Testing of quantum phase in matter-wave optics

Various phase concepts may be treated as special cases of the maximum likelihood estimation. For example, the discrete Fourier estimation that actually coincides with the operational phase of Noh, Foug\`eres, and Mandel is obtained for continuous Gaussian signals with phase modulated mean. Since signals in quantum theory are discrete, a prediction different from that given by the Gaussian hypothesis should be obtained as the best fit assuming a discrete Poissonian statistics of the signal. Although the Gaussian estimation gives a satisfactory approximation for fitting the phase distribution of almost any state, the optimal phase estimation offers in certain cases a measurably better performance. This has been demonstrated in a neutron-optical experiment.