Synchronized Bell protocol for detecting non-locality between modes of light

In the following paper, we discuss a possible detection of non-locality in two-mode light states in the Bell protocol, where the local observables are constructed using displacement operators, implemented by Mach-Zender Interferometers fed by strong coherent states. We report numerical results showing that maximizing the Braunstein-Caves Chained Bell (BCCB) inequalities requires equal phases of displacements. On the other hand, we prove that non-locality cannot be detected if the phases of displacements are unknown. Hence, the Bell experiment has to be equipped with a synchronization mechanism. We discuss such a mechanism and its consequences.