  This paper is concerned with the resolution of an inverse problem related to
the recovery of a scalar (potential) function $V$ from the source to solution
map, of the semi-linear equation $(\Box_{g}+V)u+u^3=0$ on a globally hyperbolic
Lorentzian manifold $(M,g)$. We first study the simpler model problem where the
geometry is the Minkowski space and prove the uniqueness of $V$ through the use
of geometric optics and a three-fold wave interaction arising from the cubic
non-linearity. Subsequently, the result is generalized to globally hyperbolic
Lorentzian manifolds by using Gaussian beams.
