Thouless Pumping for Dark Solitons

• The topic is defined as dark solitons experiencing quantized transport in periodically modulated lattices, where topological invariants govern displacement.
• Adiabatic modulation in continuous and discrete models facilitates both integer and fractional pumping via underlying Wannier function movements.
• Robustness to disorder and nonlinear effects positions this phenomenon for applications in robust signal transport and topological device design.

Thouless pumping for dark solitons refers to the quantized (and, in some cases, fractionally quantized) transport of phase-defect nonlinear excitations—specifically, dark solitons—in periodically modulated one-dimensional lattices. This phenomenon combines nonlinear science, topological band theory, and soliton dynamics, demonstrating that localized phase defects can be pumped across a lattice by adiabatically varying a system parameter over one cycle, with their displacement directly tied to the topological properties of the underlying linear bands (Tao et al., 9 Aug 2025). Recent theoretical advances have established both integer and fractional pumping for dark solitons in continuous Gross–Pitaevskii systems and discrete tight-binding models, showing that their net displacement per cycle can be highly robust and governed by the movement of Wannier functions associated with the linear Hamiltonian.

1. Theoretical Framework and Models

The central analytic tools are the continuous defocusing Gross–Pitaevskii equation

$$i\partial_t \Psi(x,t) = \left[-\frac{1}{2}\partial_x^2 + V(x,\phi) + g|\Psi(x,t)|^2 \right] \Psi(x,t)$$

with a time-dependent lattice potential

$$V(x,\phi) = -V_1 \cos\left(\frac{2\pi x}{p_1}\right) - V_2 \cos\left(\frac{2\pi x}{p_2} - \phi\right)$$

and its discrete tight-binding analog, the discrete nonlinear Schrödinger equation (DNLS),

$$i \partial_t \Psi_x = \sum_{x'} H_{x,x'}^{(\mathrm{lin})}(\phi) \Psi_{x'} + g|\Psi_x|^2\Psi_x$$

where the hopping amplitudes are periodically modulated (e.g., in the off-diagonal AAH model) via

$$J_x(\phi) = J_0 + J_a \cos\left(\phi + \delta\phi - \frac{2\pi q(x-1)}{p}\right)$$

with $p$ sites per unit cell and $q$ integer.

A dark soliton is defined as a localized intensity dip on a nonlinear background solution (usually a Bloch state at $k=0$) with an abrupt phase change of approximately $\pi$. The dynamical procedure involves slowly (adiabatically) ramping the modulation parameter $\phi$ over a pump period $T$, amounting to a topological pump cycle.

2. Topological Quantization and Wannier Tracking

The quantization of soliton displacement is dictated by the topology of the linear bands—in particular, the Chern number,

$$C_n = \frac{1}{2\pi} \int_0^{2\pi} d\phi \int_0^{2\pi} dk\, \Omega_n(k,\phi)$$

with Berry curvature $$\Omega_n(k,\phi) = i \left[\langle \partial_k u_{nk} | \partial_\phi u_{nk} \rangle - \langle \partial_\phi u_{nk} | \partial_k u_{nk} \rangle\right]$$ and $u_{nk}$ the periodic part of the Bloch function.

A key insight is that, although the dark soliton is not itself a simple Wannier function (due to its phase-kink structure and localization on a nonlinear background), its dynamics are "dragged" by the underlying movement of Wannier centers as $\phi$ is cycled. This is formalized via a mapping to a bright-soliton–like wave packet,

$$\psi_b(x,\phi) = \psi_{\mathrm{bg}}(x,\phi) - |\psi(x,\phi)|$$

where $\psi_{\mathrm{bg}}$ is the background nonlinear Bloch state at $k=0$. The center-of-mass of the dark soliton is then

$$x_d(\phi) = \frac{\int x [|\psi_{\mathrm{bg}}|^2 - |\psi|^2] dx}{\int [|\psi_{\mathrm{bg}}|^2 - |\psi|^2] dx}$$

which closely tracks the center of a Wannier function, itself shifted by the Chern number after a cycle. The process thus realizes a quantized Thouless pump for the dark soliton (Tao et al., 9 Aug 2025).

