Rabi-coupled countersuperflow in binary Bose-Einstein condensates

We show theoretically that periodic density patterns are stabilized in two counterpropagating Bose-Einstein condensates of atoms in different hyperfine states under Rabi coupling. In the presence of coupling, the relative velocity between the two components is localized around density depressions in quasi-one-dimensional systems. When the relative velocity is sufficiently small, the periodic pattern reduces to a periodic array of topological solitons as kinks of the relative phase. According to our variational and numerical analyses, the soliton solution is well characterized by the soliton width and density depression. We demonstrate the dependence of the depression and width on the Rabi frequency and the coupling constant of the intercomponent density-density interactions. The periodic pattern of the relative phase transforms continuously from a soliton array to a sinusoidal pattern as the period becomes smaller than the soliton width. These patterns become unstable when the localized relative velocity exceeds a critical value. The stability-phase diagram of this system is evaluated with a stability analysis of countersuperflow, by taking into account the finite-size effect owing to the density depression.

Figure 1

The stability-phase diagram of the bulk state (8) in Rabi-coupled condensates (a) and a uniform countersuperflow (b). The phase boundaries in (a) and (b) represent Eqs. (10) and (19), respectively.

Figure 2

A typical numerical solution (black dashed curve) for the single soliton with $\gamma =0.4$ and $\mathrm{\Omega }/{\mathrm{\Omega }}_0=0.04$. The gray dashed curves show the analytical result obtained from the variational analysis.

Figure 3

The soliton width ${\sigma }_a$ (a) and the density depression $\mathrm{\Delta }$ (b) as functions of $\gamma$ for different $\mathrm{\Omega }$. The numerical and analytical results are displayed with symbols and lines, respectively. The gray solid lines show the stability-phase boundaries of the single-soliton solution, as discussed in Sec. 3b.

Figure 4

Typical development from left to right of densities (a), phases (b), and momentum (c) in the numerical computation of the imaginary-time propagation.

Figure 5

The stability-phase diagram of the single soliton. The gray area surrounded by the black curve represents numerical results for the parameter region where the single soliton is stable. The gray dashed curve shows criterion (37) with $n_{\min }=n_0(1-{\mathrm{\Delta }}_v)$. The black dashed curve shows criterion (37) with numerical values of $n_{\min }$. Our analysis can be used below the gray line [see Eq. (26)].

Figure 6

The $\mathrm{\Omega }$ dependence of $V_R$ for $d/{\sigma }_a=0.25,0.5,1,2.$

Figure 7

Numerical solutions with $\gamma =0.4$ and $\mathrm{\Omega }/{\mathrm{\Omega }}_0=0.04$. The density difference $|n_1-n_2|$ is zero, and the total phase ${\theta }_{+}$ is constant (not shown). The parameters are set as (a) $m\xi V_R/\hslash =2.51\times {10}^{-1}$, where $d/{\sigma }_a=1.13$, and (b) $m\xi V_R/\hslash =9.22\times {10}^{-1}$, where $d/{\sigma }_a=3.07\times {10}^{-1}$.

Figure 8

(a) The stability-phase diagram of the Rabi-coupled countersuperflow. The gray solid curve shows the stability-phase boundary in Fig. 5. The gray dashed curve represents the stability criterion (19) of uniform countersuperflow. (b) A two-dimensional plot of (a). The black dashed curves show the criterion (45) with numerical data for $n_{\min }$. The gray dashed curves show the criterion (46).