Scattering of two-dimensional dark solitons by a single quantum vortex in a Bose-Einstein condensate

We study the process of scattering of two-dimensional dark solitons, and their vortex-antivortex pairs as a specific case, by a single quantum vortex in a Bose-Einstein condensate with repulsive interaction between atoms. An asymptotic theory describing the dynamics of such solitonlike structures in a smoothly inhomogeneous flow of ultracold Bose gas is developed. An analytical expression for the angle of scattering of two-dimensional dark solitons (including vortex pairs) by a single-phase singularity is obtained in the limit of large impact parameters. All theoretical concepts are confirmed by numerical calculations performed directly within the Gross-Pitaevskii equation. It is shown that for small impact parameters, the considered solitonlike structures interact inelastically with the core of a single quantum vortex, scattering over large angles and radiating sound waves.

Figure 1

Distributions $\left(\mathrm{a}\right)$ of the density and $\left(\mathrm{b}\right)$ of the phase of the BEC wave function in the presence of a single quantum vortex with $\varkappa =+1$ located at the center of the Cartesian coordinate system $x,y$. $\left(\mathrm{c}\right)$ Gives a comparison of the Padé approximation (7) (red solid line) for the dependence $n_v\left(r\right)$ with the results of numerical calculation (unshaded blue markers) using Eq. (4) and the boundary conditions ${\psi }_v\left(0\right)=0$ and ${\psi }_v\left(\infty \right)=1$.

Figure 2

Distributions $\left(\mathrm{a}\right)$ of the density and $\left(\mathrm{b}\right)$ of the phase of the wave function of the BEC confined in the axially symmetric harmonic potential (8) with ${\omega }_r\approx 0.0736$ in the presence of a single quantum vortex with $\varkappa =+1$ at the trap center. $\left(\mathrm{c}\right)$ Gives a comparison of the approximation (9) (red solid line) for the dependence $n_v\left(r\right)$ with the results of numerical calculation (unshaded blue markers) directly within the GP equation (1).

Figure 3

Distribution $\left(\mathrm{a}\right)$–$\left(\mathrm{c}\right)$ of the condensate density and $\left(\mathrm{d}\right)$–$\left(\mathrm{f}\right)$ of the phase of the BEC wave function in 2D dark solitons for three values of the soliton velocity $\overline{v}$, namely, $\left(\mathrm{a}\right)$, $\left(\mathrm{d}\right) \overline{v}=0.3,\left(\mathrm{b}\right)$, $\left(\mathrm{e}\right) \overline{v}=0.5$, and $\left(\mathrm{c}\right)$, $\left(\mathrm{f}\right) \overline{v}=0.7$. For $\overline{v}=0.3$ and $0.5$, these solitons have the form of vortex pairs, in which the condensate density at two points vanishes [see fragments $\left(\mathrm{a}\right)$ and $\left(\mathrm{b}\right)$], and the phase of the BEC wave function ${\mathrm{\Psi }}_s\left(\xi ,y\right)$ when going around these points changes to $\pm 2\pi$ [see fragments $\left(\mathrm{d}\right)$ and $\left(\mathrm{e}\right)$]. For the velocity $\overline{v}=0.7$, a 2D dark soliton is vortex free, i.e., the density of ultracold Bose gas in it nowhere vanishes [see fragment $\left(\mathrm{c}\right)$], while the phase of the wave function ${\mathrm{\Psi }}_s\left(\xi ,y\right)$ changes gradually [see fragment $\left(\mathrm{f}\right)$].

Figure 4

Snapshots of the BEC density at (a) $t=20$, (b) $t=70$, and (c) $t=120$, which illustrate the process of scattering of a 2D dark soliton by a single quantum vortex (with a topological charge $\varkappa =+1$) excited in the initially homogeneous ultracold Bose gas and located at the origin of the Cartesian coordinate system $x,y$ (unshaded green circle). At the time $t=0$ the solitonlike structure starts from the point $x_0=-25.6,y_0=-16$ (green cross) in the positive direction of the $x$ axis and has the normalized velocity ${\overline{v}}_0=0.3$, i.e., is initially a vortex pair. The condensate density distributions were obtained by direct numerical simulation performed immediately within the framework of the GP equation (1). The solid black line shows the trajectory of a composite quasiparticle, which is calculated using canonic equations (28) and (29). The dashed red line shows the trajectory of this quasiparticle, which is calculated in the small-angle approximation (53).

