Soliton Thouless pumping engineered by inter-site nonlinearities

TL;DR

The paper addresses how inter-site nonlinearities modify soliton Thouless pumping in a diagonal Aubry–André–Harper lattice. By employing an extended discrete nonlinear Schrödinger framework with a time-dependent phase $\theta(t)=\omega t$, the authors reveal that inter-site nonlinearities can yield quantized soliton displacements up to $D=4$ per pumping cycle, surpassing the linear band CN of $+1$, while solitons remain localized to a single bifurcating band. A driving-frequency window balancing soliton instability and adiabaticity is identified, establishing lower and upper bounds on the sweep rate and enabling multi-cycle pumping under suitable conditions. The study also shows that soliton energy levels, not Wannier-state CNs, govern transport for solitons bifurcated from different bands, and that nonlinear interactions among multiple solitons can undermine pumping robustness, offering new degrees of freedom for topological control. These results present a pathway to engineer nonlinear topological transport with potential applications in photonics and cold-atom systems.

Abstract

We study soliton Thouless pumping in an extended diagonal Aubry-André-Harper model with on-site nonlinearities and inter-site nonlinearities. We show that the inter-site nonlinearities can make solitons acquire anomalous transport distances far beyond the ones predicted by the linear bands, and the quantized displacements can be engineered well. We uncover that nonlinear instabilities require lower limits on sweeping rates for soliton pumping, challenging the common notion that slower modulation enables a more favorable realization of topological transport. The nonlinear interactions between solitons make multi-soliton pumping generally lack the robustness characteristic of Thouless pumping as linear systems. Our results provide many possibilities to engineer topological pumping by nonlinearities, and further make a step for applications of soliton pumping.

Figures (6)

• Figure 1: (a) Schematic of the diagonal extended AAH model. The dashed box labels a unit cell (indexed by $n$) containing three sites $(a_1, a_2, a_3)$, each subject to a modulated on-site potentials $J_{n,i}^{\prime}\left( \theta \right) =\sin \left[ \theta -\left( i-1 \right) 2\pi /3 \right] +\sum_j{\frac{g_{ij}}{2}\left| \psi _{n,i} \right|^2\left| \psi _{n,j} \right|^2}$ and $\psi_{n,i}$ denotes the wave-function at site $a_i$ in the $n$-th unit cell, with $\theta = \omega t$. Here, $J_4$ and $J_5$ represent the strength of linear hopping. $g_{ij}$ denotes the strength of nonlinear interactions.
• Figure 2: Phase diagram of the peak displacement (in units of cell) of solitons bifurcating from the lowest band over a cycle. From left to right, as $g_{13}$ increases, the solitons displacement per cycle exhibits quantized increments. From top to bottom, as $\omega$ decreases, the solitons fails to complete transport within a cycle, with the black numbers in the diagram indicating the displacement. In the upper-left region of the diagram, the solitons also fail to undergo transport as a result of non-adiabatic (NA) effects. Parameters: $n = 20, \mu = -1.1$.
• Figure 3: Evolution of varied solitons over three cycles. The red tick marks indicate the soliton peak positions after three cycles. In the inset, the white solid lines represent the soliton density distribution within the elliptical frame, with red dots corresponding to the same lattice site. (a) Normal transport: the soliton displacement per cycle exactly matches the CN of the bifurcating linear band, with $g_{13} = 0.25$. (b-d) Varied transport: the soliton displacement per cycle corresponds to 2, 3, and 4 unit cells. The marker shown in the lower-right corner represents the same parameters as indicated in Fig.\ref{['fig2']}. The parameters $g_{13}$ are 1.1, 3, and 6.5, respectively. Common parameters: $n = 20$, $\mu = -1.1$, and $\omega = 0.005$.
• Figure 4: (a) Soliton destruction induced by weak instability. (b) Left: COM motion after soliton breakdown; right: linear stability analysis of the soliton within one cycle, showing the maximum growth rate at each instant, with $\omega = 0.0002$. (c) Soliton transport over one cycle, with the fidelity between the initial and final states shown for different driving frequencies. The fidelity fluctuations in the light yellow region on the right arise from NA, whereas the light red region on the left corresponds to soliton destruction caused by instability. Common parameters: $n = 20$, $g_{13} = 1.1$, and $\mu = -1.1$.
• Figure 5: (a) Colored lines represent gap solitons bifurcating from different linear energy bands, all with the same particle number $N_B = 0.95$. The black regions indicate the linear energy bands. (b) Colored solid lines show the COM transport of solitons (colors correspond to those in (a) under the same driving frequency $\omega = 0.005$. The dashed lines of the same color indicate the corresponding COM transport of the Wannier states. Parameter : $g_{ij}=1, i,j = 1,2,3$