Domain walls and bubble droplets in immiscible binary Bose gases

The existence and stability of domain walls (DWs) and bubble-droplet (BD) states in binary mixtures of quasi-one-dimensional ultracold Bose gases with inter- and intraspecies repulsive interactions is considered. Previously, DWs were studied by means of coupled systems of Gross-Pitaevskii equations (GPEs) with cubic terms, which model immiscible binary Bose-Einstein condensates (BECs). We address immiscible BECs with two- and three-body repulsive interactions, as well as binary Tonks–Girardeau (TG) gases, using systems of GPEs with cubic and quintic nonlinearities for the binary BEC, and coupled nonlinear Schrödinger equations with quintic terms for the TG gases. Exact DW solutions are found for the symmetric BEC mixture, with equal intraspecies scattering lengths. Stable asymmetric DWs in the BEC mixtures with dissimilar interactions in the two components, as well as of symmetric and asymmetric DWs in the binary TG gas, are found by means of numerical and approximate analytical methods. In the BEC system, DWs can be easily put in motion by phase imprinting. Combining a DW and anti-DW on a ring, we construct BD states for both the BEC and TG models. These consist of a dark soliton in one component (the “bubble”), and a bright soliton (the “droplet”) in the other. In the BEC system, these composite states are mobile, too.

Figure 1

The existence region (shaded) for families of stationary exact domain-wall solutions given by Eqs. (4) and (7) under condition (6), with the quintic coefficient, $\alpha$, varying in the interval of $[0.01,1]$ . Continuous curves from top to bottom refer to $\alpha =0.01,0.1,0.5,1.0$, respectively.

Figure 2

Typical profiles of stationary domain walls in the binary immiscible BEC generated by Eq. (12) with values of the quintic coefficient $\alpha =0.01,0.1,$ and $0.05$ (purple dashed, blue continuous, and red dotted curves, respectively). The parameters are taken as per Eq. (14), so as to have $A=1$ in Eq. (7). We only show the profiles of the first component, the other one being produced by specular reflection with respect to the vertical axis.

Figure 3

A moving domain wall produced by phase imprinting with initial velocity $v=0.8$ (see the text).

Figure 4

(Left) Typical density profiles of an asymmetric stationary domain wall in the binary immiscible BEC with ${\gamma }_1=1,{\gamma }_2=0.9,{\gamma }_{12}=1.08$ and equal coefficients of the quintic nonlinearity, ${\alpha }_1={\alpha }_2=0.1$. The short horizontal line with the vertical arrow shows the mismatch between the left and right background levels. (Right) The same as in the left panel, but for ${\gamma }_1=1,{\gamma }_2=0,{\alpha }_1=0,{\alpha }_2=0.05$ in Eq. (12), and different strengths of the interspecies cubic interaction: ${\gamma }_{12}=0.5$ (continuous curves), ${\gamma }_{12}=1.1$ (dash-dotted curves), and asymptotic (left and right) densities $A_1=0.916,A_2=2.325$. In both panels, the blue and red lines depict profiles of the first and second components, respectively.

Figure 5

The DW-anti-DW state in Eq. (18) of the repulsive binary BEC with $\alpha =0.1,A=1,\lambda =2\sqrt{2\alpha /3}$ and separation $x_0=30$ (blue lines) or $x_0=8$ (red lines). Thin and thick lines refer to the first and second components, respectively. Filled blue and red regions depict, severally, the bubble and droplet parts of the patterns with $x_0=30$ and $x_0=8$ (they are shown for different values of $x_0$ to avoid confusing overlap between them).

Figure 6

A moving bubble-droplet solution obtained by the phase imprinting, with initial velocity $v=1.2$ velocity, onto the stationary profiles shown in Fig. 5 for $x_0=8$.

Figure 7

(Top panels) The evolution of densities of the first and second (left and right columns) components of the BD state for the same parameters as in the right panel of Fig. 4, except for ${\gamma }_{12}=0.4$, which is taken above the miscibility threshold. (Bottom panels) The same as above, but for ${\gamma }_{12}=0.3$, taken just below the miscibility threshold.

Figure 8

(Top) Density profiles of a bubble-drop state, produced by Eq. (22) as a DW-anti-DW complex, in the binary TG gas with equal masses and fully symmetric interactions [see Eq. (21)], i.e., exactly at the miscibility threshold. (Bottom) The stability test by real-time simulations of the configuration depicted in the top panel. The density profiles remain stationary up to $t\simeq 400$. Subsequently, they slowly decay into the uniformly mixed background with oscillations.

Figure 9

DW density profiles produced by Eq. (19) for equal masses $m_1=m_2=1$ and interspecies interaction strength $\chi =2$. Other parameters are fixed as ${\alpha }_1={\alpha }_2=1$ (left panel) and ${\alpha }_1=2,{\alpha }_2=1$ (right panel). Blue and red lines refer to first and second components, respectively, while the black dashed lines denote VA results.