Maximal quantum Fisher information in the semi-classical Rabi model

To improve the precision of parameter estimation in semi-classical Rabi model, we study the maximal quantum Fisher information (MQFI) of the parameters in the system by the counter-rotating hybridized rotating-wave approximation (CHRWA). The validity of this approximation is discussed by the operator fidelity method, and the applicable effective parameter zone is found out. To get the MQFI of the system, we generalize the analytical expression of the MQFI for all unitary parametrization processes in two-level system. Then we apply our results to get the MQFI of the semi-classical Rabi model, and give the condition to reach the limit of parameter estimation's precision. Finally, we discuss the relationship between the MQFI and the Bloch Siegert (BS) shift in this model.