Exponential decrease in phase uncertainty

The phase probability curve of a recently proposed photon state consists of a broad background with a sharp central peak [J. H. Shapiro, S. R. Shepard, and N. Wong, Phys. Rev. Lett. 62, 2377 (1989)]. These authors argue that the inverse peak-height phase uncertainty {δ}cphi of this distribution decreases inversely as the square of the mean photon number 〈m〉---an improvement over either coherent or highly squeezed states. We show that the width {Δ}cphi of the best-fitting Gaussian to the central peak--a measure of phase uncertainty tailored to this narrow feature--decreases exponentially with increasing 〈m〉. The importance of this result may be offset by the observation that the area under this peak also vanishes very rapidly.