The Polymer Bouncer

TL;DR

This paper applies Polymer Quantization to analyze a quantum particle bouncing under gravity, revealing energy spectrum shifts and setting bounds on the fundamental length scale based on experimental data.

Contribution

It introduces an analytical solution for the polymer Schrödinger equation in the gravitational bouncing scenario and constrains the fundamental length scale from experimental bounds.

Findings

Polymer energy spectrum shows a negative shift compared to quantum bouncer.

Upper bound for fundamental length scale is set at λ ≪ 0.6 Å.

Polymer corrections affect transition probabilities and spontaneous emission rates.

Abstract

Polymer Quantization (PQ) is a background independent quantization scheme that is deployed in Loop Quantum Gravity. This framework leads to a new short-distance (discretized) structure characterized by a fundamental length. In this paper we use PQ to analyze the problem of a particle bouncing on a perfectly reflecting surface under the influence of Earth's gravitational field, what we have called "\textit{The Polymer Bouncer}". In this scenario, deviations from the usual quantum effects are induced by the spatial discreteness, but not by a new short-range gravitational interaction. We solve the polymer Schr\"odinger equation in an analytical fashion, and we evaluate numerically the corresponding energy levels. We find that the polymer energy spectrum exhibits a negative shift compared to the obtained for the quantum bouncer. The comparison of our results with those obtained in the GRANIT experiment leads to an upper bound for the fundamental length scale, namely $\lambda \ll 0.6 \buildrel _{\circ} \over {\mathrm{A}}$. We find polymer corrections to the probability of transitions between levels, induced by small vibrations, together with the probability of spontaneous emission in the quadrupole approximation.