3. Integer and Fractional Pumping Regimes

Dark soliton transport manifests as either integer or fractional pumping, depending on the occupation of the underlying bands:

• Integer pumping arises when the mapped bright-soliton wave packet occupies predominantly a single band with Chern number $C_1$; the net dark soliton displacement per period is one unit cell ($\Delta x = C_1$).
• Fractional pumping occurs if the soliton's mapped component has significant, typically equal, occupancy in multiple bands (e.g., two bands with $C_1,C_2$); then, the displacement per period becomes a rational fraction, such as $(C_1+C_2)/2$ unit cells. For instance, when two bands each contribute half, the dark soliton is transported by half a unit cell per cycle, or one unit cell over two cycles.

Pumping is robust and quantized as long as the soliton remains localized and continues to track the instantaneous Wannier center(s). The connection with the Chern number holds in both continuous and tight-binding systems.

4. Effect of Nonlinearity and Symmetry

Nonlinearity is crucial: it stabilizes the soliton and modifies the on-site potentials via $g_x|\psi_x(\theta)|^2$, effectively dressing the linear Hamiltonian and in some cases dynamically inducing nontrivial topology (as in models where all linear bands are trivial, yet the soliton exhibits quantized or fractional pumping) (Tao et al., 10 Feb 2025). The nonlinear modification, especially in the presence of strong or spatially inhomogeneous nonlinearities, can create effective supercells with nonzero effective Chern numbers.

Symmetries (such as parity-time symmetry in the model Hamiltonian) can protect fractional pumping plateaux, ensuring that, e.g., the center-of-mass is shifted exactly by $n/2$ unit cells at half a cycle, a consequence of functional relations like

$$x_c(m\Theta - \theta) + x_c(\theta) = \text{constant}$$

with $m\Theta/2$ the midpoint of the pump cycle (Tao et al., 10 Feb 2025).

5. Experimental Implementation and Extensions

Experimental realizations are feasible on both photonic and ultracold atom platforms. For example, arrays of coupled optical waveguides, fabricated to implement off-diagonal AAH models and capable of supporting defocusing Kerr nonlinearity, can be used to observe dark soliton pumping. In these implementations, the propagation distance plays the role of time. Typical parameters yield pump cycles of order $10$–$50$ meters (Tao et al., 10 Feb 2025).

Controlling parameters such as the number of sites per unit cell ($p$), the modulation profile of the hopping ($J_x(\theta)$), and the nonlinearity $g_x$, experiments can realize both integer and fractional plateaux in soliton transport, with displacements of $1$, $1/2$, $1/3$, $1/4$ unit cells per cycle, etc.

Theoretical proposals indicate that these phenomena are not restricted to bright or gap solitons—dark solitons, vortex solitons, ring dark solitons, and potentially even higher-dimensional defects are all eligible for topological pumping if the system supports appropriate nonlinear stationary states embedded in a topologically nontrivial framework (Tao et al., 9 Aug 2025).

6. Topological Protection and Applications

The topological quantization of pumping implies robustness to disorder, impurities, and imperfections in the background potential or nonlinearity. The displacement per cycle is set by the global topology (Chern number) of the underlying (possibly dressed) band structure, not by local details. This makes Thouless-pumped dark solitons attractive for robust signal transport, soliton-based logic, quantum information encoding and routing, and other device applications seeking error-resilient information carriers.

A significant implication is the extension of the bulk-edge correspondence and topological invariants to nonlinear eigenvalue problems, as established for fractional Thouless pumping of solitons in more complex band structures with next-nearest-neighbor hopping (Bai et al., 7 Jul 2025). Nonlinear eigenvalue nonlinearity can generate phenomena—such as fractional pumping or emergent edge modes—not seen in linear systems, enabling the controlled design of novel topological phases and edge state manipulation.

7. Outlook and Generalizations

The confirmation of Thouless pumping for dark solitons advances the frontiers of nonlinear topological physics, establishing that topologically protected transport is possible for phase-kink defects fundamentally distinct from bright solitons or linear wave packets. Immediate research directions include:

• Exploring pumping of more complex nonlinear defects (vortex, ring, multi-component dark solitons).
• Determining the limits of robustness in the presence of strong interactions, disorder, or non-adiabatic driving.
• Leveraging topologically protected dark soliton transport for metrology, atomtronics, or photonic circuit applications.
• Studying higher-dimensional generalizations and systems with additional degrees of freedom (e.g., spinor condensates, nonlocal media) (Ye et al., 8 Jul 2025, Cao et al., 7 Nov 2024).

The rigorous connection between soliton dynamics, band topology, and quantized transport established for dark solitons thus opens rich possibilities in both fundamental physics and the design of functional nonlinear topological devices.