Exact dynamics of multicomponent Bose-Einstein condensates in optical lattices in one, two, and three dimensions

Numerous exact solutions to the nonlinear mean-field equations of motion are constructed for multicomponent Bose-Einstein condensates on one-, two-, and three-dimensional optical lattices. We find both stationary and nonstationary solutions, which are given in closed form. Among these solutions are a vortex-antivortex array on the square optical lattice and modes in which two or more components slosh back and forth between neighboring potential wells. We obtain a variety of solutions for multicomponent condensates on the simple cubic lattice, including a solution in which one condensate is at rest and the other flows in a complex three-dimensional array of intersecting vortex lines. A number of physically important solutions are stable for a range of parameter values, as we show by direct numerical integration of the equations of motion.

Figure 1

(Color online) Solutions on a square optical lattice: (a) Grayscale plot of the square optical lattice potential $V\left(x,y\right)$. Regions of high (low) potential are shown in white (black). (b) The current density for the one-component solution (74) with the upper sign. The direction (size) of the arrows indicates the direction (magnitude) of the current flow. This solution is a vortex-antivortex array. (c)–(f) Grayscale plots of the density of the first component $n_1\left(x,y\right)$ for the two-component solution to the Manakov case with $a_0=a_1$, $\rho =1$, and $\theta =3\pi /4$. The plots are for times $t=T/8$ [panel (c)], $t=3T/8$ [panel (d)], $t=5T/8$ [panel (e)], and $t=7T/8$ [panel (f)]. Regions of high (low) $n_1$ are shown in white (black). Each plot shows the region with $-\lambda \le x\le \lambda$ and $-\lambda \le y\le \lambda$.

Figure 2

(Color online) Division of the lattice of potential minima into four square sublattices. The lattice of potential minima are the vertices of the grid. The vertices in sublattices 1, 2, 3, and 4 are indicated by circles, squares, diamonds, and stars, respectively. The origin is at the center of the figure.

Figure 3

(Color online) Two-component condensate on a rectangular optical lattice with ${\lambda }_1=2{\lambda }_2$: (a) Grayscale plot of the 2D optical potential $V\left(x,y\right)$. Regions of high (low) potential are shown in white (black). (b) and (c) Density of the first component $n_1\left(x,y,t\right)$ at times $t=0$ and $T/2$, respectively; regions of high (low) $n_1$ are shown in white (black). (d) Current density of the first condensate component at time $t=T/4$, illustrating the flow from sublattice $A$ to sublattice $B$. The direction (size) of the arrows indicates the direction (magnitude) of the current flow. Each of the four plots shows the region with $-{\lambda }_1\le x\le {\lambda }_1$ and $-{\lambda }_2\le y\le {\lambda }_2$.

Figure 4

(Color online) Sublattices of the simple cubic lattice: The vertices of the grid are sites of the lattice of potential minima. The gray sites belong to the sc sublattice $0+$, while the white sites belong to the sc sublattice $0-$. The gray and white sites together make up the bcc sublattice 0. Sites of all eight sc sublattices are labeled at the corners of the cube in which $x$, $y$, and $z$ all lie between 0 and $\lambda /2$. The sites of sc sublattices $1+$, $1-$, $2+$, $2-$, $3+$, and $3-$ are colored red, cyan, green, magenta, blue, and yellow, respectively.

Figure 5

(Color online) Instability-onset times $t_i$ for a one-component PC solution on a square optical lattice. The dimensionless potential-strength parameter $\mathcal{V}/E_R$ and the corresponding particle density are shown on the horizontal axes. The solution is stable at both low and high potential strengths.

Figure 6

(Color online) Regions of stability and instability for a PC solution having two components on a one-dimensional optical lattice. The points denote calculated boundaries between stable and unstable behavior, the lines connecting them serving as guides to the eye. The inaccessible region has no PC solutions of the form (119).

Figure 7

(Color online) Instability-onset times for a two-component PC solution on a square optical lattice. The dimensionless potential-strength parameter $\mathcal{V}/E_R$ and corresponding particle density are shown on the horizontal axes. The solution is stable in three regions, having low, high, and intermediate potential strengths.

Figure 8

(Color online) Regions of stability and instability for the PC solution, Eq. (121), having two components on a square optical lattice. The points denote calculated boundaries between stable and unstable behavior, the lines connecting them serving as guides to the eye. The inaccessible region has no PC solutions of the form (121). The fine line parallel to the boundary of the inaccessible region is the track followed by the graph of instability-onset times in Fig. 7.

Figure 9

(Color online) Regions of stability and stability for a PC solution having three components on a one-dimensional optical lattice. The points denote calculated boundaries between stable and unstable behavior, the lines connecting them serving as guides to the eye. The inaccessible region has no PC solutions of the form (124).

Figure 10

(Color online) Instability-onset times for the two-component solution on a square lattice considered in Sec. 5C3. The solid curve is that of Fig. 7, calculated with a grid of 32 points in each direction. Crosses and circles denote the same solution with grids of 16 and 64 points, respectively.