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Physical Review A

Entanglement and bifurcation in the integrable dimer

Xi-Wen Hou, Jinghua Chen, Bambi Hu

Published Mar 10, 2005• 37 citations
Publisher

Abstract

In this Brief Report the properties of both dynamical and static entanglement in the integrable quantum dimer are studied in terms of the reduced-density linear entropy and von Neumann entropy with various coupling parameters, total boson numbers, and initial states. The mean entanglement, which is defined to be averaged over time, is used to describe the influence of the classical separatrix on the behavior of entanglement. It is shown that the mean entanglement exhibits a maximum near the position of the corresponding classical separatrix energy and that the static entanglement of the state with the largest eigenvalue of the quantum spectrum displays a maximum near the bifurcation point. For weak coupling and larger total boson number the maximum entanglement state is exactly at the position of the classical separatrix and bifurcation. In strong coupling all initial states have nearly the same mean entanglement.

DOI
10.1103/physreva.71.034302

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In this Brief Report the properties of both dynamical and static entanglement in the integrable quantum dimer are studied in terms of the reduced-density linear entropy and von Neumann entropy with various coupling parameters, total boson numbers, and initial states. The mean entanglement, which is defined to be averaged over time, is used to describe the influence of the classical separatrix on the behavior of entanglement. It is shown that the mean entanglement exhibits a maximum near the position of the corresponding classical separatrix energy and that the static entanglement of the state with the largest eigenvalue of the quantum spectrum displays a maximum near the bifurcation point. For weak coupling and larger total boson number the maximum entanglement state is exactly at the position of the classical separatrix and bifurcation. In strong coupling all initial states have nearly the same mean entanglement.
The entanglement of the N-harmonium system (i.e., a completely-integrable model of N particles where both the confinement and the two-particle interaction are har-monic) is shown to be analytically determined in terms of N and the relative interaction strength. First, the entanglement of the N-boson system via the von Neumann entropy is computed by use of the corresponding natural occupation numbers. Below a critical value Nc of N it is found that, for positive values of the coupling constant the entan-glement grows when the number of particles is increasing; opposite situation occurs for N> Nc. Then the entanglement of the N-fermion system is investigated by means of the linear entropy, which is a linearized form of the von Neumann entropy of the one-body reduced density matrix. We have shown that the entanglement grows when the number of particles is increasing in the region of negative coupling. For positively increasing values of the coupling the entanglement as a function of N varies according to various regimes. Moreover, in the strong coupling regime the entropy of the system decreases when N is growing. For fixed N, the entanglement is found (a) to decrease (increase) for negatively (positively) increasing values of the coupling constant, and (b) to grow when the energy is increasing. Finally, let us point out that the spatial and spin contributions to the total entanglement are of comparable size.
We analyze the classical and quantum properties of the integrable dimer problem. The classical version exhibits exactly one bifurcation in phase space, which gives birth to permutational symmetry broken trajectories and a separatrix. The quantum analysis yields all tunneling rates (splittings) in leading order of perturbation. In the semiclassical regime the eigenvalue spectrum obtained by numerically exact diagonalization allows to conclude about the presence of a separatrix and a bifurcation in the corresponding classical model.
We analyze the classical and quantum properties of the integrable dimer problem. The classical version exhibits exactly one bifurcation in phase space, which gives birth to permutational symmetry broken trajectories and a separatrix. The quantum analysis yields all tunneling rates (splittings) in leading order of perturbation. In the semiclassical regime the eigenvalue spectrum obtained by numerically exact diagonalization allows one to conclude about the presence of a separatrix and a bifurcation in the corresponding classical model.
The dynamical properties of quantum entanglement in the Dicke model without rotating-wave approximation are investigated in terms of the reduced-density linear entropy. The characteristic time of decoherence process in the early-time evolution is numerically obtained and it is shown that the characteristic time decreases as the coupling parameter increases. The mean entanglement, which is defined to be averaged over time, is employed to describe the influences of both quantum phase transition and corresponding classical chaos on the behavior of entanglement. For a given energy, initial conditions are taken to be minimum uncertainty wave packets centered at regular and chaotic regions of the classical phase space. It is shown that the entanglement has a distinct change at the quantum phase transition, and that the entanglement for regular initial conditions is smaller than that for chaotic ones in the case of weak coupling, while it fluctuates with small amplitude in strong coupling and for chaotic initial conditions.
The ground-state entanglement of a single particle of the N-harmonium system (i.e., a completely-integrable model of NNN particles where both the confinement and the two-particle interaction are harmonic) is shown to be analytically determined in terms of NNN and the relative interaction strength. For bosons, we compute the von Neumann entropy of the one-body reduced density matrix by using the corresponding natural occupation numbers. There exists a critical number NcN_cNc​ of particles so that below it, for positive values of the coupling constant, the entanglement grows when the number of particles is increasing; the opposite occurs for N>NcN > N_cN>Nc​. For fermions, we compute the one-body reduced density matrix for the closed-shell spinned case. In the strong coupling regime, the linear entropy of the system decreases when NNN is growing. For fixed NNN, the entanglement is found (a) to decrease (increase) for negatively (positively) increasing values of the coupling constant, and (b) to grow when the energy is increasing. Moreover, the spatial and spin contributions to the total entanglement are found to be of comparable size.