Disordered Nonlinear Pumps

• Disordered nonlinear pumps are systems where optical pumping, spatial disorder, and nonlinear interactions combine to yield unique light localization and amplification phenomena.
• Theoretical models use variants of the nonlinear Schrödinger equation with spatially modulated gain or nonlinearity to capture disorder effects and topological transport.
• Experimental studies in multimode fiber amplifiers and Thouless pumps demonstrate robust control of light propagation through engineered disorder and nonlinear tuning.

Disordered nonlinear pumps are systems in which optical pumping, nonlinear interactions, and spatial disorder interplay to produce distinctive light localization, amplification, and transport phenomena. These systems span a diverse set of physical implementations, including multimode fiber amplifiers and lasers with complex gain landscapes, nonlinear Thouless pumps of solitons in disordered lattices, and photonic structures where disorder and nonlinearity drive localization and transmission effects beyond what is seen in purely linear disordered or homogeneous nonlinear media. The paper of disordered nonlinear pumps illuminates the mechanisms by which structured or random disorder—when coupled with nonlinear gain or refractive index changes—yields optically controllable behavior, robust topological transport, and new classes of instability thresholds and phase diagrams inaccessible in linear or ordered settings.

1. Theoretical Models for Disordered Nonlinear Pumps

The physics of disordered nonlinear pumps is governed by variants of the nonlinear Schrödinger equation (NLSE) with complex spatial modulation of gain or nonlinearity and, in some cases, explicit time- or propagation-dependent pump configurations.

In multimode fiber amplifiers and lasers, the scalar, paraxial NLSE with a spatially varying gain term $g(x,y,z)$ is used:

$$i \frac{\partial E}{\partial z} + \frac{1}{2k_0} \nabla_\perp^2 E + k_0 n_2 |E|^2 E = i [g(x,y,z)/2] E$$

where $k_0=2\pi n_0/\lambda$ and $n_2$ is the Kerr nonlinearity coefficient. The gain profile $g(x,y,z)$ is itself a functional of the pump intensity and spatial disorder, often structured via spatial light modulators (SLMs) (Sperber et al., 2020).

In Thouless pumps for solitons, the model is a time-dependent NLSE or Gross-Pitaevskii equation subject to both a temporally modulated (pumping) potential and local disorder (impurity):

$$i \hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + g |\psi|^2 + V_\text{pump}(x,t) + V_\text{imp}(x) \right] \psi$$

where $V_\text{pump}(x,t)$ is a spatially and temporally periodic lattice, and $V_\text{imp}(x)$ captures the disorder, ranging from $\delta$-function to Gaussian impurity models (Cao et al., 5 Mar 2024, Chaudhari et al., 12 Dec 2025).

In quadratic ($\chi^{(2)}$) or cubic ($\chi^{(3)}$) nonlinear media with disordered nonlinear susceptibility, the governing equations become coupled-mode equations with spatially random nonlinear coefficients:

• For $\chi^{(2)}$:

$$i\partial_z A_1 + \partial_x^2 A_1 + d(x) A_1^* A_2 = 0, \quad i\partial_z A_2 + \frac{1}{2}\partial_x^2 A_2 + \delta k A_2 + \frac{1}{2} d(x) A_1^2 = 0$$

with $d(x)$ as the random susceptibility (Folli et al., 2013).

2. Disorder–Nonlinearity–Pump Interplay

The key phenomenology of disordered nonlinear pumps emerges from the way spatial disorder, nonlinear response, and dynamic pumping shape the gain or refractive index landscape.

• In multimode fibers, disorder arises from the complex modal structure: the superposition of many waveguide modes yields “speckle-like” pump intensity patterns, and thus a spatially heterogeneous gain $g(x,y,z)$. SLMs allow phase shaping of the pump to configure this disorder adaptively, thus sculpting the propagation and amplification landscape for the signal field (Sperber et al., 2020).
• In nonlinear Thouless pumps, on-site or coupling disorder modulates the local energy landscape, creating impurity or defect regions; nonlinear self-localization (solitons) permits robust quasi-particle transport under topological pumping protocols. Disorder modifies local spectral gaps and localization, but nonlinear topology can protect quantized displacement (Cao et al., 5 Mar 2024, Chaudhari et al., 12 Dec 2025).
• In random nonlinear media, disorder in $\chi^{(2)}$ or $\chi^{(3)}$ enhances spatial localization of waves and enables disorder-induced parametric amplification. Localization length and gain become tunable via the pump intensity, unlike conventional Anderson localization, which is independent of the excitation level (Folli et al., 2013).

3. Experimental Implementations and Key Observables

Multimode Fiber Amplifiers and Lasers

• The gain medium is typically a few meters of rare-earth-doped graded-index fiber supporting many transverse modes.
• The pump is structured using a phase-only SLM, producing complex intensity patterns within the core.
• Two main operational regimes are considered: (i) the amplifier, where signal transmission matrices show sensitivity to the pump disorder, and (ii) the laser, where spectral lines and mode competition are controlled by pump shaping.
• Key observables are output speckle decorrelation, spectral selectivity, modal confinement, and lasing threshold shifts.

Nonlinear Thouless Pumps

• Implemented in optical waveguide arrays or atomic Bose-Einstein condensates.
• The transport of bright solitons across disorder is tracked under adiabatic pumping cycles, monitoring center-of-mass displacement per cycle and overlap with ideal (target) wavepackets.
• Topological soliton pumping is quantified through phase diagrams of output overlap $\mathcal{O}$ versus normalized nonlinear power and disorder strength, demonstrating robustness regimes and critical disorder thresholds.

