Cavity-Assisted Electrostatic Braiding and Readout of Fractional Quantum Hall Anyons

We introduce a cavity-assisted near-field electrostatic architecture for conditional braiding of localized fractional quantum Hall quasiholes and heterodyne probing of braid-sensitive signatures, with particular relevance to candidate non-Abelian quasiholes at filling factor $\nu=5/2$. Counter-propagating cavity branches select oppositely rotating electrostatic trap trajectories, so the photonic state controls braid orientation. We derive the leading cavity cross-coherence and the conditions for bounded quasihole following, including two-dimensional confinement, mode selectivity, adiabaticity, and phase stability. For Ising anyons, the symmetric inverse-branch protocol yields only a calibrated phase for every pure braid and state, and is therefore blind to non-Abelian information. An asymmetric branch/reference protocol instead accesses a single braid power; normalized two-mode heterodyne cross-coherence reveals the braid through suppression of the braid-run first moment relative to a resolved nonzero matched trivial reference, when the Abelian phase is stabilized or tracked. A computational-fusion-basis-diagonal preparation suffices for the minimal Ising contrast; coherent fusion-space extensions require additional state control. A cylindrical-Hankel boundary-value calculation resolves the downstream trap profile and surface-to-2DEG transfer over a radius--depth window for a prescribed slow annular harmonic, while its microscopic cavity--gate generation remains device specific. Within the stated models, the framework provides testable predictions for electrostatic transfer, cavity back-action, and matched-control cross-coherence, while complete device feasibility requires independent establishment of the remaining implementation requirements.