  We propose a method to establish the rapid stabilization of the bilinear
Schr\"odinger control system and its linearized system, and the finite time
stabilization of the linearized system using the Grammian operators. The
analysis of the rapid stabilization involves a new quantity (variable) which is
inspired by the adjoint state in the optimal control theory and is proposed in
our recent work on control systems associated with strongly continuous group.
The analysis of the finite time stabilization follows the strategy introduced
by Coron and Nguyen in the study of the finite time stabilization of the heat
equation and incorporate a new ingredient involving the estimate of the cost of
controls of the linearized system in small time derived in this paper.
