From trilobite molecules to tight-binding models

When a Rydberg atom and a ground state "perturber" atom encounter one another, they interact via an oscillatory potential mediated by the scattering of the Rydberg electron off of the perturber. The wells in this potential can trap the perturber, binding the two atoms together into a molecule. This same mechanism can also cause trimers, tetramers, and larger clusters to form. As the number of perturbers increases, it becomes impractical to describe this Rydberg composite within the framework of molecular physics, as its properties start to mirror those of a solid-state system. A clear example of this is the modified spectrum of the Rydberg electron in the presence of a dense environment of immobile perturbers. Because this spectrum of an isolated Rydberg atom is highly degenerate due to its SO(4) symmetry, an exact mapping exists between the perturbed Rydberg states and the states of a particle in a tight-binding lattice. Using this mapping, we demonstrate how to realize a thermodynamic limit in the Rydberg electron. This thermodynamic limit and the control over the properties of the tight-binding lattice through the arrangement of the perturbers allow us to prove that the Rydberg electron can undergo Anderson localization in a disordered perturber landscape. The confluence of the infinite-ranged Coulomb potential and the zero-range electron-atom potentials leads to a plethora of possible lattice parameters, ranging from nearest-neighbor hopping to all-to-all coupling between sites.