Three-Dimensional Nonlinear Lattices: From Oblique Vortices and Octupoles to Discrete Diamonds and Vortex Cubes

We construct a variety of novel localized topological structures in the 3D discrete nonlinear Schrödinger equation. The states can be created in Bose-Einstein condensates trapped in strong optical lattices and crystals built of microresonators. These new structures, most of which have no counterparts in lower dimensions, range from multipole patterns and diagonal vortices to vortex “cubes” (stack of two quasiplanar vortices) and “diamonds” (formed by two orthogonal vortices).

Figure 1

Stable multipoles. The top row depicts stable tight dipoles ($d=1$): (a) straight, (b) oblique, and (c) diagonal ones. (d),(e) Quadrupoles in the $n=0$ plane, with internal separation $d=1$ and $d=2$, respectively. (f),(g) Octupoles with $d=1$ and $d=2$. Panel (h) displays the stability threshold $C_{\mathrm{cr}}^{(3\mathrm{D},d)}$ as a function of the internal distance $d$ for (from top to bottom) diagonal, oblique, and straight dipoles, octupoles, and quadrupoles. The horizontal dashed line corresponds to the stability threshold for the fundamental discrete soliton. Note that, for the quadrupole (bottom graph), $C_{\mathrm{quad}}^{(3\mathrm{D},d)}$ behaves linearly for small $d$ (see the dashed line with slope 0325 for guidance). In panels (a)–(g), level contour corresponding to $\mathrm{Re}(u_{l,m,n})=\pm 0.5$ are shown in blue (dark gray) and red (gray), respectively. All these states are stable (for the case shown, with $\Lambda =2$ and $C=0.1$).

Figure 2

Vortex cubes for $\Lambda =2$ and $C=0.1$. Panel (a) shows a stable vortex cube, built of two quasiplanar vortices with equal vorticities, $S_1=S_2=1$, and a phase shift of $\pi$. Panel (b) shows an unstable cube formed by vortices with opposite charges, $S_1=-S_2$, and panel (c) shows a snapshot (at $t=200$) of its evolution, clearly demonstrating that the phase coherence is lost. The real level contours are as in Fig. 1, and the imaginary ones, $\mathrm{Im}(u_{l,m,n})=\pm 0.5$, are shown by green (light gray) and yellow (very light gray) hues, respectively.

Figure 3

Diagonal and oblique vortices. Panel (a) shows an unstable diagonal vortex, with the axis along the direction $(1,1,1)$, for $\Lambda =2$ and $C=0.1$. Panel (b) shows an unstable oblique vortex with the axis oriented along the direction $(1,1,0)$, and panel (c) shows a stable oblique vortex of a modified form (see text) for $\Lambda =2$ and $C=0.01$.

Figure 4

The vortex diamond for $\Lambda =2$ and $C=0.1$. Panel (a) displays an unstable diamond, and panel (b) depicts its evolution, in terms of the field’s magnitude (top) and phase (bottom) at the main six lattice sites. Initially ($t<300$), pairs of sites (opposite vertices of the diamond) cyclically change the phase, and subsequently ($t>350$) the phase coherence is lost and the field magnitude oscillates erratically about its initial level.