Triple Quantum-Enhanced Trapped-Ion Sensing*

The vibrational motions of trapped ions have been demonstrated as quantum-enhanced sensors to detect extremely weak forces and electric fields. Further improvements should enable searches for dark matter. Here we present a general quantum sensing protocol of trapped-ion crystals to detect displacements and electric fields by exploiting nonclassical states from both the motional mode and the spin qubits. Our protocol reduces to existing protocols in special cases. We show that a displacement signal can be enhanced by three separate quantum mechanical effects in one sensing protocol enabled by the noiseless, parametrically amplified state, the collective spin-motion entanglement, and the spin-spin entanglement, respectively. We present a general theory of our trapped-ion sensing protocol and optimize the enhancement under practical considerations, including sources of decoherence that can limit the performance of the protocol.