Algorithms for the numerical integration of Langevin equations are compared in detail from the point of view of their accuracy, numerical efficiency, and stability to assess them as potential candidates for molecular-dynamics simulations of polymeric systems. Some algorithms are symplectic in the deterministic frictionless limit and prove to stabilize long time-step integrators. They are tested against other popular algorithms. The optimal algorithm depends on the main goal: accuracy or efficiency. The former depends on the observable of interest. A recently developed quasisymplectic algorithm with great accuracy in the position evaluation exhibits better overall accuracy and stability than the other ones. On the other hand, the well-known BrünGer-Brooks-Karplus [Chem. Phys. Lett. 105, 495 (1982)] algorithm is found to be faster with limited accuracy loss but less stable. It is also found that using higher-order algorithms does not necessarily improve the accuracy. Moreover, they usually require more force evaluations per single step, thus leading to poorer performances.

We report time-resolved measurements of Landau-Zener tunneling of Bose-Einstein condensates in accelerated optical lattices, clearly resolving the steplike time dependence of the band populations. Using different experimental protocols we were able to measure the tunneling probability both in the adiabatic and in the diabatic bases of the system. We also experimentally determine the contribution of the momentum width of the Bose condensates to the temporal width of the tunneling steps and discuss the implications for measuring the jump time in the Landau-Zener problem.