Optimal states and almost optimal adaptive measurements for quantum interferometry

We derive the optimal N-photon two-mode input state for obtaining an estimate φ of the phase difference between two arms of an interferometer. For an optimal measurement [B. C. Sanders and G. J. Milburn, Phys. Rev. Lett. 75, 2944 (1995)], it yields a variance (Δφ)² ≃ π²/N², compared to O(N^-1) or O(N^-1/2) for states considered by previous authors. Such a measurement cannot be realized by counting photons in the interferometer outputs. However, we introduce an adaptive measurement scheme that can be thus realized, and show that it yields a variance in φ very close to that from an optimal measurement.