Dynamics of a quantum phase transition in the Bose-Hubbard model: Kibble-Zurek mechanism and beyond

In this paper, we study the dynamics of the Bose-Hubbard model by using time-dependent Gutzwiller methods. In particular, we vary the parameters in the Hamiltonian as a function of time, and investigate the temporal behavior of the system from the Mott insulator to the superfluid (SF) crossing a second-order phase transition. We first solve a time-dependent Schrodinger equation for the experimental setup recently done by Braun et al. [Proc. Natl. Acad. Sci. USA 112, 3641 (2015)] and show that the numerical and experimental results are in fairly good agreement. However, these results disagree with the Kibble-Zurek scaling. From our numerical study, we reveal a possible source of the discrepancy. Next, we calculate the critical exponents of the correlation length and vortex density in addition to the SF order parameter for a Kibble-Zurek protocol. We show that beside the freeze time (t) over cap, there exists another important time, t(eq), at which an oscillating behavior of the SF amplitude starts. From calculations of the exponents of the correlation length and vortex density with respect to a quench time tau(Q), we obtain a physical picture of a coarsening process. Finally, we study how the system evolves after the quench. We give a global picture of dynamics of the Bose-Hubbard model.