NOON-state formation from Fock-state Bose-Einstein condensates

NOON states (states of the form $|N>_{a}|0>_{b}+|0>_{a}|N>_{b}$ where $a$ and $b$ are single particle states) have been used for predicting violations of local realism (Greenberger-Horne-Zeilinger violations) and are valuable in metrology for precision measurements of phase at the Heisenberg limit. We show theoretically how the use of two Fock state Bose-Einstein condensates as sources in a modified Mach-Zehnder interferometer can lead to the creation of the NOON state in which $a$ and $b$ refer to arms of the interferometer and $N$ is a subset of the total number of particles in the two condensates. The modification of the interferometer involves making {}"side" measurements of a few particles near the sources. These measurements put the remaining particles in a superposition of two phase states, which are converted into NOON states by a beam splitter if the phase states are orthogonal. When they are not orthogonal, a {}"feedforward" correction circuit is shown to convert them into proper form so a NOON results. We apply the NOON to the measurement of phase. Here the NOON experiment is equivalent to one in which a large molecule passes through two slits. The NOON components can be recombined in a final beam splitter to show interference.