Topologically Quantized Soliton-Like Pumping using Synthetic Nonlinearity

Abstract

The interplay between nonlinear and topological physics has led to intriguing emergent phenomena, such as quantized and fractionally quantized Thouless pumping of solitons dictated by the topological invariants of the underlying band structure. Unlike linear Thouless pumping, which requires excitation of a Wannier function of a uniformly filled band, quantized soliton pumping is observed even with localized excitations that do not represent Wannier functions. Here, we show that similar soliton-like quantized pumping can be observed in Aubry-Andre-Harper (AAH) model by introducing a synthetic nonlinearity in the form of a cutoff on the coupling strengths between lattice sites. More importantly, we reveal that the localized excitations driving quantized soliton pumping are precisely the Wannier functions of the uniformly filled bands of the effectively nonlinear lattice, thus restoring consistency with linear Thouless pumping. We extend this approach to multi-band systems and show that the nonlinearity introduces a degeneracy between bands, subsequently leading to the observation of fractionally quantized pumping. Our approach of introducing a synthetic nonlinearity is general and could be extended to reveal soliton dynamics in other nonlinear topological systems.

Figures (6)

• Figure 1: a. Schematic of a 1D coupled waveguide array implementing the AAH model with periodicity $N = 3$. b. Modulated coupling strengths as a function of the propagation distance $z$. The y-axis represents the positions of different waveguides of the unit cell; colors distinguish the three waveguides in a unit cell. c. Band structure of the linear AAH lattice, with periodic boundary condition (PBC), as a function of the propagation distance $z$. d. Spatial evolution and quantized pumping in the linear AAH lattice, for excitation of a Wannier function corresponding to the second bulk band. e. Coupled power in a pair of nonlinear waveguides, showing a sharp transition with increasing nonlinear strength $g$. f. The nonlinearity-induced transition in e is mimicked by introducing a synthetic nonlinearity, in the form of a threshold in coupling strength $J$. g-i Schematic of the waveguide array, modulation of coupling strengths, and band structure of the modified, effectively nonlinear lattice. j Soliton-like pumping in the nonlinear lattice by two unit cells, corresponding to the second band with Chern number 2. k. Comparison of linear and soliton-like pumping, both showing the same displacement over a drive period.
• Figure 2: Wannier functions corresponding to Band 2, for a, linear lattice, and b, effectively nonlinear lattice. The shaded red circles show the excitation wavefunction. c. Quantized pumping for Band 2 by 2 unit cells per period. The solid blue line shows the displacement of the CoM. d. Instantaneous Wannier function intensity in the lattice, showing displacement by Chern number $+2$. e-h. Corresponding results for Band 1, showing pumping by one unit cell, but in the opposite direction. i-l Results for Band 3, also showing pumping by one unit cell.
• Figure 3: a, Variation of intensity difference between input and output soliton state, and b, the number of pumped lattice sites as a function of $J_{Th}$.
• Figure 4: Wannier functions corresponding to Band 3, for a, linear lattice, and b, effectively nonlinear lattice. c. Quantized pumping for Band 3 by 2 unit cells per period. d Wannier function for Band 2. e Composite Wannier function for degenerate bands 1 and 2. f. Fractional pumping by one unit cell in two periods. g. Wannier function for Band 1 of the linear lattice. h, i Wannier function and fractional pumping for the second Wannier function of degenerate bands 1 and 2. j,k Energy eigenvalues at $z=0$ for the linear lattice and the nonlinear lattice showing degeneracy between bands 1 and 2.
• Figure 5: Eigenvalues for a 9-band AAH model, with PBC and 21 cells, for a.$J_{Th} = 0$, b.$J_{Th} \simeq 1.232$, c.$J_{Th} \simeq 1.754$. 5 bands and 7 bands are degenerate in b and c, respectively, at zero energy. d. One of the Wannier functions corresponding to the degenerate 5 bands of b, showing occupation of a single lattice site. e. Observed pumping of 2 unit cells in 5 periods. f. Wannier function for the 7 degenerate bands of c, with the same occupation as d. g. Pumping by 1 unit cell in 7 periods.