In this paper we consider a system of identical three two-level atoms interacting at resonance with a single-mode of the quantized field in a lossless cavity. The initial cavity field is prepared in the coherent state while the atoms are taken initially to be either in the uppermost excited state "|eee " or The GHZ-state or the W-state. For this system we investigate different kinds of atomic inversion and entanglement, which arise between the different parts of the system due to the interaction. Also the relationship, between entanglement and some other nonclassical effects in the statistical properties, such as collapses and revivals in the atomic inversion where superharmonic effects appear, is discussed. The Q-functions for different cases are discussed. Most remarkably it is found that the GHZ-state is more robust against energy losses, showing almost coherent trapping and Schrödinger-cat states can not be produced from such state. Also the entanglement of GHZ-state is more robust than the W-state. Another interesting feature found is that the state which has no pairwise entanglement initially will have a much improvement of such pairwise entanglement through the evolution. Sudden death and sudden revival of atoms-pairwise entanglement are produced with the W-state.PACS numbers: 37.30.+i, 03.67.*
INTRODUCTIONThe quantum entanglement phenomenon is not only one of the most interesting features of the quantum theory [1], that signifies it from the classical theory, but also lies at the heart of the new rapidly developing area known as the quantum information processing [2]. It is one of the crucial resources required in the applications in this new area of science, which include, quantum computation [3], quantum teleportation [4], quantum dense coding [5] and quantum cryptography [6]. In quantum optics domain, the interaction of an atom with a quantized electromagnetic field mode described by the Jayens-Cumming model [7, 8]leads to an entanglement of these two systems such that the total state vector cannot be written as a product of the time-dependent atomic and field component vectors [9, 10, 11]. To quantify entangled states, one should know whether they are pure or mixed states. Thus, if the entangled state is in a pure state, then it is sufficient to use von Neumann entropy as a measure of entanglement. Many efforts have been devoted to quantify entanglement, particularly for mixed states of a bipartite system, and a number of measures have been proposed, such as entanglement of formation, relative entropy of entanglement and negativity. The Peres-Horodecki criterion for separability [12, 13] leads to a natural computable measure of entanglement, called negativity [14, 15, 16]. It has been proved that the negativity N (ρ) is an entanglement monotone and therefore can be used as a good measure of entanglement [16].On the other hand the cooperative nature of the interaction of a quantized radiation field with a system of two-level atoms has first been treated by Dicke [17]. A particular case of the Dicke ...