Spectra of Neutron Wave Functions in Earth's Gravitational Field

TL;DR

This paper analyzes the quantum states of neutrons in Earth's gravitational field, examining their spectra, phase space distributions, and the effects of hypothetical Yukawa-like forces, providing insights into quantum gravitational interactions.

Contribution

It introduces a detailed analysis of neutron wave functions in Earth's gravity, including spectra, phase space, and perturbations from hypothetical forces, extending understanding of quantum gravitational effects.

Findings

Space and momentum spectra are characterized using Airy functions.

Wigner functions reveal phase space structures of neutron states.

Yukawa-like forces cause measurable modifications in distributions.

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 Schr\"odinger 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 eventual 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.