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Communications in Theoretical Physics

Degree of Entanglement for Some Bipartite Entangled Bosonic Systems

Xian-Ting Liang

Published Oct 15, 2004• 2 citations
Publisher

Abstract

We calculate the degree of entanglement for some bipartite entangled states of continuous variables. These states include common two-mode squeezed vacuum state, thermal vacuum state of a free single particle (where the fictitious tilde system is regarded as another particle), and the squeezed vacuum state of two coupling harmonic oscillators. The degree of entanglement for these quantum systems are shown clearly by using the technique of integration within an ordered product of operators.

DOI
10.1088/0253-6102/42/4/535

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We calculate the degree of entanglement for some bipartite entangled states of continuous variables. These states include common two-mode squeezed vacuum state, thermal vacuum state of a free single particle (where the fictitious tilde system is regarded as another particle), and the squeezed vacuum state of two coupling harmonic oscillators. The degree of entanglement for these quantum systems are shown clearly by using the technique of integration within an ordered product of operators.
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We study two continuous variable systems (or two harmonic oscillators) and investigate their entanglement evolution under the influence of non-Markovian thermal environments. The continuous variable systems could be two modes of electromagnetic fields or two nanomechanical oscillators in the quantum domain. We use the quantum open system method to derive the non-Markovian master equations of the reduced density matrix for two different but related models of the continuous variable systems. The two models both consist of two interacting harmonic oscillators. In model A, each of the two oscillators is coupled to its own independent thermal reservoir, while in model B the two oscillators are coupled to a common reservoir. To quantify the degrees of entanglement for bipartite continuous variable systems in Gaussian states, logarithmic negativity is used. We find that the dynamics of the quantum entanglement is sensitive to the initial states, the oscillator-oscillator interaction, the oscillator-environment interaction and the coupling to a common bath or to different, independent baths.
We analyze density operator, Q function, photon statistics, quadrature squeezing, and Cv entanglement that holds true for a pair of superposed two-mode photon system. We also present a slightly modified definition of entanglement and squeezing for a pair of super- posed two-mode system and new definition for degree of entanglement. In order to carry out our analysis, we consider a quantum system with Gaussian variables with zero mean. It is found that maximum degree of entanglement as well as squeezing occurs at steady-state and threshold.
Mode entanglement in a two-mode squeezed state is studied. The entanglement spectrum is derived and interpreted as Boltzmann coefficients of harmonic oscillator eigenstates at thermal equilibrium. The effective temperature is expressed in terms of squeezing parameters and is equal to zero when the state is unsqueezed and increases without bound as the state is infinitely squeezed. It is demonstrated that the reduced density operator can be identified as a canonical ensemble of single harmonic oscillator at some effective temperature Te. The Renyi entanglement entropy is calculated and found to increase without bound as the squeezing parameter increases. The single-copy entanglement, which is a limiting value of the Renyi entropy, is interpreted as a partition function.