Quantum Fisher information and polaron picture for the identification of transition coupling in the quantum Rabi model

The quantum Rabi model (QRM) is a fundamental model for light-matter interactions. A fascinating feature of the QRM is that it manifests a quantum phase transition which is applicable for critical quantum metrology (CQM). Effective application for CQM needs the exact location of the transition point, however the conventional expression for transition coupling is only valid in the extreme limit of low frequency, while apart from zero frequency an accurate location is still lacking. In the present work we conversely use the quantum Fisher information (QFI) in the CQM to identify the transition coupling, which reveals that transition coupling indeed deviates greatly from the conventional one once a finite frequency is turned on. Polaron picture is applied to analytically reproduce the numeric QFI. An accurate expression for the transition coupling is obtained by inspiration from the revealed fractional-power-law effect of polaron frequency renormalization. From the QFI in the polaron picture we find that the transition occurs around a point where the effective velocity and the susceptibility of the expected value of mode quadrature reach maximum. Our result provides an accurate reference of transition couplings for quantum metrology at nonzero frequencies. The formulation of the QFI in the polaron picture also prepares an analytic method with an accurate compensation for the parameter regime difficult to access for the numerics. Besides the integer/fractional-power-law analysis to extract the underlying physics of transition, the QFI/velocity relation may also add some insight in bridging the QFI and transition observables.