Quantum entanglement is one of the most outstanding features of quantum mechanics

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 can
be described by the Jaynes-Cummings model (\texttt{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 effects
of the atomic motion and the field mode structure \cite{key27,key28}.\\