Fast Generation of Metrologically Relevant Fock State Mixtures

We propose a fast laser pulse sequence for the generation of non-thermal Fock state mixtures of the motion of a trapped ion, targeted at displacement metrology beyond the standard quantum limit. Using a polaron-frame description of the ion-laser interaction, we identify a resonant operating point-zero detuning and a Rabi frequency matching the trap frequency-at which selective population trapping survives strong driving, enabling preparation speeds beyond the weak-driving limit of previous protocols without requiring ground-state cooling. We trace the residual infidelity at large Lamb-Dicke parameter $\eta$ to a single coherent process, the counter-rotating blue-sideband term neglected in the rotating-wave approximation, and show that it is suppressed by two routine calibrations: a percent-level refocusing of the pulse duration and a small compensating Bloch-Siegert detuning. Numerical simulations of the full sequence show that this refinement keeps the preparation error at or below the $10\%$ level up to $\eta\approx0.5$ and restores the displacement-sensing Fisher information that the uncorrected protocol loses at strong coupling, recovering up to 9 dB relative to the nominal sequence.