Applicability of symplectic integrator to classically unstable quantum dynamics

Applicability of symplectic integrator (SI) to classically unstable (chaotic) quantum systems is examined, and accuracy and efficiency as a numerical integrator of Schrödinger equation are demonstrated. The second order SI is well known as the split operator method to molecular scientists. Recently, construction of higher order SIs has been developed by several authors. In the present paper, we compare systematically various higher order SI schemes by applying them to a simple quantum chaos system and emphasize the necessity of introducing higher order schemes. Although the higher order SIs have originally been invented for high precision computation of classical trajectories, it will more promisingly be applied to quantum systems. This is because the exponential instability is absent in quantum systems, but an extensive numerical test reveals that the accumulation of error accompanying a long time integration by SI reflects the stability of the system in the classical limit, and closer attention must be payed for classically chaotic quantum system in the semiclassical regime. It is in such a regime that higher order SI schemes work efficiently.