Resonance Cascades and Number Theory

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.