arXiv:0811.4633v2 [quant-ph] 1 May 2009 Breakdown of the rotating-wave approximation in the description of entanglement of spin anticorrelated states Jun Jing 1,2* , Zhi-Guo L¨ u 3 , and Zbigniew Ficek 4 1 Department of Physics, Shanghai University, Shanghai 200444, China 2 The Shanghai Key Lab of Astrophysics, Shanghai 200234, China 3 Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China 4 Department of Physics, The University of Queensland, Brisbane 4072, Australia (Dated: May 1, 2009) It is well established that an entanglement encoded in the Bell states of a two-qubit system with correlated spins exhibits completely different evolution properties than that encoded in states with the anti-correlated spins. A complete and abrupt loss of the entanglement, called the entanglement sudden death, can be found to occur for the spin correlated states, but the entanglement evolves without any discontinuity or decays asymptotically for the spin anti-correlated states. We consider the evolution of an initial entanglement encoded in the spin anti-correlated states and demonstrate that the asymptotic behavior predicted before occurs only in the weak coupling limit or equiva- lently when the rotating-wave approximation (RWA) is made on the interaction Hamiltonian of the qubits with the field. If we do not restrict ourselves to the RWA, we find that the entanglement undergoes a discontinuity, the sudden death phenomenon. We illustrate this behavior by employing an efficient scheme for entanglement evolution between two cold-trapped atoms located inside a single-mode cavity. Although only a single excitation is initially present in the system, we find that the two-photon excited state, which plays the key role for the discontinuity in the behavior of the entanglement, gains a population over a short time of the evolution. When the RWA is made on the interaction, the two-photon excited state remains unpopulated for all times and the discontinuity is absent. We attribute this phenomenon to the principle of complementarity between the evolution time and energy, and the presence of the counter-rotating terms in the interaction Hamiltonian. PACS numbers: 03.65.Ud, 03.67.-a, 42.50.Pq An understanding of entanglement evolution and en- tanglement transfer between qubits is of fundamental in- terest in quantum information processing [1, 2]. The controlled transfer that preserves initial entanglement is crucially important. Transfer processes are susceptible to decoherence and dissipation due to the inevitable cou- pling of the qubits and the transfer channels to an ex- ternal environment. Therefore, in order to minimize de- coherence effects and to achieve the perfect fidelity, fast transfer processes or transfer operations performed over a very short time scale are highly desirable. It has been recognized that the entanglement evolution depends on the state in which it is encoded. For a simple system of two qubits, the basis states for entanglement are four mutually orthogonal Bell states [3]. The states can be divided into two groups, one involving linear su- perpositions of the spin correlated states and the other involving spin anti-correlated states. The states belong- ing to these groups are often called two-photon and one- photon entangled states, respectively. The qubits can be prepared in a spin correlated state by a transfer of two- photon entangled states from quantum-correlated light fields produced e.g., in a nonlinear process of parametric down conversion [4]. Preparation of a spin anti-correlated state is more sophisticated as it involves a single excita- tion ”shared” between two qubits. In principle, it can * Email: jungen@shu.edu.cn be achieved, for example by applying a short single laser pulse either in a running or in a standing wave configura- tion [5]. This will result in the qubits prepared either in a symmetric or in an antisymmetric combination of the spin anti-correlated states. Dynamics of an entanglement encoded in spin corre- lated states have been extensively studied since the pio- neering work of Yu and Eberly [6, 7], who showed that an initial entanglement encoded in two separate qubits interacting with local environments can decay to zero in a finite time [8]. When the qubits are subjected of the interaction with each other through the coupling to the same environment, the already dead entanglement may revival after a finite time [9]. The interaction between the qubits induces a population difference between the symmetric and antisymmetric combinations of the spin anti-correlated states, which results in a nonzero entan- glement. A completely different conclusion applies to the evo- lution of entanglement initially encoded in a spin anti- correlated state. It was pointed out by Jamr´ oz [10] that an initial entanglement encoded in a spin anti-correlared state of two independent qubits interacting with local en- vironments decays asymptotically in time without any discontinuity. This prediction agrees with Yonac and Eberly’s work [11], and also with other results [12]. The same conclusion applies to the case of two qubits mutu- ally interacting through the coupling to a common en- vironment [13]. In these papers, the analysis were re- stricted to the RWA and this raises the question on the