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In this paper, the entanglement between two atoms and squeezed field via four photon process is investigated. The dynamical behavior of the entanglement between two atoms and a squeezed field is analyzed. In particular, the effects of the atomic motion, the initial atomic state and the field squeezing are examined. A high amount of entanglement is generated by increasing the field squeezing. Furthermore, we show that a sudden death and sudden birth emerge when the moving atoms are initially prepared in the excited state.

In recent years, the studies focus on the quantum entanglement concept or so called non-local correlation [

Quantum and classical correlations are at the heart of quantum information theory. It is seen a great deal of attention pointed to a subtler but more general kind of quantum correlations. In this regard, there are different entanglement measures and quantifiers have been used for the pure and mixed states such as concurrence [

Recently, much attention has been focused on the properties of the Tavis-Cummings model (TCM) when the time dependent coupling with field is considered. The entanglement between two atoms (qubits) is discussed in the case of open and closed quantum system [

The paper is prepared in the following order: In Section 2, the Hamiltonian system and the solution for the system under consideration will be given. In Section 3, we introduce the von Neumann entropy measure and discussed the results. The main conclusion is summarized in Section 4.

We consider two identical atoms moving in a cavity of length L, and they have the same velocity i.e.

where p₁ and p₂ are the atomic motion parameters, well, if we put p₁ = 1 and p_{2} = 0, then the shape function takes the form

which means, there is no atomic motion inside the cavity, but if p₁ = 0 and p₂ = 1, the shape function will be

hence, the atomic motion exists inside the cavity. The interaction Hamiltonian of moving two atoms interacting with a strong squeezed field via four-photon process with a closed system (In other words, there’s not any external fields or external effects can act on the system) in the rotating wave approximation RWA can be written as [

where

where

where r is the squeeze parameter. Using the above initial conditions, the wave function of the whole system

where

and

Thus, one can obtain the atomic density matrix

where

are an atomic occupation probabilities of levels

In the following section, we use the relations obtained above to investigate the atoms-field entanglement through a statistical aspect, such as von Neumann entropy measurement for the system under consideration.

Due to the crucial role of entanglement in quantum information processes, the study of entanglement has attracted a lot of interest in recent years. Among the various studies on entanglement, the first question which may be asked is how to know that a quantum state is entangled. For a pure bipartite state, the Schmidt decomposition [

This is zero for all pure states, i.e., for states that satisfy the condition

where

Since the excited identical atoms interacting with a squeezed field in the absence of atomic center-of-mass motion i.e.

However, if the atoms initially prepared in Bell state by setting

maximum values for von Neumann entropy equal one, i.e.

By considering the atomic center-of-mass motion in

In summary, we have investigated the moving and non-moving two two-level identical atoms interacting with a squeezed coherent field via four-photon process in a closed system. The von Neumann entropy is used as a measure of the two atoms-field entanglement. The results show that some new important and interesting features of entanglement such as entanglement sudden birth and sudden death which can be obtained when the moving two atoms are initially in the excited state. Also, the entropy has a chaotic behavior in the presence or absence squeezing and atomic motion parameters. On the other hand, the initial state setting for the atomic position θ and squeeze parameter r playing an important role to get a high amount of entanglement since the atoms initially in Bell’s state. Our recommendation as an important future investigation will be the study of two atoms interacting with a squeezed field in the presence of the decoherence and the dipole-dipole interaction effects.