Phase-coherent sensing of the center-of-mass motion of trapped-ion crystals

Trapped ions are sensitive detectors of weak forces and electric fields that excite ion motion. Here measurements of the center-of-mass motion of a trapped-ion crystal that are phase coherent with an applied weak external force are reported. These experiments are conducted far from the trap motional frequency on a two-dimensional trapped-ion crystal of approximately 100 ions, and determine the fundamental measurement imprecision of our protocol free from noise associated with the center-of-mass mode. The driven sinusoidal displacement of the crystal is detected by coupling the ion crystal motion to the internal spin degree of freedom of the ions using an oscillating spin-dependent optical dipole force. The resulting induced spin precession is proportional to the displacement amplitude of the crystal, and is measured with near-projection-noise-limited resolution. A 49 pm displacement is detected with a signal-to-noise ratio of 1 in a single experimental determination, which is an order-of-magnitude improvement over prior phase-incoherent experiments. This displacement amplitude is 40 times smaller than the zero-point fluctuations. With our repetition rate, an $8.4 \text{pm}/\sqrt{\text{Hz}}$ displacement sensitivity is achieved, which implies $12 \left(\text{yN/ion}\right)/\sqrt{\text{Hz}}$ and $77 \left(\mathrm{micro}\text{V/m}\right)/\sqrt{\text{Hz}}$ sensitivities to forces and electric fields, respectively. This displacement sensitivity, when applied on-resonance with the center-of-mass mode, indicates the possibility of weak force and electric field detection below 10^-3 yN/ion and 1 nV/m, respectively.