Learning unified control of internal spin squeezing in atomic qudits for magnetometry

Abstract

Generating and preserving metrologically useful quantum states is a central challenge in quantum-enhanced atomic magnetometry. In multilevel atoms operated in the low-field regime, the nonlinear Zeeman (NLZ) effect is both a resource and a limitation. It nonlinearly redistributes internal spin fluctuations to generate spin-squeezed states within a single atomic qudit, yet under fixed readout it distorts the measurement-relevant quadrature and limits the accessible metrological gain. This challenge is compounded by the time dependence of both the squeezing axis and the effective nonlinear action. Here we show that physics-informed reinforcement learning can transform NLZ dynamics from a source of readout degradation into a sustained metrological resource. Using only experimentally accessible low-order spin moments, a trained agent identifies, in the $f=21/2$ manifold of $^{161}\mathrm{Dy}$, a unified control policy that rapidly prepares strongly squeezed internal states and stabilizes more than $4\,\mathrm{dB}$ of fixed-axis spin squeezing under always-on NLZ evolution. Including state-preparation overhead, the learned protocol yields a single-atom magnetic sensitivity of $13.9\,\mathrm{pT}/\sqrt{\mathrm{Hz}}$, corresponding to an advantage of approximately $3\,\mathrm{dB}$ beyond the standard quantum limit. Our results establish learning-based control as a practical route for converting unavoidable intrinsic nonlinear dynamics in multilevel quantum sensors into operational metrological advantage.