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Chinese Physics B

Thermal vacuum state for the two-coupled-oscillator model at finite temperature: Derivation and application

Xue-xiang Xu, Liyun Hu, Qin Guo, Hong-Yi Fan

Published Sep 1, 2013• 3 citations
Publisher

Abstract

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.

DOI
10.1088/1674-1056/22/9/090302

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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.
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.
By using the partial trace method and the technique of integration within an ordered product of operators we obtain the explicit expression of the generalized thermal vacuum state (GTVS) for an RLC circuit instead of using the Takahashi–Umezawa approach. According to thermal field dynamics (TFD), namely, the expectation value of physical observables in this GTVS is equivalent to their ensemble average, based on GTVS we successfully derive the quantum fluctuations at nonzero temperature and the thermodynamical relations for the mesoscopic RLC circuit. Our results show that the higher the temperature is, the more quantum noise the RLC circuit exhibits.
Perturbation theory in thermofield dynamics (a real time finite temperature quantum field theory) is formulated in the interaction representation and a diagrammatic expansion for the expectation value of time ordered products is derived solely on the basis of the thermal state condition and the algebraic properties of the operators. This new formulation has a much wider domain of applicability than those presented previously. The method is applicable even in the case of purely quantum mechanical systems. When certain bosonic operators, which commute with the unperturbed Hamiltonian, contribute to the renormalization of the thermal state vacuum the vacuum diagrams do not cancel in the usual way. It will be shown how these diagrams arise and how they may be computed.