Optimal states and almost optimal adaptive measurements for quantum interferometry

We derive the optimal N-photon two-mode input state for obtaining an estimate straight phi 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 (Deltastraight phi)(2) approximately pi(2)/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 straight phi very close to that from an optimal measurement.