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Journal of Physics A Mathematical and General

Separability dynamics of two-mode Gaussian states in parametric conversion and amplification

Alexandre V. Dodonov, V. V. Dodonov, Salomon Sylvain Mizrahi

Published Dec 24, 2004• 52 citations
Publisher

Abstract

We give a simplified form of Simon's separability criterion for two-mode Gaussian states, showing that for systems whose unitary evolution is governed by arbitrary time-dependent quadratic Hamiltonians, the separability dynamics is completely described in terms of the determinant of the cross-covariance matrix. As concrete examples, we consider the evolution of the 'inverse negativity coefficient' (which gives a quantitative estimation of the 'degree of entanglement') for two initially uncoupled modes (each being in a squeezed thermal state) in the cases of parametric converter, parametric amplifier and for a cavity whose boundary oscillates in resonance with two field modes.

DOI
10.1088/0305-4470/38/3/008

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We give a simplified form of Simon's separability criterion for two-mode Gaussian states, showing that for systems whose unitary evolution is governed by arbitrary time-dependent quadratic Hamiltonians, the separability dynamics is completely described in terms of the determinant of the cross-covariance matrix. As concrete examples, we consider the evolution of the 'inverse negativity coefficient' (which gives a quantitative estimation of the 'degree of entanglement') for two initially uncoupled modes (each being in a squeezed thermal state) in the cases of parametric converter, parametric amplifier and for a cavity whose boundary oscillates in resonance with two field modes.
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 the 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 the 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 takes place. We determine the asymptotic Gaussian maximally entangled mixed states (GMEMS) and their corresponding asymptotic maximal logarithmic negativity.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuousvariable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment.Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the generation and evolution of entanglement in terms of the covariance matrix for an arbitrary Gaussian input state.For some values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled.In other cases, entanglement generation or entanglement collapse (entanglement sudden death) take place or even a periodic collapse and revival of entanglement.We show that for certain classes of environments the initial state evolves asymptotically to an entangled equilibrium bipartite state, while for other values of the coefficients describing the environment, the asymptotic state is separable.We calculate also the logarithmic negativity characterizing the degree of entanglement of the asymptotic state.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, a description of the continuous-variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment was given. Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, the generation and evolution of entanglement in terms of the covariance matrix for an arbitrary Gaussian input state was described. For some values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a periodic collapse and revival of entanglement take place. The time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state was analyzed.
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.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous-variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment. Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the generation and evolution of entanglement in terms of the covariance matrix for an arbitrary Gaussian input state. For some values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation or entanglement collapse (entanglement sudden death) take place or even a periodic collapse and revival of entanglement. We show that for certain classes of environments the initial state evolves asymptotically to an entangled equilibrium bipartite state, while for other values of the coefficients describing the environment, the asymptotic state is separable. We calculate also the logarithmic negativity characterizing the degree of entanglement of the asymptotic state.
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.