Bright solitary-matter-wave collisions in a harmonic trap: Regimes of solitonlike behavior

Systems of solitary waves in the one-dimensional Gross-Pitaevskii equation, which models a trapped atomic Bose-Einstein condensate, are investigated theoretically. To analyze the soliton nature of these solitary waves, a particle analogy for the solitary waves is formulated. Exact soliton solutions exist in the absence of an external trapping potential, which behave in a particlelike manner, and we find the particle analogy we employ to be a good model also when a harmonic trapping potential is present up to a gradual shift in the trajectories when the harmonic trap period is short compared with the collision time of the solitons. We find that the collision time of the solitons is dependent on the relative phase of the solitons as they collide. In the case of two solitons, the particle model is integrable, and the dynamics are completely regular. In the case of a system of two solitary waves of equal norm, the solitons are shown to retain their phase difference for repeated collisions. This phase preservation can be used to find regimes where there is agreement between the wave and particle models. This also implies that soliton regimes may be found in three-dimensional geometries where solitary waves can be made to repeatedly collide out of phase, stabilizing the condensate against collapse. The extension to three particles supports both regular and chaotic regimes. The trajectory shift observed for two solitons carries over to the case of three solitons. This shift aside, the agreement between the particle model and the wave dynamics remains good, even in chaotic regimes.

Figure 1

(Color online) Ground-state solution for harmonically trapped soliton of 5000 particles (solid line). Corresponding soliton solution to the homogeneous equation (dotted-dashed line), which is used as an ansatz in the particle model. The ground state of the linear Schrödinger equation (dashed line) is given for comparison. $x$ is measured in units of ${\hslash }^2/m\left|g_{1\text{D}}\right|N$. The parameters of the system are taken to be similar to those of a recent experiment [13]. The axial trapping frequency is 1.59 Hz, the radial trap frequency is 127.32 Hz, atomic mass and scattering length of ${}^7\text{L}\text{i}$. A unit of $x$ is hence equal to $7.19\times {10}^{-6}\text{m}$.

Figure 2

(Color online) Two-soliton collision taking place when (a) $\Delta {\varphi }_{\text{col}}=0$, (b) $\Delta {\varphi }_{\text{col}}=\pi /2$, (c) $\Delta {\varphi }_{\text{col}}=\pi$, and (d) $\Delta {\varphi }_{\text{col}}=3\pi /2$. $x$ is measured in units of ${\hslash }^2/m\left|g_{1\text{D}}\right|N$ and $t$ in units of ${\hslash }^3/m{\left|g_{1\text{D}}\right|}^2{N}^2$. The parameters of the system are taken to be similar to those of a recent experiment [13]. The axial trapping frequency is 1.59 Hz, the radial trap frequency is 127.32 Hz, atomic mass and scattering length of ${}^7\text{L}\text{i}$, and 5000 particles per soliton. The unit of $x$ is then equal to $3.6\mu \text{m}$, and a unit of $t$ is equal to 1.4 ms.

Figure 3

(Color online) Poincaré sections for the two-soliton system corresponding to the momentum $p_1$ and position $q_1$ of one soliton, while the other soliton has coordinates $q_2=0$, $p_2<0$. $q_1$ and $q_2$ are measured in units of ${\hslash }^2/m\left|g_{1\text{D}}\right|N$, and $p_1$ and $p_2$ in units of $\left|g_{1\text{D}}\right|N/\hslash \eta$. The value of center-of-mass energy $E$ is given by the color scale. (a) Total energy $H=5\times {10}^{-4}$; (b) $H\approx 5.6\times {10}^{-3}$, the star corresponds to the trajectory in Fig. 4; (c) $H\approx 8.1\times {10}^{-3}$, the upper trajectory corresponds to that in Fig. 5, the lower to that in Fig. 6; (d) $H\approx 2.2\times {10}^{-2}$, the star corresponds to the trajectory in Fig. 7. The figures correspond to regimes where the solitons have equal effective masses, the axial trapping frequency is 1.59 Hz, and the other parameters (radial trap frequency of 127.32 Hz, atomic species mass and scattering length of ${}^7\text{L}\text{i}$, and 5000 particles per soliton) are comparable to those in a recent experiment [13].

Figure 4

(Color online) Trajectories in the particle model (lines) plotted over density distributions predicted by 1D GPE dynamics, corresponding to the trajectory marked in Fig. 3b. The relative phase of the solitons in the wave dynamics is zero in (a), and $\pi$ in (b). $x$ is measured in units of ${\hslash }^2/m\left|g_{1\text{D}}\right|N$ and $t$ in units of ${\hslash }^3/m{\left|g_{1\text{D}}\right|}^2{N}^2$. The figures correspond to regimes where the solitons have equal effective masses, the axial trapping frequency is 1.59 Hz, and the other parameters (radial trap frequency of 127.32 Hz, atomic species mass and scattering length of ${}^7\text{L}\text{i}$, and 5000 particles per soliton) are comparable to those in a recent experiment [13]. The unit of $x$ is then equal to $3.6\mu \text{m}$, and a unit of $t$ is equal to 1.4 ms.

