Particle-number scaling of the phase sensitivity in realistic Bayesian twin-mode Heisenberg-limited interferometry

We investigate the scaling of the phase sensitivity of a nonideal Heisenberg-limited interferometer with the particle number N, in the case of the Bayesian detection procedure proposed by Holland and Burnett [ Phys. Rev. Lett. 71, 1355 (1993) ] for twin boson input modes. Using Monte Carlo simulations for up to 10 000 bosons, we show that the phase error of a nonideal interferometer scales with the Heisenberg limit if the losses are of the order of N-1. Greater losses degrade the scaling which is then in N-1/2, such as the beam-splitter shot-noise limit, yet the sensitivity stays sub-shot-noise as long as photon correlations are present. These results give the actual limits of Bayesian detection for twin-mode interferometry and prove that it is an experimentally feasible scheme, contrary to what is implied by the coincidence-detection analysis of Kim [ Phys. Rev. A 60, 708 (1999) ].