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arXiv:1601.06132 · Physical review. D/Physical review. D.

Gravitationally induced quantum transitions

Alexandre Landry, Manu B. Paranjape

Published Jun 16, 2016• 9 citations
arXivHTMLPDF

Abstract

In this paper, we calculate the probability for resonantly inducing transitions in quantum states due to time-dependent gravitational perturbations. Contrary to common wisdom, the probability of inducing transitions is not infinitesimally small. We consider a system of ultracold neutrons, which are organized according to the energy levels of the Schr\"odinger equation in the presence of the Earth's gravitational field. Transitions between energy levels are induced by an oscillating driving force of frequency $\ensuremath{\omega}$. The driving force is created by oscillating a macroscopic mass in the neighborhood of the system of neutrons. The neutron lifetime is approximately 880 sec while the probability of transitions increases as ${t}^{2}$. Hence, the optimal strategy is to drive the system for two lifetimes. The transition amplitude then is of the order of $1.06\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$, and hence with a million ultracold neutrons, one should be able to observe transitions.

DOI
10.1103/physrevd.93.122006

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In this paper, we calculate the probability for resonantly inducing transitions in quantum states due to time-dependent gravitational perturbations. Contrary to common wisdom, the probability of inducing transitions is not infinitesimally small. We consider a system of ultracold neutrons, which are organized according to the energy levels of the Schr"odinger equation in the presence of the Earth's gravitational field. Transitions between energy levels are induced by an oscillating driving force of frequency \ensuremathω\ensuremath{\omega}\ensuremathω. The driving force is created by oscillating a macroscopic mass in the neighborhood of the system of neutrons. The neutron lifetime is approximately 880 sec while the probability of transitions increases as t2{t}^{2}t2. Hence, the optimal strategy is to drive the system for two lifetimes. The transition amplitude then is of the order of 1.06\ifmmode×\else\texttimes\fi10\ensuremath−51.06\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}1.06\ifmmode×\else\texttimes\fi10\ensuremath−5, and hence with a million ultracold neutrons, one should be able to observe transitions.
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.
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.
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.
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.
Gravity is the weakest of all four known forces in the universe. Quantum states of an elementary particle due to such a weak field is certainly very shallow and would therefore be an experimental challenge to detect. Recently an experimental attempt was made by V. V. Nesvizhevsky et al., Nature 415, 297 (2002), to measure the quantum states of a neutron, which shows that ground state and few excited states are \sim 10^{-12}eV. We show that the energy of the ground state of a neutron confined above Earth's surface should be \sim 10^{-37}eV. The experimentally observed energy levels are 10^{25} times deeper than the actual energy levels it should be and thus certainly not due to gravitational effect of Earth. Therefore the correct interpretation for the painstaking experimental results of Ref. \cite{nes1} is due to the confinement potential of a one dimensional box of length L \sim 50μm, generated from the experimental setup as commented before \cite{hansoon}. Our results thus creates a new challenge to the experimentalist to resolve the shallow energy levels of the neutron in Earth's gravitational field in future.
The authors study quantum effects resulting from mass-energy equivalence using ultracold neutrons that are confined gravitationally by considering a quantum correction to the neutron's mass due to the neutron spin in the presence of an external magnetic field. The authors show that the neutron's additional weight due to the quantum treatment of mass-energy equivalence will cause a small shift in the neutron's eigenenergies and eigenstates, which could provide signatures in an interferometric experiment.