  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 the 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 finds out that transition coupling indeed much deviates 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 the inspiration from the
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 single-photon absorption
rate reach maximum. Our result provides an accurate reference of transition
couplings for quantum metrology at non-zero 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.
