Quantum-enhanced multi-parameter sensing in a single mode: simultaneous measuring displacements in position and momentum

Quantum metrology is often constrained by measurement backaction when jointly estimating two non-commuting observables—a canonical example is position and momentum. Here, we experimentally demonstrate a single-mode displacement sensor that estimates small shifts in both position and momentum with precision below the standard quantum limit (SQL). Our approach measures commuting modular position and modular momentum observables, enabled by highly nonclassical grid states (Gottesman–Kitaev–Preskill–type) prepared in the mechanical vibrational mode of a single trapped ion. Using a phase-estimation–based measurement, we extract both parameters in one experiment while evading the backaction that would ordinarily couple the estimates. We further extend the protocol to another naturally non-commuting pair—number and phase—and again observe sub-SQL performance.