Optimal States and Almost Optimal Adaptive Measurements for Quantum Interferometry

We derive the optimal $N$-photon two-mode input state for obtaining an estimate $\phi$ 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 $(\Delta \phi {)}^2\simeq {\pi }^2/N^2$, compared to $O\left(N^{-1\mathrm{}}\right)$ or $O\left(N^{-1\mathrm{}/2}\right)$ 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 $\phi$ very close to that from an optimal measurement.