Strongly Coupled Chameleons and the Neutronic Quantum Bouncer

We consider the potential detection of chameleons using bouncing ultracold neutrons. We show that the presence of a chameleon field over a planar plate would alter the energy levels of ultra cold neutrons in the terrestrial gravitational field. When chameleons are strongly coupled to nuclear matter, $\beta\gtrsim 10^8$, we find that the shift in energy levels would be detectable with the forthcoming GRANIT experiment, where a sensitivity of order one percent of a peV is expected. We also find that an extremely large coupling $\beta\gtrsim 10^{11}$ would lead to new bound states at a distance of order 2 microns, which is already ruled out by previous Grenoble experiments. The resulting bound, $\beta\lesssim 10^{11}$, is already three orders of magnitude better than the upper bound, $\beta\lesssim 10^{14}$, from precision tests of atomic spectra.