Adaptive quantum measurements of a continuously varying phase
Dominic W. Berry, Howard M. Wiseman · Physical Review A · 2002
We analyze the problem of quantum-limited estimation of a stochastically varying phase of a continuous beam (rather than a pulse) of the electromagnetic field. We consider both nonadaptive and adaptive measurements, and both dyne detection (using a local oscillator) and interferometric detection. We take the phase variation to be $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}=\sqrt{\ensuremath{\kappa}}\ensuremath{\xi}(t),$ where $\ensuremath{\xi}(t)$ is \ensuremath{\delta}-correlated Gaussian noise. For a beam of power P, the important dimensionless parameter is $N=P/\ensuremath{\Elzxh}\ensuremath{\omega}\ensuremath{\kappa},$ the number of photons per coherence time. For the case of dyne detection, both continuous-wave (cw) coherent beams and cw (broadband) squeezed beams are considered. For a coherent beam a simple feedback scheme gives good results, with a phase variance $\ensuremath{\simeq}{N}^{\ensuremath{-}1/2}/2.$ This is $\sqrt{2}$ times smaller than that achievable by nonadaptive (heterodyne) detection. For a squeezed beam a more accurate feedback scheme gives a variance scaling as ${N}^{\ensuremath{-}2/3},$ compared to ${N}^{\ensuremath{-}1/2}$ for heterodyne detection. For the case of interferometry only a coherent input into one port is considered. The locally optimal feedback scheme is identified, and it is shown to give a variance scaling as ${N}^{\ensuremath{-}1/2}.$ It offers a significant improvement over nonadaptive interferometry only for N of order unity.