Dual-Frequency Topological Pumping

• Topologically pumping dual-frequency waveforms is defined as the quantized transfer of energy and physical observables through two distinct modulations governed by system topology.
• It employs a Floquet lattice framework with synthetic dimensions where Chern numbers and Berry curvature dictate universal transport properties across diverse platforms.
• Experimental implementations in photorefractive crystals, superconducting circuits, and atomic lattices confirm robust, metrologically relevant quantization under nonlinear and nonadiabatic conditions.

Topologically pumping dual-frequency waveforms refers to the quantized transfer of wave energy, soliton momentum, or physical observables—mediated by system topology and driven by two distinct temporal (or spatial) frequencies. This phenomenon generalizes Thouless pumps and quantum Hall analogues to systems modulated at two frequencies, often incommensurate, and manifests crucially in nonlinear optics, photorefractive crystals, driven quantum circuits, and classical metamaterials. The dual-frequency context introduces rich topological structures, universal transport, and distinctive nonlinear responses fundamentally governed by Berry curvature and Chern invariants.

1. Mathematical Frameworks and Model Hamiltonians

Dual-frequency topological pumping is formulated using dynamically modulated potentials with two distinct frequencies. In quadratic optical media, a typical model for the fundamental (FF) and second-harmonic (SH) waves employs coupled evolution equations:

$$\begin{align*} i\,\frac{\partial E_1}{\partial z} &= -\frac{1}{2}\,\frac{\partial^2 E_1}{\partial x^2} + V_1(x,z) E_1 + \kappa E_2^* e^{-i\Delta k z}, \ i\,\frac{\partial E_2}{\partial z} &= -\frac{1}{4}\,\frac{\partial^2 E_2}{\partial x^2} + V_2(x,z) E_2 + \kappa E_1^2 e^{+i\Delta k z}. \end{align*}$$

Here, $V_1(x,z)$ and $V_2(x,z)$ are dynamic optical potentials composed of two sliding sublattices (periods $d_1$, $d_2$) that move relative to each other with velocity $v$. This framework generalizes to photorefractive systems, acoustic metamaterials, and driven lattice models, in which temporal variables are replaced by a pair of periodic drives or quasiperiodic modulations.

The Floquet (synthetic-momentum) approach is essential when the drives are rapidly modulated, mapping the system to a higher-dimensional "Floquet lattice." For two drive frequencies $(\omega_1, \omega_2)$, the effective lattice is two-dimensional, governed by phase variables $\varphi_i = \omega_i t$, and the system dynamics become periodic in both coordinates. The resulting Hamiltonian exhibits a band structure with Chern numbers defined over the synthetic Brillouin zone (Martin et al., 2016).

2. Topological Quantization and Chern Numbers

A central feature of dual-frequency pumping is quantization arising from nontrivial topology—encoded in Chern invariants associated to the system's Floquet bands. For each band $\nu$ in the Bloch-Floquet spectrum, the space-time Chern number is:

$$C_\nu = \frac{i}{2\pi} \int_0^Z \int_{\mathrm{BZ}} \Big[ \langle \partial_z u_\nu | \partial_k u_\nu \rangle - \langle \partial_k u_\nu | \partial_z u_\nu \rangle \Big] \, dk dz,$$

where $u_\nu(k, z)$ is the normalized instantaneous Bloch wave. This invariant determines the quantized displacement, charge, or energy pumped per cycle. In quadratic media, both FF and SH fields inherit identical Chern numbers for topologically equivalent sublattices—verified by spectral homotopy (Kartashov et al., 15 Jan 2025). In photorefractive setups with dual sliding lattices, Chern quantization persists for each rational approximant in the case of quasi-periodic (irrational) frequency ratios, yielding quantized center-of-mass shifts for each supercell (Peng et al., 5 Sep 2025).

In quantum circuits, the Berry curvature two-form $F^{(\nu)}_{\theta_1 \theta_2}$ integrated over the torus gives the Chern number $C^{(\nu)}$ which controls quantized power transfer between classical modes (Luneau et al., 2021). For driven quantum systems, the pumping rates are strictly tied to topological invariants: $P_{12} = \frac{C}{2\pi} \hbar \omega_1 \omega_2$ (Martin et al., 2016).

3. Nonlinear Effects and Sharp Threshold Transitions

Unlike linear Thouless pumps, dual-frequency topological pumping in nonlinear (e.g., $\chi^{(2)}$) media displays a sharp power-dependent transition from a non-topological to a topological phase. As soliton power $U$ increases, the net shift per cycle exhibits a step-like behavior centered at a critical threshold $U_{th}(v)$ (dependent on sublattice velocity $v$):

$$U_{th}(v) \simeq U_0 + \alpha v^\mu, \quad \mu \lesssim 1,$$

where below $U_{th}$ the soliton's spectrum is too narrow, resulting in zero net motion, whereas above this threshold, the spectrum fills the full band and quantized transport $Δx = C X$ is established (Kartashov et al., 15 Jan 2025). The transition sharpens for higher $v$ and stronger nonlinearity.

