Entanglement of multimode BEC via decay

Three decaying boson systems A, B and C, which give rise to only two independent relative phases are analyzed. Two different ways of observing these phases are considered. In the first case, one beam splitter has the decay products of A and B in its input channels, and another one mixes the decay products of B and C. In this case, the relative phases π_AB and π_BC are measured. In the second case, a third beam splitter is added that is sensitive to the relative phase π_CA of C and A, which is determined by the other two. Also in this case, the build-up of entanglement and relative phases by detection of particles in the output channels are calculated. The three beam splitters overdetermine the system. This implies that the characteristics of the detection histories for the detection channels will strongly depend on the settings of the phases detected by the beam splitters. A specific detection history tends to converge the relative phases to specific values only when the observed values are compatible.