Bright-dark soliton complexes in spinor Bose-Einstein condensates

We consider vector solitons of mixed bright-dark types in quasi-one-dimensional spinor $\left(F=1\right)$ Bose-Einstein condensates. Using a multiscale expansion technique, we reduce the corresponding nonintegrable system of three coupled Gross-Pitaevskii equations (GPEs) to an integrable Yajima-Oikawa system. In this way, we obtain approximate solutions for small-amplitude vector solitons of dark-dark-bright and bright-bright-dark types, in terms of the $m_F=+1,-1,0$ spinor components, respectively. By means of numerical simulations of the full GPE system, we demonstrate that these states indeed feature soliton properties, i.e., they propagate undistorted and undergo quasielastic collisions. It is also shown that in the presence of a parabolic trap the bright component(s) is (are) guided by the dark one(s) and, as a result, the small-amplitude vector soliton as a whole performs quasiharmonic oscillations. The oscillation frequency is found as a function of the spin-dependent interaction strength for both small-amplitude and large-amplitude solitons.

Figure 1

(Color online) The two top panels show contour plots of the densities of the ${\psi }_{\pm 1}$ (left panel) and ${\psi }_0$ (right panel) components of the spinor condensate with $\delta =0.0314$ in the homogeneous system $\left({\Omega }_{\text{tr}}=0\right)$. The ${\psi }_{\pm 1}$ components contain a pair of dark pulses, initially placed at $x_0=\pm 15$, that, together with the bright components in the ${\psi }_0$ field coupled to them, undergo a quasielastic collision at $t\approx 19$, and propagate unscathed afterward. The parameters are $\mu =2$, $\xi =1.54$, $\eta =3.091$, and $\nu =1.2$. The four bottom panels show snapshots of the densities observed at $t=0,10$ (before the collision), $t=19$ (when the collision occurs), and $t=40$ (after the collision).

Figure 2

(Color online) The two top panels show contour plots of the densities of the ${\psi }_{\pm 1}$ (left) and ${\psi }_0$ (right) fields confined in the harmonic trap with ${\Omega }_{\text{tr}}=0.05$ (other parameters are the same as in Fig. 1). Initially, each of the Thomas-Fermi profiles of the ${\psi }_{\pm 1}$ components carries a dark soliton, while the ${\psi }_0$ component is a bright soliton (the initial position is at the trap’s center $x=0$). The four bottom panels show snapshots of the densities observed at $t=0$, 444, 937, and 987.

Figure 3

(Color online) Top panel: The spatial dependence of the soliton velocity $\tilde{v}$ for $\delta =0.0314$. Dots correspond to results produced by the numerical simulations, while the (red) solid line represents the best fit, which is found to be $0.028\sqrt{2n_0\left(X\right)}=0.028\sqrt{\mu -\left(1/2\right){\Omega }_{\text{tr}}^2{\tilde{X}}^2}$ (the values of the chemical potential and trap’s strength are $\mu =2$ and ${\Omega }_{\text{tr}}=0.05$, as before). Bottom panel: Contour plots of the effective density of the shallow dark soliton ${\left|{\psi }_{\pm 1}\right|}^2-2n_0\left(X\right)$ for $\delta =0.0314$, with initial amplitude $\nu =0.13$; the other parameters are $\eta =1$, $\xi =0.5$, $\mu =2$, and ${\Omega }_{\text{tr}}=0.05$. The soliton performs oscillations at frequency ${\omega }_{\text{osc}}\approx 0.03433$, which is almost identical to the analytical prediction ${\Omega }_{\text{osc}}=0.0344$ (the error is $\approx 0.24\%$).

Figure 4

(Color online) The oscillation frequency of the small-amplitude spinor DDB soliton (with $\nu /\mu =0.15$) normalized to the characteristic value ${\Omega }_{\text{tr}}/\sqrt{2}$ as a function of $\delta$. The thick (red) solid line corresponds to the semianalytical prediction given by Eq. (46), while the piecewise linear line (black) with dots represents results obtained by the direct numerical integration of the Gross-Pitaevskii Eqs. (4, 5). The vertical dotted line indicates the value of $\delta =0.0314$ corresponding to the ${}^{23}\text{N}\text{a}$ spinor BEC.

Figure 5

(Color online) The relative deviation of the numerically found soliton oscillation frequency from the value predicted by Eq. (46), as a function of the relative dark-soliton depth $\nu /\mu$ for $\delta =0.0314$. The region of $\nu /\mu ⪡1$ corresponds to shallow solitons, while $\nu /\mu =1$ corresponds to a black soliton. It is seen that the analytical prediction is fairly good for shallow solitons (for $\nu /\mu <0.4$, the error is below 5%), but becomes worse for the solitons with moderate and large amplitudes (for $\nu /\mu >0.8$, the error becomes $\sim 10\%$).

Figure 6

(Color online) The oscillation frequency of the large-amplitude spinor DDB (dark-dark-bright) soliton, with $\nu /\mu =0.95$, normalized to the value ${\Omega }_{\text{osc}}$ given in Eq. (47), as a function of $\delta$. The piecewise linear line (black) with dots represents the results obtained by the direct numerical integration of Eqs. (4, 5), while the thick (red) solid line depicts the best fit based on Eq. (49). The vertical dashed line indicates the value of $\delta =0.0314$ corresponding to the spinor BEC in ${}^{23}\text{N}\text{a}$.

Figure 7

(Color online) Same as Fig. 2, but for $\delta =0.2$. The soliton parameters are $\xi =0.61$, $\eta =1.22$, and $\nu =1.2$, while the trap strength and chemical potential are ${\Omega }_{\text{tr}}=0.05$ and $\mu =2$. This choice makes the initial soliton densities identical to those shown in the second-row left panel of Fig. 2. The two bottom panels are snapshots of the densities at $t=311$ and 967.