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arXiv:0910.5793 · Journal of Physics A Mathematical and Theoretical

Non-Markovian entanglement dynamics between two coupled qubits in the same environment

Wei Cui, Zairong Xi, Yu Pan

Published Mar 24, 2009• 32 citations
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Abstract

We analyze the dynamics of the entanglement in two independent non-Markovian channels. In particular, we focus on the entanglement dynamics as a function of the initial states and the channel parameters like the temperature and the ratio $r$ between $ω_0$ the characteristic frequency of the quantum system of interest, and $ω_c$ the cut-off frequency of Ohmic reservoir. We give a stationary analysis of the concurrence and find that the dynamic of non-markovian entanglement concurrence $\mathcal{C}_ρ(t)$ at temperature $k_BT=0$ is different from the $k_BT>0$ case. We find that "entanglement sudden death" (ESD) depends on the initial state when $k_BT=0$, otherwise the concurrence always disappear at finite time when $k_BT>0$, which means that ESD must happen. The main result of this paper is that the non-Markovian entanglement dynamic is fundamentally different from the Markovian one. In the Markovian channel, entanglement decays exponentially and vanishes only asymptotically, but in the non-Markovian channel the concurrence $\mathcal{C}_ρ(t)$ oscillates, especially in the high temperature case. Then an open-loop controller adjusted by the temperature is proposed to control the entanglement and prolong the ESD time.

DOI
10.1088/1751-8113/42/15/155303

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We analyze the dynamics of the entanglement in two independent non-Markovian channels. In particular, we focus on the entanglement dynamics as a function of the initial states and the channel parameters like the temperature and the ratio rrr between ω0ω_0ω0​ the characteristic frequency of the quantum system of interest, and ωcω_cωc​ the cut-off frequency of Ohmic reservoir. We give a stationary analysis of the concurrence and find that the dynamic of non-markovian entanglement concurrence Cρ(t)\mathcal{C}_ρ(t)Cρ​(t) at temperature kBT=0k_BT=0kB​T=0 is different from the kBT>0k_BT>0kB​T>0 case. We find that "entanglement sudden death" (ESD) depends on the initial state when kBT=0k_BT=0kB​T=0, otherwise the concurrence always disappear at finite time when kBT>0k_BT>0kB​T>0, which means that ESD must happen. The main result of this paper is that the non-Markovian entanglement dynamic is fundamentally different from the Markovian one. In the Markovian channel, entanglement decays exponentially and vanishes only asymptotically, but in the non-Markovian channel the concurrence Cρ(t)\mathcal{C}_ρ(t)Cρ​(t) oscillates, especially in the high temperature case. Then an open-loop controller adjusted by the temperature is proposed to control the entanglement and prolong the ESD time.
Abstract In this paper, we numerically explore the quantum correlations in a system consisting of two identical qubits, each of them locally interacting with a bosonic reservoir under non-Markovian regime, dissipative quantum system. The influences of Ohmic reservoir with Lorentz–Drude regularization on the quantum discord and entanglement are studied. We analyze the quantum correlations rate in high temperature, intermediate temperature, and low temperature reservoirs, respectively. The dependence of the quantum correlations on the ratio r of the reservoir cutoff frequency and the system oscillator frequency, and temperature k B T shows that the decreasing of the frequency ratio r enhances the amount of quantum correlations between the two qubits, while the increasing of the temperature damages the amount of correlations. More precisely, we find a sudden death of quantum entanglement, that is, the entanglement vanishes as time becomes large for different temperatures. On the other hand, we show in contrast to what happens to entanglement, the quantum discord does not vanish even at high temperatures.
We analyze the dynamics of quantum correlations between two qubits coupled to a linear chain of oscillators. The chain mediates interactions between the qubits and acts as a non-Markovian reservoir. The model is amenable to an analytical solution when the initial state of the chain is Gaussian. We study the dynamics of the qubits' concurrence starting from a separable state and assuming that the chain spectrum is gapped and the chain is initially in a thermal state. We identify three relevant regimes that depend on the strength of the qubit-chain coupling in relation to the spectral gap. These are (i) the weak-coupling regime, where the qubits are entangled at the asymptotics, (ii) the strong-coupling regime, where the concurrence can exhibit collapses followed by revivals with exponentially attenuated amplitude, and (iii) the thermal damping regime, where the concurrence rapidly vanishes due to the chain's thermal excitations. In all cases, if entanglement is generated, this occurs after a finite time has elapsed. This time scale depends exponentially on the qubits' distance and is determined by the spectral properties of the chain. Entanglement irreversible decay, on the other hand, is due to the dissipative effect induced by the coupling with the chain and is controlled by the coupling strength between the chain and qubits. This study unravels the basic mechanisms leading to entanglement in a non-Markovian bath and allows one to identify the key resources for realizing quantum coherent dynamics of open systems.
We study the non-Markovian effect on the dynamics of the quantum discord by exactly solving a model consisting of two independent qubits subject to two zero-temperature non-Markovian reservoirs, respectively. Considering the two qubits initially prepared in Bell-like or extended Werner-like states, we show that there is no occurrence of the sudden death, but only instantaneous disappearance of the quantum discord at some time points, in comparison to the entanglement sudden death in the same range of the parameters of interest. This implies that the quantum discord is more useful than the entanglement to describe the quantum correlation involved in quantum systems.
We investigate the sudden birth and sudden death of entanglement of two qubits interacting with un-correlated structured reservoirs. The system is initially prepared in two-qubit extended Werner-like state. We work out the dependence of the entanglement dynamics on both non-Markovian environments and the purity of initial state, and show that non-Markovian environments and the purity can control the time of the two-qubit entanglement sudden death and the reservoirs' entanglement sudden birth. Furthermore, under the conditions of different purity and initial entanglement, the revival of qubits' entanglement can manifest before, simultaneously or even after the disentanglement of their corresponding reservoirs.
We investigate the entanglement dynamics of the bipartite quantum system between two qubits with the dissipative environment. We begin with the standard Markovian master equation in the Lindblad form and the initial state which is prepared in the extended Werner-like state: ρ Φ AB (0). We examine the conditions for entanglement sudden death (ESD) and calculate the corresponding ESD time by the Wootters concurrence. We observe that ESD is determined by the parameters such as the mean occupation number of the environment N , amount of initial entanglement α and the purity r . For N = 0, we get the analytical expressions of both ESD condition and ESD time. For N > 0 we give a theoretical analysis that ESD always occurs, and simulate the concurrence as a function of γ 0 t and one of the parameters N , α and r .