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Physical Review A

Networks of dissipative quantum harmonic oscillators: A general treatment

M. A. de Ponte, Salomon Sylvain Mizrahi, M. H. Y. Moussa

Published Sep 4, 2007• 26 citations
Publisher

Abstract

We present a general treatment of a bosonic dissipative network: a chain of coupled dissipative 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 a general more realistic scenario where each oscillator is coupled to its own reservoir, we also discuss the case where all the network oscillators are coupled to a common reservoir. We obtain the master equation governing the dynamic of the network states and the associated evolution equation of the Glauber-Sudarshan $P$ function. With these instruments we briefly show how to analyze the decoherence and the evolution of the linear entropy of general states of the network. We also show how to obtain the master equation for the case of distinct reservoirs from that of a common one.

DOI
10.1103/physreva.76.032101

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We present a general treatment of a bosonic dissipative network: a chain of coupled dissipative 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 a general more realistic scenario where each oscillator is coupled to its own reservoir, we also discuss the case where all the network oscillators are coupled to a common reservoir. We obtain the master equation governing the dynamic of the network states and the associated evolution equation of the Glauber-Sudarshan PPP function. With these instruments we briefly show how to analyze the decoherence and the evolution of the linear entropy of general states of the network. We also show how to obtain the master equation for the case of distinct reservoirs from that of a common one.
In this article we extend the results presented in Ref. [Phys. Rev. A 76, 032101 (2007)] to treat quantitatively the effects of reservoirs at finite temperature in a bosonic dissipative network: a chain of coupled harmonic oscillators whichever 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. We also present a technique to estimate the decoherence time of network states by computing separately the effects of diffusion and the attenuation of the interference terms of the Wigner function. A detailed analysis of the diffusion mechanism is also presented through the evolution of the Wigner function. The interesting collective diffusion 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.
In this work we analyze the coherence dynamics and estimate decoherence times of quantum states in a network composed of NNN coupled dissipative quantum oscillators. We assume a symmetric network where all oscillators are coupled to each other with the same coupling strength. Master equations are derived for regimes of both weak and strong coupling between the oscillators. The strong coupling regime is characterized by the coupling strength between the oscillators or by the number of oscillators in the network. The decoherence times of particular states of the network are computed and the results are clarified by analyzing the processes of state swap and recurrence of reduced states of the network together with the linear entropies of the joint and reduced systems.
In this paper we derive an exact master equation for two coupled quantum harmonic oscillators interacting via bilinear coupling with a common environment at arbitrary temperature made up of many harmonic oscillators with a general spectral density function. We first show a simple derivation based on the observation that the two harmonic oscillator model can be effectively mapped into that of a single harmonic oscillator in a general environment plus a free harmonic oscillator. Since the exact one harmonic oscillator master equation is available [B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 45, 2843 (1992)], the exact master equation with all its coefficients for this two harmonic oscillator model can be easily deduced from the known results of the single harmonic oscillator case. In the second part we give an influence functional treatment of this model and provide explicit expressions for the evolutionary operator of the reduced density matrix which are useful for the study of decoherence and disentanglement issues. We show three applications of this master equation: on the decoherence and disentanglement of two harmonic oscillators due to their interaction with a common environment under Markovian approximation, and a derivation of the uncertainty principle at finite temperature for a composite object, modeled by two interacting harmonic oscillators. The exact master equation for two, and its generalization to N, harmonic oscillators interacting with a general environment are expected to be useful for the analysis of quantum coherence, entanglement, fluctuations, and dissipation of mesoscopic objects toward the construction of a theoretical framework for macroscopic quantum phenomena.
In this work we analyze the coherence dynamics and estimate decoherence times of quantum states in a network where a central dissipative oscillator is coupled with $N\ensuremath{-}1$ peripheral noninteracting dissipative oscillators. The results obtained here are compared with those in part I of this work where a symmetric network was considered. This comparison helps us to understand the influence of the topology of a network on the coherence dynamics of quantum superposition states. As in part I, master equations are derived for regimes of both weak and strong coupling between the oscillators. Decoherence times of particular states of the network are computed and the results are analyzed in the light of state swap and recurrence processes of reduced states of the network. The linear entropies of the joint and reduced systems are also analyzed.
We investigate how to model Markovian evolution of coupled harmonic oscillators, each of them interacting with a local environment. When the coupling between the oscillators is weak, dissipation may be modeled using local Lindblad terms for each of the oscillators in the master equation, as is commonly done. When the coupling between oscillators is strong, this model may become invalid. We derive a master equation for two coupled harmonic oscillators that are subject to individual heat baths modeled by a collection of harmonic oscillators and show that this master equation in general contains nonlocal Lindblad terms. We compare the resulting time evolution with that obtained for dissipation through local Lindblad terms for each individual oscillator and show that the evolution is different in the two cases. In particular, the two descriptions give different predictions for the steady state and for the entanglement between strongly coupled oscillators. This shows that when describing strongly coupled harmonic oscillators, one must take great care in how dissipation is modeled and that a description using local Lindblad terms may fail. This may be particularly relevant when attempting to generate entangled states of strongly coupled quantum systems.
We calculate the reduced density matrix for a system of coupled harmonic oscillators in a bosonic heat bath using the Born-Markov approximation and show that the expectation values of position and momentum evolve like classical quantities. We consider the cases where every oscillator is coupled to its own reservoir and where all oscillators are coupled to the same heat bath, pointing out the differences between these two models. In particular, for systems coupled to a common reservoir, we present an uncommon dissipation mechanism, which disappears, when the dynamics of the system are gouverned by the low frequency modes of the environment (Caldeira-Leggett limit). We also address the interesting phenomenon of reservoir-induced interactions.