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The standard technique for measuring the phase of a single mode field isheterodyne detection. Such a measurement may have an uncertainty far above theintrinsic quantum phase uncertainty of the state. Recently it has been shown[H. M. Wiseman and R. B. Killip, Phys. Rev. A 57, 2169 (1998)] that an adaptivetechnique introduces far less excess noise. Here we quantify this difference byan exact numerical calculation of the minimum measured phase variance for thevarious schemes, optimized over states with a fixed mean photon number. We alsoanalytically derive the asymptotics for these variances. For the case ofheterodyne detection our results disagree with the power law claimed byD'Ariano and Paris [Phys. Rev. A 49, 3022 (1994)].Comment: 9 pages, 2 figures, minor changes from journal versio

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