Localization of Two-Component Bose-Einstein Condensates in Optical Lattices

We study nonlinear localization of a two-component Bose-Einstein condensate (BEC) in a one-dimensional optical lattice. Our theory shows that spin-dependent optical lattices can be used to effectively manipulate the nonlinear interactions between the BEC components, and to observe composite localized states of a BEC in both bands and gaps of the matter-wave spectrum.

Figure 1

Schematics of the chemical potential vs momentum of the BEC components, relative to the lattice band-gap structure, necessary to achieve the (a) attractive-attractive, (b) repulsive-repulsive, and (c) repulsive-attractive effective nonlinear interactions. In (b) and (c), for a given location of the $|a⟩$ component either of the $|b⟩$ states can be chosen.

Figure 2

Existence domain for the bright-bright atomic gap solitons in the $\{{\mu }_a,{\mu }_b\}$ plane ($V_0=4$). Shaded: two lowest Bloch bands $B{1}_{a,b}$ and $B{2}_{a,b}$ exhibited by the condensate components $|a⟩$ and $|b⟩$. Solid: existence line ${\mu }_a={\mu }_b$ predicted by the envelope theory and effective potential calculations for $\theta =0$. Dashed: borders of the existence domain for $\theta =\pi /4$. Top panels: condensate wave functions at the marked points, found by solving time-independent Eqs. (1). Lattice potentials are shown by dotted/green lines.

Figure 3

Existence domains for the dark-bright atomic solitons ($V_0=4$). Shaded: two lowest Bloch bands $B{1}_{a,b}$ and $B{2}_{a,b}$ exhibited by the condensate components $|a⟩$ and $|b⟩$. Solid: borders of the existence domains in repulsive-repulsive regime for $\theta =0$. Dotted: borders of the existence domains for $\theta =\pi /4$. Top (bottom) panels: condensate wave functions in the repulsive-repulsive (attractive-repulsive) regime at the marked points, found by solving the time-independent Eqs. (1).

Figure 4

Temporal evolution of a dark-bright localized state in the repulsive-repulsive regime obtained by solving Eqs. (1) with both components initially within (a),(b) the second band, at ${\mu }_a=4.2$, ${\mu }_b=4.18$, and (c),(d) the first band at ${\mu }_a=1.62$, ${\mu }_b=1.6$. Shown are (a),(c) initial spatial density profiles and (b),(d) profiles of the coupled state and (bold) a spreading bright component decoupled from the dark one at (b) $t=1.35\mathrm{ms}$ and (d) $t=2.7\mathrm{ms}$. Note the slower expansion for the lower density state (c).