Scattering of a massive quantum vortex-dipole from an obstacle

TL;DR

This study addresses the scattering of two-dimensional ($2$D) massive vortex dipoles in binary Bose-Einstein condensates against a disk-like obstacle, a problem pertinent to $2$D quantum turbulence and trajectory control. It develops a variational, point-like description based on a two-component Lagrangian, predicting two robust scattering channels—fly-by and go-around—and reveals a massless subregime far from the obstacle. An analytic characterization links the transition to minimal approach distances and deflection angles $\\phi$ as a function of impact parameter $b$, with a critical geometry signaling go-around. The PLM is benchmarked against coupled Gross-Pitaevskii equations, showing good qualitative agreement and exposing boundary-induced, beyond-point-like effects such as nonuniform density and infilling-component tunneling, with implications for steering vortex-dipole dynamics in quantum fluids.

Abstract

In binary mixtures of Bose-Einstein condensates, massive-vortex dipoles can arise, and undergo scattering processes against obstacles. These show an intriguing dynamics, governed by the strongly nonlinear character of the quantum vortex motion, where we are able to highlight the effects of the boundaries. We first characterize such scattering dynamics via some point-like models, for the cases of an unbounded plane and a confined geometry. Within this framework, we find two fundamental scattering behaviors of a vortex dipole, the "fly-by" and the "go-around" processes. By plotting the deflection angle of the dipole versus the impact parameter we are able to quantify the transition between different scattering behaviors. We then are able to introduce an analytical distinction of the two scenarios, basing on the point-like model for the plane geometry. Furthermore, another interesting result shows the emergence of an on-average massless dynamics whenever the nonlinear interactions with the obstacle become negligible. Alongside, we investigate the quantum dipole scattering via the numerical simulation of two coupled Gross-Pitaevskii equations, describing the quantum mixture at a mean-field level. In this way, we benchmark the point-like model against the mean-field simulations.

Figures (19)

• Figure 1: Trajectory of a vortex dipole in an unbounded domain with a central obstacle: in case of symmetric initial positions (black dots) the recombination of the dipole occurs symmetrically with respect to both the $x$- and $y$-axes, as shown by the final positions (squares).The evolution time is $t=1.5$$s$.
• Figure 2: The left (right) panel shows a go-around (fly-by) type of scattering, for a dipole in an unbounded domain with asymmetrical initial vortex positions. While the vortex is kept fixed, the initial position of the antivortex varies in the two panels. Note that a smaller dipole size corresponds to a higher velocity. Overall evolution time: $t =1.5$$s$.
• Figure 3: The left (right) panel presents the fly-by (go-around) of a vortex dipole in its scattering against an obstacle in an unbounded domain. In the left panel, the vortex is kept at the same initial position of the vortex in the right panel of Fig. \ref{['fig:PL_asymmetrical1']}, while the antivortex is placed closer to the obstacle. This choice still preserves the fly-by process, thus increasing the dipole's overall velocity. On the other hand, keeping now fixed the antivortex, and moving the vortex closer to the obstacle, induces a go-around type of scattering, which involves an asymmetric recombination of the dipole (right panel). Evolution time: $t=1$$s$.
• Figure 4: Comparison of the scattering dynamics of a massive dipole (continuous lines) and a massless dipole (dotted lines) against a disk in a go-around scenario and in an unbounded domain. Left panel sees trajectories for the usual core boson number of $N_b=10^2$, where the slight variation in the dipole translation line after the recombination is attributable to the nonlinear interaction between the obstacle and the core masses. Apart from this, the agreement in the trajectories of the two dynamics shows how, within the asymptotic regimes, it is possible to reduce the massive dynamics to its massless counterpart. The right panel shows greater differences due to an increase vortex mass ($N_b=2.5\times 10^2$): the deflection angle is increased due to the stronger centrifugal effect caused by the heavier cores. Evolution time: $t = 2.5$$s$.
• Figure 5: Deflection angle $\phi$ of the dipole trajectory, after a scattering event in an unbounded domain, as a function of the impact parameter $b$, for two different values of the dipole size $r_{21}$, and for $N_b=10^2$. The initial velocities are, for the case at $r_{21}=4R_1$ ($r_{21}=8R_1$), $\dot{x}_1(0)=\dot{x}_2(0)=0$, $\dot{y}_1(0)=\dot{y}_2(0) \simeq 7.3\times 10^{-5}$$m/s$ ($\dot{y}_1(0)=\dot{y}_2(0) \simeq 3.7\times 10^{-5}$$m/s$).