Fractatomic Physics: An Invitation with Atomic Stability and Rydberg States in Fractal Spaces

TL;DR

The work extends quantum theory to fractal spaces by formulating a fractal Laplacian with $\Delta \propto r^{-2(\mathscr{D}_{v}-\mathscr{D}_{s})}$ and a central potential $U(r) \propto - r^{-\,\kappa}$, to study Hydrogen-like atoms. It identifies scale-free boundaries at $\kappa = 2(\mathscr{D}_{v}-\mathscr{D}_{s})$, yielding fractal thresholds such as $\mathscr{D}_{v}/\mathscr{D}_{s}=4/3$ (full fractal) and $\mathscr{D}_{v}-\mathscr{D}_{s}=1/2$ (embedded). Through a WKB approach with a generalized Langer modification, it derives Rydberg-state scaling laws and shows that near the threshold the atomic size grows as $\tilde{r}_{\max}\propto n^{1/0^+}$, enabling strong long-range interactions. The findings point to feasible laboratory realizations and suggest fractal-space atoms as a platform for quantum technologies.

Abstract

We explore the physical quantum properties of atoms in fractal spaces, both as a theoretical generalization of normal integer-dimensional Euclidean spaces and as an experimentally realizable setting. We identify the threshold of fractality at which Ehrenfest atomic instability emerges, where the Schrödinger equation describing the wave-function of a single electron orbiting around an atom becomes scale-free, and discuss the potential of observing this phenomena in laboratory settings. We then study the Rydberg states of stable atoms using the Wentzel-Kramers-Brillouin approximation, along with a proposed extension for the Langer modification, in general fractal dimensionalities. We show that fractal space atoms near instability explode in size even at low-number excited state, making them highly suitable to induce strong entanglements and foster long-range many-body interactions. We argue that atomic physics in fractal spaces -- ``fractatomic physics'' -- is a rich research avenue deserving of further theoretical and experimental investigations.

Figures (2)

• Figure 1: A fractal lattice and an artificial atom created in the lab.(A1) A two-dimensional slice of a fractal "blueprint" from normal three-dimensional natural soil, with fractality $\left( \mathscr{D}_{\text{v}},\mathscr{D}_{\text{v}}\right) = \left( 1.79,1.48 \right)$phan2024vanishinggimenez1997fractal; the black area represents the open space. (A2) The corresponding fractal lattice embedded in three-dimensional Euclidean space, in which dots are lattice sites and hopping between neighbor sites serves to establish an effective continuity within the material. (B) An artificial nucleus -- made of five Calcium dimers placed on a two-dimensional graphene surface -- created in laboratory setting wang2013observing; the bright color representing the high $dI/d\phi$ signal (differential current with respect to electrical potential change) when probing the material.
• Figure 2: Numerical investigations of atomic properties at different fractalities. The row (A) are figures from the full fractal space scenario. (A1) The estimated energy $| \tilde{E}|$ as a function of excited state number $n$ (from ground state $n=1$ to $n=50$) in log-log scale, following from solving Eq. \ref{['WS_quant']} numerically. We then compare them with their expected asymptotic behavior, given by Eq. \ref{['R_E_asymp']}. The fractal dimensionalities $\left( \mathscr{D}_{\text{v}},\mathscr{D}_{\text{s}} \right)$ we choose to investigate are $\left(3,2 \right)$, $\left(2.5,1 \right)$, and $\left(2.1,1.4 \right)$; their numerical values are given by triangle markers pointing right, pointing left, and pointing up, while their theoretical asymptotic behaviors are shown with continuous, dot-dot, and dot-dash lines. (A2) Rydberg energy exponent and (A3) Rydberg size exponent as a function of fractalities, from Eq. \ref{['Ryd_gen']}. The row (B) are figures from the embedded fractal space scenario, in which (B1-3) are of similar descriptions with (A1-3).