Hubbard-like Interactions and Emergent Dynamical Regimes Between Modulational Instability and Self-Trapping

TL;DR

This work analyzes modulational instability in a discrete 2D nonlinear lattice described by a third-order nonlinear Schrödinger equation with diagonal Hubbard-like detuning $U$. Using numerical integration of a uniform background plus small perturbations, the authors map how the MI-driven transition to self-trapping evolves as functions of the nonlinear strength $χ$ and the diagonal detuning $U$, revealing a direct transition for $U=0$ but intermediate breathing regimes and two distinct self-trapped states for $U>0$. A phase diagram in the $χ$–$U$ plane shows that Hubbard-like interactions shrink the uniform-state stability domain and induce bound (diagonal) and unbound (off-diagonal) self-trapped regimes, along with regular and chaotic-like breathing dynamics. The findings offer routes to control nonlinear localization in photonic lattices and have implications for optical switching and photonic quantum simulation in engineered nonequilibrium states.

Abstract

We investigate the modulational instability of uniform wave packets governed by a discrete third-order nonlinear Schrödinger equation in finite square lattices, modeling light propagation in two-dimensional nonlinear waveguide arrays. We analyze how initially stable uniform distributions evolve into self-trapped (localized) regimes and the influence of a refractive index detuning selectively applied along the diagonal waveguides on this transition. This detuning effectively emulates the effect of on-site Hubbard-like interactions $U$ in photonic analogs of interacting particles in a one-dimensional lattice. While for $U = 0$ the system exhibits the known direct transition from stable uniform states to asymptotically localized profiles, we show that $U > 0$ induces an emergence of intermediate dynamical regimes. These regimes include coherent breathing modes that can be either confined along diagonal or off-diagonal waveguides, as well as chaotic-like propagation patterns. At higher nonlinearities, we identify distinct self-trapped regimes characterized by diagonal or off-diagonal localized modes, depending on the strength of $U$. The critical nonlinear strengths separating the existing regimes are shown in the phase diagram, underscoring the competing trends imposed by the Hubbard-like interaction on the optical field.

Figures (6)

• Figure 1: Time evolution of the marginal intensity profile $I_{n}(t)=|c_{n}(t)|^2$ along a single axis of a square waveguide lattice with $L = 100$, for representative nonlinear strengths $\chi$. In the absence of the on-site Hubbard-like interaction (panels a-d), a direct transition is observed from a regime of stable uniform light distribution to a stationary, strongly localized optical mode. This transition occurs around $\chi\approx 19.7$, consistent with previous reports. When a weak on-site Hubbard interaction is introduced (panels e-h), the direct transition is replaced by the emergence of intermediate breathing regimes. These results suggest that Hubbard-like interactions significantly narrow the range of $\chi$ over which the uniform optical distribution remains stable.
• Figure 2: Time evolution of the normalized participation ratio for a single transverse direction [$\overline{P}_{n}(t) = P_{n}(t)/L$], analyzed for the same settings of $\chi$ and $U$ as in Fig. \ref{['fig:1']}. In a stable uniform light distribution, the intensity profile remains fully extended over the lattice, yielding $\overline{P}_{n}(t) = 1$. Conversely, in a self-trapped regime, light becomes strongly localized, driving $\overline{P}_{n}(t) \approx 0$. The transition between these regimes is influenced by the presence of the diagonal Hubbard-like coupling, which induces intermediate breathing dynamics, both regular and chaotic-like, and modifies the critical nonlinear threshold required to achieve a stable uniform regime.
• Figure 3: Minimum of the normalized participation function for a single transverse direction as a function of the nonlinear strength $\chi$ in waveguide arrays with representative Hubbard-like interactions ($U$). For $U=0$, the direct transition from stable uniform field distribution to self-trapped regimes with increasing $\chi$ is confirmed. In contrast, panels where $U>0$ signal this transition mediated by distinct breathing regimes, including regular and chaotic-like dynamics. A complementary analysis of the temporal evolution reveals the emergence of additional breathing modes to regular breathing (see Fig. 3g), posing challenges for mapping the chaotic-like regime.
• Figure 4: Time evolution of wave-packet centroids along both lattice directions and their corresponding intensity profiles, depicting two distinct breathing regimes arising from the competition between existing diagonal and off-diagonal modes. (a) Bound breathing regime: The field undergoes periodic amplitude modulation (breathing) while remaining localized at a diagonal waveguide ($n=m$). (b) Unbound breathing regime: The field exhibits a breathing pattern characterized by a periodic spatial shift between symmetric off-diagonal waveguides.
• Figure 5: Time evolution of the marginal intensity profiles for representative values of $\chi$ and $U$, illustrating distinct dynamical regimes: (a) Chaotic-like dynamics, characterized by irregular spatiotemporal fluctuations of field across the array; (b) Bound self-trapping, in which the optical field remains spatially localized along the diagonal waveguides after an initial transient; (c) Unbound self-trapping, described by persistent localization of the field at off-diagonal waveguides.