Phase-locking transition of Josephson coupled Bose–Einstein condensates in wood-pile geometry

We consider theoretically phase-locking transition of ultra-cold neutral atoms trapped in optical lattice where the system can be realized as the array of individual Bose–Einstein condensates of elongated vertical and horizontal N rods in a wood-pile form. In this geometry every horizontal (vertical) rod of a condensate is linked to its vertical (horizontal) counterpart, so that the number of nearest neighbors z of a given rod in this system is z=N, implying that the system is fully connected. For this arrangement we implement a model Hamiltonian of the Josephson array and show that in the thermodynamic limit (N→∞) the model allows exact determination of the free energy. With this result we calculate the critical temperature for the phase-locking transition in the array, caused by the Josephson tunneling of bosons, and discuss the result in the context of system parameters and possible experiments.