Resonance Cascades and Number Theory

TL;DR

This paper explores how quantum resonance cascades in a one-dimensional potential relate to a specific number theory statement, demonstrating that the cascade's existence depends on the validity of the statement about powers of 3.

Contribution

It establishes a novel connection between quantum resonance phenomena and a fundamental number theory statement, using a tailored potential and spectral analysis.

Findings

Resonance cascades occur along a logarithmically spaced energy level subsequence.

Removing a specific energy level halts the cascade.

Numerical experiments confirm the dependence on the number theory statement.

Abstract

In this article, we are interested in situations where the existence of a contiguous cascade of quantum resonant transitions is predicated on the validity of a particular statement in number theory. The setting is a tailored one-atom one-dimensional potential with a prescribed spectrum, under a weak periodic perturbation. The former is, by now, an experimental reality [D. Cassettari, G. Mussardo and A. Trombettoni, PNAS Nexus {\bf 2}, pgac279 (2022)]. As a case study, we look at the following trivial statement: "Any power of $3$ is an integer." Consequently, we "test" this statement in a numerical experiment where we demonstrate an unimpeded upward mobility along an equidistant, $\ln(3)$-spaced subsequence of the energy levels of a potential with a log-natural spectrum, under a frequency $\ln(3)$ time-periodic perturbation. We further show that when we "remove" $9$ from the set of integers -- by excluding the corresponding energy level from the spectrum -- the cascade halts abruptly.