Disordered Nonlinear Media

• Measurement focuses on parametric gain, localization length of generated modes, and control of these via input pump intensity.
• Localization is verified by exponential growth and spatial profile analysis, with direct measurement or simulation of the scaling relations between gain, localization, and pump intensity.

System	Disorder Source	Pump Control Scheme	Key Observable(s)
MMF amplifiers/lasers	Modal structure	SLM-shaped pump intensity	Speckle decorrelation, spectrum
Thouless soliton pumps	On-site/coupling	Temporal lattice modulation	Quantized soliton displacement
Random $\chi^{(2)}$/$\chi^{(3)}$ media	Nonlinear susceptibility	Uniform, high-intensity pump	Localization, parametric gain

4. Disorder-Robustness, Nonlinear Control, and Topological Effects

Robustness and Quantization

• In nonlinear Thouless pumps, displacement quantization survives errors and moderate disorder for $gP/J_m\gtrsim1$ and $V_m<V_\text{cr}(gP)$, with $V_\text{cr}/J_m$ increasing as nonlinearity increases. Quantized pumping fails when disorder closes the relevant spectral gap or exceeds critical strength (Chaudhari et al., 12 Dec 2025).
• In MMF lasers, spectral selectivity and single-mode stabilization can be achieved and maintained by pump shaping, with over 90% of output power in the targeted line even amid complex mode competition (Sperber et al., 2020).

Tunability and Control

• In purely nonlinear disorder-induced localization, gain and localization length follow power-law scaling with pump intensity:

$$\Gamma \propto I_2^{2/3}, \quad L_{\text{loc}} \propto I_2^{-1/3}$$

(for $\chi^{(2)}$ media, with analogous scaling in $\chi^{(3)}$) (Folli et al., 2013).

• In MMF systems, the “control window” for gain shaping is quantified by $\Delta g$, scaling with the ratio of emission to absorption cross-sections and rare-earth concentration, which can be optimized for device performance (Sperber et al., 2020).

Topological Protection

• The quantized shift of a soliton in a nonlinear Thouless pump is determined by the Chern number of the underlying band structure. This protection persists even in the presence of strong disorder as long as the drive remains adiabatic and the soliton remains within a single nonlinear band (Cao et al., 5 Mar 2024, Chaudhari et al., 12 Dec 2025).
• Topologically protected transport—unaffected by local impurities—enables concepts such as disorder-immune “soliton shift registers.”

5. Breakdown Phenomena and Phase Diagrams

• In open, driven-dissipative systems with strong nonlinearity and disorder (e.g., Jaynes–Cummings–Hubbard cavity arrays), even weak disorder combined with losses eliminates mean-field bistability. The ensemble-averaged response becomes smooth, and the phase distribution of the local field is randomized, destroying coherent multistable steady states (Kulaitis et al., 2012).
• Phase diagrams succinctly express the competitive balance between nonlinearity and disorder in maintaining quantized transport, localization, and spectral stability. In Thouless pumps, three regimes emerge: quantized transport, self-trapping, and mixed/partial trapping as functions of nonlinearity and impurity strength (Cao et al., 5 Mar 2024, Chaudhari et al., 12 Dec 2025).
• A plausible implication is that in many classes of disordered nonlinear pumps, the disorder-induced destruction of phase coherence can be the limiting factor in achieving robust multistability or coherent switching, unless system symmetries or topological invariants are invoked.

6. Extensions, Applications, and Future Directions

• Disordered nonlinear pump control has implications for integrated photonics, random lasers, topological photonic circuits, and robust information processing platforms employing optical solitons or speckle-sensitive amplifiers.
• Design principles derived from topologically robust quantized pumps support the realization of non-reciprocal light transport and routing, with device compactness and disorder tolerance enhanced by intrinsic nonlinearities (Chaudhari et al., 12 Dec 2025).
• Generalization to semiconductor disk lasers, quantum-cascade lasers, and microresonator architectures is feasible wherever spatially addressable gain or refractive index can be implemented (Sperber et al., 2020).
• The formalism of stochastic mean-field theory underpins a systematic approach to exploring phase diagrams and steady-state distributions in open nonlinear photonic lattices subject to disorder and pumping (Kulaitis et al., 2012).
• Further research is directed at extending disorder-immune transport to higher dimensions, exploring interplay with many-body physics in quantum simulators, and exploiting engineered disorder for ultrafast optical control in complex photonic networks.

7. Summary of Key Quantitative Findings

• Amplifier (MMF): up to $\sim 40\%$ pattern decorrelation at constant gain; up to $\sim 23\%$ focal-spot confinement enhancement in simulations. Laser: single-mode fractions $>$90%; threshold shifts sufficient to invert mode ordering (Sperber et al., 2020).
• Thouless pumps: critical disorder strength $V_\text{cr}/J_m$ scales with normalized power $gP/J_m$, enabling robust quantized soliton transport for moderate to high power solitons and compact propagation lengths $LJ_m \sim 10^1$–$10^2$ (versus $10^3$ in linear systems) (Chaudhari et al., 12 Dec 2025).
• Disordered $\chi^{(2)}$/$\chi^{(3)}$ media: parametric gain and localization length tunable via pump intensity, with universal scaling exponents, facilitating optically programmable disorder-induced amplification and confinement (Folli et al., 2013).