Photon-Based Vacuum-Response Tomography: An Inverse-Problem Framework

This work introduces a rigorous methodological framework for Photon-based Vacuum-Response Tomography, reformulating the study of nonlinear quantum vacuum response as a formal inverse problem. In the low-energy regime, the effective vacuum state is characterized by Euler-Heisenberg-type nonlinearities in the CP-even sector inducing state-dependent birefringence and an axion-type magnetoelectric response (θ-electrodynamics) in the CP-odd sector. The framework demonstrates that spatial gradients or θ-interfaces behave as lossless Hall sheets, which fundamentally modify photon polarization. The analytical core of the manuscript is the derivation of a unified forward operator that integrates eikonal ray tracing with polarization transport (Jones calculus and Stokes parameters) in effective uniaxial media. This operator accounts for complex scattering at topological interfaces, including TE/TM mode mixing at oblique incidence. Addressing the critical challenge of parameter identifiability, the work proposes multi-dimensional measurement protocols utilizing frequency, angle, and state diversification to decouple vacuum signals from background-field uncertainties. The theoretical framework is further grounded in a microscopic closure where Kaluza-Klein towers determine the nonlinear coupling coefficients, treating the Wilson-line holonomy and radion as dynamical vacuum variables. Additionally, a gravi-optical extension is outlined, coupling the vacuum’s electromagnetic response to weak-field metric lensing. The stability of the proposed reconstruction is verified through synthetic benchmarks and uncertainty quantification, providing a robust algorithmic foundation for the next generation of high-precision QED experiments.