Synchronized Bell protocol for detecting nonlocality between modes of light

In the following paper, we discuss a possible detection of nonlocality in two-mode light states in the Bell protocol, where the local observables are constructed using displacement operators, implemented by Mach-Zehnder interferometers fed by strong coherent states. We report numerical results showing that maximizing the Braunstein-Caves chained Bell inequalities requires equal phases of displacements. On the other hand, we prove that nonlocality 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.