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Journal of Physics A Mathematical and Theoretical

Thermal purification and thermal entanglement of the two-resonant-coupled-oscillator system at finite temperature

Heng-Mei Li, Xue-xiang Xu, Hong-Yun Huang, Ye‐Jun Xu

Published Aug 29, 2014• 0 citations
Publisher

Abstract

Following the spirit of thermo field dynamics, we adopt the partial trace method to explicitly derive the thermal vacuum state (TVS) for the Hamiltonian H of the two-resonant-coupled-oscillator system coupled to the thermal equilibrium environment. The TVS is a pure state in the four-dimensional extended Hilbert space. Based on the TVS, the Wigner function of this quantum system is derived conveniently. From the known Gaussian Wigner function, we obtain the covariance matrix of the quadrature component and quantify the degree of thermal entanglement for the TVS by using logarithmic negativity. It is found that entanglement takes place only at low temperature.

DOI
10.1088/1751-8113/47/37/375301

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Following the spirit of thermo field dynamics, we adopt the partial trace method to explicitly derive the thermal vacuum state (TVS) for the Hamiltonian H of the two-resonant-coupled-oscillator system coupled to the thermal equilibrium environment. The TVS is a pure state in the four-dimensional extended Hilbert space. Based on the TVS, the Wigner function of this quantum system is derived conveniently. From the known Gaussian Wigner function, we obtain the covariance matrix of the quadrature component and quantify the degree of thermal entanglement for the TVS by using logarithmic negativity. It is found that entanglement takes place only at low temperature.
Following the spirit of thermo field dynamics initiated by Takahashi and Umezawa, we employ the technique of integration within an ordered product of operators to derive the thermal vacuum state (TVS) for the Hamiltonian H of the two-coupled-oscillator model. The ensemble averages of the system are derived conveniently by using the TVS. In addition, the entropy for this system is discussed based on the relation between the generalized Hellmann—Feynman theorem and the entroy variation in the context of the TVS.
Using the theory of open systems based on completely positive quantum dynamical semigroups, we describe the dynamics of entanglement for a system consisting of two uncoupled harmonic oscillators interacting with a thermal reservoir. Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the evolution of entanglement in terms of the covariance matrix for a Gaussian input state. For some values of the temperature of environment, the state keeps for all times its initial type - separable or entangled. In other cases, entanglement generation, entanglement sudden death or a repeated collapse and revival of entanglement take place. We analyze also the time evolution of the logarithmic negativity, which characterizes the degree of quantum entanglement.
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.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the dynamics of entanglement for a system consisting of two uncoupled harmonic oscillators interacting with a thermal environment. Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the evolution of entanglement in terms of the covariance matrix for a Gaussian input state. For some values of the temperature of environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a repeated collapse and revival of entanglement take place. We analyze also the time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state.
We describe the evolution of the quantum entanglement in a system composed of two interacting bosonic modes immersed in a thermal reservoir, in the framework of the theory of open systems based on completely positive quantum dynamical semigroups. The evolution of entanglement is described in terms of the covariance matrix for Gaussian initial states. We calculate the logarithmic negativity and show that for separable initial squeezed thermal states entanglement generation may take place, for definite values of squeezing parameter, average photon numbers, temperature of the thermal bath, dissipation constant and the strength of interaction between the two modes. After its generation one can observe temporary suppressions and revivals of the entanglement. For entangled initial squeezed thermal states, entanglement suppression takes place, for all temperatures of the reservoir, and temporary revivals and suppressions of entanglement can be observed too. In the limit of infinite time the system evolves asymptotically to an equilibrium state which may be entangled or separable.
Using the thermal Green's function approach we propose a general method to investigate entanglement of the vacuum state or thermal ground states in an arbitrary dimensional space-time. As an application we show quantum separability of the massive thermal scalar field vacuum in the 1+1 dimensional cylindrical space-time. Separability is demonstrated using the positive partial transpose criterion for effective two-mode Gaussian states of collective operators. In this case, for all mass and temperature values entanglement is absent between the collective operators.