Tutorial: Calculation of Rydberg interaction potentials

TL;DR

This tutorial comprehensively reviews methods for calculating Rydberg interaction potentials, emphasizing the importance of full interaction models for high-precision experiments and providing open-source software tools.

Contribution

It introduces a complete approach to calculating Rydberg interaction potentials, including symmetry considerations and electromagnetic field effects, with practical software implementation.

Findings

Full interaction calculations are crucial for accurate Rydberg atom experiments.

Symmetry and selection rules significantly simplify Hamiltonian diagonalization.

Open-source software enables accessible high-order potential calculations.

Abstract

The strong interaction between individual Rydberg atoms provides a powerful tool exploited in an ever-growing range of applications in quantum information science, quantum simulation, and ultracold chemistry. One hallmark of the Rydberg interaction is that both its strength and angular dependence can be fine-tuned with great flexibility by choosing appropriate Rydberg states and applying external electric and magnetic fields. More and more experiments are probing this interaction at short atomic distances or with such high precision that perturbative calculations as well as restrictions to the leading dipole-dipole interaction term are no longer sufficient. In this tutorial, we review all relevant aspects of the full calculation of Rydberg interaction potentials. We discuss the derivation of the interaction Hamiltonian from the electrostatic multipole expansion, numerical and analytical methods for calculating the required electric multipole moments, and the inclusion of electromagnetic fields with arbitrary direction. We focus specifically on symmetry arguments and selection rules, which greatly reduce the size of the Hamiltonian matrix, enabling the direct diagonalization of the Hamiltonian up to higher multipole orders on a desktop computer. Finally, we present example calculations showing the relevance of the full interaction calculation to current experiments. Our software for calculating Rydberg potentials including all features discussed in this tutorial is available as open source.