Quench-induced dynamical phase transitions and π -synchronization in the Bose-Hubbard model

We investigate the nonequilibrium behavior of a fully connected (or all-to-all coupled) Bose-Hubbard model after a Mott to superfluid quench, in the limit of large boson densities and for an arbitrary number $V$ of lattice sites, with potential relevance in experiments ranging from cold atoms to superconducting qubits. By means of the truncated Wigner approximation, we predict that crossing a critical quench strength the system undergoes a dynamical phase transition between two regimes that are characterized at long times either by an inhomogeneous population of the lattice (i.e., macroscopical self-trapping) or by the tendency of the mean-field bosonic variables to split into two groups with phase difference $\ensuremath{\pi}$, that we refer to as $\ensuremath{\pi}$-synchronization. We show the latter process to be intimately connected to the presence, only for $V\ensuremath{\ge}4$, of a manifold of infinitely many fixed points of the dynamical equations. Finally, we show that no fine-tuning of the model parameters is needed for the emergence of such $\ensuremath{\pi}$-synchronization, that is in fact found to vanish smoothly in presence of an increasing site-dependent disorder, in what we call a synchronization crossover.