Chaos,Bifurcation And Quantum Entanglement In Ultra-cold Atom Systems

In recent years, quantum entanglement (referred to hereafter as entanglement) has been regarded as a physical resource which can be exploited to perform many useful tasks in quantum information processing. Latterly, a new emphasis has emerged that entanglement can be related to the properties of a many-body system. For example, one important issue is whether there exists any relation between entanglement and quantum phase transitions. Many studies have shown that entanglement can be used to identify quantum phase transitions. Along these lines, we relate quantum chaos, bifurcation to quantum entanglement in periodically kicked two-component Bose-Einstein condensates. The fourth and sixth chapter involve the author's partial original work.The two-component Bose-Einstein condensates that behave collectively as a spin system obeying the dynamics of a quantum kicked top. Depending on the nonlinear interaction between atoms in the classical limit, the kicked top exhibits both regular and chaotic dynamical behavior. Since the individual bosons are not physically accessible and distinguishable subsystems of the kicked two-component BEC system, we need to consider other possible decompositions into subsystems. While we cannot measure which mode a specific particle is in, the occupation number of a given mode is physical observable. In our case, the two modes differ in the internal quantum number, and are a clearly distinguishable subsystem. We can thus regard the two coupled BECs as a bipartite system of the modes. Here the von Neumann entropy of the subsystem is employed to measure the mode entanglement. Our mainly conclusion is as follows. The von Neumann entropy increases more rapidly for an initialstate localized in the chaotic region than for one centered on a periodical orbit. Furthermore, the entropy for a fixed point or stable orbit displays a more clearly periodic modulation after a fast increase for a short time, which is an indicator of the underlying regular classical dynamics. If the initial states are in the chaotic region, the classical chaos leads the entropy to more rapidly rise and to arrive at a saturation of the quantum entanglement in a chaotic oscillatory manner. Finally, the phase coherence between the two BECs suppresses the increase of entanglement induced by chaos.We find two different types of bifurcation existing in this system: one is the saddle-node bifurcation and the other the supercritical pitchfork bifurcation. Our results show that the classical bifurcations are closely associated with a topological change in the structure of the corresponding quantum energy levels. The saddle-node bifurcationis related to the avoided level-crossing, and the supercritical pitchfork bifurcation is linked to the near degeneracy of the structure of quantum energy levels. Such a structure change can be manifested in the entanglement properties of the corresponding quantum states. The entanglement of the quantum states displays some peaks near the classical bifurcation points.