Bright Matter-Wave Soliton Collisions in a Harmonic Trap: Regular and Chaotic Dynamics

Collisions between bright solitary waves in the 1D Gross-Pitaevskii equation with a harmonic potential, which models a trapped atomic Bose-Einstein condensate, are investigated theoretically. A particle analogy for the solitary waves is formulated and shown to be integrable for a two-particle system. The extension to three particles is shown to support chaotic regimes. Good agreement is found between the particle model and simulations of the full wave dynamics, suggesting that the dynamics can be described in terms of solitons both in regular and chaotic regimes, presenting a paradigm for chaos in wave mechanics.

Figure 1

Trajectories in the particle model (lines) plotted over density distributions predicted by 1D dynamics in the homogeneous GPE. The trajectories correspond to solitons colliding with a relative phase of the golden ratio $\varphi =(1+\sqrt{5})/2$ in (a) and (c), and a relative phase of $\varphi =\pi (1+\sqrt{5})/2$ in (b) and (d). In (a) and (b), the incoming speeds of the solitons are $-0.1$ $|g_{1\mathrm{D}}|N/\hslash$ and 0.2 $|g_{1\mathrm{D}}|N/\hslash$, and in (c) and (d), the incoming speeds are $-0.1$ $|g_{1\mathrm{D}}|N/\hslash$ and 0.3 $|g_{1\mathrm{D}}|N/\hslash$. The solitons have equal effective masses, and other parameters (radial trap frequency of 800 Hz, atomic mass and scattering length of ${}^7\mathrm{Li}$, and 5000 particles per soliton) are taken to agree with recent experiment [8]. The unit of $x$ is then $=3.6\mu \mathrm{m}$, and a unit of $t$ $=1.4\mathrm{ms}$.

Figure 2

Poincaré section of the three-soliton system with $\tilde{H}=10$. Regions corresponding to trajectories in Figs. 3a, 3b, 3c are labeled and highlighted using darker points. The section corresponds to the momentum $p_r$ and position $q_r$ of the “asymmetric stretch” mode when the stretch mode variables $q_c=0$, $p_c<0$. The figure corresponds to the regime where the solitons have equal effective masses, the axial trapping frequency is 10 Hz, and the other parameters (radial trap frequency of 800 Hz, mass and scattering length of ${}^7\mathrm{Li}$, and 5000 particles per soliton) correspond to recent experiment [8].

Figure 3

Trajectories in the particle model (lines) plotted over density distributions predicted by 1D GPE dynamics, corresponding to (a) a regular orbit, (b) a chaotic orbit, and (c) a bound state, as shown in Fig. 2. In the center of the bound state, the density increases to $\sim 0.5$ normalized units, but our scale is pinned at 0.14 units in order to resolve the low density regions better. Here $\tilde{H}=10$, the solitons have equal effective masses, the axial trapping frequency is 10 Hz, and other parameters (radial trap frequency of 800 Hz, atomic mass and scattering length of ${}^7\mathrm{Li}$, and 5000 particles per soliton) correspond to recent experiment [8]. The unit of $x$ is then $=2.4\mu \mathrm{m}$, and a unit of $t$ $=0.6\mathrm{ms}$.