Lower bounds on phase sensitivity in ideal and feasible measurements

The detection of the phase shift between a single mode of the em field and a local reference oscillator is analyzed in the proper framework of quantum estimation theory. Such a fully quantum treatment clarifies the meaning of the ``operational approach'' suggested by Noh, Foug\'eres, and Mandel Phys. Rev. A 45, 424 (1992), in which different measurement schemes correspond to different phase operators. We show that the phase shift is actually measured in the form of the polar angle between two real output photocurrents, namely through a joint measurement of two conjugated quadratures of the field. This scheme is the only feasible one for detecting the quantum phase, and it is equivalently performed by either heterodyne detection or double-homodyne detection of the field. On the contrary, the customary homodyne detection (and, generally, any kind of measurement of a single phase-dependent variable) is more properly a zero-point phase measurement. As a definition of sensitivity we consider the output rms noise of the detection scheme, the only relevant one for actual experiments, in contrast with many other different notions currently adopted in the literature. We show that the rms phase sensitivity versus the average photon number n is bounded by the ideal limit ^-1, whereas for the feasible schemes the bound is ^-2/3, in between the shot-noise level ^-1/2 and the ideal bound. The latter can acutally be achieved by single homodyne detection of the suitable squeezed states, but only in the neighborhood of a fixed zero-phase working point. The phase sensitivity bound ^-2/3 can be reached using coherent states with only 2% of squeezing photons, in contrast with the homodyne-detection bound ^-1 which is reached with 50%. The uncertainty product of two conjugated phase quadratures largely exceeds the Heisenberg limit, even for the ideal joint measurement. We also show that no sizeable improvement in sensitivity is found for detection schemes which involve wideband states, at least for the case of nonentangled states. At the end of the paper we give an extensive table of asymptotic sensitivities at large n for both ideal and feasible schemes.