Entanglement and U(D)-spin squeezing in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin-Meshkov-Glick D-level atom models

TL;DR

This paper generalizes spin coherent states and entanglement measures to symmetric multi-quDit systems with U(D) symmetry, analyzing their quantum phase transitions and proposing SU(D) spin squeezing as an entanglement marker.

Contribution

It introduces U(D)-spin coherent states, extends spin squeezing to SU(D), and applies these concepts to analyze quantum phases in multi-level atom models.

Findings

Entanglement and squeezing characterize quantum phase transitions.

Ground states exhibit distinct entanglement patterns at critical points.

SU(D) spin squeezing correlates with pairwise entanglement in multi-level systems.

Abstract

Collective spin operators for symmetric multi-quDit (namely, identical $D$-level atom) systems generate a U$(D)$ symmetry. We explore generalizations to arbitrary $D$ of SU(2)-spin coherent states and their adaptation to parity (multicomponent Schr\"odinger cats), together with multi-mode extensions of NOON states. We write level, one- and two-quDit reduced density matrices of symmetric $N$-quDit states, expressed in the last two cases in terms of collective U$(D)$-spin operator expectation values. Then we evaluate level and particle entanglement for symmetric multi-quDit states with linear and von Neumann entropies of the corresponding reduced density matrices. In particular, we analyze the numerical and variational ground state of Lipkin-Meshkov-Glick models of $3$-level identical atoms. We also propose an extension of the concept of SU(2) spin squeezing to SU$(D)$ and relate it to pairwise $D$-level atom entanglement. Squeezing parameters and entanglement entropies are good markers that characterize the different quantum phases, and their corresponding critical points, that take place in these interacting $D$-level atom models.