Matter-Wave Gap Solitons in Atomic Band-Gap Structures

We demonstrate that a Bose-Einstein condensate in an optical lattice forms a reconfigurable matter-wave structure with a band-gap spectrum, which resembles a nonlinear photonic crystal for light waves. We study in detail the case of a two-dimensional square optical lattice and show that this atomic band-gap structure allows nonlinear localization of atomic Bloch waves in the form of two-dimensional matter-wave gap solitons.

Figure 1

Dispersion diagram for atomic Bloch waves in a 2D lattice in the reduced zone representation: shaded, first three energy bands; numbered, band gaps ($V_0=5.0$). Dashed, the line $\mu =V_0$. Right top: contour plot of the lattice potential in Cartesian space, black shading corresponds to potential minima. Right bottom: the first Brillouin zone of the 2D lattice in the reciprocal lattice space; marked are the high-symmetry points of the irreducible zone.

Figure 2

Band-gap structure of the atomic Bloch-wave spectrum as a function of the lattice amplitude $V_0$. Shaded: Bloch-wave bands; numbered, gaps. Dashed, the line $\mu =V_0$.

Figure 3

Peak amplitude, ${\varphi }_0\equiv \varphi (0,0)$, of the fundamental gap soliton in the first gap ($V_0=5.0$); shaded, the first band; dashed, the line $\mu =V_0$. Below: Spatial structure of the gap solitons (a) near the high-energy edge of the first band, $\mu =3.5$, and (b) at the top of the lattice potential, $\mu =5.0$. Contour plots on the 3D plots (left) show the structure of the lattice potential, darker contours correspond to potential minima. Potential contours on the right are guides to the eye. Insets: the cross sections of the localized modes and the potential along the line $(x,0)$.

Figure 4

Spatial structure of the higher-order gap solitons formed by two (left) out-of-phase and (right) in-phase fundamental modes ($V_0=5.0$, $\mu =4.0$). Filled contours show the lattice potential; dark shading corresponds to potential minima.