Quantum-enhanced sensing of displacements and electric fields with large trapped-ion crystals*

We theoretically analyze the protocol used for implementation of a quantum enhanced sensor of displacements and electric fields in a large crystal of trapped ions (N=150). The protocol uses the center of mass vibrational mode (COM) of the crystal as a high-Q mechanical oscillator and lets it interact with the ions' collective electronic spin, which works as the readout degree of freedom. We derive an effective model of coupled oscillators encompassing both the spins and the COM of mode and demonstrate that the spin-boson entanglement at the core of the protocol can be understood as the squeezing dynamics of this pair of oscillators. This allows us to explain experimental observations that use this system for electric field and displacement measurements with a sensitivity that surpasses both the standard quantum limit and typical thermal bounds for uncorrelated initial states. This provides both fundamental and practical advantages with respect to the use of purely classical resources.