On observation of neutron quantum states in the Earth’s gravitational field

Observation of neutron gravitational quantum states $E_n=mgz_n$ in the peV energy range ($z_1$ is about 10 μm 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 $T(z_a)$, namely, a horizontal flux of relatively fast neutrons ($k\gg k_z$ in wavelength terms) passing through a slit of variable height $z_a$ 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 $T(z_a)$ and the probability density distribution $P(z,z_a)$ 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 $T(z_a)$ 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 $10^5$ times greater than eigenvalues $E_n$. The latter are related to the neutron transverse mode $k_z$ 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.