We propose an adiabatic Ramsey interferometry technique for detecting weak nonlinearities with trapped ions. The method relies on using the quantum Rabi model, which is sensitive to nonlinear symmetry-breaking perturbations, as a probe. We show that the couplings which arise either from anharmonic terms of the trapping potential or from higher-order terms in the Coulomb interaction expansion can be efficiently estimated by measuring the spin-state probabilities alone. We show that the spin signal is amplified by the mean-phonon excitations, which results in the estimation precision reaching the super-Heisenberg limit. Notably, achieving such high-precision estimation does not require specific entangled-state preparation and can be reached even for the initial thermal-motion state. Furthermore, we show that the super-Heisenberg scaling can be observed even in the presence of weak spin dephasing.

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• Quantum metrology
• Quantum parameter estimation
• Trapped ions