Figure 5

Same as Fig. 4, but at the times $\left(\mathrm{a}\right)t=30,\left(\mathrm{b}\right)t=70$, and $\left(\mathrm{c}\right)t=110$ for the 2D dark soliton which was initially specified in the form of a vortex pair with normalized velocity ${\overline{v}}_0=0.5$ and started from the point with the coordinates $x_0=-25.6$ and $y_0=12.8$.

Figure 6

Same as Fig. 4, but at the times $\left(\mathrm{a}\right)t=20,\left(\mathrm{b}\right)t=40$, and $\left(\mathrm{c}\right)t=60$ for the initially vortex-free 2D dark soliton with normalized velocity ${\overline{v}}_0=0.7$ which started from the point with the coordinates $x_0=-25.6$ and $y_0=-9.6$.

Figure 7

Same as Fig. 4, but at the times $\left(\mathrm{a}\right)t=20,\left(\mathrm{b}\right)t=40,\left(\mathrm{c}\right)t=60,\left(\mathrm{d}\right)t=80,\left(\mathrm{e}\right)t=100$, and $\left(\mathrm{f}\right)t=120$ for the 2D dark soliton, which is initially specified in the form of a vortex pair with the normalized velocity ${\overline{v}}_0=0.3$ and starts from the point $x_0=-25.6,y_0=-4.8$. The impact distance $y_0=-4.8$ is smaller than the bifurcation value ${y_0}_a\left({\overline{v}}_0=0.3\right)\approx -3.44$, which corresponds to the flyby scattering mode of a solitonlike structure on a single quantum vortex.

Figure 8

Same as Fig. 4, but at the times $\left(\mathrm{a}\right)t=25,\left(\mathrm{b}\right)t=60,\left(\mathrm{c}\right)t=95,\left(\mathrm{d}\right)t=130,\left(\mathrm{e}\right)t=165$, and $\left(\mathrm{f}\right)t=200$ for the 2D dark soliton, which is initially specified in the form of a vortex pair with the normalized velocity ${\overline{v}}_0=0.3$ and starts from the point $x_0=-25.6,y_0=9.6$. The impact parameter $y_0=9.6$ lies in the interval ${y_0}_a<y_0<{y_0}_b$, where ${y_0}_a\left({\overline{v}}_0=0.3\right)\approx -3.44$ and ${y_0}_b\left({\overline{v}}_0=0.3\right)\approx 13.65$, i.e., the exchange scattering mode of the solitonlike structure with a single quantum vortex is implemented.

Figure 9

The dependencies of the minimum distances ${r_s}_{\min }\left(y_0,{\overline{v}}_0\right)$ to which the quasiparticles starting from the points $x_0=-25.6,y_0$ and having different initial normalized velocities ${\overline{v}}_0$, namely, ${\overline{v}}_0=0.3$ (red solid curve $A$), ${\overline{v}}_0=0.5$ (green solid curve $B$), and ${\overline{v}}_0=0.7$ (blue solid curve $C$), approach the scattering center, which were calculated using the asymptotic theory developed in Sec. 2d. For each value of ${\overline{v}}_0$ there is an interval from ${y_0}_a\left({\overline{v}}_0\right)$ to ${y_0}_b\left({\overline{v}}_0\right)$, inside which the transcendental algebraic equation connecting ${r_s}_{\min }$ and impact parameters $y_0$ does not have real-valued solutions. These nested intervals correspond to the regions shown by colors of different shades. The contrast of these regions increases with increasing ${\overline{v}}_0$. Unshaded markers show the minimum distances to which the 2D dark solitons with ${\overline{v}}_0=0.3$ (red circles), ${\overline{v}}_0=0.5$ (green squares), and ${\overline{v}}_0=0.7$ (blue diamonds) approach the origin of the Cartesian coordinates $x y$, which were obtained by direct numerical calculations within the framework of the GP equation (1).

