Realization of versatile and effective quantum metrology using a single bosonic mode

Quantum metrology offers the potential to surpass its classical counterpart, pushing the boundaries of measurement precision toward the ultimate Heisenberg limit. This enhanced precision is normally attained by utilizing large squeezed states or multi-particle entangled quantum states, both of which are often challenging to implement and prone to decoherence in real quantum devices. In this work, we present a versatile and on-demand protocol for deterministic parameter estimation that leverages two efficient state-transfer operations on a single bosonic mode. Specifically, we demonstrate this protocol in the context of phase estimation using the superposition of coherent states in the bosonic circuit quantum electrodynamics (cQED) platform. With low average photon numbers of only up to 1.76, we achieve quantum-enhanced precision approaching the Heisenberg scaling, reaching a metrological gain of 7.5(6) dB. Importantly, we show that the gain or sensitivity range can be further enhanced on the fly by tailoring the input states, with different superposition weights, based on specific system constraints. The realization of this versatile and efficient scheme affords a promising path towards practical quantum-enhanced sensing, not only for bosonic cQED hardware but also readily extensible to other continuous-variable platforms.