Figure 5

(Color online) Trajectories in the particle model (lines) plotted over density distributions predicted by 1D GPE dynamics. The trajectories correspond to those given in Fig. 4, but with additional center-of-mass displacements. The trajectories also correspond to the upper trajectory marked in Fig. 3c. The relative phase of the solitons in the wave dynamics is zero in (a), and $\pi$ in (b). $x$ is measured in units of ${\hslash }^2/m\left|g_{1\text{D}}\right|N$ and $t$ in units of ${\hslash }^3/m{\left|g_{1\text{D}}\right|}^2{N}^2$. The solitons have equal effective masses, the axial trapping frequency is 1.59 Hz, and the other parameters (radial trap frequency of 127.32 Hz, atomic species mass and scattering length of ${}^7\text{L}\text{i}$, and 5000 particles per soliton) are comparable to those in a recent experiment [13]. The unit of $x$ is then equal to $3.6\mu \text{m}$, and a unit of $t$ is equal to 1.4 ms.

Figure 6

(Color online) Trajectories in the particle model (lines) plotted over density distributions predicted by 1D GPE dynamics, corresponding to the trajectory marked in Fig. 3d. The relative phase of the solitons in the wave dynamics is zero in (a), and $\pi$ in (b). $x$ is measured in units of ${\hslash }^2/m\left|g_{1\text{D}}\right|N$ and $t$ in units of ${\hslash }^3/m{\left|g_{1\text{D}}\right|}^2{N}^2$. The solitons have equal effective masses, the axial trapping frequency is 1.59 Hz, and the other parameters (radial trap frequency of 127.32 Hz, atomic species mass and scattering length of ${}^7\text{L}\text{i}$, and 5000 particles per soliton) are comparable to those in a recent experiment [13]. The unit of $x$ is then equal to $3.6\mu \text{m}$, and a unit of $t$ is equal to 1.4 ms.

Figure 7

(Color online) Trajectories in the particle model (lines) plotted over density distributions predicted by 1D GPE dynamics, corresponding to the trajectory marked in Fig. 3d. The relative phase of the solitons in the wave dynamics is zero in (a), and $\pi$ in (b). $x$ is measured in units of ${\hslash }^2/m\left|g_{1\text{D}}\right|N$ and $t$ in units of ${\hslash }^3/m{\left|g_{1\text{D}}\right|}^2{N}^2$. The solitons have equal effective masses, the axial trapping frequency is 1.59 Hz, and the other parameters (radial trap frequency of 127.32 Hz, atomic species mass and scattering length of ${}^7\text{L}\text{i}$, and 5000 particles per soliton) are comparable to those in a recent experiment [13]. The unit of $x$ is then equal to $3.6\mu \text{m}$, and a unit of $t$ is equal to 1.4 ms.

Figure 8

(Color online) (a) Difference between soliton peak trajectory from wave simulation and particle model. The trajectories are those of the left-hand soliton in Fig. 2 going into and reemerging from a collision in the case of $\Delta \varphi =\pi$ (dark line) and $\Delta \varphi =0$ (light line). Zero is indicated by the dotted line. The difference $\Delta q$ is equal to the particle trajectory $q_1$ minus the position of the peak of the left-hand soliton, or equivalently the position of the peak of the right-hand soliton minus the particle trajectory $q_2$. Position is measured in units of ${\hslash }^2/m\left|g_{1\text{D}}\right|N$ and $t$ in units of ${\hslash }^3/m{\left|g_{1\text{D}}\right|}^2{N}^2$. (b) Difference between the curves in (a). Note that the separation between the trajectories in the particle model and GPE dynamics is always larger in the $\pi$-phase case.

Figure 9

(Color online) Poincaré section of the three-soliton system with (a) $\tilde{H}=60$; (b) $\tilde{H}=10$, regions corresponding to trajectories in Figs. 2a, 2b, 2c are labeled, and highlighted using larger, darker points; (c) Poincaré section of the system with $\tilde{H}=2$; (d) $\tilde{H}=-2$; (e) $\tilde{H}=-5$; (f) $\tilde{H}=-10$ The section corresponds to the momentum $p_r$ and position $q_r$ of the “asymmetric stretch” mode when the “stretch” mode coordinates $q_c=0$, $p_c<0$. $q_c$ and $q_r$ are measured in units of $\eta {\hslash }^2/m\left|g_{1\text{D}}\right|N$, and $p_c$ and $p_r$ are measured in units of $|g_{1\text{D}}N/\hslash \eta \phantom{|}$. The figures correspond to the regime where the solitons have equal effective masses, the axial trapping frequency is 1.59 Hz, and the other parameters (radial trap frequency of 127.32 Hz, atomic species mass and scattering length of ${}^7\text{L}\text{i}$, and 5000 particles per soliton) correspond to a recent experiment [13].

Figure 10

(Color online) Trajectories in the particle model (lines) plotted over density distributions predicted by 1D GPE dynamics, corresponding to a regular orbit. The parameters of the system are $\tilde{H}=10$, the solitons have equal effective masses, the axial trapping frequency is 1.59 Hz, and other parameters (radial trap frequency of 127.32 Hz, atomic mass and scattering length of ${}^7\text{L}\text{i}$, and 5000 particles per soliton) correspond to the recent experiment [13]. The unit of $x$ is then equal to $2.4\mu \text{m}$, and a unit of $t$ is equal to 0.6 ms.