Interaction of solitons in one-dimensional dipolar Bose-Einstein condensates and formation of soliton molecules

The interaction between two bright solitons in a one-dimensional dipolar Bose-Einstein condensate (BEC) is investigated, with the aim of finding the regimes where they form a stable bound state, known as the soliton molecule. To study soliton interactions in BECs we employed a method similar to that used in experimental investigation of the interaction between solitons in optical fibers. The idea consists in creating two solitons at some spatial separation from each other at initial time $t_0$ and then measuring the distance between them at a later time $t_1>t_0$. Depending on whether the distance between solitons has increased, decreased, or remained unchanged, compared to its initial value at $t_0$, we conclude that the soliton interaction was repulsive, attractive, or neutral, respectively. We propose an experimentally viable method for estimating the binding energy of a soliton molecule, based on its dissociation at critical soliton velocity. Our theoretical analysis is based on the variational approach, which appears to be quite accurate in describing the properties of soliton molecules in dipolar BECs, as reflected in the good agreement between the analytical and the numerical results.

Figure 1

Left: A molecular-type potential associated with VA, Eq. (10), for different strengths of the contact interaction. Right: The shape of a two-soliton molecule in a pure dipolar BEC with $N=2, q=0, g=20$ as predicted by VA with the trial function, (6), for $A=3.538, a=0.553, N=1.876$ [solid (red) line], predicted by two antiphase Gaussian functions with parameters given in Eq. (7) [dashed (blue) line], and found from the numerical optimization procedure, applied to the GPE, (1) [dash-dotted (brown) line]. The minimum of the effective potential $U\left(a\right)$ is attained at $a_0=0.53$, and the equilibrium distance between center-of-mass positions of pulses predicted by Eq. (7) is ${\mathrm{\Delta }}_0=4a_0/\sqrt{\pi }\simeq 1.2$, while the GPE optimization gives ${\mathrm{\Delta }}_0\simeq 1.3$.

Figure 2

(a) Stable propagation of a two-soliton molecule composed of two antiphase ($\phi =\pi$) Gaussian pulses with parameters $A_0=1.196, a_0=0.394$, placed at their equilibrium positions $x_0=\pm 0.657$. (b) When the solitons are slightly displaced (by 20%) from equilibrium positions, they perform oscillations. The density plot ${\left|\psi \right|}^2$ is obtained by numerical solution of the GPE, (1). The dashed line corresponds to calculations according to the VA equation, (10), which shows increasing phase shift with respect to GPE. (c) Periodic exchange of atoms between two solitons when the initial phase difference is slightly decreased.

Figure 3

Dynamics of the centers of masses of two solitons, forming the molecule, when the two solitons are set in motion at different velocities in opposite directions. When the velocity is less than critical $v<v_{\mathrm{cr}}=0.91$, solitons perform oscillations near equilibrium positions. At critical velocity the molecule dissociates into freely moving individual solitons [solid (red) lines].

Figure 4

Left: The potential of interaction between two solitons, retrieved from numerical GPE simulation, for two values of the molecule's norm. Similarity to the VA-predicted potential $U\left(a\right)$ in Fig. 1 is evident. Right: The stationary half-separation between solitons of the molecule $x_0$ [dashed (blue) line] and its binding energy $E_b$ [solid (red) line] as a function of the molecule's norm $N$. Symbols correspond to values found from numerical simulations of the GPE, (1), and lines are interpolating curves for visual convenience.

Figure 5

Left: The character of soliton interactions for antiphase and inphase solitons. Antiphase solitons attract each other at large separations and repel at small separations [solid (red) line]. There exists a stationary separation, shown by the filled (red) circle, where attraction changes to repulsion. In-phase solitons always attract and collide [dashed (blue) line]. Middle: The initial separation between two solitons is varied (vertical axis) and the density profiles ${\left|\psi \right|}^2$ at the final time ($t=t_1$) are measured as a function of $x$. Two antiphase solitons can form a stable soliton molecule at the appropriate initial separation ${\mathrm{\Delta }}_0\simeq 1.3$. Right: Similar to the middle panel, but for in-phase solitons. Two in-phase solitons always collide and do not form a stable bound state.