  The dynamics of a closed quantum system is often studied with the direct
evolution of the Schrodinger equation. In this paper, we propose that the gauge
choice (i.e. degrees of freedom irrelevant to physical observables) of the
Schrodinger equation can be generally non-optimal for numerical simulation.
This can limit, and in some cases severely limit the time step size. We find
that the optimal gauge choice is given by a parallel transport formulation.
This parallel transport dynamics can be simply interpreted as the dynamics
driven by the residual vectors, analogous to those defined in eigenvalue
problems in the time-independent setup. The parallel transport dynamics can be
derived from a Hamiltonian structure, thus suitable to be solved using a
symplectic and implicit time discretization scheme, such as the implicit
midpoint rule, which allows the usage of a large time step and ensures the long
time numerical stability. We analyze the parallel transport dynamics in the
context of the singularly perturbed linear Schrodinger equation, and
demonstrate its superior performance in the near adiabatic regime. We
demonstrate the effectiveness of our method using numerical results for linear
and nonlinear Schrodinger equations, as well as the time-dependent density
functional theory (TDDFT) calculations for electrons in a benzene molecule
driven by an ultrashort laser pulse.
