Multiple Domain Formation Induced by Modulation Instability in Two-Component Bose-Einstein Condensates

The dynamics of multiple domain formation caused by the modulation instability of two-component Bose-Einstein condensates in an axially symmetric trap are studied by numerically integrating the coupled Gross-Pitaevskii equations. The modulation instability induced by the intercomponent mean-field coupling occurs in the out-of-phase fluctuation of the wave function and leads to the formation of multiple domains that alternate from one domain to another, where the phase of one component jumps across the density dips where the domains of the other exist. This behavior is analogous to a soliton train, which explains the origin of the long lifetime of the spin domains observed by Miesner et al. [Phys. Rev. Lett. 82, 2228 (1999)].

Figure 1

Time development of the condensate densities $|{\psi }_1{|}^2$ (solid curve), $|{\psi }_2{|}^2$ (dashed curve), and total density $n_T=|{\psi }_1{|}^2+|{\psi }_2{|}^2$ (dotted curve) along the $z$ axis at $r=0$ for $N_1=N_2={10}^6$, $a_1=a_{12}=2.75\mathrm{n}\mathrm{m}$, $a_2=2.65\mathrm{n}\mathrm{m}$, and $\lambda =1/70$. Time of each figure is (a) $t=40\mathrm{m}\mathrm{s}\mathrm{e}\mathrm{c}$, (b) $t=80\mathrm{m}\mathrm{s}\mathrm{e}\mathrm{c}$, (c) $t=120\mathrm{m}\mathrm{s}\mathrm{e}\mathrm{c}$, and (d) $t=160\mathrm{m}\mathrm{s}\mathrm{e}\mathrm{c}$. The spatial grid is $[N_r,N_z]=[50\hspace{0.17em}4400]$ with the mesh size ${\Delta }_r={\Delta }_z=0.14\mu \mathrm{m}$. Figure (e) shows the time development of condensate densities at $r=0$ and $z=0$.

Figure 2

The density profile (solid curves) and phase profile (dotted curves) for the multidomain structure of ${\psi }_1$ and ${\psi }_2$ at $t=225\mathrm{m}\mathrm{s}\mathrm{e}\mathrm{c}$ in the simulation of Fig. 1.

Figure 3

The decay of the total energy in the numerical simulation of Eq. (1) with the phenomenological dissipation $\gamma =0.025$ (solid line) and $\gamma =0.05$ (dashed line) starting from the multiple domain state of Fig. 1d. Snapshots of the condensate densities for $\gamma =0.025$ are shown at times (a) – (d). Inset on right shows the densities of the ground state with $E=12.64\hslash {\omega }_{\perp }$.