Quantized Soliton Pumping in Nonlinear Periodic Systems

• Quantized soliton pumping is the phenomenon where a localized nonlinear wave (soliton) shifts its center-of-mass by an exact, topologically quantized amount as a pump parameter is cycled adiabatically.
• The process extends traditional Thouless pumping by incorporating nonlinear effects, leading to anomalous and even fractional displacement in both topologically nontrivial and trivial bands.
• Robust quantization has been verified through models like the Gross–Pitaevskii and discrete nonlinear Schrödinger equations, with experiments in photonic, ultracold atom, and superconducting platforms confirming the theory.

Quantized soliton pumping is the phenomenon by which a nonlinear, spatially localized excitation—typically a soliton formed via mean-field interactions in a periodic potential—undergoes a precisely quantized shift of its center-of-mass as a system parameter (the "pump parameter") is adiabatically cycled over one period. While the classical Thouless pump requires uniform band filling and yields Chern-number quantized charge pumping, quantized soliton pumping generalizes these concepts to the regime of nonlinear, few-mode occupation where the wavepacket is no longer a linear Wannier function. Recent theoretical and experimental studies have demonstrated rich new phenomena in this setting, including anomalous integer and fractional quantized soliton transport, nonlinearity-induced pumping even in topologically trivial bands, and robust behavior in the presence of disorder and dissipation.

1. Mathematical Formulation and Models

The canonical framework is the mean-field nonlinear Schrödinger (Gross–Pitaevskii) equation or its tight-binding discrete nonlinear Schrödinger (DNLS) reduction. For a two-component setting on a 1D lattice with unit cells labeled by $x$ and sublattice index $\sigma = 1,2$, the DNLS equation takes the form: $$i\,\partial_t\psi_{\sigma,x} = \sum_{\sigma',x'}H^{\rm lin(\theta)}_{\sigma x,\sigma'x'}\,\psi_{\sigma',x'} + (g\,|\psi_{\sigma,x}|^2+g_{12}\,|\psi_{\bar\sigma,x}|^2)\,\psi_{\sigma,x},$$ where $H^{\rm lin(\theta)}$ is the Bloch Hamiltonian with $k$-dependent hopping amplitudes and pseudo-spin terms: $$H^{\rm lin}(k;\theta) = (m_z+J_1\cos k)\,\sigma_z + J_1'\,\sin k\,\sigma_y + J_2\,\sigma_x,$$ with parameters $m_z=m_0+\cos\theta$, $J_1=J_1'=1$, $J_2=\sin\theta$, and where $\theta(t)$ is varied slowly across $[0,2\pi]$ over a pump cycle.

Nonlinearity arises via $g$ (intra-component) and $g_{12}$ (inter-component) coefficients, controlling the interaction-induced self-focusing or defocusing.

Cold-atom and photonic realizations often begin from a continuous model (e.g., the Gross–Pitaevskii equation) and project onto Wannier orbitals for the tight-binding limit, importing the nonlinearity into the hopping and on-site terms.

2. Soliton Displacement, Wannier Center Flow, and Topology

The key to quantized pumping is the evolution of maximally localized Wannier functions of the instantaneous linear Hamiltonian, denoted $\{W_n(x;\theta)\}$. The Wannier center is: $X_W(\theta) = \sum_{n,x} n |W_n(x;\theta)|^2,$ with $n$ labeling unit cells. For linear, uniformly band-filled systems, the Thouless formula gives the center-of-mass shift per pump cycle: $$\Delta X_W = \int_0^{2\pi} d\theta\, \frac{dX_W}{d\theta} = \int_0^{2\pi} d\theta\, \mathcal{F}_{k\theta} = C,$$ where $\mathcal{F}_{k\theta}$ is the Berry curvature and $C$ the Chern number of the band. In adiabatic nonlinear pumping, for a soliton adiabatically following a single Wannier-branch,

$\Delta x = \Delta X_W = C.$

This persists even for bright solitons with localization over a few sites and for dark solitons (which effectively map to a vacancy on a nonlinear Bloch background), provided the system remains in the single-band or weakly hybridized regime.

