Nonlinear Optical Thouless Pumps

• Nonlinear optical Thouless pumps are photonic platforms that combine Kerr nonlinearity with adiabatic lattice modulations to yield quantized soliton transport and topologically nontrivial behavior.
• They demonstrate both integer and fractional quantization, with soliton displacement governed by Chern numbers and influenced by nonlinear self-interaction and bifurcation dynamics.
• Robustness against disorder and dissipation, achieved through engineered waveguide arrays and circuit QED setups, makes these systems attractive for advanced non-reciprocal and optical switching applications.

Nonlinear optical Thouless pumps are photonic systems in which the interplay of periodic modulations and Kerr-type nonlinearity endows soliton excitations with quantized transport, generalizing the original linear Thouless pump paradigm into the nonlinear and topologically nontrivial regime. Their dynamics are governed by discrete (or continuum) nonlinear Schrödinger equations, incorporating both adiabatically modulated lattice potentials and nonlinear self-interaction. The quantized displacement of solitons per pump cycle is dictated—though sometimes anomalously altered—by the topological invariants (Chern numbers) of the underlying Bloch bands, and can exhibit both integer and fractional quantization. Robustness to disorder, capacity for non-reciprocal photonic device design, and ability to control transport regimes via parameters such as nonlinearity strength, disorder amplitude, dissipation, and multi-component effects are established features.

1. Mathematical Models, Soliton Dynamics, and Topological Invariants

The prototypical mathematical formulation employs the discrete nonlinear Schrödinger equation (DNLS) for an array of single-mode waveguides with Kerr nonlinearity and longitudinal modulation:

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

Here, $J_n(z)$ are z-dependent hopping amplitudes, $g$ is the nonlinear coefficient, and $V_n$ represents static disorder or defect sites (Chaudhari et al., 12 Dec 2025). The pump cycle is realized via periodic modulation of waveguide positions, inducing spatially varying $J_n(z)$ with a unit cell of three sites, yielding Bloch bands of nontrivial topology. The quantum numbers governing soliton displacement arise from first Chern numbers computed over the $(k, z/L)$ torus for each band; e.g., in Rice–Mele-like modulations, bands carry $\{-1, +2, -1\}$ (Jürgensen et al., 2021).

Nonlinearity modifies the instantaneous Hamiltonian by adding self-consistent potentials, $-g|\phi_n|^2$, resulting in the formation of spatially localized solitons. For weak nonlinearity, soliton center-of-mass evolution tracks the Wannier center of the bifurcation band, leading to displacement per cycle $\Delta X_{\rm sol} = C$, where $C$ is the band’s Chern number (Jürgensen et al., 2021, Mostaan et al., 2021). In strongly nonlinear regimes, spontaneous saddle-node or pitchfork bifurcations can produce fractional transport and anomalous plateaux (Jürgensen et al., 2022, Wu et al., 10 Jun 2025, Tao et al., 29 Sep 2024).

A unified topological invariant for pumped displacement reads: $\mathcal{I} = \begin{cases} C^{(\mathrm{Ab})}, & \text{isolated band, weak nonlinearity} \ \frac{1}{N}C^{(\mathrm{nAb})}, & \text{strong nonlinearity, %%%%9%%%%-fold band braiding} \end{cases}$ where $C^{(\mathrm{Ab})}$ is the Abelian Chern number (Berry curvature of an isolated band) and $C^{(\mathrm{nAb})}$ is the non-Abelian Wilczek–Zee invariant over a degenerate multiplet (Wu et al., 10 Jun 2025).

2. Integer and Fractional Quantization, Bifurcation Mechanisms

For moderate nonlinearity and adiabatic protocols, bright and dark solitons in both discrete and continuum models exhibit integer displacement per cycle—the canonical scenario—with the center-of-mass matching the quantized shift of the corresponding Wannier state (Jürgensen et al., 2021, Tao et al., 9 Aug 2025).

At intermediate nonlinear powers, bifurcation analysis reveals the emergence of fractional pumping. When soliton branches become degenerate or intertwine with multiple bands, the transport per cycle can be $1/2$, $1/3$, or $1/4$ unit cells, determined by the symmetry and number of branches visited over multiple cycles (Tao et al., 10 Feb 2025, Jürgensen et al., 2022, Bai et al., 7 Jul 2025). Fractional pumping is attributed to soliton-induced modification of local potentials, rendering the effective linear Hamiltonian topologically nontrivial even when the original bands are trivial (Tao et al., 10 Feb 2025, Bai et al., 7 Jul 2025). Fractional plateaux correspond to persistent symmetry-protected or parity-time symmetric trajectories.

Table: Typical Fractional Pumping Scenarios

Configuration	Cycles $m$	Displacement $n$	Per-cycle Shift $n/m$
$p=3, q=1$ (single band)	1	1	1
$p=5, q=2$ (two band)	2	1	1/2
$p=7, q=3$	3	1	1/3
$p=9, q=4$	4	1	1/4

The breakdown of quantized transport occurs via nonlinear spectral loop structures ("swallowtails"), which mark saddle-node bifurcations, closing topological band gaps and destroying the adiabatic soliton path (Tuloup et al., 2022, Fu et al., 2021). Beyond a critical $g_c$, transport is arrested and trapping ensues. In certain models, intricate band population dynamics enable fractional quantization even above threshold via persistent Rabi oscillations between bands with compensating Chern numbers (Fu et al., 2021).

