Heterodyne and adaptive phase measurements on states of fixed mean photon number

The standard technique for measuring the phase of a single-mode field is he terodyne detection. Such a measurement may have an uncertainty far above th e intrinsic quantum phase uncertainty of the state. Recently it has been sh own [H. M. Wiseman and R. B. Killip, Phys. Rev. A 57, 2169 (1998)] that an adaptive technique introduces far less excess noise. Here we quantify this difference by an exact numerical calculation of the minimum measured phase variance for the various schemes, optimized over states with a fixed mean p hoton number. We also analytically derive the asymptotics for these varianc es. For the case of heterodyne detection our results disagree with the powe r law claimed by D'Ariano and Paris [Phys. Rev. A 49, 3022 (1994)]. [S1050- 2947(99)04009-3].