Quantization of Physical Interaction Strengths via Singular Moduli

Abstract

Since the 2019 redefinition of the SI units, precision metrology has sought to anchor all physical quantities to fundamental constants and integer invariants. While the optical frequency comb revolutionized timekeeping by discretizing the continuum of light into countable teeth, and the Quantum Hall Effect standardized resistance via topological invariants, a comparable standard for interaction strength remains elusive. Currently, measuring the coupling constant ($g$) between quantum systems is an estimation problem, inherently subject to drift, noise, and fabrication variance. Here, we introduce Interaction Metrology, a protocol that transforms the measurement of coupling strengths from an analog estimation into a topological counting problem. By engineering a specific class of algebraic catastrophe -- the Unimodal $X_9$ singularity -- in a driven-dissipative lattice, we prove that the system's interaction moduli are topologically forced to take discrete, quantized values, forming a "Geometric $k$-Comb." We derive the universality class of this quantization, showing that it arises from the discrepancy between the Milnor ($μ$) and Tjurina ($τ$) numbers of the effective potential, a strictly non-Hermitian effect forbidden in standard quantum mechanics. Finally, we provide an ab-initio blueprint for a silicon nitride implementation, demonstrating that this quantization is robust against disorder levels exceeding current foundry tolerances. This discovery establishes a universal standard for force sensing and quantum logic gates, enabling the calibration of interaction strengths with topological certainty.

Figures (5)

• Figure 1: Concept of the Geometric k-Space Comb. (a) Physical Lattice: A chain of coupled Kerr Dimers (Resonators A and B). The cross-coupling strength $J_{cross}$ acts as the "Interaction Modulus" $a$. (b) Modulus Dispersion: Just as energy varies with momentum in a band structure, the effective coupling $a(k)$ varies continuously across the Brillouin zone (Purple curve). (c) The geometric k-Space Comb: The underlying topology acts as a selection rule, permitting stable solutions only at effectively quantized levels (Horizontal lines). The k-Comb "teeth" emerge exactly where the continuous physics intersects these topological stability conditions, creating a robust, discretized spectrum.
• Figure 2: The Moduli Stability Diagram. (a) Arnold Tongues: Stability regions in the Detuning ($\Delta$) vs. Modulus ($a$) plane. The system is stable (colored lobes) only in the vicinity of the quantized topological levels ($n=6,7,8,9,10$). (b) Exceptional Point Surface: The 3D surface shows the Pump Power ($P$) required for stability. Deep "Valleys" align with the topological moduli, indicating that the k-Space Comb states have a minimized lasing threshold, providing a natural energetic advantage.
• Figure 3: Topological Robustness. (a) Fidelity Phase Diagram: Comparison of Spectral Fidelity vs. Lattice Disorder. The Standard Microcomb (Red) degrades exponentially. The Geometric k-Comb (Green) exhibits a Topological Plateau, maintaining high fidelity up to a critical breakdown at $\sim 15\%$ disorder. (b) Spectral Pinning: Superposition of 15 disorder realizations. Trivial states (Red) drift and broaden continuously. The Topological k-Comb lines (Green) remain pinned to the quantized momenta, exhibiting only amplitude fluctuations but no frequency shift.
• Figure 4: Proposed Photonic Architecture. An illustrative blueprint of the SiN integrated circuit. The array consists of coupled Kerr Racetrack Dimers ($\sim 100$ sites). The evanescent coupling $a(k)$ is structurally engineered via the gap distance, while TiN heaters provide dynamic control over the topological phase transition.
• Figure 5: Predicted Performance. (a) Coherence Prediction: Comparison of the simulated RF beat note for standard (Red) vs. topological (Green) designs. The theory predicts a significant linewidth narrowing, confirming the suppression of phase noise. (b) Predicted Spectrum: Simulation of the output comb in the topological phase ($\mu=9$), showing a robust soliton envelope.