  We report on precision resonance spectroscopy measurements of quantum states
of ultracold neutrons confined above the surface of a horizontal mirror by the
gravity potential of the Earth. Resonant transitions between several of the
lowest quantum states are observed for the first time. These measurements
demonstrate, that Newton's inverse square law of Gravity is understood at
micron distances on an energy scale of~$10^{-14}$~eV. At this level of
precision we are able to provide constraints on any possible gravity-like
interaction. In particular, a dark energy chameleon field is excluded for
values of the coupling constant~$\beta > 5.8\times10^8$ at~95% confidence
level~(C.L.), and an attractive (repulsive) dark matter axion-like spin-mass
coupling is excluded for the coupling strength $g_sg_p >
3.7\times10^{-16}$~($5.3\times10^{-16}$)~at a Yukawa length of~$\lambda =
20$~{\textmu}m~(95% (C.L.).
