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Journal of Computer Chemistry Japan

Decoherence of Entanglement in Markov Approximation in Terms of Rotating Wave Approximation

Kenji Mishima, Koichi Yamashita

Published Jan 1, 2012• 0 citations
Publisher

Abstract

In this paper, we extend our theoretical work on the pure state entanglement of bipartite or two-mode physical and chemical systems interacting with general static entangling interactions and external electromagnetic fields to include the mixed state entanglement. It is assumed that the Markov approximation is valid in our model system. Based on the partitioning of the entangling interaction matrix, we investigate the effect of the entangling interaction matrix elements and the relaxation constants on the decoherence of the Bell states. Our calculations indicate that only a few elements of the entangling interaction matrix and relaxation constants are dominant for the decoherence dynamics of the Bell states.

DOI
10.2477/jccj.2011-0016

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In this paper, we extend our theoretical work on the pure state entanglement of bipartite or two-mode physical and chemical systems interacting with general static entangling interactions and external electromagnetic fields to include the mixed state entanglement. It is assumed that the Markov approximation is valid in our model system. Based on the partitioning of the entangling interaction matrix, we investigate the effect of the entangling interaction matrix elements and the relaxation constants on the decoherence of the Bell states. Our calculations indicate that only a few elements of the entangling interaction matrix and relaxation constants are dominant for the decoherence dynamics of the Bell states.
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We study the non-equilibrium dynamics of a pair of qubits made of two-level atoms separated in space with distance r and interacting with one common electromagnetic field but not directly with each other. Our calculation makes a weak coupling assumption, but no Born or Markov approximation. We derived a non-Markovian master equation for the evolution of the reduced density matrix of the two-qubit system after integrating out the electromagnetic field modes. It contains a Markovian part with a Lindblad type operator and a nonMarkovian contribution, the physics of which is the main focus of this study. We use the concurrence function as a measure of quantum entanglement between the two qubits. Two classes of states are studied in detail: Class A is a one parameter family of states which are the superposition of the highest energy |I〉 ≡ |11〉 and lowest energy |O〉 ≡ |00〉 states, υiz, |A〉≡p|I〉+(1−p)|O〉, with 0 ≤ p ≤ 1; and Class B states |B〉 are linear combinations of the symmetric |+〉=12(|01〉+|10〉) and the antisymmetric |−〉=12(|01〉−|10〉) Bell states.
We study the non-Markovian dynamics of a qubit made up of a two-level atom interacting with an electromagnetic field (EMF) initially at finite temperature. Unlike most earlier studies where the bath is assumed to be fixed, we study the complete evolution of the combined qubit-EMF system, thus allowing for the coherent backaction from the bath on the qubit and the qubit on the bath in a self-consistent manner. In this way we can see the development of quantum correlations and entanglement between the system and its environment, and how that affects the decoherence and relaxation of the system. We find nonexponential decay for both the diagonal and nondiagonal matrix elements of the qubit's reduced density matrix in the pointer basis. The former shows the qubit relaxing to thermal equilibrium with the bath, while the latter shows the decoherence rate beginning at the usually predicted thermal rate, but changing to the zero-temperature value as the qubit and bath become entangled. The decoherence and relaxation rates are comparable, as in the zero-temperature case. On the entanglement of a qubit with the EMF we calculated the fidelity and the von Neumann entropy, which is a measure of the purity of the density matrix. The present more accurate non-Markovian calculations predict lower loss of fidelity and purity as compared with the Markovian results. Generally speaking, with the inclusion of quantum correlations between the qubit and its environment, the non-Markovian processes tend to slow down the drive of the system to equilibrium, prolonging the decoherence and better preserving the fidelity and purity of the system.
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Considering two identical two-level atoms interacting with a single-model dissipative coherent cavity field without rotating wave approximation, we explore the entanglement dynamics of the two atoms prepared in different states using concurrence. Interestingly, our results show that the entanglement between the two atoms that initially disentangled will come up to a large constant rapidly, and then keeps steady in the following time or always has its maximum when prepared in some special Bell states. The model considered in this study is a good candidate for quantum information processing especially for quantum computation as steady high-degree atomic entanglement resource obtained in dissipative cavity.
The interaction of qubits with quantized modes of electromagnetic fields has been largely addressed in the quantum optics literature under the rotating wave approximation (RWA), where rapid oscillating terms in the qubit-mode interaction picture Hamiltonian can be neglected. At the same time, it is generally accepted that, provided the interaction is sufficiently strong or for long times, the RWA tends to describe physical phenomena incorrectly. In this work, we extend the investigation of the validity of the RWA to a more involved setup where two qubit-mode subsystems are brought to interaction through their harmonic coordinates. Our treatment is all analytic thanks to a sequence of carefully chosen unitary transformations, which allows us to diagonalize the Hamiltonian within and without the RWA. By also considering qubit dephasing, we find that the purity of the two-qubit state presents non-Markovian features which become more pronounced as the coupling between the modes gets stronger and the RWA loses its validity. In the same regime, there occurs fast generation of entanglement between the qubits, which is also not correctly described under the RWA. The setup and results presented here clearly show the limitations of the RWA in a scenario amenable to exact description and free from numerical uncertainties. Consequently, it may be of interest for the community working with cavity or circuit quantum electrodynamic systems in the strong coupling regime.