Quantum sensing of the phase-space-displacement parameters using a single trapped ion

We introduce a quantum sensing protocol for detecting the parameters characterizing the phase-space displacement by using a single trapped ion as a quantum probe. We show that, thanks to the laser-induced coupling between the ion's internal states and the motion mode, the estimation of the two conjugated parameters describing the displacement can be efficiently performed by a set of measurements of the atomic state populations. Furthermore, we introduce a three-parameter protocol capable of detecting the magnitude, the transverse direction, and the phase of the displacement. We characterize the uncertainty of the two-and three-parameter problems in terms of the Fisher information and show that state projective measurement saturates the fundamental quantum Cramer-Rao bound.