Non equilibrium dynamics of quantum Ising chains in the presence of transverse and longitudinal magnetic fields

There has been much recent interest in the study of out of equilibrium quantum systems due to experimental advances in trapped cold atomic gases that allow systematic studies of these effects. In this work we study a quantum Ising model in the presence of transverse and longitudinal magnetic fields. The model lacks an analytical solution in the most general case and has to be treated numerically. We first investigate the static properties of the model, including the ground state energy and the zero temperature phase diagram, using the Density Matrix Renormalization Group (DMRG) method. We identify ordered and disordered phases separated by a quantum phase transition. We then consider the time evolution of the system following an abrupt quench of the fields. We study the response of the order parameter to the quenches within the ordered and disordered phases as well as quenches between the phases using the time-dependent DMRG method. The results are in qualitative agreement with experiments and the implemented numerical method in this work can be extended to allow for more general interactions.