Optimal Phase Sensing in Trapped Ions with One-Axis-Twisting Hamiltonian

Trapped ions are a well-developed platform to estimate a phase imprinted on their internal quantum states. However, an optimal quantum-enhanced protocol to robustly read out the phase under practical conditions in trapped ions has not been explored. Here, we study such a protocol via interaction-based readout in trapped ions with a one-axis twisting Hamiltonian considering a practical experimental situation. We start from a more fundamental spin-motion coupling in trapped ions, mediated by laser beams, to explore the dependence of the phase sensitivity on the specific interaction. We evaluate the decoherence effects of both Rayleigh scattering and Raman scattering due to spontaneous emission of the laser beams. We find that the phase sensitivity is mostly robust to decoherence when the system is operated in the so-called “squeezing regime”. Interestingly, we find that larger Raman scattering leads to slightly better phase sensitivity for a fixed total decoherence rate in this regime. We further analyze the effects of the frequency fluctuations from the center-of-mass (COM) mode in the phase sensing protocol and find that they are negligible under practical experimental parameters. Our work lays the theoretical foundation for estimating a collective phase rotation using entangled spin states in trapped ions.