Effect of the detector efficiency on the phase sensitivity in a Mach-Zehnder interferometer

The effect of quantum efficiencies of detectors on the phase sensitivity of a Mach-Zehnder is analyzed for the Fock state input The phase uncertainty of two different interference experiments, the second order and the fourth order, are compared with two practical detectors with less than unit quantum efficiencies. It is found out that in the second-order interference experiment we cannot beat the classical limit in the phase sensitivity, even with ideal detectors. We show that almost ideal detectors are required to get the Heisenberg limit in the phase sensitivity of the fourth-order interference for two Fock state light inputs. There exists an optimum angle given by the efficiency of the detectors.