| As we all know, when the boson system is below one critical temperature, all of the bosons will occupy the same quantum state, this physical effect is named by Bose-Einstein Condensation (BEC). Einstein predicted this special state in 1924, however the predication is not approved because of the hard experiment condition until JILA, Rice university and MIT groups firstly realized this macroscopical quantum state by cooling alkali metal atoms in 1995. The density of atoms in experiments is so dilute that the system still remains in gaseous state at extremely low temperature. The interaction between atoms is so weak that we can simplify the theoretical model of BEC. In another hand, with the development of the technology of the laser cooling and trapping of atoms we can simulate many physical models by boson gas, for example, the crystal lattices and the Hubbard model in condensed matter physics. The achievement of BEC has opened a new chapter for studying the property of atoms.In early experiment the dilute alkali atoms are cooled and trapped by magnetic field, and only one internal state are occupied because of the magnetic confinement. After the application of the optical potential, the external field does not confine the freedom of bosons again and more internal states can be saved in one potential. It is called by multi-component BEC. The object of our paper is just to study the property of one multi-component BEC which is composed of two species of pseudospin-1/2 Bose gases.In chapter 1, we first review the history of BEC and the process of how to realize the condensation by the the technology of the laser cooling. Secondly, we introduce the property of the ground state and the elementary excitation through the mean-field theory and nonmean-field theory. At last, the vortex in single-component BEC is mentioned.In chapter 2, the physical property of multi-component system is the main content. We review the elementary excitation and the vortex lattice in two-component system in detail. The mean and nonmean-field theory are introduced to study the property of the Spinor-1 bose system. At last we describe the general theory of arbitrary spinor BEC.In chapter 3, we first introduce the mixture of two species of 1/2 pseudospin bose system. Through integrating the space wave function, the Hamiltonian of system can be transformed into two coupled giant spins. After elucidating that the entanglement peak is indeed located at a quantum phase transition point, we study how the ground state of the collective spins affects the elementary excitations of the orbital wave func-tions in which EBEC occurs. Furthermore, we find that in the vicinity of this quantum phase transition, the energy gap of the gapped orbital elementary excitation is strikingly different from those of disentangled ground states. Away from the quantum phase tran-sition point, the elementary excitations tend to approach those of a disentangled ground state.In chapter 4, we mainly analyze the classical dynamics and the quantum dynamics of this mixture of bose gases. All of the fixed points are found and their stability regions are determined by the stability theory in nonlinear mathematics. By numerical simulation, we show that the similar character between the quantum dynamics and the classical dynamics. When the numbers of two species of atoms are equivalent, the entanglement of the ground state is maximal at the bifurcation point, however, when the numbers of atoms are not equivalent, the entanglement of the ground state is not maximal at bifurcation point. At last, we demonstrate the existence of the gap between the classical state and the ground state in one parameter region, the gap is still a limited value even when the numbers of atoms tends to infinity. It is shown that the ground state is not direct product between two subsystems by solving the approximate Hamiltonian, the entanglement of the ground state is a large value in this parameter region.
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