Demonstrating optical control of complex phases in a trapped-ion qubit

Practical and robust control of the complex phases of qubit superposition states is necessary for high-fidelity quantum gates. Here we experimentally demonstrate such control in a trapped ultracold ⁴⁰Ca⁺ ion, where the phase of qubit ground state is modified by optical driving to an excited state. Using a third spectator quantum level as reference, we experimentally measure the phases generated on the ground state amplitude by different driving schemes. While geometric phases exhibit intrinsic robustness to systematic control errors due to their dependence on global geometric properties (e.g., Berry curvature) rather than dynamical details, their resilience to decoherence remains a topic requiring critical clarification. In this work, we show that geometric phases are not in general immune to environment-induced decoherence, and their practical robustness, including to decoherence, depends on active error-suppression strategies. We implement different driving schemes and find that minimizing excited state decay is more decisive for their robustness on decoherence than whether the phase is of a geometric or dynamical character. This work established and demonstrated a new control strategy that mitigates decoherence effects in geometric/complex phase systems, compensating for their susceptibility to decoherence. We experimentally benchmark four phase-imprinting protocols under a tunable engineered dissipation channel and controlled systematic errors in a single trapped-ion qubit. Our central quantitative finding is that the sensitivity to decoherence is governed by the time-integrated excited-state population, rather than by whether the accumulated phase is labeled geometric or dynamical.