Self-trapping and skin solitons in two-dimensional non-Hermitian lattices

Abstract

Two-dimensional non-Hermitian photonic lattices with asymmetric couplings offer rich possibilities for controlling wave localization, through the emergence of the non-Hermitian skin effect at lattice corners or sides. Incorporating optical nonlinearity fundamentally alters these boundary-localization characteristics. Here we show that in a two-dimensional Hatano-Nelson lattice with Kerr nonlinearity, the interplay between self-trapping and directional propagation leads to position dependent amplitude thresholds. Single-site excitations having above a critical amplitude become confined to their initial position, with lower thresholds near the position where the linear eigenmodes are localized and higher thresholds within the lattice's bulk. Additionally, we study the differences of this dynamical interplay, for wider initial excitations, between the focusing and defocusing Kerr-nonlinearity regimes. Lastly, we identify skin soliton solutions in a variety of two-dimensional lattice geometries featuring coupling asymmetry.

Figures (10)

• Figure 1: Schematic representation of a two-dimensional Hatano-Nelson lattice. The arrows represent the asymmetric couplings: $e^{h_x}$ and $e^{-h_x}$ along the $n_x$-direction, and $e^{h_y}$ and $e^{-h_y}$ along the $n_y$-direction. The figure identifies the four geometrically distinct corners of the lattice, labeled as (A), (B$_x$), (B$_y$), and (C). The spatial distribution of four indicative linear eigenmodes, for a case with $h_x=h_y$, is included as inset.
• Figure 2: Propagation in a 2D HN lattice for the linear (top row) and Kerr-nonlinear (bottom row) cases, with non-Hermiticity parameter $h = 0.2$ and initial condition $\psi_{n_x,n_y}(z=0) = A\delta_{n_x,13} \delta_{n_y,13}$. (a)–(c) Normalized complex amplitude $|\psi_{n_x, n_y}(z)|$ for the linear case at propagation distances (a) $z = 1$, (b) $z = 3.5$, and (c) $z = 6$. (e)–(g) Normalized complex amplitude $|\psi_{n_x, n_y}(z)|$ for the nonlinear case with input amplitude $A=4$, shown at the same propagation distances as in the top row. Panels (d) and (h) show the evolution of the optical power $\mathcal{P}_{\text{HN}}$ (red lines) and the optical intensity at the site of initial excitation $|\psi_{13,13}|^2$ (blue lines) for the linear and nonlinear cases, respectively.
• Figure 3: Evolution of mean positions and uncertainties in a 2D HN lattice with non-Hermiticity parameter $h= 0.2$, for the initial condition $\psi_{n_x,n_y}(z=0) = A\delta_{n_x,13} \delta_{n_y,13}$. (a) Mean positions $\langle n_x \rangle$ and $\langle n_y \rangle$ and (b) position uncertainties $\Delta n_x$ and $\Delta n_y$ for the linear case (gray line) and nonlinear cases with excitation amplitudes $A=1$ (red line), $A=4$ (blue line), and $A=6$ (green line).
• Figure 4: Comparison of the self-trapping tendency between (a) a Hermitian ($h=0$) and (b) a non-Hermitian ($h=0.2$) lattice, for single-site excitation at the center, i.e., $\psi_{n_{x},n_{y}}(z=0) = A\,\delta_{n_{x},13} \delta_{n_{y},13}$. The normalized amplitudes $|\psi_{n_{x},n_{y}}(z=5)|$ are plotted for pertinent values of $A$ (left panels) and the $IPR$ for $z=5$ is shown as a function of $A$ (right panels). Red dots correspond to the $A$ values used in the left panels.
• Figure 5: Comparison of the self-trapping tendency for different locations of excitation channel, under fixed $h=0.4$. Normalized amplitudes $|\psi_{n_{x},n_{y}}(z=5)|$ for various values of $A$ (top row) under single-site excitation (a) at the center, $n_{x_0}=n_{y_0}=13$; (b) near corner (A), $n_{x_0}=n_{y_0}=23$; and (c) near corner (B), $n_{x_0}=23$ and $n_{y_0}=3$. $IPR$ for $z=5$ is shown as a function of $A$ in bottom row for all cases. Red dots indicate the values of $IPR$ for the corresponding values of $A$ in top row.