3. Novel Deviations: Anomalous and Nonlinearity-Induced Quantized Pumping

Two primary non-conventional behaviors have been established (Tao et al., 29 Sep 2024):

• Anomalous doubled displacement: For a range of parameters (e.g., $m_0=1$, $C=-1$, $g>0$, $g_{12} \to 0$), the soliton can track not a single Wannier branch but a composite branch connecting neighboring Wannier centers. As a result, the soliton's center-of-mass advances by $2C$ unit cells ($\Delta x = 2C$), in contrast to the standard $C$.
• Nonlinearity-induced integer pumping in a trivial band: By modulating all parameters so that the instantaneous linear Chern number is zero—i.e., the band is topologically trivial—but varying $g_{12}(\theta)$ non-adiabatically, a soliton can still shift by $\pm1$ per pump cycle. This is enabled by emergent multi-Wannier branches in the nonlinear eigenproblem.

At $\theta = \pi$, the linear system becomes equivalent to an SSH chain with Wannier functions at half-integer cell positions. The nonlinear system hosts two types of localized solutions: (i) usual single-Wannier center solitons, (ii) new mixed-Wannier branches,

$\phi(x) \approx w_\ell W_n(x) + w_u W_{n+1}(x),$

with the mixing angle $\varphi$ determined by a transcendental equation depending on $g$, $g_{12}$, and $N$. When this mixed branch is stable, it enforces the $2C$-cell jump.

These effects are confirmed by:

• Stability analysis via the Bogoliubov–de Gennes formalism: all relevant eigenvalues satisfy $|\Im \omega| < 10^{-7}$ in the regime of interest.
• Direct time-evolution (adiabatic driving) of the DNLS, matching the center-of-mass trajectory predicted by the instantaneous nonlinear eigenstate construction.
• Simulations in both discrete and continuum models with neutral-atom and photonic parameters.

4. Generalization: Synthetic Nonlinearity, Fractional Pumping, and Multi-Band Physics

The paradigm can be further generalized in "synthetic nonlinearity" protocols (Maisuriya et al., 4 Nov 2025), where a threshold cutoff $J_{\rm th}$ is imposed on inter-site couplings of a modulated Aubry–André–Harper (AAH) model. In this synthetic-nonlinear lattice, exact correspondence is restored between quantized soliton pumping and Wannier function transport of the underlying band: $\Delta x = C_n,$ for a Wannier function (or highly localized soliton-like state) of band $n$.

In cases where multiple bands become degenerate (due to synthetic nonlinearity or strong real nonlinearity), one constructs composite Wannier functions for the degenerate subspace. Pumping becomes fractionally quantized,

$f = \frac{C_{\text{total}}}{N_d},$

with $C_{\text{total}}$ the sum of Chern numbers of the $N_d$ degenerate bands. This has been observed experimentally and numerically as robust, fractionally quantized shifts per cycle—e.g., $f = -1/2, 2/5, 1/7$—depending on the number and topology of degenerate bands.

This uncovering of a unified approach was further advanced using non-Abelian Berry connections and corresponding non-Abelian Chern numbers as universal invariants in multi-band scenarios (Wu et al., 10 Jun 2025).

5. Experimental Platforms and Physical Regimes

Quantized soliton pumping is directly accessible in multiple experimental platforms:

• Photonic waveguide arrays using Kerr nonlinearity, where system parameters (hopping, on-site energies) are modulated using femtosecond laser inscription and operating in the regime $|\dot{\theta}| \ll \Delta$ (the spectral gap).
• Ultracold atom gases in optical lattices with tunable mean-field interactions (Feshbach resonance control), using optical superlattices or spin-dependent lattices to implement pump cycles. In these systems, $gN$ is typically set near the bandwidth of the underlying tight-binding model for stability of new soliton branches.
• Superconducting circuits and LC resonator arrays, by dynamically tuning couplers to mimic synthetic nonlinearity.

