Nonlinear Thouless Pumping

• Nonlinear Thouless pumping is a topological transport phenomenon where cyclic modulation in nonlinear systems induces quantized soliton displacement.
• The process achieves robust, integer or fractional shifts by harnessing the interplay between nonlinearity, topology, and disorder under adiabatic protocols.
• Experimental realizations in photonic waveguides, circuit-QED, and ultracold atoms validate theoretical models and open pathways for advanced device applications.

Nonlinear Thouless pumping is a topological transport phenomenon in classical and quantum many-body systems with nonlinear interactions, where the quantized, unidirectional displacement of localized nonlinear excitations—most notably solitons—emerges in response to an adiabatic, cyclic modulation of system parameters. In contrast to traditional (linear) Thouless pumps, where quantized transport is dictated by Chern numbers of filled bands, nonlinear Thouless pumps are governed by the interplay between nonlinearity, topology, and (often) disorder. Uniquely, nonlinearity not only modifies the topological transport—leading to quantized, fractional, or even anomalous shifts—but can also induce robust transport in settings where the underlying linear system is topologically trivial. This synthesis unifies perspectives from nonlinear Schrödinger-type models, photonics, driven-dissipative platforms, and correlated photon states.

1. Theoretical Foundations and Model Formulation

The prototype for nonlinear Thouless pumping is the adiabatically modulated, discrete nonlinear Schrödinger (DNLS) chain,

$$i\,\partial_z\phi_n(z) = J_n(z)\,\phi_{n+1} + J_{n-1}(z)\,\phi_{n-1} - g\,|\phi_n|^2\,\phi_n + V_n \phi_n,$$

where $z$ is the propagation coordinate (or time), $J_n(z)$ a $z$-periodic hopping amplitude (engineered by modulating waveguide positions or onsite energies), $g$ the Kerr nonlinearity coefficient, and $V_n$ a disorder/defect potential (Chaudhari et al., 12 Dec 2025). In atomic and excitonic implementations, analogous Gross–Pitaevskii or Bose–Hubbard dynamics apply (Mostaan et al., 2021, Fu et al., 2021, Ravets et al., 2024, Tangpanitanon et al., 2016).

A defining aspect is the pump protocol: periodic, slow (adiabatic) modulation of $J_n(z)$ or onsite parameters over a spatial or temporal cycle. In the linear regime ($g=0$), this reduces to the celebrated Rice–Mele or Aubry–André–Harper pump (Mostaan et al., 2021), where quantized charge or photon transport over a cycle is controlled by the Chern number of an occupied Bloch band.

In the nonlinear regime, the system generically supports localized stationary solutions (solitons) $u_n(z)$ satisfying

$$\Lambda u_n = J_n(z)\,u_{n+1} + J_{n-1}(z)\,u_{n-1} - g u_n^3,$$

where $\Lambda$ is a nonlinear eigenvalue and the soliton family adiabatically follows the drive as long as the spectrum remains gapped from linear bands (Chaudhari et al., 12 Dec 2025, Jürgensen et al., 2021).

2. Quantized Soliton Transport: Integer and Fractional Pumping

Nonlinear Thouless pumps exhibit robust, quantized displacement of soliton centers: $$\Delta n = \sum_n n|\phi_n(L)|^2 - \sum_n n|\phi_n(0)|^2,$$ with $L$ the length of one pump period. For a soliton bifurcating from a band with Chern number $C$, the quantized shift per cycle is $C$ lattice sites, i.e.,

$\Delta n = C$

for integer (topologically protected) pumping (Chaudhari et al., 12 Dec 2025, Mostaan et al., 2021, Jürgensen et al., 2021).

At intermediate powers, multiband and bifurcation effects induce fractional pumping: the soliton returns to itself only after multiple pump cycles, with a net shift of $p$ sites over $q$ cycles ($\Delta n = p/q$ per cycle) (Chaudhari et al., 12 Dec 2025, Jürgensen et al., 2022, Tao et al., 10 Feb 2025). These fractional plateaux correspond to emergent degeneracies of nonlinear eigenbranches and interband mixing, distinct from linear (fermionic) fractionalization mechanisms. Notably, nonlinearity can induce integer or fractional quantized pumping in systems whose linear bands are all topologically trivial, due to soliton-induced modification of the effective bandstructure and Chern topology of the soliton's host (Tao et al., 10 Feb 2025).

3. Topological Invariants and Nonlinear Band Structure

While the linear pump's quantization is dictated by the Berry curvature of filled Bloch bands, in nonlinear Thouless pumps, the relevant topological invariant derives from the adiabatic flow of Wannier centers associated with instantaneous, soliton-modified bands (Mostaan et al., 2021, Jürgensen et al., 2021). The soliton's center-of-mass is locked to the instantaneous Wannier flow: $\Delta X_{\text{soliton}} = \Delta X_W = C\,a,$ with $a$ the lattice constant and $C$ the effective nonlinear Chern number. In the weak-nonlinearity, single-band regime, this remains strictly integer and robust to perturbations, provided interband mixing is negligible and soliton stability persists (Jürgensen et al., 2021).

