Four-dimensional photonic lattices and discrete tesseract solitons

We theoretically study discrete photonic lattices in more than three dimensions and point out that such systems can exist in continuous three-dimensional space. We study discrete diffraction in the linear regime and demonstrate the existence of four-dimensional (4D) tesseract solitons in nonlinear 4D periodic photonic lattices. Finally, we propose a design towards a potential realization of such periodic 4D lattices in experiments.

Figure 1

Discrete diffraction in 4D lattices and tesseract solitons. (a) Evolution of the inverse participation ratio $I\left(t\right)$ during discrete diffraction for linear lattices with $D=1$, 2, 3, and 4 (bottom up). The base of the logarithm is $e$, i.e., we plot $\ln I\left(t\right)$. Red dashed lines depict $I\left(t\right)$ for an infinite lattice, whereas blue solid lines correspond to a finite size system with $N=2^{24}$ sites and periodic boundary conditions. For $D=4$ the boundary is reached at $t\approx 15$ and the IPR starts to oscillate. (b) A schematic illustration of a tesseract soliton. The phases at nearest-neighbor nodes differ by $\pi$, $\gamma /J=630$. (c) Inverse participation ratio dynamics of a tesseract soliton with (black dotted line) and without (red dashed line) noise demonstrating stability (the two lines are on top of each other). When the nonlinearity is turned off, the tesseract initial condition diffracts (solid blue).

Figure 2

Illustration of the idea for creating photonic lattices in more than three dimensions via complex networks of optical resonators. (a) A schematic illustration of a (finite size) 2D lattice of coupled resonators. (b) The coupling scheme which is topologically fully equivalent to the one in panel (a), despite the fact that resonators are located on a 1D line. By the same token a 4D (discrete) lattice can be embedded in a continuous 3D space, see text for details.

Figure 3

Proposal for the design of a 4D photonic lattice. (a) Illustration of a simple-cubic 4D lattice. Waveguides enable coupling between distant resonators along the edges of 3D cubes. Evanescent coupling (hopping) along the diagonal enables construction of the fourth dimension for the discrete lattices, see encircled region enlarged in panel (b). For clarity only two sublattices are displayed in panel (a); the number of sublattices that can be added along the fourth dimension depends on the size of each individual cavity and the lattice size $a_l$ of every 3D sublattice. See text for details.