The Critical Point Entanglement and Chaos in the Dicke Model

Lina Bao,Feng Pan,Jerry Draayer,Jing Lu

doi:10.3390/e17075022

Lina Bao, Feng Pan + Show 2 more

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https://doi.org/10.3390/e17075022

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Journal: Entropy	Publication Date: Jul 16, 2015
Citations: 65	License type: CC BY 4.0

Affiliation: Liaoning Normal University, Louisiana State University

Abstract

Ground state properties and level statistics of the Dicke model for a finite number of atoms are investigated based on a progressive diagonalization scheme (PDS). Particle number statistics, the entanglement measure and the Shannon information entropy at the resonance point in cases with a finite number of atoms as functions of the coupling parameter are calculated. It is shown that the entanglement measure defined in terms of the normalized von Neumann entropy of the reduced density matrix of the atoms reaches its maximum value at the critical point of the quantum phase transition where the system is most chaotic. Noticeable change in the Shannon information entropy near or at the critical point of the quantum phase transition is also observed. In addition, the quantum phase transition may be observed not only in the ground state mean photon number and the ground state atomic inversion as shown previously, but also in fluctuations of these two quantities in the ground state, especially in the atomic inversion fluctuation.

Highlights

It is well known that the Dicke model [1] exhibits a “superradiant” quantum phase transition (QPT) in the thermodynamic limit as shown by Hepp and Lieb [2]
The purpose of this paper is to show that, for a finite number of atoms, the entanglement measure defined in terms of the normalized von Neumann entropy of the reduced density matrix of the atoms reaches its maximum value at the critical point of the QPT, at which the system is most chaotic
Ground state properties and the level statistics of the Dicke model for a finite number of atoms are investigated based on the progressive progressive diagonalization scheme (PDS)

Summary

Introduction

It is well known that the Dicke model [1] exhibits a “superradiant” quantum phase transition (QPT) in the thermodynamic limit as shown by Hepp and Lieb [2]. Instead of the Dicke basis, the Dicke Hamiltonian can be diagonalized in a shifted boson basis where convergence is reached with a smaller number of shifted boson states across the whole coupling regime [21] This approach was used to study the quantum criticality, the finite size effects, fidelity and the order parameter in the Dicke model [22]. An application of the PDS to the Dicke model can be used for a determination of ground state properties, such as photon number, the atomic inversion, fluctuations of the latter two quantities, an entanglement measure, the Shannon information entropy, etc., and, in addition, for the study of level statistics with a larger, but finite number of atoms. The QPT may be observed in the ground state mean photon number and the ground state atomic inversion, as shown in [10], and in fluctuations of these two quantities in the ground state, especially in the atomic inversion fluctuation

The PDS for the Dicke Model

The QPT and Entanglement

Level Statistical Properties

Conclusions