Field-Tunable Meissner-Levitated Ferromagnetic Microsphere Sensor for Cryogenic Casimir and Short-Range Gravity Tests

Abstract: Near-field force measurements at submicron separations can probe Casimir effects and hypothetical short-range interactions, but require cryogenic operation and stable, \textit{in situ} control of separation-dependent backgrounds. We propose a self-calibrating quantum force-gradient sensor in which a ferromagnetic microsphere is Meissner-levitated above a type-I superconducting plane, while a bias magnetic field reproducibly tunes the equilibrium gap for in situ separation scans without mechanical approach. The force gradient is encoded as a resonance-frequency shift tracked by a phase-locked loop, and the motion is read out with a SQUID-coupled, flux-tunable microwave resonator that provides adjustable measurement strength without optical heating. Using the input--output formalism, we derive the conditions for reaching the standard quantum limit (SQL) and identify a counterintuitive scaling law: because displacement-to-flux transduction increases with microsphere size, larger microspheres require fewer photons to reach the SQL, enabling a pathway to macroscopic quantum metrology. We quantify the trade-off between suppression of electrostatic patch potentials (via Au coating) and eddy-current dissipation, project force sensitivities of $\sim 10^{-19}\,\rm{N\,Hz^{-1/2}}$ at millikelvin temperatures, and outline protocols to extract Casimir pressure and constrain Yukawa-type deviations from Newtonian gravity over $0.1$--$10\,μ\mathrm{m}$.