Figure 10

The trajectories of quasiparticles for three initial normalized velocities ${\overline{v}}_0$, namely, $\left(\mathrm{a}\right){\overline{v}}_0=0.3,\left(\mathrm{b}\right){\overline{v}}_0=0.5$, and $\left(\mathrm{c}\right){\overline{v}}_0=0.7$, which were calculated using canonic equations (28) and (29). The quasiparticles start in the positive direction of the $x$ axis from the points $x_0=-25.6,y_0$, for which the impact distance $y_0$ takes the following values: $y_0=\pm 16$ (green dotted lines), $y_0=\pm 9.6$ (blue dashed lines), and $y_0=4.8$ (red solid line). If the impact parameter $y_0$ enters the interval ${y_0}_a\left({\overline{v}}_0\right)<y_0<{y_0}_b\left({\overline{v}}_0\right)$, to which its region shown by a color corresponds in each fragment $\left(\mathrm{a}\right),\left(\mathrm{b}\right)$, or $\left(\mathrm{c}\right)$, then the quasiparticle is trapped by the scattering center located at the origin of the Cartesian coordinates $x,y$ and approaches the scattering center by curling.

Figure 11

Snapshots $\left(\mathrm{a}\right)$–$\left(\mathrm{c}\right)$ of the condensate density and $\left(\mathrm{d}\right)$–$\left(\mathrm{f}\right)$ of the phase of the BEC wave function at the times $\left(\mathrm{a}\right)$, $\left(\mathrm{d}\right) t=30,\left(\mathrm{b}\right)$, $\left(\mathrm{e}\right) t=50$, and $\left(\mathrm{c}\right)$, $\left(\mathrm{f}\right) t=70$ for the case where a 2D dark soliton, which starts at $t=0$ from the point $x_0=-25.6,y_0=-6.4$ (green cross) in the positive direction of the $x$ axis and has the normalized velocity ${\overline{v}}_0=0.5$, i.e., initially represents a vortex pair, is incident on the initially isolated topological defect (with $\varkappa =+1$) located at the center of the Cartesian coordinate system $x,y$ (unshaded green circle). Spatial distributions of the density and phase were obtained by direct numerical simulation performed immediately within the framework of the GP equation (1). A solid black curve shows the trajectory, which was calculated using canonic equations (28) and (29), of a composite quasiparticle corresponding to the solitonlike structure. In the active-interaction region [$\left(\mathrm{b}\right)$ and $\left(\mathrm{e}\right)$] with single-phase singularity, the vortex pair transforms into a vortex-free soliton, which when moving away from the topological defect converts again into a vortex pair [$\left(\mathrm{c}\right)$ and $\left(\mathrm{f}\right)$].

Figure 12

Snapshots $\left(\mathrm{a}\right)$–$\left(\mathrm{f}\right)$ of the condensate density and $\left(\mathrm{g}\right)$–$\left(\mathrm{i}\right)$ of the phase of the BEC wave function at the times $\left(\mathrm{a}\right) t=27.5,\left(\mathrm{b}\right)$, $\left(\mathrm{g}\right) t=47.5,\left(\mathrm{c}\right)$, $\left(\mathrm{h}\right) t=50,\left(\mathrm{d}\right)$, $\left(\mathrm{i}\right) t=52.5,\left(\mathrm{e}\right) t=55$, and $\left(\mathrm{f}\right) t=75$ for the case where a 2D dark soliton, which starts at $t=0$ from the point $x_0=-25.6,y_0=-0.8$ (green cross) in the positive direction of the $x$ axis and has the normalized velocity ${\overline{v}}_0=0.5$, i.e., is initially a vortex pair, is incident on the initially isolated topological defect (with $\varkappa =+1$) located at the origin of the Cartesian coordinate system $x,y$ (unshaded green circle). Spatial distributions of the density and phase were obtained by direct numerical simulation performed immediately within the framework of the GP equation (1). The value of the impact parameter $y_0=-0.8$ falls in the interval from ${y_0}_a\left({\overline{v}}_0=0.5\right)\approx -2.33$ to ${y_0}_b\left({\overline{v}}_0=0.5\right)\approx 8.74$. Therefore, the solitonlike structure has an exchange interaction with a single quantum vortex. This is demonstrated by $\left(\mathrm{b}\right)$–$\left(\mathrm{d}\right)$ and $\left(\mathrm{g}\right)$–$\left(\mathrm{i}\right)$. In this case, the radiative losses due to the sound-wave emission are small.

