Temperature effects on a network of dissipative quantum harmonic oscillators: collective damping and dispersion processes

In this paper we extend the results presented in (de Ponte, Mizrahi and Moussa 2007 Phys. Rev. A 76 032101) to treat quantitatively the effects of reservoirs at finite temperature in a bosonic dissipative network: a chain of coupled harmonic oscillators whatever its topology, i.e., whichever the way the oscillators are coupled together, the strength of their couplings and their natural frequencies. Starting with the case where distinct reservoirs are considered, each one coupled to a corresponding oscillator, we also analyze the case where a common reservoir is assigned to the whole network. Master equations are derived for both situations and both regimes of weak and strong coupling strengths between the network oscillators. Solutions of these master equations are presented through the normal ordered characteristic function. These solutions are shown to be significantly involved when temperature effects are considered, making difficult the analysis of collective decoherence and dispersion in dissipative bosonic networks. To circumvent these difficulties, we turn to the Wigner distribution function which enables us to present a technique to estimate the decoherence time of network states. Our technique proceeds by computing separately the effects of dispersion and the attenuation of the interference terms of the Wigner function. A detailed analysis of the dispersion mechanism is also presented through the evolution of the Wigner function. The interesting collective dispersion effects are discussed and applied to the analysis of decoherence of a class of network states. Finally, the entropy and the entanglement of a pure bipartite system are discussed.