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arXiv:1203.3819 · Physical Review A

Dynamics of Gaussian discord between two oscillators interacting with a common environment

José Nahuel Freitas, Juan Pablo Paz

Published Mar 16, 2012• 22 citations
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Abstract

We analyze the evolution of the Gaussian discord between two resonant harmonic oscillators coupled to a common environment. For this, we use the same tools we applied before to fully characterize the evolution of the entanglement in this system [Paz and Roncaglia, Phys. Rev. Lett. 100, 220401 (2008)]. The asymptotic value of Gaussian discord is obtained as a function of parameters characterizing the environment (temperature, couplings, etc.) and the initial state of the system (initial squeezing, initial purity, etc.) The type of Gaussian measurement optimizing the extraction of information between the oscillators is fully characterized by means of a phase diagram. Such a diagram (with phases corresponding to homodyne or heterodyne measurements) has similar topology to the one describing dynamical phases for the entanglement. We present evidence pointing to the fact that Gaussian discord is not always a good approximation of true discord as the asymptotic value of the former is shown to be a nondecreasing function of temperature (in the high-temperature regime), reaching an asymptotic value of $\mathrm{ln}(2)$ for a pure initial state (and lower values for mixed initial states).

DOI
10.1103/physreva.85.032118

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We analyze the evolution of the Gaussian discord between two resonant harmonic oscillators coupled to a common environment. For this, we use the same tools we applied before to fully characterize the evolution of the entanglement in this system [Paz and Roncaglia, Phys. Rev. Lett. 100, 220401 (2008)]. The asymptotic value of Gaussian discord is obtained as a function of parameters characterizing the environment (temperature, couplings, etc.) and the initial state of the system (initial squeezing, initial purity, etc.) The type of Gaussian measurement optimizing the extraction of information between the oscillators is fully characterized by means of a phase diagram. Such a diagram (with phases corresponding to homodyne or heterodyne measurements) has similar topology to the one describing dynamical phases for the entanglement. We present evidence pointing to the fact that Gaussian discord is not always a good approximation of true discord as the asymptotic value of the former is shown to be a nondecreasing function of temperature (in the high-temperature regime), reaching an asymptotic value of ln(2)\mathrm{ln}(2)ln(2) for a pure initial state (and lower values for mixed initial states).
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous-variable quantum entanglement and quantum discord for a system consisting of two noninteracting modes embedded in a thermal environment. Entanglement and discord are used to quantify the quantum correlations of the system. For all values of the temperature of the thermal reservoir, an initially separable Gaussian state remains separable for all times. We study the time evolution of logarithmic negativity, which characterizes the degree of entanglement, and we show that in the case of an entangled initial Gaussian state, entanglement suppression (entanglement sudden death) takes place for non-zero temperatures of the environment. Only for zero temperature of the thermal bath the initial entangled state remains entangled for finite times. We analyze time evolution of Gaussian quantum discord, which is a measure of all quantum correlations in the bipartite state, including entanglement, and we show that quantum discord decays asymptotically in time under the effect of thermal bath. This is in contrast with the sudden death of entanglement. Before the suppression of entanglement, the qualitative evolution of quantum discord is very similar to that of the entanglement. We describe also time evolution of the degree of classical correlations and of quantum mutual information, which measures the total correlations of quantum system.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous-variable quantum entanglement and quantum discord for a system consisting of two noninteracting modes embedded in a thermal environment. Entanglement and discord are used to quantify the quantum correlations of the system. For all values of the temperature of the thermal reservoir, an initial separable Gaussian state remains separable for all times. We study the time evolution of logarithmic negativity, which characterizes the degree of entanglement, and show that in the case of an entangled initial Gaussian state, entanglement suppression (entanglement sudden death) takes place, for non-zero temperatures of the environment. Only for a zero temperature of the thermal bath the initial entangled state remains entangled for finite times. We analyze the time evolution of the Gaussian quantum discord, which is a measure of all quantum correlations in the bipartite state, including entanglement, and show that quantum discord decays asymptotically in time under the effect of the thermal bath. This is contrast with the sudden death of entanglement. Before the suppression of the entanglement, the qualitative evolution of quantum discord is very similar to that of the entanglement. We describe also the time evolution of the degree of classical correlations and of quantum mutual information, which measures the total correlations of the quantum system.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of continuous variable quantum entanglement and quantum discord for a system consisting of two non-interacting non-resonant bosonic modes embedded in a thermal environment. We study the time evolution of logarithmic negativity, which characterizes the degree of entanglement, and show that in the case of an entangled initial squeezed thermal state, entanglement suppression takes place for all temperatures of the environment, including zero temperature. We analyze the time evolution of the Gaussian quantum discord, which is a measure of all quantum correlations in the bipartite state, including entanglement, and show that discord decays asymptotically in time under the effect of the thermal bath. We describe also the time evolution of classical correlations.
In this work, we calculate the exact asymptotic quantum correlations between two interacting nonresonant harmonic oscillators in a common Ohmic bath. We derive analytical formulas for the covariances, fully describing any Gaussian stationary state of the system, and use them to study discord and entanglement in the strong- and weak-dissipation regimes. We discuss the rich structure of the discord of the stationary separable states arising in the strong-dissipation regime. Also under strong dissipation, when the modes are not mechanically coupled, these may entangle only through their interaction with the common environment. Interestingly enough, this stationary entanglement assisted by dissipation is only present within a finite band of frequencies and increases with the dissipation rate. Robust entanglement between detuned oscillators is also observed at low temperature.
In the framework of the theory of open systems, we give a description of quantum entanglement and quantum discord for two noninteracting modes embedded in a thermal environment. We describe the evolution of entanglement in terms of the covariance matrix for Gaussian input states. For all values of the temperature of the thermal reservoir, an initial separable squeezed thermal state remains separable for all times. In the case of an entangled initial squeezed thermal state, entanglement suppression (entanglement sudden death) takes place for nonzero temperatures of the environment. The Gaussian quantum discord, which is a measure of all quantum correlations in the state, including entanglement, decays asymptotically in time.
We study the dynamics of the entanglement between two oscillators that are initially prepared in a general two-mode Gaussian state and evolve while coupled to the same environment. In a previous paper, we showed that there are three qualitatively different dynamical phases for the entanglement in the long-time limit: sudden death, sudden death and revival, and no sudden death [Paz and Roncaglia, Phys. Rev. Lett. 100, 220401 (2008)]. Here we generalize and extend those results along several directions: We analyze the fate of entanglement for an environment with a general spectral density providing a complete characterization of the evolution for Ohmic, sub-Ohmic, and super-Ohmic environments. We consider two different models for the interaction between the system and the environment (one where the coupling is through position and another where the coupling is symmetric in position and momentum). Finally, we show that for nonresonant oscillators, the final entanglement is independent of the initial state and may be nonzero at very low temperatures.