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 δcphi 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 Δcphi 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.