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 [p.r.l. 71, p. 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, like the 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 et al. [p.r.a. 60, p. 708 (1999)].