Dispersion control for matter waves and gap solitons in optical superlattices

We present a numerical study of dispersion manipulation and formation of matter-wave gap solitons in a Bose-Einstein condensate trapped in an optical superlattice. We demonstrate a method for controlled generation of matter-wave gap solitons in a stationary lattice by using an interference pattern of two condensate wave packets, which mimics the structure of the gap soliton near the edge of a spectral band. The efficiency of this method is compared to that of gap soliton generation in a moving lattice recently demonstrated experimentally by Eiermann et al. [Phys. Rev. Lett., 92, 230401 (2004)]. We show that, by changing the relative depths of the superlattice wells, one can fine-tune the effective dispersion of the matter waves at the edges of the minigaps of the superlattice Bloch-wave spectrum and, therefore, effectively control both the peak density and the spatial width of the emerging gap solitons.

Figure 1

The structure of the superlattice potential described by Eq. (2) for different values of $\epsilon$ and a fixed potential height $V_0=1$: single-periodic optical lattice for (a) $\epsilon =0$ and (d) $\epsilon =1$, and double-periodic superlattices for (b) $\epsilon =0.3$ and (c) $\epsilon =0.7$.

Figure 2

(a) The band-gap spectra for a single-periodic optical lattice $(\epsilon =0)$ and a double-periodic superlattice with weak modulation $(\epsilon =0.05)$. In both cases, the lattice height is set to $V_0=5$. The first minigap is shaded. (b) The spectrum around the first minigap for a strongly modulated superlattice at $V_0=1$ and $\epsilon =0.3$.

Figure 3

The effective group-velocity dispersion $D$ (a) at the edges of the first gap of a single-periodic optical lattice of varying strength and (b) at the edges of the first minigap for a superlattice of height $V_0=1$ and period $d=\pi$, as the shape of the superlattice is changed by varying $\epsilon$.

Figure 4

Soliton generation by (a) a single Gaussian wave packet accelerated to the edge of the first minigap $(k=1)$ in a moving superlattice and (d) interference of two counterpropagating Gaussian wave packets in a stationary superlattice. Both superlattices have $\epsilon =0.3$ and $V_0=1$. (a) is plotted in the lattice frame moving at $v_{\mathrm{rec}}$. Initially the wave packet contains 500 atoms and has a width parameter of $w=100$. In (d) the initial parameters are $w=81.6$ and $N=500$. (b),(c) and (e),(f) capture the spatial structure of the wave packets at different evolution times.

Figure 5

Interference patterns of two Gaussian wave packets as described by Eq. (5) with $A=1.3$, $k=1$, $w=10$, and (a) ${\theta }_1=0$, ${\theta }_2=0$, and (b) ${\theta }_1=3\pi ∕2$, ${\theta }_2=\pi ∕2$. Dotted lines show the superlattice potential ($\epsilon =0.3$ and $V_0=1$).

Figure 6

Linear $(g_{1D}=0)$ evolution of wave packets formed by the interference of two Gaussian wave packets described by Eq. (5) in a superlattice potential ($\epsilon =0.3$ and $V_0=1$). Number of atoms is $N=1000$, $k=1$, and $w=30$. In (a) the phases of the two wave packets are ${\theta }_1=0$ and ${\theta }_2=0$. In (b) the phases are ${\theta }_1=3\pi ∕2$ and ${\theta }_2=\pi ∕2$, respectively.

Figure 7

Evolution of wave packets described by Eq. (5) with repulsive interaction $g_{1D}=0.001$, $N=1000$, $k=1$, and $w=30$. In (a)–(c) ${\theta }_1=0$ and ${\theta }_2=0$ (i.e., the wave packet is at the bottom of the first minigap). In (d) ${\theta }_1=3\pi ∕2$ and ${\theta }_2=\pi ∕2$ (i.e., the wave packet is at the top of the first minigap). Each of the superlattice potentials have the same period and lattice height $V_0=1$. The shape of each superlattice, however, varies: (a) $\epsilon =0.2$, (b), (d) $\epsilon =0.3$, and (c) $\epsilon =0.5$.

Figure 8

Evolution of a wave packet formed by the interference of two Gaussian wave packets as described by Eq. (5) in a superlattice potential with $V_0=1$ and $\epsilon =0.3$ for $N=1000$, $k=1$, and $w=30$. In contrast to Figs. 7b, 7d, we use an attractive nonlinearity, $g_{1D}=-0.001$. In (a) ${\theta }_1=0$ and ${\theta }_2=0$ (i.e., the wave packet is at the bottom of the first minigap). In (b) ${\theta }_1=3\pi ∕2$ and ${\theta }_2=\pi ∕2$ so that the wave packet is at the top of the first minigap.

Figure 9

Left: density profiles of the wave packets in Figs. 7a, 7b, 7c. The initial conditions are identical in each case, but the superlattice potential varies with (a) $\epsilon =0.2$, (b) $\epsilon =0.3$, and (c) $\epsilon =0.5$. In each case the lattice height is $V_0=1$. Right: exact stationary gap soliton solutions obtained by numerically solving the time-independent version of the full model equation (1) for the same superlattice parameters as on the left.