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arXiv:1404.7834 · New Journal of Physics

Exact solvability, non-integrability, and genuine multipartite entanglement dynamics of the Dicke model

Shu He, Liwei Duan, Qing‐Hu Chen

Published Apr 16, 2015• 16 citations
arXivHTMLPDF

Abstract

In this paper, the finite-size Dicke model of arbitrary number of qubits is solved analytically in a unified way within extended coherent states. For the or Dicke models ( k is an integer), the G -function, which is only an energy dependent determinant, is derived in a transparent manner. The regular spectrum is completely and uniquely given by stable zeros of the G -function. The closed-form exceptional eigenvalues are also derived. The level distribution controlled by the pole structure of the G -functions suggests non-integrability for model at any finite coupling in the sense of recent criteria found in the literature. A preliminary application to the exact dynamics of genuine multipartite entanglement in the finite- N Dicke model is presented using the obtained exact solutions.

DOI
10.1088/1367-2630/17/4/043033

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In this paper, the finite-size Dicke model of arbitrary number of qubits is solved analytically in a unified way within extended coherent states. For the or Dicke models ( k is an integer), the G -function, which is only an energy dependent determinant, is derived in a transparent manner. The regular spectrum is completely and uniquely given by stable zeros of the G -function. The closed-form exceptional eigenvalues are also derived. The level distribution controlled by the pole structure of the G -functions suggests non-integrability for model at any finite coupling in the sense of recent criteria found in the literature. A preliminary application to the exact dynamics of genuine multipartite entanglement in the finite- N Dicke model is presented using the obtained exact solutions.
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We construct a set of criteria detecting genuine multipartite entanglement in arbitrary dimensional multipartite systems. These criteria are optimally suited for detecting multipartite entanglement in nnn-qubit Dicke states with mmm excitations, as shown in exemplary cases. Furthermore, they can be employed to detect multipartite entanglement in different states related to quantum cloning, decoherence-free communication, and quantum secret sharing. In a detailed analysis, we show that the criteria are also more robust to noise than any other criterion known so far, especially with increasing system size. Furthermore, it is shown that the number of required local observables scales only polynomially with size, thus making the criteria experimentally feasible.
The quantum dynamics of initial coherent states is studied in the Dicke model and correlated with the dynamics, regular or chaotic, of their classical limit. Analytical expressions for the survival probability, i.e. the probability of finding the system in its initial state at time t, are provided in the regular regions of the model. The results for regular regimes are compared with those of the chaotic ones. It is found that initial coherent states in regular regions have a much longer equilibration time than those located in chaotic regions. The properties of the distributions for the initial coherent states in the Hamiltonian eigenbasis are also studied. It is found that for regular states the components with no negligible contribution are organized in sequences of energy levels distributed according to Gaussian functions. In the case of chaotic coherent states, the energy components do not have a simple structure and the number of participating energy levels is larger than in the regular cases.
For every N-qubit density matrix written in the computational basis, an associated "X-density matrix" can be obtained by vanishing all entries out of the main- and anti-diagonals. It is very simple to compute the genuine multipartite (GM) concurrence of this associated N-qubit X-state, which, moreover, lower bounds the GM-concurrence of the original (non-X) state. In this paper, we rely on these facts to introduce and benchmark a heuristic for estimating the GM-concurrence of an arbitrary multiqubit mixed state. By explicitly considering two classes of mixed states, we illustrate that our estimates are usually very close to the standard lower bound on the GM-concurrence, being significantly easier to compute. In addition, while evaluating the performance of our proposed heuristic, we provide the first characterization of GM-entanglement in the steady states of the driven Dicke model at zero temperature.