# Universal momentum-to-real-space mapping of topological singularities

## TL;DR

This paper demonstrates a universal method to map topological properties from momentum space to real space in various physical systems, revealing new insights into topological charge conversion in photonic and atomic platforms.

## Contribution

It introduces a universal mapping technique for topological singularities from momentum to real space, applicable across different physical systems including photonic lattices and ultracold atomic gases.

## Key findings

- Topological charge conversion follows the rule l to l+2s in photonic lattices.
- The mapping accounts for all initial excitation conditions with pseudospin-orbit interaction.
- The topological mapping persists even in deformed lattices where angular momentum isn't conserved.

## Abstract

Topological properties of materials, as manifested in the intriguing phenomena of quantum Hall effect and topological insulators, have attracted overwhelming transdisciplinary interest in recent years. Topological edge states, for instance, have been realized in versatile systems including electromagnetic-waves. Typically, topological properties are revealed in momentum space, using concepts such as Chern number and Berry phase. Here, we demonstrate a universal mapping of the topology of Dirac-like cones from momentum space to real space. We evince the mapping by exciting the cones in photonic honeycomb (pseudospin-1/2) and Lieb (pseudospin-1) lattices with vortex beams of topological charge l, optimally aligned for a chosen pseudospin state s, leading to direct observation of topological charge conversion that follows the rule of l to l+2s. The mapping is theoretically accounted for all initial excitation conditions with the pseudospin-orbit interaction and nontrivial Berry phases. Surprisingly, such a mapping exists even in a deformed lattice where the total angular momentum is not conserved, unveiling its topological origin. The universality of the mapping extends beyond the photonic platform and 2D lattices: equivalent topological conversion occurs for 3D Dirac-Weyl synthetic magnetic monopoles, which could be realized in ultracold atomic gases and responsible for mechanism behind the vortex creation in electron beams traversing a magnetic monopole field.

---
Source: https://tomesphere.com/paper/1908.05633