  The semiclassical Schr\"odinger equation with time-dependent potentials is an
important model to study electron dynamics under external controls in the
mean-field picture. In this paper, we propose two multiscale finite element
methods to solve this problem. In the offline stage, for the first approach,
the localized multiscale basis functions are constructed using sparse
compression of the Hamiltonian operator at the initial time; for the latter,
basis functions are further enriched using a greedy algorithm for the sparse
compression of the Hamiltonian operator at later times. In the online stage,
the Schr\"odinger equation is approximated by these localized multiscale basis
in space and is solved by the Crank-Nicolson method in time. These multiscale
basis have compact supports in space, leading to the sparsity of stiffness
matrix, and thus the computational complexity of these two methods in the
online stage is comparable to that of the standard finite element method.
However, the spatial mesh size in multiscale finite element methods is $
H=\mathcal{O}(\epsilon) $, while $H=\mathcal{O}(\epsilon^{3/2})$ in the
standard finite element method, where $\epsilon$ is the semiclassical
parameter. By a number of numerical examples in 1D and 2D, for approximately
the same number of basis, we show that the approximation error of the
multiscale finite element method is at least two orders of magnitude smaller
than that of the standard finite element method, and the enrichment further
reduces the error by another one order of magnitude.
