Dynamics of dark-bright vector solitons in Bose-Einstein condensates

We analyze the dynamics of two-component vector solitons, namely, dark-bright solitons, via the variational approximation in Bose-Einstein condensates. The system is described by a vector nonlinear Schrödinger equation appropriate to multicomponent Bose-Einstein condensates. The variational approximation is based on a hyperbolic tangent (hyperbolic secant) for the dark (bright) component, which leads to a system of coupled ordinary differential equations for the evolution of the ansatz parameters. We obtain the oscillation dynamics of two-component dark-bright solitons. Analytical calculations are performed for same-width components in the vector soliton, and numerical calculations extend the results to arbitrary widths. We calculate the binding energy of the system and find it to be proportional to the intercomponent coupling interaction and numerically demonstrate the breakup or unbinding of a dark-bright soliton. Our calculations explore observable eigenmodes, namely, the internal oscillation eigenmode and the Goldstone eigenmode. We find analytically that the number of atoms in the bright component is required to be less than the number of atoms displaced by the dark soliton in the other component in order to find the internal oscillation eigenmode of the vector soliton and support the existence of the dark-bright soliton. This outcome is confirmed by numerical results. Numerically, we find that the oscillation frequency is amplitude independent. For dark-bright solitons in ${}^{87}\mathrm{Rb}$ we find that the oscillation frequency range is 90 to 405 Hz and therefore observable in multicomponent Bose-Einstein condensate experiments.

Figure 1

Oscillation frequency of the two components in the dark-bright soliton versus the interaction coefficients $g$. We set $N_1\approx 0.503\times {10}^5$ atoms and width $w=1$, where $\omega$ and $g$ are unitless. The range of the values in $g$ is from 2.4 to 5.8, matching the range of $g$ in the numerical calculations.

Figure 2

Coupling energy versus the distance between the two components $l\left(t\right)$ when $t=0$. Here we normalize the interaction coefficients to unity and set $N_1\approx 0.503\times {10}^5$ atoms. The solid blue line represents Eq. (19), and the dashed red line represents Eq. (20).

Figure 3

Approach to the Manakov case. Ground-state density of a two-component BEC when the interaction coefficients $g_1,g_2$ are equal to unity versus the coupling interaction coefficient $g$. The bright (dark) component is the dashed blue (solid red) line. In (a)–(h) $g=0.0,0.2,0.4,0.6,0.8,0.95,1.0,1.2$, respectively. We allow the relative particle number between the two components to fluctuate, and past the Manakov point at $g=1$ the lowest-energy solution places all atoms in one component.

Figure 4

Dark-bright solitons through the miscible-immiscible phase transition. We take $g_1=2.0$ and $g_2=2.7$. In (a)–(h) $g=0.0,0.4,0.8,1.2,1.6,2.0,2.4,2.8$, respectively. The phase transition occurs at $g=2.3$, leading to well-localized bright solitons in the immiscible domain in the last two panels.

Figure 5

Amplitude of the bright and dark components versus the coupling interaction $g$. We measure the amplitudes of the two components at the ground state with different values of $g_1$, $g_2$, and $g$. (a) $g_1=1.0,g_2=1.5$ and (b) $g_1=2.0,g_2=2.7$.

Figure 6

Collective velocity of dark-bright soliton after phase imprint. We take a phase imprint of $\varphi =0.5$ and two different cases for $g_1$ and $g_2$ in the immiscible domain when $g>\sqrt{g_1{g}_2}$. See Sec. 3d for converted units. Note that a dark-bright soliton can be created as we get very close to this line from the miscible domain. The amplitude of the bright soliton controls the rate of the velocity of the dark-bright soliton. As we increase the intercomponent coupling interaction $g$, the amplitude of the bright soliton decreases as shown in Fig. 5, and therefore the density of the bright soliton decreases too. Imprinting a phase on the low-density bright soliton will have a small effect on dragging the dark soliton and therefore will result in a low velocity of the dark-bright soliton.

Figure 7

Trends in internal dynamics. Oscillation frequency of the two components in the dark-bright soliton, with $g_1=2.0$, $g_2=2.7$, and $\varphi =0.7$ and 1.0, obtained from numerical integration of Eq. (3) versus the oscillation frequency obtained from the analytical calculations, Eq. (18). Numerically, the oscillation frequency of the two components versus the coupling coefficient $g$ for different values of $\varphi$ shows that the oscillation frequency is amplitude independent in the case explored. We also plot the result from Eq. (18) to compare the two outcomes from the analytical and numerical calculations. The discrepancy between numerics and the model is due to the restricted ansatz (equal soliton widths) in the variational calculation.

Figure 8

Oscillation of the two-component wave function $|u\left(x,t\right){|}^2$ and $|v\left(x,t\right){|}^2$ in the immiscible domain with $g_1=2.0$, $g_2=2.7$, $g=3.2$, and $\varphi =0.7$. (a), (b), (c), and (d) represent the density and the phase of the bright and dark components, respectively. In (e), (f), (g), and (h) we plot the preceding images with small time and space intervals to show the oscillations.

Figure 9

Unbinding of a dark-bright soliton. We demonstrate the breakup of the dark-bright soliton by imprinting different values of the phase $\varphi$ on the bright component with interaction coefficients $g_1=2.0$, $g_2=2.7$, $g=2.6$. (a)–(d), (e)–(h), and (i)–(l) use phase imprintings of $\varphi =2$, 6, and 10, respectively. The left (right) panels show the density (phase) of the bright and dark components. The box dimension is $L=100$.

Figure 10

Percentage density loss of the bright component in the dark-bright soliton for different phase-imprinting values. Below the critical value mentioned above (i.e., $\varphi =0$, 1, and 2) the dark-bright soliton maintains its internal structure, and the bright-soliton component density is almost intact; see the inset. Above the critical value (i.e., $\varphi =6$, 8, and 10) we see that the bright soliton loses density due to the relatively strong kick that allows for a significant portion of the density to escape. The inset also highlights the stability of the dark-bright soliton at long times for small enough phase imprinting.