Fig. 3
Probability and Fisher information analysis. a Average transition probability (\(p\)) as a function of \({\omega }_{{\rm{r}}}t\) for different values of \({\delta }_{{\rm{s}}}\): \({\delta }_{{\rm{s}}}t=\pi\) (blue, solid line), \({\delta }_{{\rm{s}}}t=1.8\pi\) (orange, dashed line) and \({\delta }_{{\rm{s}}}t=2\pi\) (dotted, yellow line). For every \({\delta }_{{\rm{s}}},\frac{{\mathrm{{d}}}p}{{\mathrm{{d}}}{\omega }_{{\rm{r}}}}=0\) for \({\omega }_{{\rm{r}}}=0\). Hence a finite \({I}_{{\rm{r}}}\) can be achieved only if \(p=0\). This requirement is fulfilled when \({\delta }_{{\rm{s}}}t=2\pi n\). b FI about \({\omega }_{{\rm{r}}}\) (\({I}_{{\rm{r}}}\)) as a function of \({\delta }_{{\rm{s}}}t\). Clear peaks can be observed whenever \({\delta }_{{\rm{s}}}t=2\pi n\). The width of the peaks is illustrated in the inset: for \({\omega }_{{\rm{r}}}=0,\) the width vanishes; however finite \({\omega }_{{\rm{r}}}\) leads to a finite width (given \({\omega }_{{\rm{r}}}t\ll 1\) this width goes as \({\omega }_{{\rm{r}}}\), see section “Limitations and imperfections”). For this illustration: \({\omega }_{{\rm{r}}}t=0.001\) (red, solid line), \({\omega }_{{\rm{r}}}t=0.05\) (black, dashed line), \({\omega }_{{\rm{r}}}t=0.1\) (blue, dotted line)