Figure 13

Same as Fig. 12, but at the times $\left(\mathrm{a}\right) t=25,\left(\mathrm{b}\right)$, $\left(\mathrm{g}\right) t=35,\left(\mathrm{c}\right)$, $\left(\mathrm{h}\right) t=37.5,\left(\mathrm{d}\right)$, $\left(\mathrm{i}\right) t=40,\left(\mathrm{e}\right) t=42.5$, and $\left(\mathrm{f}\right) t=52.5$ for the initially vortex-free 2D dark soliton with the normalized velocity ${\overline{v}}_0=0.7$. The value of the impact parameter $y_0=-0.8$ falls in the interval from ${y_0}_a\left({\overline{v}}_0=0.7\right)\approx -1.77$ to ${y_0}_b\left({\overline{v}}_0=0.7\right)\approx 6.16$. Therefore, the exchange interaction mode takes place. In this case, the initially vortex-free solitonlike structure as it approaches the phase singularity converts at first into a vortex pair [$\left(\mathrm{b}\right)$ and $\left(\mathrm{g}\right)$]. Then, the topological defect with $\varkappa =+1$, which is included in this pair, becomes isolated, so that the antivortex and the initially single quantum vortex form a new vortex pair [$\left(\mathrm{c}\right)$ and $\left(\mathrm{h}\right)$], which later converts into a vortex-free 2D dark soliton [$\left(\mathrm{d}\right)$ and $\left(\mathrm{i}\right)$]. Comparing $\left(\mathrm{a}\right)$ and $\left(\mathrm{f}\right)$, it can easily be verified that the new vortex-free solitonlike structure is a wider and less deep drop of the BEC density than the initial 2D dark soliton. This is due to the fairly intense emission of sound waves which can be noticed in $\left(\mathrm{f}\right)$.

Figure 14

Dependencies of the radiative losses $\mathrm{\Delta }\overline{\mathcal{E}}\left(y_0,{\overline{v}}_0\right)$ on the impact parameter $y_0$, which accompany the scattering by the initially single quantum vortex of 2D dark solitons that start in the positive direction of the $x$ axis from the points with the coordinates $x_0=-25.6,y_0$ and at the time $t=0$ have different normalized velocities ${\overline{v}}_0$, namely, ${\overline{v}}_0=0.3$ (solid red curve connecting unshaded red circles), ${\overline{v}}_0=0.5$ (solid green curve connecting unshaded green squares), and ${\overline{v}}_0=0.7$ (solid blue curve connecting unshaded blue diamonds). For all three values of ${\overline{v}}_0$, the functions $\mathrm{\Delta }\overline{\mathcal{E}}\left(y_0,{\overline{v}}_0\right)$ have local maxima near the boundaries ${y_0}_a\left({\overline{v}}_0\right)$ and ${y_0}_b\left({\overline{v}}_0\right)$ of the impact-parameter interval within which the solitonlike structure has an exchange interaction with the initially isolated topological defect. These nested intervals correspond to the regions shown by different-shade colors whose contrast increases with increasing ${\overline{v}}_0$.

Figure 15

$\left(\mathrm{a}\right)$ Shows the dependencies of the angle $\beta \left(y_0,{\overline{v}}_0\right)$, to which a 2D dark soliton that starts from the points $x_0=-26.6,y_0$ in the positive direction of the $x$ axis with the initial normalized velocity ${\overline{v}}_0$ is scattered, on the impact distance $y_0$. The solid curves connecting the unshaded markers correspond to three different values of the initial normalized velocity ${\overline{v}}_0$ of a soliton: ${\overline{v}}_0=0.3$ (red curve connecting red circles), ${\overline{v}}_0=0.5$ (green curve connecting green squares), and ${\overline{v}}_0=0.7$ (blue curve connecting blue diamonds). The nested intervals ${y_0}_a\left({\overline{v}}_0\right)<y_0<{y_0}_b\left({\overline{v}}_0\right)$ of the impact parameters $y_0$, for which the collisions of solitonlike structures with a solitary quantum vortex are exchange ones, correspond to the region indicated by different-shade colors, whose contrast increases with increasing ${\overline{v}}_0$. $\left(\mathrm{b}\right),\left(\mathrm{c}\right)$ Include the parts illustrating the asymptotic behavior of the function $\beta (y_0,{\overline{v}}_0)$ for large negative and positive $y_0$, respectively. Here, the unshaded markers show the results obtained by direct numerical simulation immediately within the framework of the GP equation (1) and the dashed lines show the law determined by the analytical formula (54) for three different values ${\overline{v}}_0$: ${\overline{v}}_0=0.3$ (red curve $A$), ${\overline{v}}_0=0.5$ (green curve $B$), and ${\overline{v}}_0=0.7$ (blue curve $C$).