Geometric representation of spin correlations and applications to ultracold 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.

New phase space formulations and quantum dynamics approaches

“…Phase space with coordinate‐momentum variables is a fundamental concept and offers a convenient tool to describe statistics as well as dynamics in classical mechanics. In comparison to other equivalent interpretations of quantum mechanics, phase space formulations offer more insight and understanding between quantum and classical counterpart concepts, which are widely used in chemical and biological dynamics and spectroscopy, 1–60 quantum optics, 51,61–70 cryogenic physics/chemistry, 71–75 quantum information and computation, 76–87 and so forth.…”

Analysis of continuous and discrete Wigner approximations for spin dynamics

“…We are able to gain insight into the workings of TWA and DTWA and isolate the nuanced differences between them by utilizing the correlation matrix visualization (CMV) technique, which was recently introduced in Ref. [58] building on geometrical visualization techniques in Refs. [59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78].…”

“…The correlation matrix C ij is a 3×3 matrix with components C µν ij , µ, ν ∈ {x, y, z}. Reference [58] introduced a geometric tool to visualize C ij using a three-dimensional contour called a CMV. We use this tool to analyze the results of the Wigner approximations.…”

Shielding Property in Higher Dimensions

“…Then, the quantum Ising model is often used as a benchmark for checking the solution of other models or for verifying the effectiveness of new approximation techniques [8,9]. It is a paragmatic model to study, introduce or illustrate many features and concepts of quantum many body systems, such the relation between entanglement and phase transitions [10], decoherence of open quantum systems [11], definitions of work in quantum thermodynamics [12], and geometric and topological characterizations of many body models [13,14].…”