Phase properties of Kerr media via variance and entropy as measures of uncertainty

The phase properties of an initially coherent field propagating through a Kerr medium are examined using two different measures of phase uncertainty: (i) variance of periodic functions and (ii) the entropy. The variance of a periodic function of phase (with period 2\ensuremath{\pi}/m) is found to be minimized only at the times when the field evolves into a discrete superposition of n distinguishable coherent states (Schr\"odinger-cat states) where n is a factor of m. The phase entropy is found to be more sensitive, as it is minimized at the times when the field evolves into a superposition of an arbitrary number of distinguishable coherent states. The sum of the photon number and phase entropies is found to display behavior very similar to the Wehrl entropy for the evolving field, i.e., the sum is minimal at the times the cat states occur, and at these times it has a value that indicates the number of component coherent states in the cat states.