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arXiv:0704.1965 · Physical Review A

Quantum entanglement of symmetrically decohered two-mode squeezed states in an absorbing and amplifying environment

Phoenix S. Y. Poon, C. K. Law

Published Jul 31, 2007• 11 citations
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Abstract

We investigate the properties of quantum entanglement of two-mode squeezed states interacting with symmetric linear baths with general gain and loss parameters. By explicitly solving for $\ensuremath{\rho}$ from the master equation, we determine analytical expressions of eigenvalues and eigenvectors of ${\ensuremath{\rho}}^{{T}_{A}}$ (the partial transposition of density matrix $\ensuremath{\rho}$). In Fock space, ${\ensuremath{\rho}}^{{T}_{A}}$ is shown to maintain a block diagonal structure as the system evolves. In addition, we discover that the decoherence induced by the baths would break the degeneracy of ${\ensuremath{\rho}}^{{T}_{A}}$, and leads to a set of eigenvectors for the construction of entanglement witness operators. Such eigenvectors are shown to be time independent, which is a signature of robust entanglement of two-mode squeezed states in the presence of noise.

DOI
10.1103/physreva.76.012333

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We investigate the properties of quantum entanglement of two-mode squeezed states interacting with symmetric linear baths with general gain and loss parameters. By explicitly solving for \ensuremathρ\ensuremath{\rho}\ensuremathρ from the master equation, we determine analytical expressions of eigenvalues and eigenvectors of \ensuremathρTA{\ensuremath{\rho}}^{{T}_{A}}\ensuremathρTA​ (the partial transposition of density matrix \ensuremathρ\ensuremath{\rho}\ensuremathρ). In Fock space, \ensuremathρTA{\ensuremath{\rho}}^{{T}_{A}}\ensuremathρTA​ is shown to maintain a block diagonal structure as the system evolves. In addition, we discover that the decoherence induced by the baths would break the degeneracy of \ensuremathρTA{\ensuremath{\rho}}^{{T}_{A}}\ensuremathρTA​, and leads to a set of eigenvectors for the construction of entanglement witness operators. Such eigenvectors are shown to be time independent, which is a signature of robust entanglement of two-mode squeezed states in the presence of noise.
We study a system consisting of two identical non-interacting single-mode cavity fields coupled to a common vacuum environment and provide general, explicit, and exact solutions to its master equation by means of the characteristic function method. We analyze the entanglement dynamics of two-mode squeezed thermal state in this model and show that its entanglement dynamics is strongly determined by the two-mode squeezing parameter and the purity. In particular, we find that two-mode squeezed thermal state with the squeezing parameter is extremely fragile and almost does not survive in a common vacuum environment. We investigate the time evolution of nonlocality for two-mode squeezed thermal state in such an environment. It is found that the evolved state loses its nonlocality in the beginning of the evolution, but after a time, the revival of nonlocality can occur.
In this work we find explicitly the decoherence-free subspace (DFS) for a two two-level system in a common squeezed vacuum bath. We also find an orthogonal basis for the DFS composed of a symmetrical and an antisymmetrical (under particle permutation) entangled state. For any initial symmetrical state, the master equation has one stationary state which is the symmetrical entangled decoherence-free state. In this way, one can generate entanglement via a common squeezed bath of the two systems. If the initial state does not have a definite parity, the stationary state depends strongly on the initial conditions of the system and it is a statistical mixture of states which belong to DFS. We also study the effect of the coupling between the two-level systems on the DFS.
For the first time we derive the dissipating result of an initial two-mode squeezed pure vacuum state passing through a two-mode amplitude dissipative channel described by the direct product of two independent single-mode master equations. Although these two master equations do not mix the two modes (there is no coupling between them), since the two-mode squeezed state is simultaneously an entangled state, the final state which emerges from passing this channel is a two-mode mixed density operator. The compact expression of the outcoming state is obtained, which manifestly shows that as time evolves, the squeezing effect decreases.
We investigate the entanglement dynamics of two modes in a common squeezed thermal environment by means of the characteristic function method and the logarithmic negativity. When the two modes are prepared initially in separable squeezed states, we find that the external environment can not only induce but also enhance the two-mode entanglement if their initial squeezing parameters are suitably chosen. When the two modes are initially in a two-mode squeezed vacuum state, we find that whether the initial entanglement is degraded or increased depends strongly on the difference between the squeezing parameters of the considered bath and of a quantum state. Finally, we give a condition of all two-mode entangled Gaussian states to be the decoherence-free state in a given squeezed bath.
Continuous-variable (CV) entanglement is a valuable resource in the field of quantum information. One source of CV entanglement is the correlations between the position and momentum of photons in a two-mode squeezed state of light. In this paper, we theoretically study the generation of squeezed states, via spontaneous parametric down conversion, inside a two-mode lossy cavity that is pumped with a classical optical pulse. The dynamics of the density operator in the cavity is modeled using the Lindblad master equation, and we show that the solution to this model is the density operator for a two-mode squeezed thermal state, with a time-dependent squeezing amplitude and average thermal photon number for each mode. We derive an expression for the minimum correlation variance inside the cavity that depends crucially on the difference in the losses between the two modes. We apply our analytic solution to the important example of a microring resonator that is pumped with a Gaussian pulse. The expressions that we derive will help researchers optimize CV entanglement in lossy cavities.
As one of the most remarkable features of quantum mechanics, quantum coherence is regarded as an important quantum resource in the quantum information processing. The one-mode squeezed state and the two-mode squeezed state (Einstein-Podolsky-Rosen (EPR) entangled states) as the most representative examples of nonclassical states both have quantum coherence. The squeezing property of the squeezed state is described by the variance of quadrature components, and the positive partial transposition (PPT) criterion is used to describe the entanglement of the EPR entangled states. The research of the quantum coherence of Gaussian states is also a bridge between the properties of squeezing and entanglement. It has been shown that the quantum coherence with infinite-dimensional systems can be quantified by relative entropy. One of the widely used effective methods to obtain the value of quantum coherence experimentally is the quantum tomography. The covariance matrices of the quantum states are reconstructed via balanced homodyne detection and then taken into quantum coherence expression to calculate the corresponding value. The main factors affecting quantum coherence are the classical and uncorrelated noise in the actual experimental generation processing and the decoherence effect caused by the coupling between quantum resources and the surrounding environment. And the quantum coherence evolution in the generation and transmission process of the quantum resources is essential for the practical applications. Therefore, we analyze in detail the influences of the impurity of quantum resource on squeezing, entanglement and quantum coherence. The evolutions of quantum coherence of these Gaussian states in the lossy channels are demonstrated experimentally. The quantum coherence is shown to be robust against the loss in the lossy channels, which is similar to the case of squeezing and entanglement. The quantum coherences of the squeezed states and the EPR entangled states are robust against the thermal photons in the actual experimental generation processing, although the squeezing and entanglement of Gaussian states disappear at a certain number of thermal photons. Our research results provide a reference for the practical applications of quantum coherence of the squeezed state and entangled states in the lossy environment.