In summer 1999 an experiment at ILL, Grenoble was conducted. So-called ultra-cold neutrons (UCN) were trapped in the vertical direction between the Fermi-potential of a smooth mirror below and the gravitational potential of the earth above [Ne00, Ru00]. If quantum mechanics turns out to be a sufficiently correct description of the phenomena in the regime of classical, weak gravitation, one should observe the forming of quantized bound states in the vertical direction above a mirror. Already in a simplified view, the data of the experiment provides strong evidence for the existence of such gravitationally bound quantized states. A successful quantum-mechanical description would then provide a convincing argument, that the socalled first quantization can be used for gravitation as an interaction potential, as this is widely expected. Furthermore, looking at the characteristic length scales of about 10 mikron of such bound states formed by UCN, one sees, that a complete quantum mechanical description of this experiment additionally would enable one to check for possible modifications of Newtonian gravitation on distance scales being one order of magnitude below currently available tests [Ad00]. The work presented here deals mainly with the development of a quantum mechanical description of the experiment.
The experiment performed with ultra-cold neutrons at the Laue-Langevin Institute, Grenoble, is analyzed in view of the claim that ``neutron quantum states in Earth gravitational field'' are observed. Our conclusion is that the above claim is neither theoretically nor experimentally substantiated. We also criticize the statement that ``observation of the gravitationally bound quantum states of neutrons and the related experimental techniques provide a unique tool for a broad range of investigations in fundamental physics of particles and fields''.
Quantum states in the Earth's gravitational field were observed, when ultra-cold neutrons fall under gravity. The experimental results can be described by the quantum mechanical scattering model as it is presented here. We also discuss other geometries of the experimental setup which correspond to the absence or the reversion of gravity. Since our quantum mechanical model describes, particularly, the experimentally realized situation of reversed gravity quantitatively, we can practically rule out alternative explanations of the quantum states in terms of pure confinement effects.
Observation of neutron gravitational quantum states
En=mgzn in the peV energy range (
z1 is about $10\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$ in the vertical direction) in the experiment conducted at Laue-Langevin Institute, Grenoble, with ultracold neutrons was recently reported in a series of publications. The purpose of the present work is to analyze the experiment. The experimental apparatus is designed to measure a transmission function
T(za), namely, a horizontal flux of relatively fast neutrons ($k\ensuremath{\gg}{k}_{z}$ in wavelength terms) passing through a slit of variable height
za of upper absorbing wall. The quantum states in question are defined by the so-called Airy functions, which are solutions to the stationary 1D equation for a neutron ``bouncing'' above the perfect mirror in a linear potential field. The Airy functions describe the quantum bouncer (QB), the concept of which is subject to theoretical study of toy 1D models of gravitationally bound particles in nonrelativistic quantum mechanics (QM). This is essentially different from the 3D nonstationary QM object, ``the running QB,'' investigated in the experiment. The authors assume that there is a connection between
T(za) and the probability density distribution
P(z,za) for QB states. They devised the ``phenomenological model,'' in which the quantum pattern should be visible in the transmission curve. We argue, however, that the measured curve
T(za) is not sensitive to QB states. Instead, it is sensitive to dynamics of neutron horizontal transport inside the absorbing slit for neutrons of energy values about
105 times greater than eigenvalues
En. The latter are related to the neutron transverse mode
kz and cannot be termed ``energies of neutron gravitational quantum states.'' We conclude that the experiment setup and real conditions are not adequate to the claimed objective, and the methodology of measured data treatment is flawed. The authors' claim that ``neutron gravitational quantum states are observed'' is neither theoretically nor experimentally substantiated. Final, statistically significant results of the experiment are consistent with our physical reasoning that the experiment is not sensitive to ``neutron gravitational quantum states'' (in terms of Airy mode) and does not prove even their existence in rigorous quantum-mechanical terms.
We studied the neutron quantum states in the potential well formed by the Earth's gravitational field and a horizontal mirror. The estimated characteristic sizes of the neutron wave functions in two lowest quantum states correspond to their expectations with an accuracy of ~25%. A spatial density distribution in a standing neutron wave above a mirror was measured for a set of a few lowest quantum states. A position-sensitive neutron detector with an extra high spatial resolution of 1-2 mu m was developed and tested for this particular task. Although this experiment was not designed or optimized to search for an additional short-range force, nevertheless it allowed us to slightly improve the published boundary in the nanometer range of characteristic distances. We studied systematical uncertainties in the chosen "flow-through" method as well as the feasibility to improve further the accuracy in this experiment