  Neutrons and neutrinos are natural probes for new physics. Since they carry
no conserved gauge quantum numbers, both can easily mix with the fermions from
hidden sectors. A particularly interesting effect is the oscillation of a
neutron or a neutrino into a fermion propagating in large extra dimensions. In
fact, such a mixing has been identified as the possible origin of small
neutrino mass. In this paper, we study neutron oscillations into an
extra-dimensional fermion and show that this effect provides a resonance
imaging of the Kaluza-Klein tower. The remarkable feature of this phenomenon is
its generic nature: because of a fine spacing of the Kaluza-Klein tower,
neutrons at a variety of energy levels, both free or within nuclei, find a bulk
oscillation partner. In particular, the partner can be a Kaluza-Klein mode of
the same species that gives mass to the neutrino. The existence of bulk states
matching the neutron energy levels of nuclear spectra gives rise to tight
constraints as well as to potentially observable effects. For a free neutron,
we predict recurrent resonant oscillations occurring with the values of the
magnetic field correlated with the KK levels. We derive bounds on extra
dimensions from ultra-cold neutron experiments and suggest signatures for
refined measurements, which, in particular, can probe the parameter space
motivated by the Hierarchy Problem. Ultra-cold neutron experiments offer a
unique way of Kaluza-Klein spectroscopy.
