In this paper, the quantum entanglement between a single mode binomial field and a cascade three-level atom is calculated mechanically without the rotating wave approximation. The numerical results indicate that the quantum entanglement at the first few periods is reduced notably due to the fact that the atom is initially in the superposition state. With increasing field parameter , the period of the entanglement evolution becomes obvious and the quantum decoherence phenomenon emerges in a short time.binomial state field, quantum entanglement, field entropy

We obtain the exact solution of energy spectrum and dynamics for the two-site Hubbard-Holstein model by the coherent states orthogonal expansion method. The influences of coupling strength g, the average number of phonons n and the initial electronic state on the evolution of system entanglement and von Neumann entropy are discussed. Numerical results are as follows. (a) Entanglement evolution with time shows a good periodicity. When the other parameters are fixed, the evolution period decreases as the coupling strength g goes up but it is independent of the average number of phonons n. (b) The von Neumann entropy of the system demonstrates strict synchronia with the electronic state occupancy probability. (c) Under the weak coupling strength and low average number of phonons, the initial electronic state c2+ c2+|Oe or c1+ c1+|Oe shows larger maximum von Neumann entropy during its evolution than that of c1+c2+c1+c2++ but they gradually approach to each other with the increase of coupling strength or average number of phonons.

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Under the non-rotating wave approximation, the quantum evolution of entanglement property of a two-qubit and oscillator coupling system is accurately investigated by the method of coherent-state orthogonalization expansion. The property of the ground state for qubit-oscillator system and the difference between qubit-oscillator entanglement and qubit-qubit entanglement when resonant vibration occurs are discussed. The calculation results show that when the external field is not taken into consideration, the qubit-qubit entanglement reduces from 1 to 0 rapidly with the increase of coupling strength, indicating the strong sensitivity of the entanglement to the coupling strength. On the contrary, with the increase of the coupling the qubit-oscillator entanglement rises from 0, but does not reach the maximum value 2. At the beginning, when the two qubits do not entangle, the vacuum field does not lead to the entanglement in weak coupling. However, the strong coupling can induce the sudden appearance of the entanglement.