door to analytic studies of vortices in dipolar quantum matter and establish a new platform for exploring vortex dynamics and turbulence in these systems.

Figures (5)

• Figure 1: Analytic phase of an elliptic vortex. (a) Solid lines show the vortex phase obtained from the dipolar GPE with interaction strength $\varepsilon_\text{dd}=0.9$ at fixed radius $R = 10\,$µm; dashed lines indicate the corresponding solution to Eq. \ref{['eqn:elliptic_phase']}, using only the fitted ellipticity $\lambda$ extracted from the vortex density. The polarization angle $\alpha$ increases in steps of $\pi/6$, and each curve is vertically offset by $\pi$ for clarity. (b,c) Example solutions for the normalized vortex density $n(x,y)$ and phase at $\alpha = 0$ and $\alpha = \pi/2$, respectively. The colored circle marks the location from which the phase is extracted in (a). (d) Relative energy error: GPE vs. vortex Ansatz (Eq. \ref{['eqn:density_Ansatz']}, solid), and GPE vs. isotropic Ansatz ($\lambda = 1$, dashed). Speed $|\nabla S|$ with $\alpha=\pi/2$ from the (e) GPE and (f) analytic solution, with contours at steps of 20µm/s.
• Figure 2: Comparison of vortex pair dynamics. (a-c) Vortex-vortex orbits with initial condition $x_j=0$ and $y_1=-y_2=5$µm in the DPVM (top row) and GPE (bottom row) for $\varepsilon_\text{dd}=0.9$ and (a) $(\alpha,\lambda)=(\pi/6,1.08)$ (b) $(\pi/3,1.15)$ and (c) $(\pi/2,1.3)$. Color represents relative deviation from the local speed to the average speed $v$, i.e. $(|\textbf{v}(x,y)| -v)/v$. (d) Same as (a-c) with $y_1=-y_2=0.6$µm, with fixed $\alpha=\pi/2$ and $(\varepsilon_\text{dd},\lambda)=(0.3,1.03)$ (0.6,1.08) (0.9,1.15). (e-f) Traveling vortex-antivortex pairs with the vector between them (e) perpendicular and (f) parallel to the magnetic field and $(\varepsilon_\text{dd},\alpha,\lambda)=(0.9,\pi/2,1.3)$. The DPVM vortex (blue, $q=1$) and antivortex (red, $q=-1$) are shown with the corresponding GPE solutions (black dashed lines) overlaid, normalized to the same maximum traveled distance in (f). We fix $\xi_v = 20.3a_\text{dd}$ throughout.
• Figure 3: Elliptic orbits of three vortices. Vortices are initially placed at the vertices of an equilateral triangle with circumradius 1 and evolved using (a) the DPVM and (b) the GPE. Vortex trajectories are color- and line-styled as follows: blue, solid; red, dashed; yellow, dotted. Initial vortex positions are marked in (a), while the background in (b) shows the initial condensate density. Parameters: $\alpha=\pi/2$, $\varepsilon_\text{dd}=0.6,~\lambda=1.16$, others match Fig. \ref{['fig:one_vortex']}.
• Figure 4: Contributions to the dipolar vortex velocity. Panels show the gradient of the corresponding terms appearing in Eq. \ref{['eqn:analytic_phase']} with $\lambda=2$.
• Figure 5: Analytic phase with spatially varying $\lambda$. (a) Speed from Eq. \ref{['eqn:elliptic_phase']} with $\lambda(\rho)$, with same parameters as Fig. \ref{['fig:one_vortex']}(f). (b) Function used in (a), with the length scale $l=1$µm.