Bloch-Landau-Zener Oscillations in Moiré Lattices

TL;DR

This work develops a multimode theory of two-dimensional Bloch-Landau-Zener oscillations for localized modes in incommensurate moiré lattices under a weak transverse gradient, where mobility edges replace band gaps and energy-space tunneling accompanies real-space mode transfer. By introducing two localized-mode bases connected by a unitary transform and formulating a selection rule that combines spatial proximity and quasi-resonance of propagation constants, the authors predict and classify BLZ dynamics, including two-mode, multimode, and orientation-dependent oscillations, as well as robustness to moderate disorder. They extend the framework to nonlinear regimes, showing that weak focusing or defocusing Kerr-type nonlinearities largely preserve the few-mode character while modifying the oscillation period and amplitude, and that higher powers can induce new resonances or suppress existing ones. The results provide a path for experimental observation in photonic moiré lattices and generalize to other aperiodic systems, including cold atoms in moiré potentials and related quasi-periodic structures.

Abstract

We develop a theory of two-dimensional Bloch-Landau-Zener (BLZ) oscillations of wavepackets in incommensurate moiré lattices under the influence of a weak linear gradient. Unlike periodic systems, aperiodic lattices lack translational symmetry and therefore do not exhibit a conventional band-gap structure. Instead, they feature a mobility edge, above which (in the optical context) all modes become localized. When a linear gradient is applied to a moiré lattice, it enables energy transfer between two or several localized modes, leading to the oscillatory behavior referred to as BLZ oscillations. This phenomenon represents simultaneous tunneling in real space and propagation constant (energy) space, and it arises when quasi-resonance condition for propagation constants and spatial proximity of interacting modes (together constituting a selection rule) are met. The selection rule is controlled by the linear gradient, whose amplitude and direction play a crucial role in determining the coupling pathways and the resulting dynamics. We derive a multimode model describing BLZ oscillations in the linear regime and analyze how both attractive and repulsive nonlinearities affect their dynamics. The proposed framework can be readily extended to other physical systems, including cold atoms and Bose-Einstein condensates in aperiodic potentials.

Figures (9)

• Figure 1: (a) Profile of the moiré lattice for $\bm\alpha=0$. (b) Positions of the centers of mass ${\bm r}_{i}^{(0)}={\bm r}_{\bm{n}}^{(0)}$ of different localized modes with $\chi\ge 0.3$, corresponding to the ordering $\bm n\to i$ described in the text, for the lattice without gradient. The color code is used to indicate the form-factors $\chi^{1/2}$ of corresponding modes. The indices $i$ of the modes used below for dynamical simulations are highlighted by the numbers. The white circle in panel (a) and the dotted black circle in panel (b) with radius $\rho=30$ indicate the cutoff used to restrict the number of modes in the subsequent numerical analysis of the dynamics. This ensures that only modes sufficiently far from the boundaries of the simulation domain are included. (c) Propagation constants $\beta$ plotted as a function of mode index $i$ for $\bm\alpha=0$ (black circles). The corresponding squared form-factors $\chi$ are shown as red squares (right vertical axis). Blue and green labels indicate the specific modes chosen as initial conditions for dynamical simulations. Here and in all figures below, $p_1=p_2=4$, $d=2.5$, $w=0.5$, and $\theta=\pi/6$.
• Figure 2: (a, b) Propagation constants as functions of $|{\bm \alpha}|$ for the angles $\gamma=\pi/3$ (a) and $\gamma=-\pi/5$ (b). The branches used below to illustrate the evolution are highlighted in color. (c, d) Positions of the centers of mass ${\bm r}_i^{({\bm \alpha})}=\left(x_c (\alpha),y_c (\alpha)\right)$ of modes $124$ and $129$ (c) and modes $127$, $135$, and $137$ (d) as a function of $\alpha$ for $\gamma=\pi/3$. (e) Mode profiles $|\phi_{124}^{({\bm \alpha})}|$ and $|\phi_{129}^{({\bm \alpha})}|$ corresponding to the points marked by numbers 1 to 6 in panels (a) and (c).
• Figure 3: Variation of the overlap integrals $S_{ij}^{({\bm \alpha})}$ with $|{\bm \alpha}|$ for ${j}=124$ (a) and ${j}=137$ (b). Centers ${\tilde{\beta}}_i^{({\bm \alpha})}$ (solid lines) of the Gershgorin intervals (shaded regions) as functions of $|{\bm \alpha}|$ for the modes $124$, $129$ (c) and $127$, $135$, and $137$ (d). In all panels $\gamma=\pi/3$. Dashed vertical lines indicate the values of $|{\bm \alpha}|$ corresponding to the dynamic examples shown in Figs. \ref{['fig4']} and \ref{['fig5']}.
• Figure 4: Evolution of weights $|c_i |^2$ of all linear modes of the system at $|{\bm \alpha}|=0$ (top) and field modulus distributions at different values of $z$ corresponding to the dots and vertical dashed lines in the top panels (bottom) for the angle $\gamma=\pi/3$ (a) and $\gamma=-\pi/5$ (b) for $|{\bm \alpha}|=0.003$. The initial state is mode $n=124$ of the system with $|{\bm \alpha}|=0$. Arrows indicate the direction of the gradient. The dashed lines show the analytical solution.
• Figure 5: Evolution of weights $|c_n |^2$ of all linear modes of the system at $|{\bm \alpha}|=0$ for the angle $\gamma=\pi/3$ (top) and field modulus distributions at different values of $z$ corresponding to the dots and vertical dashed lines in the top panels (bottom) for $|{\bm \alpha}|=0.001$ (a), $|{\bm \alpha}|=0.005$ (b), and $|{\bm \alpha}|=0.010$ (c). The initial state is mode $n=137$ of the system with $|{\bm \alpha}|=0$. Arrows indicate the direction of the gradient. The dashed lines show the analytical solution.