Adaptive Frequency Scanning Mechanism

• AFSM is a method that adaptively controls frequency scanning using context-driven adjustments, integrating strategies like phase modulation and dwell scheduling.
• It leverages both hardware and algorithmic approaches to ensure linear tuning, low noise, and efficient spectral coverage across diverse physical and computational domains.
• Applications range from high-resolution atomic spectroscopy and dark-matter experiments to frequency-aware neural sequence models, demonstrating its versatility and impact.

The Adaptive Frequency Scanning Mechanism (AFSM) is a methodological principle and architectural strategy employed across diverse domains—including atomic physics, nonlinear dynamical systems, dark-matter detection, and neural sequence modeling—to enable continuous, efficient, and adaptively controlled frequency scanning. AFSM leverages context-sensitive adaptation of frequency, path, or dwell strategies, often integrating slow–fast dynamical systems theory or frequency-domain signal priors, to optimize signal tracking, spectral coverage, or feature extraction. Implementations range from hardware-based spectral tuning via phase modulation in optics to algorithmic control in oscillators and parameter scheduling in experimental frequency scans, as well as subgraph-oriented traversals in neural architectures (Rajasree et al., 2021, Righetti et al., 2021, Zhang et al., 2023, Pan et al., 3 Dec 2025).

1. AFSM in Frequency-Stabilized Atomic and Molecular Spectroscopy

In optical spectroscopy, the AFSM enables continuous, wideband frequency tuning of a target laser locked to a distant reference, essential for high-resolution atomic, molecular, and Rydberg-state spectroscopy. In the method demonstrated by Yudkoff et al., a three-level ladder EIT resonance in $^{87}$Rb vapor phase-coherently links two lasers separated by $\sim 300$ nm. The target (482 nm) laser is locked via EIT to the reference (780 nm) laser. By phase-modulating the reference laser with a broadband waveguide EOM, the AFSM drags the EIT resonance—and thus the locked target laser frequency—over a $>1.6$ GHz range with sub-MHz stability and constant output power. The scan is realized by sweeping the RF drive to generate and select the appropriate sideband, shifting the resonance via

$$\Delta\nu_{\rm tgt} = -m\;\frac{\Delta\omega_m}{2\pi}$$

The approach yields linear tuning (with slope $\sim0.9$–0.95), low residual nonlinearity ($<2\%$), and stable output unaffected by amplitude noise due to pure phase modulation. The scan range is ultimately limited by the Doppler-broadened atomic transition, while feedback control ensures robust resonance tracking as the drive frequency is swept (Rajasree et al., 2021).

2. AFSM in Slow–Fast Adaptive Oscillator Networks

The AFSM as a dynamical systems mechanism was formalized to address limitations in standard oscillators: finite synchronization region and loss of learned information after driving removal. In this context, the AFSM augments a phase oscillator's dynamics with an adaptive internal frequency parameter $\omega(t)$, modulated according to the sign changes (zero-crossings) of a driving input $F(t)$. The governing equations are

$$\begin{aligned} \dot\phi & = \lambda\,\omega - K\,\sin\phi\,F(t) \ \dot\omega & = -K\,\sin\phi\,F(t) \end{aligned}$$

for gain $\lambda$ and coupling $K\gg 1$. Fast–slow decomposition reveals a layered structure of invariant slow manifolds $M_k^\epsilon$ corresponding to phase–frequency relationships:

$\Omega + \omega = k\pi, \qquad F(\theta) \neq 0$

The adaptation proceeds via exponential convergence on each attracting sheet, punctuated by fast jumps as $F(t)$ crosses zero, causing the system to switch manifolds. This process enables:

• Infinite synchronization region: The frequency range to which adaptation can occur is unbounded for $K>0$.
• Exponential adaptation: Rapid locking of $\omega$ to the dominant $F(t)$ frequency.
• Memory retention: On termination of $F(t)$, the learned frequency persists indefinitely.

In oscillator networks with amplitude adaptation, the AFSM enables real-time spectral decomposition of external inputs, with each oscillator locking to and cancelling a distinct harmonic component (Righetti et al., 2021).

