Effect Of Counter-Rotating Term On The Entanglement Dynamics Of Two Superconducting Qubits Coupling To Quantum Oscillators

Two-level system (qubit) is highly relevant to the quantum computing, quan-tum information science, condensed matter physics, and even quantum biology, and has attracted considerable attentions. Recently, the dynamics of two-level system coupling with a cavity field is studied extensively in both theory and in experiments, because its dynamic evolution play a important role in quan-tum computing and quantum communications. Quantum entanglement is a very important quantum information resource, describing nonlocal property of a quan-tum system. It has broad applications in modern science and technologies such as quantum codes, quantum information transfer, quantum cryptography.Jaynes-Cummings model is a ideal simple model describing a qubit and single-mode cavity field. But in the previous cavity electrodynamic system, be-cause the coupling of the system and cavity is very weak, the rotating-wave approximation (RWA) was usually made, i.e. neglecting the counter-rotating wave terms. These terms describe the the energy non-conserved process, and results in the non-conservation of the total bosonic number, so it is impossible to solve the model analytically. Without the counter-rotating wave terms, there are simple analytical solutions in the Jaynes-Cummings model, and one can obtain very concise eigensolutions. In the weak coupling system, the entanglement of two independent Jaynes-Cummings atoms is very simple and has been studied widely.Recently, superconduting qubits and LC oscillator systems in quantum joseph-son junctions has been artificially produced, which is called now as circuit elec-trodynamic system. It is now experimentally realized that the coupling is around0.1, entering ultrastrong coupling regime. The properties of spectrum can not be explained qualitatively in the framework of the RWA, so the it is necessary to study beyond the RWA.Without the RWA, only numerically exact solutions are available, or some generalized RWA analytical ones. In the study of entanglement dynamics of two Jaynes-Cummings without interaction, two initial states are usually chosen, they are Bell correlated and anti-correlated spin states. Based on previous numeri-cally exact eigensolutions, there are some progress achieved in the entanglement evolution. Especially for Bell anti-correlated state, the previous picture based on the RWA has been modified considerably, such as the entanglement sudden death is enhanced dramatically. But some analytical or semi-analytical studies without the RWA have not been reported before.In this paper, by using the recent developed two analytical approaches with-out the RWA, such as generalized RWA and corrected RWA, the entanglement dynamics from different initial states are studied. It is shown that in the wide coupling regime, the results based on the corrected RWA is more close to the ex-act ones than those by the generalized RWA. Furthermore, initiated from thje bell anti-correlated states, the entanglement sudden death emerges as the coupling increases, different from the observation under the RWA. And the entanglement disappear periodically with the increasing coupling. While from the correlated states, the entanglement sudeen death always exists, and its persistent time is prolonged with the coupling, and entanglement also disappear periodically with the increasing coupling.