Multimode Fock-State Measurements using Dispersive Shifts in a Trapped Ion

Trapped ions naturally host multiple motional modes alongside long-lived spin qubits, providing a scalable multimode bosonic register. Efficiently characterizing such bosonic registers requires the ability to access many motional modes with limited spin resources. Here we introduce a single-spin, multimode measurement primitive using dispersive shifts in the far-detuned multimode Jaynes-Cummings interaction. We implement a Ramsey sequence that maps phonon-number-dependent phases onto the spin, thereby realizing a multimode spin-dependent rotation (SDR). We also introduce a selective-decoupling scheme that cancels the phase induced by the carrier AC-Stark shift while preserving the phonon-number-dependent phase induced by the dispersive shift. Using this SDR-based Ramsey sequence on a single trapped ion, we experimentally extract two-mode Fock-state distributions, perform parity-based filtering of two-mode motional states, and realize a nondestructive single-shot measurement of a single-mode Fock state via repeated filtering steps.