But why neutrons?

As gravitation is a very weak force, quantum phenomena can only be observed using very small masses and without any external perturbation.
Ultra cold neutrons (UCNs) are therefore very suitable because their mass is small, their zero charge shields them from electromagnetic perturbations, their lifetime is long enough to separate the quantum states and finally because they can be selected so that their energy is sufficiently low to make the phenomenon observable.
We show here the principle of this ground-breaking experiment. GRANIT is based on the same approach but with much improved instrument setup and neutron flux.
The experiment consisted in making the neutron enter a gravitational potential well and checking that settles in discrete quantum states. The movie shows how this happens.