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.
Abstract The time evolution of a quantum wave packet in the linear gravity potential is known as Quantum Bouncing Ball. The qBounce collaboration recently observed such a system by dropping wave packets of ultracold neutrons by a height of roughly 30 microns. In this article, space and momentum spectra as well as Wigner functions of the neutron wave functions in the gravitational field of the Earth are analyzed. We investigate the quantum states in the “preparation region”, into which they transition after exiting a narrow double-mirror system and where we would expect to observe free fall and bounces in classical physics. For this, we start from the stationary solutions and eigenvalues of the Schrödinger equation in terms of Airy functions and their zeros. Subsequently, we examine space and momentum distributions as well as Wigner functions in phase space for pure and mixed quantum states. The possible influence of Yukawa-like forces for small distances of several micrometers from the mirror is included through first order perturbation calculations. Those allow us to study the resulting modifications of space and momentum distributions, and phase space functions.
We propose to apply Ramsey's method of separated oscillating fields to the spectroscopy of the quantum states in the gravity potential above a horizontal mirror. This method allows a precise measurement of quantum mechanical phaseshifts of a Schr\"odinger wave packet bouncing off a hard surface in the gravitational field of the Earth. Measurements with ultracold neutrons will offer a sensitivity to Newton's law or hypothetical short-ranged interactions, which is about 21 orders of magnitude below the energy scale of electromagnetism.
Ultracold neutrons confined in the Earth's gravitational field display quantized energy levels that have been observed for over a decade. In recent resonance spectroscopy experiments [T. Jenke et al., Nat. Phys. 7, 468 (2011)], the transition between two such gravitational quantum states was driven by the mechanical oscillation of the plates that confine the neutrons. Here we show that by applying a sinusoidal modulation with slowly varying frequency (chirp), the neutrons can be brought to higher excited states by climbing the energy levels one by one. The proposed experiment should make it possible to observe the quantum-classical transition that occurs at high neutron energies. Furthermore, it provides a technique to realize superpositions of gravitational quantum states, to be used for precision tests of gravity at short distances.
The quantum bouncer (QB) concept is a known QM textbook example of confined particle, namely, a solution to the 1D Schroedinger equation for a linear potential (the so-called Airy equation). It would be a great methodological challenge to create such a QM object in laboratory. An attempt of observation of the QB ``running'' in the horizontal direction was recently made by the international team at the Laue-Langevin Institute, Grenoble. The experiment was performed with ultra-cold neutrons. In this paper, the experiment is analyzed in view of the authors' claim that ``neutron quantum states in Earth gravitational field'' are observed. The experimental apparatus is designed for measurements of horizontal flux of neutrons passing through an absorbing wave guide with a variable height of absorber. From our analysis, it follows, however, that in such a layout measured data are not sensitive to quantum probability density in the vertical direction. The overall conclusion is made that the experimental data do not contain sufficient information to justify the claim.