Researches On Phase Separation And Anderson Localization Of A Two-component Bose-einstein Condensate

Interacting Bose-Einstein condensate(BEC) is an open question of ultra-cold atomic quantum gases. Recently, a dipolar BEC which was observed in experiments has motivated many theoretical studies. The long-range and anisotropic feature of dipolar interaction provides novel and fascinating properties of dipolar BEC. Exploration of properties of a two-component dipolar BEC has been one of the fundamental areas in dipolar BEC. Additionally, Anderson localization of a weak interacting BEC which was recently realized in experiments is a milestone of studies on disorder systems. The interplay of interactions in atomic gases and disorder is the most essential factor in localization of BEC. Clarifying this interplay is a fundamental issue of understanding many quantum systems including superfluid helium, superconductor and so on. In this dissertation, mean-field theory, finite difference method and Crank-Nicolson algorithm have been used to study quantum behaviors of a two-component dipolar BEC as well as Anderson localization of a two-component BEC in disorder potentials. The following results are concluded:(1) A two-component BEC which contains atoms with magnetic dipole moments aligned along the z direction(labled as component 1) and nonmagnetic atoms(labled as component 2) is considered. In the mean-field frame, we use exact numerical simulations to study the effect of dipole-dipole interaction on phase separation by the means of finite difference method. Before performing the numerical calculations, we reduce the two coupled three-dimensional Gross-Pitaevskii(GP) equations to the quasi-one-dimensional(quasi-1D) and quasi-two-dimensional(quasi-2D) regime. In the quasi-1D regime, the atoms in component 2 are squeezed out when the dimensionless dipolar strength parameter is small, whereas the atoms in component 1 are pushed out when the parameter is large. This is contrast to the phenomena in the quasi-2D regime. The two components are kicked out by each other in the quasi-1D regime and this phenomenon is discussed as well. Due to the presence of dipole-dipole interaction, the critical value of inter-component scattering length which refers to phase separation in a two-component BEC needs to be modified. The immiscible criterion for the strength of the repulsive inter-component interaction is independent of the atom numbers as well as the trap strength.(2) We consider a weakly interacting two-component BEC in a 2D quasi-periodic bichromatic optical lattice(BOL). This problem is studied by means of split-step Crank-Nicolson method. For different laser waves in BOL, the key feature of Anderson localization which is exponential decay of the atomic density shows up. The effects of weak intra- and inter-component interactions on localization of a two-component BEC are investigated in detail. It is shown that in the quasi-2D regime, due to the enhanced disorder, there is no symmetry breaking like that in the 1D case under a sine-typed potential, while configurations of density profiles are also quite different from that in the 1D case. By modulating interactions, the interplay of disorder and weak repulsive or attractive interactions is studied in different quasi-periodic potentials. When a sine-typed potential is imposed on component 1 and a cosine-typed potential is acting on component 2, the two localized BEC components can not be in a miscible state, and they can be stably localized in certain sites even with the presence of attractive inter-component interaction. This theoretical result may shed a light on realization of stable localized BEC in experiments.(3) A weakly interacting two-component BEC in a 1D random speckle potential is considered. The problem is studied with solutions of GP equations by means of numerical method in Crank-Nicolson scheme. We investigate properties of various cases owing to the competition of disorder and repulsive interactions of a cigar-shaped two-component BEC in detail. In the central region, phase separation of a two-component BEC is not only affected by the intraand inter-component interactions, but also influenced by the strength of the random speckle potential. Due to the strong disorder of the potential, the criterion of phase separation in an ordered potential, such as a harmonic potential, is no longer available. The influence of different random numbers generated by distinct processes on localization of BEC in the random potential is also investigated, as well as the configurations of the density profiles in the tail regions. We use MATLAB and FORTRAN function to generate two different sets of random numbers respectively, then figure out that the center and shape of the localized BEC are different due to the different shape of the trough in the potential.