Quantum size effect and biased diffusion of gravitationally bound neutrons in a rough waveguide

A comprehensive theory of gravitational quantum states of ultracold neutrons in a rough waveguide is presented. The theory covers recent experiments in which the ultracold neutrons were beamed between a mirror and a rough scatterer and absorber. The results are in very good agreement with experimental data. The analysis is based on a recently developed theory of quantum transport in waveguides with rough absorbing and scattering walls. The calculation is done using two methods: an exact transport equation and a simplified model of biased scattering-driven diffusion of neutrons between quantum states. Both sets of results are in excellent agreement with each other. The exit neutron count is sensitive to the amplitude and the correlation radius (lateral size) of surface inhomogeneities and to the overall time of flight (length of the waveguide). The results indicate that it is possible to choose the waveguide parameters in such a way so to observe the quantum size effect in neutron count-the quantum steps that correspond to individual quantum states-even in a weak roughness regime. Away from the obvious limiting cases, the results are not very sensitive to the ratio of the particle energy to the absorption threshold. The main unresolved issue, which is related to a complexity of required calculations for a "real" experimental cell, is the lack of accurate information on the occupation numbers of neutrons entering the waveguide. Our analysis indicates that the initial occupancies of all gravitational states are expected to be the same except for the smallest values of the waveguide width.