Fig. 4
Estimation analysis. Maximum-likelihood estimation of \({\omega }_{{\rm{r}}}\) (beyond the resolution limit; i.e. \({\omega }_{{\rm{r}}}t\ll 1\)) with different control methods. a and b Histogram of the estimated \({\omega }_{{\rm{r}}}\) for the optimal control method: \({\delta }_{s}t=2\pi ,\) compared to the histogram obtained slightly off the resonance: \({\delta }_{s}t=1.8\pi\). When resonance is achieved, the two frequencies are clearly resolved (\({\Delta} {\omega}_{\rm{r}} < \frac{1}{10}{\omega }_{\rm{r}}\)), while off the resonance they are not resolvable (\(\Delta {\omega }_{{\rm{r}}}\,> \,{\omega }_{{\rm{r}}}\)). Note that off resonance, the standard deviation is too large; hence the probability cannot be distinguished from \(p\left({\omega }_{{\rm{r}}}=0\right)\) (see insets). For both plots \(N=1{0}^{6},\sigma t=5,{\omega }_{{\rm{r}}}t=0.01\). c The root mean square error (RMSE) as a function of \(N\) for both control methods. For \({\delta }_{s}t=2\pi\) the RMSE goes as \({\left(N{I}_{{\rm{r}}}\right)}^{-0.5}\) as expected. Off the resonance (\({\delta }_{s}t=1.8\pi\)) the FI vanishes and the RMSE goes as \({N}^{-0.25}\) (the estimation is biased)