Quadratic pumping discriminates sharply from cubic Kerr media, where high power can induce fractional or broken quantization. The parametric coupling in quadratic systems strongly suppresses interband scattering and fractional pumping, yielding robust quantization even well beyond the strict adiabatic regime (Kartashov et al., 15 Jan 2025).

4. Experimental Systems and Implementations

Topological pumping of dual-frequency waveforms is realized across diverse platforms:

• Photorefractive Crystals: Superposition of two sliding optical lattices in lithium niobate, with longitudinal periods scaled as successive Fibonacci numbers, enables dual-frequency quasi-periodic pumping. Each rational approximant yields quantized center-of-mass shifts, converging to golden ratio transport in the limit (Peng et al., 5 Sep 2025).
• Quantum Circuits: Superconducting qutrits driven by multiple modulated microwave tones implement Berry-curvature-induced quantized power transfer between cavity modes, tracked via sideband-resolved spectroscopy. Measurement protocols directly extract topological pumping plateaus corresponding to integer Chern numbers (Luneau et al., 2021).
• Atomic Lattices: Ultracold fermions in shaken optical lattices, dynamically modulated with $\omega$ and $2\omega$ drives, realize adiabatic Floquet–Thouless pumps with quantized center-of-mass displacement per cycle. Experimental procedures rely on frequency chirps and amplitude/phase cycling to prepare topological Floquet bands (Minguzzi et al., 2021).
• Acoustic Metamaterials: Aperiodic bilayered crystals with incommensurate layer spacings and time-driven phason windings topologically pump dual-frequency acoustic signals from one edge to the other, validating high-fidelity edge transport protocols (Cheng et al., 2020).

The following table summarizes representative system architectures:

Platform	Dual-Freq Drive Type	Observable
Photorefractive crystal	Sliding optical lattices	Center-of-mass shift
Superconducting qutrit circuit	Multi-tone microwave drives	Sideband power transfer
Ultracold atomic lattice	Mirror shaking at ω,$2ω$	Cloud displacement
Acoustic metamaterial	Layered phason pumping	Edge signal transmission

5. Floquet Lattices, Synthetic Dimensions, and Universal Transport

Dual-frequency pumping inherently utilizes synthetic lattices, with system parameters promoted to synthetic (momentum/phase) coordinates. For two irrational frequencies, the synthetic space becomes two-dimensional, hosting Bloch bands with topological invariants. System evolution under an effective "electric field" $E=(\omega_1,\omega_2)$ leads to transverse quantized transport analogous to the Hall effect (Martin et al., 2016).

Universal scaling arises in nonadiabatic regimes mapped to synthetic Weyl semimetals. The pumped energy per mode decomposes into topological (Fermi-function) and excitation (Gaussian) contributions, with universal functional forms:

$$\widetilde{P}_T(x) = \frac{1}{1+e^{\alpha x}}, \quad \widetilde{P}_E(x) = C \exp(-\beta x^2)$$

Here, $x$ is an adiabaticity parameter, and integration over synthetic momentum yields total pumping rates robust to disorder (Qi et al., 2021). This highlights the persistence of topological quantization under strong nonadiabaticity and spatial fluctuations.

6. Metrological and Technological Implications

Topologically quantized pumping is fundamentally robust: it is insensitive to disorder, drive imperfections, or detailed forms of the potential—provided the topological gap remains open. This robustness underpins proposals for AC current standards, quantum energy converters, and topologically protected signal transduction in photonics and acoustics (Minguzzi et al., 2021, Luneau et al., 2021).

Metrological protocols exploit the quantized center-of-mass displacement (or spectral sidebands) per pumping cycle, directly linked to topological invariants, offering high-precision implementation. The convergence of quasi-periodic and periodic dual-frequency pumps in finite systems extends topological quantization to realistic experimental settings (Peng et al., 5 Sep 2025).

7. Distinctive Features, Limitations, and Future Research Directions

Key distinctions of dual-frequency topological pumping include:

• Universality Across Domains: Demonstrated in optics, quantum circuits, atomic physics, and acoustics (Kartashov et al., 15 Jan 2025, Minguzzi et al., 2021, Luneau et al., 2021, Peng et al., 5 Sep 2025, Cheng et al., 2020).
• Absence of Broken/Fractional Quantization in Quadratic Media: Due to parametric coupling, quadratic systems do not exhibit the fractional pumping encountered in cubic/Kerr models (Kartashov et al., 15 Jan 2025).
• Finite-System and Irrational-Ratio Convergence: Bi-chromatic pumping governed by irrational frequency ratios is observable via sequences of rational approximants (Fibonacci fractions), providing a concrete path to experimental realization and quantization (Peng et al., 5 Sep 2025).
• Nonadiabatic Robustness: Both topological and excitation (heating) contributions to pumped energy survive substantial disorder and nonadiabaticity, up to universal prefactors (Qi et al., 2021).

Open questions and future avenues include further exploration of multi-frequency and quasi-periodic pumping in higher synthetic dimensions, systematic engineering of pumps in strongly interacting systems, and integration with quantum information architectures for topological energy protocols.