Ouantum Phase Transition And Dynamical Properties Of Multi-component Ultracold Bosons

Bose-Einstein condensate (BEC) is a macroscopic state of matter. It has attracted much attention since the realization of BEC in dilute bose gases. In this dissertation, we focus on the dynamical instability of BEC in optical lattices, the conversion effi-ciency of the atom-to-molecule conversion system and the quantum phase transition in the atom-to-molecule conversion system. The dissertation is organized as follows:In Chapter1, we make a brief overview of the theory, the experimental realiza-tion and progress of the atomic BEC, BEC in lattices, and spin-orbital coupling BEC. In Chapter2, we briefly introduce the realization of the molecular BEC. We mainly describe the direct cooling techniques to reach cold molecules, the indirect cooling technologies to obtain ultracold molecules and the ultracold molecules in optical lat-tice.In Chapter3, we study the dynamical stability of Bose-Einstein condensates in an optical lattice with a time-periodic modulation potential and a constant acceleration force simultaneously. We derive the explicit expressions of quasienergies and obtain the stability diagrams of the ground state in parameter spaces. For integer and non-integer (rational) ratio of the acceleration force to the modulation frequency, different dependence relationships of the critical interaction strength on the modulation am-plitude are observed. Our results are expected to help experimentalists to determine parameter regions as needed.In Chapter4, we observe the atom-to-molecule conversion in a magnetic lattice in the mean-field approximation. For the case of shallow lattice, we give the dependence of the atom-to-molecule conversion efficiency on the tunneling strength and the atomic interaction by taking a double-well as an example. We find that one can obtain a high atom-to-molecule conversion by tuning the tunneling and interaction strengths of the system. For the case of deep lattice, we show that the existence of lattice can improve the atom-to-molecule conversion for certain initial states. Our research provides a new approach to obtain high atom-to-molecule conversion efficiency in experiments: magnetic lattice. In Chapter5, we study the quantum phase transition in an atom-molecule con-version system with atomic transition between two hyperfine states. In the mean field approximation we give the phase diagram in which the phase boundary only depends on the atomic hopping strength and the atom-molecule energy detuning, but not on the atomic interactions. Such a phase boundary is further confirmed by calculating the fidelity of the ground state and the energy gap between the first-excited and the ground states. As a comparison to the mean field results, we also study the quan-tum phase transition in the full quantum method where the phase boundary depends on the particle number N. Analyzing the finite-size scaling behaviors of the energy gap, the fidelity susceptibility, and the first-order derivative of entanglement entropy, we show that in the limit of Nâ†'∞, one can obtain the same phase boundary as in the mean field approximation. Our results show a new route to manipulate the quan-tum phase transition by tuning the atomic hopping strength, not just the conventional atom-molecule energy detuning.In Chapter6, we give a brief summary of our studies and the prospects of our future research.