Collisions of Solitons and Vortex Rings in Cylindrical Bose-Einstein Condensates

Interactions of solitary waves in a cylindrically confined Bose-Einstein condensate are investigated by simulating their head-on collisions. Slow vortex rings and fast solitons are found to collide elastically contrary to the situation in the three-dimensional homogeneous Bose gas. Strongly inelastic collisions are absent for low density condensates but occur at higher densities for intermediate velocities. The scattering behavior is rationalized by use of dispersion diagrams. During inelastic collisions, spherical shell-like structures of low density are formed and they eventually decay into depletion droplets with solitary-wave features. The relation to similar shells observed in a recent experiment by Ginsberg et al. [Phys. Rev. Lett. 94, 040403 (2005)] is discussed.

Figure 1

The energy fraction $\Delta E$, defined in (4), which is radiated during collision of solitons and vortex rings of type I and II (opposite direction) as a function of their initial velocity, for three values of the coupling constant $\gamma =n_{1}a$. The velocity is normalized to the speed of sound $v_s=0.95,1.29,1.61$ (in units of $\hslash /(m{l}_{\rho })$) for $\gamma =1,3,7$, respectively. The symbols correspond to the simulations performed.

Figure 2

The excitation energy $E$ in units of $n_1{\hslash }^2/(m{l}_{\rho })$ versus impulse $Q$ in units of $n_1\hslash$ for axially symmetric solitary waves [7]. At $\gamma =1$ in panel (a) there is one family of solitons. Panel (b) for $\gamma =7$ shows two fundamental branches (solid lines) of vortex rings related by (3). The vortex ring radius has a minimum at the cusps, where $|v|=0.3v_s$. The impulse transfer during bounce-off collisions is indicated by the dashed arrow connecting initial solitary waves (filled symbols; see text) with collision products (open symbols). The vertical dotted lines indicate the energy loss.

Figure 3

Plots of the particle density $|\Psi {|}^2$ for solitary-wave collisions on a scale from white at zero to black at maximum density for each plot. Each row shows five snapshots of a solitary-wave collision with values of $\gamma =1$ (a), (b), and $\gamma =3$ (c), and velocities $v$ as indicated. The time for each snapshot is shown in the figure. The spatial dimensions of the plots are $8\times 12$ for (a), (b) and $8\times 16$ for (c). The initial separation (at time $t=0$) of the solitary waves is $d=12$ in units of $l_{\rho }$ for all cases.

Figure 4

Particle density $|\Psi {|}^2$ for solitary-wave collisions for $\gamma =7$. The frames are $8\times 12$.