3. Adaptive Frequency Scanning in Dark-Matter ALP Experiments

For axionlike particle (ALP) dark matter detection via nuclear magnetic resonance (NMR), AFSM refers to optimized scanning of the Larmor frequency across a broad range to efficiently search for resonant spin-precession signatures induced by the ALP field. The method dynamically schedules the frequency step size ($\Delta\omega$) and dwell time ($T$) at each step according to the scaling of detection bandwidth:

$$\Delta\omega = \max\left\{\frac{\omega}{Q_a},\,\frac{2}{T_2^*}\right\}$$

$T(\omega) = \kappa\,\omega^n$

where $Q_a$ is ALP coherence, $T_2^*$ is transverse relaxation, and $n$ is chosen to maximize $(\omega,g)$-space coverage (e.g., $n=0$ for $\Delta\omega\propto\omega$, $n=-1$ when $\Delta\omega$ is constant). An explicit algorithm prioritizes parameter-space area coverage per total measurement time, with continuous updating as experimental conditions change. Adaptive dwell laws have been shown to slightly increase sensitivity and log-area coverage compared to naive uniform scans, especially at low frequency where ALP linewidths dominate (Zhang et al., 2023).

4. AFSM in Frequency-Aware Neural Sequence Models

In deep learning, AFSM is implemented as a zero-parameter path-selection scheme for sequential state-space models in the frequency domain, notably in the Frequency-Aware Mamba (FA-Mamba) architecture for adverse weather image restoration. Standard spatial models apply rigid row/column ordering or treat wavelet subbands independently. AFSM exploits the intrinsic structure of subbands from a wavelet transform (LL, LH, HL, HH), scanning each with a path matched to its dominant texture orientation:

• LL, HL: horizontal scan
• LH: vertical scan
• HH: diagonal scan

By concatenating the flattened subband traversals, a single Mamba SSM processes the full spectrum of context, integrating local-global and cross-frequency cues. The method tangibly improves recovery of fine detail, as quantified by performance gains (e.g., +0.14 dB PSNR, +0.003 SSIM on Raindrop dataset) over prior 2D scan or spatial baselines, without trainable overhead. The explicit pseudocode in the original paper outlines transform, scan, SSM application, and reconstruction steps (Pan et al., 3 Dec 2025).

Domain	Physical/Algorithmic Substrate	Adaptation Modality
Laser spectroscopy	EIT-coupled lasers, EOM	RF-sideband sweep for frequency lock
Adaptive oscillators	Phase–frequency slow–fast ODEs	Input-zero–crossing-driven jumps
ALP spin-precession NMR	Field-swept NMR, acquisition loop	Dwell schedule matching bandwidth
Deep learning (FA-Mamba)	Wavelet+Mamba sequence model	Subband traversal path selection

5. Comparative Analysis and Implementation Considerations

Common to all AFSM variants is the leverage of context-driven adaptation of scan parameters—physical (frequency, phase, bias field), temporal (dwell time), or topological (sequence path)—to maximize signal relevance, stability, or efficiency. Distinct approaches include hardware phase modulation, feedback and control loops (in atomic physics), hybrid continuous–discrete dynamical system design (adaptive oscillators), algorithmic scan orchestration (dark-matter NMR), and structured permutation for neural sequence models.

Limitations vary by application: scan range may be delimited by physical linewidths or transition broadening (atomic case), bandwidth of feedback actuators, or computational constraints (deep learning). In neural architectures, AFSM does not introduce additional learnable parameters, maintaining complexity at $O(L\,m)$ with $L$ the total sequence length (Pan et al., 3 Dec 2025).

6. Application Scenarios and Future Directions

AFSM protocols are critical in contexts demanding adaptive, continuous, and efficient spectral scanning or feature extraction:

• High-resolution atomic and molecular spectroscopy (Rydberg states, phase-coherent transport, STIRAP).
• Adaptive spectral-tracking and memory-retaining oscillators for signal synthesis, observer-based control, and prostheses.
• Large-parameter-space search in dark matter and new-particle detection experiments, where efficient coverage is essential for sensitivity.
• Frequency-domain feature learning in image restoration and related tasks.

Potential future extensions include higher-order phase modulation for expanded scan range, optical phase-locked loops for fine-tuned frequency control, integration of amplitude adaptation in oscillator networks for richer spectral learning, and exploration of learned (rather than fixed) traversal paths in neural sequence models for further performance gains (Rajasree et al., 2021, Righetti et al., 2021, Zhang et al., 2023, Pan et al., 3 Dec 2025).