Machine optimized quantum metrology of concurrent entanglement generation and sensing

Entanglement is one of the key ingredients for enhancing the measurement precision of quantum sensors. Generally, there is a trade-off between state preparation and sensing within a limited coherence time. To fully exploit temporal resources, concurrent entanglement generation and sensing with designed sequence of rotations are proposed. Based on twist-and-turn dynamics, modulated rotations along only one axis may be sufficient to drive the state to the optimal one for tiny estimated parameter. However, when the estimated parameter is not tiny, it may impact the evolved state and hence degrade the final measurement precision. Here, we introduce another modulated rotations along different axis and find out the optimal control sequences by means of machine optimization. The optimal measurement precision bounds become independent on the estimated parameter, which improves the dynamic range of the machine designed sensors. Particularly, by optimizing the interaction strength for different particle number and the time-modulated rotations along two different axes via machine optimization, the Heisenberg-limited precision scaling can be attained. Our work points out a way for designing optimized quantum-enhanced metrology protocols, which is promising for developing practical quantum sensors.