Because quantum critical systems are very sensitive to the variation of parameters around the quantum phase transition (QPT), quantum criticality has been presented as an efficient resource for metrology. In this paper, we address the issue whether the divergent feature of the inverted variance is realizable in the presence of noise when approaching the QPT. Taking the quantum Rabi model (QRM) as an example, we obtain the analytical result for the inverted variance with single-photon relaxation. We show that the inverted variance may be convergent in time due to the noise. Since the precision of the metrology is very sensitive to the noise, as a remedy, we propose squeezing the initial state to improve the precision under decoherence. In addition, we also investigate the criticality-based metrology under the influence of the two-photon relaxation. Strikingly, although the maximum inverted variance still manifests a power-law dependence on the energy gap, the exponent is positive and depends on the dimensionless coupling strength. This observation implies that the criticality may not enhance but weaken the precision in the presence of two-photon relaxation, due to the non-linearity introduced by the two-photon relaxation.
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