of gravitationally bound ultracold neutrons in rough waveguides. Phys. Rev. A, 83, 033618 1-9 (2011 We investigate the propagation of ultracold neutrons through a rough waveguide in conjunction with recent experiments in which the ultracold neutrons were beamed between a perfect mirror and a rough scatterer and absorber. The main goal is to find a way to resolve the lowest gravitationally quantized discrete states in the peV range. We compare the neutron count for various types of mirrors with Gaussian, power-law, and exponential correlation functions of surface inhomogeneities. The main conclusion is that all the information about inhomogeneities, including their amplitude, correlation radius, and the rate of decay of the correlation function, enter the exit neutron count via just a single constant , which effectively renormalizes the amplitude of roughness. To observe well-defined quantum steps, one should have an experimental setup with > 40. For a wide variety of correlation functions, the constant is proportional to the square of the amplitude of the surface roughness and is inversely proportional to the square root of the correlation radius. The strong dependence of on roughness parameters and the shape of the correlation function opens a novel way for improving the resolution of gravitationally bound states by optimizing the roughness pattern without reverting to an undesirable strong roughness. We discuss how to optimize the scatterer and absorber by first generating numerically the desired roughness profile and then transferring it to the mirror. We also study the effect of beam preparation on the initial occupancies of gravitational states before the beam enters the waveguide. It turns out that there are simple ways to manipulate the beam in front of the waveguide that can help to resolve the gravitationally bound quantum states. Our results are in good agreement with available experimental data.

We report analysis of rough mirrors used as the gravitational state selectors in neutron beam and similar experiments. The key to mirror properties is its roughness correlation function (CF) which is extracted from the precision optical scanning measurements of the surface profile. To identify CF in the presence of fluctuation-driven fat tails, we perform numerical experiments with computer-generated random surfaces with the known CF. These numerical experiments provide a reliable identification procedure which we apply to the actual rough mirror. The extracted CF allows us to make predictions for ongoing GRANIT experiments. We also propose a radically new design for rough mirrors based on Monte Carlo simulations for the 1D Ising model. The implementation of this design provides a controlled environment with predictable scattering properties.

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We apply our general theory of transport in systems with random rough boundaries to gravitationally quantized ultracold neutrons in rough waveguides as in GRANIT experiments (ILL, Grenoble). We consider waveguides with roughness in both two and one dimensions (2D and 1D). In the biased diffusion approximation the depletion times for the gravitational quantum states can be easily expressed via each other irrespective of the system parameters. The calculation of the exit neutron count reduces to evaluation of a single constant which contains a complicated integral of the correlation function of surface roughness. In the case of 1D roughness (random grating) this constant is calculated analytically for common types of the correlation functions. The results obey simple scaling relations which are slightly different in 1D and 2D. We predict the exit neutron count for the new GRANIT cell.

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