  We address phase-shift estimation by means of squeezed vacuum probe and
homodyne detection. We analyze Bayesian estimator, which is known to
asymptotically saturate the classical Cramer-Rao bound to the variance, and
discuss convergence looking at the a posteriori distribution as the number of
measurements increases. We also suggest two feasible adaptive methods, acting
on the squeezing parameter and/or the homodyne local oscillator phase, which
allow to optimize homodyne detection and approach the ultimate bound to
precision imposed by the quantum Cramer-Rao theorem. The performances of our
two-step methods are investigated by means of Monte Carlo simulated experiments
with a small number of homodyne data, thus giving a quantitative meaning to the
notion of asymptotic optimality.