3. Robustness to Disorder, Dissipation, and Nonlocality

Nonlinear optical Thouless pumps exhibit quantization robustness absent in linear systems. Focusing nonlinearity not only protects quantized soliton transport against localized defects and short-range disorder, but also accelerates the pump cycle, relaxing adiabatic constraints to $L J_m \sim 10$ for solitons compared to $L J_m \gtrsim 100$ in linear (Wannier) states (Chaudhari et al., 12 Dec 2025, Cao et al., 5 Mar 2024). Integer quantization persists for disorder strengths $V_m \lesssim 0.4 J_m$, with self-trapping at higher nonlinear powers.

Dissipative extensions based on the complex Ginzburg–Landau equation feature topological phase transitions in temporal solitons. Dissipative bifurcations, e.g. via spectral filtering or nonlinear gain, control the transition from trapped to quantized drift, and enable dynamically emergent phase transitions during evolution (Cao et al., 5 Sep 2024). Robust quantized drift persists in multi-soliton states due to continual reshaping by gain and filtering.

Nonlocal nonlinear media expand the stability window for quantized pumping, preventing breakdown commonly observed at high powers in local Kerr systems (Ye et al., 8 Jul 2025). Broad, low-power solitons do not transport, while high-power, spectrally broad solitons and multipoles (dipole/tripole) exhibit stable quantized transport within finite power windows that broaden with increased nonlocality.

4. Variants: Vector Solitons, Multi-Dimensional Pumps, and Resonator Arrays

Nonlinear Thouless pumping extends naturally to two-component (spinor) and vector soliton systems. For two-component Bose–Einstein condensates with spin-dependent lattices, the relative displacement $d_r$ between spin-up and spin-down potentials acts as a control knob, enabling transitions between pumped, arrested, dynamically arrested, and revival regimes. The inter-component interaction strength and overlap tuning via $d_r$ orchestrate phase boundaries for quantized displacement (Cao et al., 7 Nov 2024). Analytical variational theory predicts the pumping–arrest threshold as $$[g + g_{12} e^{-d_r^2/(2\sigma_0^2)}] A^2 \approx 61$$.

In higher-dimensional models, e.g., separable 2D lattice potentials, solitonic and dispersive wavepackets exhibit center-of-mass displacement governed by the product of Chern numbers of constituent bands. Multi-soliton regimes and fractional Chern indices (over half-period evolution in parity-time symmetric potentials) yield nontrivial, sometimes fractional, transport (Fu et al., 2022).

Nonlinear Thouless pumps in nonlinear resonator or circuit QED arrays exploit spatial/temporal modulation of resonance frequencies and attractive Kerr nonlinearity. Many-body Fock states adiabatically follow topologically separated bands, with integer pumped shift protected by multi-photon band gaps; fractional quantization and disorder tolerance are set by anticrossing gaps (Tangpanitanon et al., 2016).

5. Experimental Realizations and Design Principles

Direct implementation uses femtosecond-laser-written waveguide arrays, with longitudinal modulation engineering the pump protocol. Material platforms include borosilicate glass, AlGaAs, SiN, chalcogenides, and nematic liquid crystals (Jürgensen et al., 2021, Jung et al., 2022). Critical experimental parameters are input power (tuning $gP/J_m$), array length (setting adiabaticity window), waveguide separation and modulation amplitude (controlling band gaps and nonlinearity-induced transitions).

Defects and disorder are introduced via localized changes to the writing protocol. Out-coupling and fluorescence imaging yield real-time center-of-mass displacement per cycle; robustness to disorder and transition to fractional pumping plateaux are directly observable (Chaudhari et al., 12 Dec 2025, Jürgensen et al., 2022).

Power-induced bifurcations enable all-optical switching in topological delay-line devices, with thresholds calibrated by the balance of Kerr shift and intersite couplings (Jung et al., 2022). Circuit QED implementations harness transmon arrays with tunable frequency offsets and capacitive/hybrid couplings for few-photon topological transport (Tangpanitanon et al., 2016).

6. Theoretical Generalizations, Open Problems, and Applications

Nonlinear Thouless pumps manifest a fundamentally different transport quantization mechanism from filled-band (fermionic) Thouless pumps, resting on the adiabatic tracking and translation invariance of nonlinear eigenstates—generalizing the concept of bulk-edge correspondence into nonlinear spectral theory. Auxiliary eigenvalue frameworks with overlap operators yield fractional Chern numbers not accessible by conventional approaches, aligning fractional soliton transport with the spectral winding of non-Hermitian problems (Bai et al., 7 Jul 2025).

Nonlinearity can induce topological pumping even in models whose linear bands are topologically trivial, provided the nonlinear self-consistent Hamiltonian undergoes topological band bending during evolution (Tao et al., 10 Feb 2025, Tao et al., 29 Sep 2024). Anomalous branches involving equal superpositions of neighboring Wannier functions lead to displacements $\Delta X = 2C$ per cycle. Inter-component interactions, disorder, dissipation and multi-band population dynamics enable engineering of quantized and fractional pumping, forming the basis for topological photonic routers, robust quantum delay lines, and power-controlled nonlinear beam switches.

Future directions include extensions to vortex and ring solitons in higher dimensions (Tao et al., 9 Aug 2025), multi-component transport control (Cao et al., 7 Nov 2024), and dissipative stabilization of nonlinear topological states (Cao et al., 5 Sep 2024). The synthesis of adiabaticity requirements, spectral design, and nonlinearity tuning will facilitate designer fractional transport devices and deeper quantum–nonlinear topological phenomenology.