Adaptive quantum measurements of a continuously varying phase

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 $\dot{\phi }=\sqrt{\kappa }\xi \mathrm{}\left(t\right),$ where $\xi \mathrm{}\left(t\right)\mathrm{}$ is δ-correlated Gaussian noise. For a beam of power P, the important dimensionless parameter is $N=P/ħ\omega \kappa ,$ 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 $\simeq N^{-1/2}/2.\mathrm{}$ 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 $N^{-2/3},$ compared to $N^{-1/2}$ 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 $N^{-1/2}.$ It offers a significant improvement over nonadaptive interferometry only for N of order unity.