Quantum-limited mirror-motion estimation.

We experimentally demonstrate optomechanical motion and force measurements near the quantum precision limits set by the quantum Cramér-Rao bounds. Optical beams in coherent and phase-squeezed states are used to measure the motion of a mirror under an external stochastic force. Utilizing optical phase tracking and quantum smoothing techniques, we achieve position, momentum, and force estimation accuracies close to the quantum Cramér-Rao bounds with the coherent state, while the estimation using squeezed states shows clear quantum enhancements beyond the coherent-state bounds.