Geometric representation of spin correlations and applications to ultracold systems

TL;DR

This paper introduces a geometric visualization method for spin correlations in ultracold systems, revealing hidden structures and simplifying the analysis of their dynamics through three-dimensional shapes.

Contribution

It establishes a one-to-one correspondence between spin correlations and 3D shapes, enabling intuitive geometric reasoning about complex correlation dynamics.

Findings

Shapes unveil hidden correlation structures

Simplifies analysis of correlation dynamics

Applied to cold atom systems

Abstract

We provide a one-to-one map between the spin correlations and certain three-dimensional shapes, analogous to the map between single spins and Bloch vectors, and demonstrate its utility. Much as one can reason geometrically about dynamics using a Bloch vector -- e.g. a magnetic field causes it to precess and dissipation causes it to shrink -- one can reason similarly about the shapes we use to visualize correlations. This visualization demonstrates its usefulness by unveiling the hidden structure in the correlations. For example, seemingly complex correlation dynamics can be described as simple motions of the shapes. We demonstrate the simplicity of the dynamics, which is obscured in conventional analyses, by analyzing several physical systems of relevance to cold atoms.