  Gravity gradient is known as a serious systematic effect in atomic tests of
the universality of free fall, where the initial central position and velocity
of atoms need to be exactly controlled. In this paper, we study quantum free
fall with high-order gravity gradients. It is shown that the cubic terms in the
Newtonian potential shall generate a new phase shift in atom interferometers,
which depends on the position and velocity uncertainties of the incident atoms.
We further investigate the nonclassicality of free fall and show that, due to
the cubic potential, the gravitational Wigner equation in phase space of
position and velocity is different from the classical Liouville equation. There
exists a mass-dependent correction in the dynamical equation regardless of
initial state. Nevertheless, this is just a quantum mechanical effect of
microparticles, which does not violate equivalence principle that inertia mass
is equal to the gravitational mass.