Measurement is typically achieved by imaging the center-of-mass of the soliton (or atomic cloud) before and after a pump cycle, or by light-beam centroid shifts in photonic chips.

Tables of key parameter requirements:

Parameter/Condition	Value, Regime, or Constraint	Experimental Control
Pump period $T$	$T \gg 1/\Delta$ (adiabaticity)	Lattice sliding frequency
Nonlinearity $gN$	$\sim$ bandwidth, moderate	Feshbach tuning, input power
Inter-component $g_{12}$	$0 \leq g_{12} \lesssim g$ for double pumps	Feshbach, chip design
Synthetic threshold $J_{\rm th}$	Bifurcation point of soliton branch	Coupling modulations
Disorder	Up to 10% (cont.), 1% (discrete): robust	Realistic fabrication

Stable anomalous and nonlinearity-induced pumping requires moderate nonlinearity—too high, and solitons can become arrested or fragmented; too low, and topological transport reverts to linear Chern-number-quantized bands.

6. Robustness and Limitations

Numerical and analytic studies confirm that quantized soliton pumping is robust against:

• Weak disorder in the underlying potential (confirmed up to 10% disorder in continuous and 1% in discrete models).
• Localized impurities: solitons traverse finite-strength impurities without loss of quantization in the pumped displacement (Cao et al., 5 Mar 2024).
• Weak-to-moderate dissipation, in both conservative and dissipative systems (e.g., with gain and spectral filtering), where a dissipative topological "force" can induce phase transitions between trapped and quantized drifting regimes (Cao et al., 5 Sep 2024).

However, breakdown of quantization occurs:

• At strong nonlinearity, where the instantaneous nonlinear bands develop loops, self-crossings, or band mergers, nonadiabatic Landau–Zener transitions occur (evidenced by plateaux and breakdown in the pumped shift) (Tuloup et al., 2022, Bohm et al., 30 May 2025, Fu et al., 2021).
• In multi-band regimes, the net displacement per cycle is determined by a holonomy or a fractional topological invariant, and quantization can vanish when all bands merge or higher-order transitions are encountered.

A key experimental consideration is that the transition between conventional, anomalous, and fractional pumping is sharp—manifested as abrupt changes in the center-of-mass shift as a function of nonlinearity or system parameters.

7. Broader Implications, Related Phenomena, and Generalizations

Quantized soliton pumping extends classical topological transport to the nonlinear, few-mode regime—a setting where the traditional notion of a filled band is not applicable. Two decisive implications are:

• Nonlinearity can drive quantized transport even outside the reach of linear band topology, including integer pumping in topologically trivial bands and fractional pumping in hybridized multi-band settings.
• Generalization to broad families of solitons: Dipole, tripole, multi-frequency quadratic solitons, and vector solitons can all undergo quantized or fractionally quantized topological pumping, with stability and quantization windows governed by power, nonlocality, and the overlap between soliton and band structure (Ye et al., 8 Jul 2025, Kartashov et al., 15 Jan 2025, Cao et al., 7 Nov 2024, Tao et al., 9 Aug 2025). In nonlocal and quadratic media, high-power robustness and absence of fractionalization at large amplitude have been noted.

Fractional quantization observed in both mean-field and full quantum many-body models is controlled by non-Abelian Chern numbers, composite or auxiliary invariants, and the merging/degeneracy of soliton bands (Bohm et al., 30 May 2025, Bai et al., 7 Jul 2025).

These results have far-reaching consequences for the engineering of robust, quantized information transport in optical, atomic, and solid-state platforms, offering new routes toward devices where topological pumping can be dynamically tuned via nonlinearity or system architecture.