Extension of Linear Response Regime in Weak-Value Amplification Technique

The achievable precision of parameter estimation plays a significant role in evaluating a strategy of metrology. In practice, one may employ approximations in a theoretical model development for simplicity, which, however, will cause systematic error and lead to a loss of precision. We derive the error of maximum likelihood estimation in the weak-value amplification technique where the linear approximation of the coupling parameter is used. We show that this error is positively related to the coupling strength and can be effectively suppressed by improving the Fisher information. Considering the roles played by weak values and initial meter states in the weak-value amplification, we also point out that the estimation error can be decreased by several orders of magnitude by averaging the estimations resulted from different initial meter states or weak values. These results are finally illustrated in a numerical example where an extended linear response regime to the parameter is observed.