Abstract
A theory of gravitational quantum states of ultracold neutrons in waveguides with absorbing/scattering walls is presented. The theory covers recent experiments in which the ultracold neutrons were beamed between a mirror and a rough scatterer/absorber. The analysis is based on a recently developed theory of quantum transport along random rough walls which is modified in order to include leaky (absorbing) interfaces and, more importantly, the low-amplitude high-aperture roughness. The calculations are focused on a regime when the direct transitions into the continuous spectrum above the absorption threshold dominate the depletion of neutrons from the gravitational states and are more efficient than the processes involving the intermediate states. The theoretical results for the neutron count are sensitive to the correlation radius (lateral size) of surface inhomogeneities and to the ratio of the particle energy to the absorption threshold in a weak roughness limit. The main impediment for observation of the higher gravitational states is the "overhang" of the particle wave functions which can be overcome only by use scatterers with strong roughness. In general, the strong roughness with high amplitude is preferable if one wants just to detect the individual gravitational states, while the strong roughness experiments with small amplitude and high aperture are preferable for the quantitative analysis of the data. We also discuss the ways to further improve the accuracy of calculations and to optimize the experimental regime.
Highlights
One of the recent discoveries in neutron physics was the experimental observation of quantization of motion of ultracold neutrons in a gravitational field1͔
Strong roughness with high amplitude is preferable if one wants just to detect the individual gravitational states, while strongroughness experiments with small amplitude and high aperture are preferable for the quantitative analysis of the data
For the gravitational states in an infinite well, uc → ρ, the wave functions2͒ should become equal to zero on the walls s = 0 ; h, and equations for the eigenvalues Eq ͑5͒ should be replaced by the following equation for the energy spectrumnhhere and below we identify physical parameters, which are calculated for a well with infinite walls, by a bar over the corresponding symbols:
Summary
One of the recent discoveries in neutron physics was the experimental observation of quantization of motion of ultracold neutrons in a gravitational field1͔. Roughness of the walls affects the quantized, in this case, gravitational, states of particles inside the quantum wells by shifting and broadening the energy levels. The same condition is necessary for the quantitative interpretation of experiments: though strong roughness can make the effect of the gravitational quantization of states more pronounced, the uncertainty in the level positions can render the results unusable for the calibration of the short-range forces. Simple measurements of the average amplitude and the lateral size of roughness are insufficient when one needs to know the shape of the correlation function of surface inhomogeneities which can be extracted, almost exclusively, from diffraction measurements8͔ Another major source of uncertainty is relatively poor information on the energy or velocity distribution of neutrons in the beam. VI contains conclusions and discussion of how to improve our theory and optimize the experiment
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