Phase uncertainties of a squeezed state

We investigate various measures of phase uncertainty in their dependence on the average number of photons for the case of a phase squeezed state. We use the following three measures: (i) the geometrical uncertainty (δ${\mathit{cphi}}_{\mathit{g}}$${)}^2$, (ii) the dispersion (δ${\mathit{cphi}}_{\mathit{d}}$${)}^2$, and (iii) the rotational width (δ${\mathit{cphi}}_{\mathit{r}}$${)}^2$. Whereas (δ${\mathit{cphi}}_{\mathit{g}}$${)}^2$ and (δ${\mathit{cphi}}_{\mathit{r}}$${)}^2$ decay quadratically with the average number of photons 〈n〉, the dependence of (δ${\mathit{cphi}}_{\mathit{d}}$${)}^2$ is only of the form ln〈n〉/〈n${\mathrm{〉}}^2$. We give physical pictures for these scaling laws.