Universal Quartic Scaling Law for Kerr-Type Interactions: Projection-Law Factorization Across Nonlinear Quantum Platforms

TL;DR

This paper derives and validates a universal projection-law for Kerr-type nonlinear interactions across various quantum platforms, enabling accurate predictions and a computational toolkit for diverse systems.

Contribution

It introduces a universal factorization of Kerr interactions into a projection coefficient and energy scale, validated across platforms with a practical computational tool.

Findings

Predicted cross-Kerr rate for superconducting device matches experimental measurement within 1.4%.

Validated the universality of the scaling law across five experiments spanning eight orders of magnitude.

Established that the main uncertainty arises from Josephson-energy extraction, not the projection factor.

Abstract

We present a rigorous derivation and numerical validation of a universal projection-law factorization for quartic nonlinear coupling rates across physically distinct platforms. The central result is that observable Kerr-type interactions -- self-Kerr, cross-Kerr, and cross-phase modulation -- factorize into a dimensionless projection coefficient and an intrinsic quartic energy scale. This structure follows from canonical quantization of a quartic interaction projected onto a finite normal-mode basis. We state this result as a formal proposition with clearly enumerated assumptions, provide a proof sketch based on second quantization of the anharmonic potential, and enumerate the domain of validity. We then implement the scaling law as a lightweight computational toolkit (UEFT-Designer) with platform-specific kernels for superconducting circuits, photonic microcavities, and epsilon-near-zero (ENZ) structures. A complete, step-by-step worked example for a superconducting quarton device -- with full uncertainty propagation -- predicts a cross-Kerr rate $\chi/2\pi = 361\pm 13\,\mathrm{MHz}$, agreeing with the independently measured value of $366\pm 0.5\,\mathrm{MHz}$ \cite{Ye2025} to within 1.4\%. A formal error-propagation analysis establishes that the dominant uncertainty is the Josephson-energy extraction uncertainty ($\sim\!2\%$), not the geometric projection factor. Cross-platform validation against five independent experiments spanning eight orders of magnitude in coupling strength confirms the universality of the factorization to within reported experimental uncertainties. The ghost-sector spectral correction introduced in earlier versions is reframed as an optional phenomenological self-energy model with no mandatory role in the validated scaling law.