Neutrons not only fall classically in the gravitational field of the Earth as potatoes and other normal objects do; at certain conditions, they reveal quantum behavior, or even form well defined gravitational quantum states. Due to the relative simplicity of quantum mechanical rules for solving this problem, we could predict analytically the motion of a neutron in a grav itational field in the vicinity of a reflecting surface. Such a bouncing neutron can have only certain, discrete values of energy. These extremely small energy values turn out to be very sensitive probes to the presence of even tiny extra interactions, in particular those between the neutron and the mirror. Moreover, several emerging fields of research profit from experi ence and knowledge gained in the recent experimental observation of gravitational quantum states of ultracold neutrons, and extend the developed methods to atoms and anti-atoms, to whispering-gallery effect and other domains, thus providing more and more application for the method. Quantum bouncing particles could play their role in quantum tests of the equivalence principle, in explorations of gravitational properties of antimatter, in searching for new funda mental short-range and other interactions beyond the Standard Model of particle physics, in surface and thin-layer physics, in chemistry, and so on. In this short note we mention results of a few previous years and project nearest developments ..
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.
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 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.
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 where 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 the 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 using scatterers with strong roughness. In general, strong roughness with high amplitude is preferable if one wants just to detect the individual gravitational states, while strong-roughness experiments with small amplitude and high aperture are preferable for the quantitative analysis of the data. We also discuss ways to further improve the accuracy of calculations and to optimize the experimental regime.