Optimal states and almost optimal adaptive measurements for quantum interferometry

We derive the optimal N-photon two-mode input state for obtaining an estima te phi of the phase difference between two arms of an interferometer. For a n optimal measurement [B. C. Sanders and G. J. Milburn, Phys. Rev. Lett, 75 , 2944 (1995)], it yields a variance (Delta phi)(2) similar or equal to pi (2)/N-2, compared to O(N-1) or O(N-1/2) for states considered by previous a uthors. Such a measurement cannot be realized by counting photons in the in terferometer outputs. However, we introduce an adaptive measurement scheme that can be thus realized, and show that it yields a variance in phi very c lose to that from an optimal measurement.