Quantum entanglement is one of the most

Quantum entanglement is one of the most outstanding features of quantum mechanics. Quantum entangled states, as a fundamental physical resource of quantum information processing~(QIP)~cite{key1}, are widely used, in quantum computation and quantum communication cite{key2}, quantum cryptography cite{key3,key4}, quantum teleportation cite{key5}, etc. Also, there is a lot of attention has been focused on information entropies as a measure of entanglement in quantum information such as von Neumann entropy cite{key6}, Linear entropy, Shannon information entropy cite{key7} and atomic Wehrl entropy cite{key8}.

On the other hand, we find that the most important problems in quantum optics are the studies of different systems interaction such as field-atom, atom-atom, and the field-field, these problems have been extensively considered in a huge number of papers; see for example cite{key9}-cite{key20} This is due to the richness of many different phenomena that have been observed in the laboratory. In fact,~Dynamics of a two-level atom interacting with a single mode of the radiation field in a lossless cavity canbe described by the Jaynes-Cummings model ( exttt{JCM})cite{key21},~and it has been discussed in detail from various points of view cite{key22}-cite{key26},~and it has been generalized or extended further to incorporate the effectsof the atomic motion and the field mode structure cite{key27,key28}.\

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