Sherrington-Kirkpatrick glassy-phase of random Josephson coupled Bose-Einstein condensates in wood-pile geometry

We study a transition to a glassy phase of neutral atoms trapped in optical lattice, where the system is realized as the array of individual Bose-Einstein condensates of N elongated vertical and horizontal N rods in a wood-pile form coupled via the random Josephson tunneling. 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. This together with randomness forms a prerequisite of the Sherrington-Kirkpatrick model for N→∞ widely employed in the theory of spin glasses. For this arrangement we solve a model Hamiltonian of the Josephson array in the thermodynamic limit (N→∞) and calculate the critical temperature for the glassy phase-locking transition, caused by the Josephson tunneling of bosons in random environment.