  We study an inverse problem related to the dynamical Schr{\"o}dinger equation
in a bounded domain of $\Rb^n,n\geq 2$. Since the concerned non-linear
Schr\"odinger equation possesses a trivial solution, we linearize the equation
around the trivial solution. Demonstrating the well-posedness of the direct
problem under appropriate conditions on initial and boundary data, it is
observed that the solution admits $\eps$-expansion. By taking into account the
fact that the terms $\Oh(|\nabla u(t,x)|^3)$ are negligible in this context, we
shall reconstruct the time-dependent coefficients such as electric potential
and vector-valued function associated with quadratic nonlinearity from the
knowledge of input-output map using the geometric optics solution and Fourier
inversion.