At strong nonlinearity or in the presence of non-Hermitian elements, the effective band structure becomes nonlinear, with auxiliary eigenvalue problems such as $H\Psi = \omega S(\omega)\Psi$, leading to fractional Chern numbers and robust but non-integer soliton motion (Zheng et al., 2 Feb 2026). In some regimes, nonlinear bifurcations yield anomalous phenomena—such as displacement per cycle twice the Chern number or non-trivial shifts even in topologically trivial bands—mediated by new branches of localized nonlinear states (Tao et al., 2024).

4. Disorder, Dissipation, and Robustness of Quantization

Nonlinear Thouless pumping is resilient to moderate disorder, in stark contrast to linear (bosonic) pumps, which are generically destroyed by infinitesimal potential fluctuations. The nonlinearity-induced gap $\Delta_s \sim gP$ spectrally isolates the soliton from extended modes, protecting quantization up to a disorder strength $V_m \lesssim 0.1$–$0.2J_m$, with higher $gP$ extending this window (Chaudhari et al., 12 Dec 2025). Similarly, in dissipative systems, such as complex Ginzburg–Landau arrays, quantized pumping persists in the presence of loss and gain, with nonlinear solitons transitioning between trapped and quantized-drift phases as dissipation parameters are varied (Cao et al., 2024).

Device-level implications include the ability to execute pumps more rapidly (i.e., less adiabatically) than in the linear case, due to the enlarged adiabatic gap associated with soliton dynamics. This enables robust, high-speed all-optical isolators and non-reciprocal devices immune to moderate disorder and fabrication imperfections (Chaudhari et al., 12 Dec 2025).

5. Generalizations: Dimensionality, Multicomponent, and Engineered Nonlinearities

Nonlinear Thouless pumping extends beyond single-component, 1D systems:

• Higher dimensions: In separable 2D optical lattices, nonlinear Thouless pumping realizes multidimensional topological transport, with integer and fractional quantization governed by vector Chern indices and inter-sublattice interactions (Fu et al., 2022).
• Vector solitons: In spinor Bose–Einstein condensates and multicomponent photonic arrays, the relative displacement of spin-component superlattices serves as a control parameter for vector soliton transport, leading to phase diagrams of arrested, pumped, and revived states controlled by both interaction strength and relative lattice offset (Cao et al., 2024).
• Nonlocal and intersite nonlinearities: Nonlocal or inter-site Kerr effects can programmatically engineer the quantized Thouless shift. Inter-site nonlinearities allow the per-cycle shift to be tuned beyond the linear Chern number sequence, with integer-multiplication or designer stepsizes (Jiang et al., 4 Jan 2026).
• Dissipation and non-Hermiticity: The inclusion of gain/loss and nonreciprocal coupling in nonlinear pumps yields new regimes where fractionally quantized topological transport survives, linked directly to auxiliary Chern numbers of the nonlinear eigenvalue problem (Zheng et al., 2 Feb 2026, Cao et al., 2024).

6. Limitations, Instabilities, and Breakdown of Quantization

Nonlinear Thouless pumping is robust only within certain parameter windows. At sufficiently high nonlinearity, loop- or swallowtail-like structures emerge in the nonlinear topological bands, leading to a breakdown of quantization as solitons undergo Rabi oscillations between multiple nonlinear branches. If the sum of Chern numbers of wavepackets populating multiple bands is zero, transport can be arrested entirely, while fractional quantized pumping emerges otherwise (Tuloup et al., 2022, Fu et al., 2021). Nonlinear instability presents lower limits on adiabatic pump rates: pumping too slowly allows modulational instabilities to destroy coherent soliton transport (Jiang et al., 4 Jan 2026).

Quantization in multi-soliton settings is generally not robust; nonlinear interactions between simultaneously pumped solitons can lead to destruction of quantized motion unless solitons remain well-separated throughout evolution (Jiang et al., 4 Jan 2026).

7. Experimental Realizations and Outlook

Nonlinear Thouless pumps have been realized in femtosecond-laser-written photonic waveguide arrays—with both fundamental and higher-order solitons—demonstrating integer and fractional quantized shifts with high stability to structural disorder (Mostaan et al., 2021, Jürgensen et al., 2022, Tao et al., 10 Feb 2025). Circuit-QED architectures of nonlinear resonator arrays, as well as ultracold atomic realizations with tunable interactions, are promising platforms for probing multiphoton or matter-wave pumping, multicomponent pumps, and strongly nonlinear regimes (Tangpanitanon et al., 2016).

Advances in engineering nonlocal, intersite, and dissipative nonlinearities open up unprecedented control of topological transport for photonic, atomic, and mechanical systems. The emergent field invites further research on higher-dimensional and non-Hermitian extensions, disorder-driven phase transitions, and the role of nonlinear dynamical instabilities, with applications anticipated in robust information transport, photonic isolation, and neuromorphic devices.

Key references: (Chaudhari et al., 12 Dec 2025, Mostaan et al., 2021, Jürgensen et al., 2022, Tao et al., 10 Feb 2025, Ravets et al., 2024, Tao et al., 2024, Jiang et al., 4 Jan 2026, Jürgensen et al., 2021, Fu et al., 2021, Zheng et al., 2 Feb 2026, Fu et al., 2022, Tuloup et al., 2022, Cao et al., 2024, Perroni et al., 2013, Tangpanitanon et al., 2016).