Dark-soliton collisions in a toroidal Bose-Einstein condensate

We study the dynamics of two gray solitons in a Bose-Einstein condensate confined by a toroidal trap with a tight confinement in the radial direction. Gross-Pitaevskii simulations show that solitons can be long-living objects passing through many collisional processes. We have observed quite different behaviors depending on the soliton velocity. Very slow solitons, obtained by perturbing the stationary solitonic profile, move with a constant angular velocity until they collide elastically and move in the opposite direction without showing any sign of lowering their energy. In this case the density notches are always well separated and the fronts are sharp and straight. Faster solitons present vortices around the notches, which play a central role during the collisions. We have found that in these processes the solitons lose energy, as the outgoing velocity turns out to be larger than the incoming one. To study the dynamics, we model the gray soliton state with a free parameter that is related to the soliton velocity. We further analyze the energy, soliton velocity, and turning points in terms of such a free parameter, finding that the main features are in accordance with the infinite one-dimensional system.

Figure 1

Black soliton isodensity contour (solid white lines) and phase distribution (colors) are shown in the left panel. The GP order parameter (solid line) and the almost superposed results from Eq. (4) with $f(r=r_0)=1$ and $k=2.785\mu {\mathrm{m}}^{-1}$ (dotted line) are displayed in the right panel.

Figure 2

Density profiles $\rho$ as functions of the radial coordinate for the ground (left panel) and black soliton (right panel) states. Solid lines correspond to the GP density, whereas dashed lines correspond to ${\rho }_{\mathrm{GM}}={\left|{\psi }_{\mathrm{GM}}\right|}^2$, where ${\psi }_{\mathrm{GM}}$ is given by Eq. (6) with $\gamma =34$, for $X=1$ (left panel) and $X=0$ (right panel).

Figure 3

Angular position (left panel) of the soliton located at $x>0$ and the phase difference $\mathrm{\Delta }\varphi$ (right panel) are depicted as functions of the time using an initial PBS profile with the black soliton slightly perturbed with $X=0.01$.

Figure 4

Snapshots before (top panels), during (middle panels), and after (bottom panels) the first soliton collision for the same initial state of Fig. 3. In the left panels we show the phase distribution (colors) and density isocontours (white solid lines), while in the right panels we depict the corresponding angular distribution of particle density (red dashed line) and phase (black solid line) at $r=4\mu \mathrm{m}$.

Figure 5

Angular position (left panel) of the soliton located at $x>0$ and the phase difference $\mathrm{\Delta }\varphi$ (right panel) are depicted as functions of the time using the initial GM profile given by Eq. (6) with $X=0.1$.

Figure 6

Snapshots before (top panels), during (middle panels), and after (bottom panels) the first soliton collision for the same initial state of Fig. 5. In the left panels we show the phase distribution (colors) and density isocontours (white solid lines), while in the right panels we depict the corresponding angular distribution of particle density (red dashed line) and phase (black solid line) at $r=4\mu \mathrm{m}$.

Figure 7

Angular position (left panel) of the soliton located at $x>0$ and the phase difference $\mathrm{\Delta }\varphi$ (right panel) are depicted as functions of the time for an initial GM profile given by Eq. (6) with $X=0.6$.

Figure 8

Snapshots before (top panels), during (middle panels), and after (bottom panels) the first soliton collision for the same initial state of Fig. 7. In the left panels we show the phase distribution (colors) and density isocontours (white solid lines), while in the right panels we depict the corresponding angular distribution of particle density (red dashed line) and phase (black solid line) at $r=4\mu \mathrm{m}$.

Figure 9

Soliton energy per particle as a function of $X$. The solid black line corresponds to Eq. (9); the black squares, blue circles, and red triangles correspond to numerical calculations using PBS, GM, and PG states, respectively. The red dashed line corresponds to an infinite 1D system.

Figure 10

The soliton initial velocity for the PBS (black squares), GM (blue circles), and PG (red triangles) states depicted as a function of $X$. The black solid line has been drawn to guide the eye on the GM results. The linear functions $c_{i}X$ are depicted as a red solid line ($i=0$) and as a dashed blue line ($i=1$), where $c_i$ is an estimate of the sound velocity at either the ground ($i=0$) or the black soliton ($i=1$) states.

Figure 11

The turning points arising from GP simulations using the GM order parameter (blue circles) and the estimate given by Eq. (11) (black solid line) are depicted as a function of $X$.