Nonlinear Dynamical Behaviors Of Atomic Spinor Bose-Enstein Condensates Confined In An Optical Lattice

Optical lattice is formed by the interference between counter-propagating laser beams. It has perfect periodical property, the lattice constant and the depth of the potential well can be well controlled by tuning the parameters of the laser beams. Now optical lattice has been widely used to control ultra-cold atoms. Since the realization of Bose-Einstein condensate (BEC), much effort has been concentrated on the fundamental research of BEC in optical lattice, which has become one of the hottest subjects in atomic and molecular physics.On the other hand, most real physical systems are strong-interacting many-body systems. When it is difficult to get the dynamical evolution of the systems through analytical or even numerical methods, one can utilize the quantum sim-ulation method. Taking solid crystal system as an example, the distance between atoms is short, so the exchange interaction between atoms and the interaction between atom and the crystal field are very strong. While in optical lattice, the distance between atoms is several hundred times longer than that in solid crystal system, and the interaction between atom and the laser field is very weak. This leads to that the variation of the dynamical characters is much slower than that in solid crystal system. Nonetheless, they share very similar characters in motion. So it is possible to simulate the phenomena of solid crystal system by utilizing the BEC confined in optical lattice. Similarly, we can also simulate other compli-cated physical phenomena encountered in condensed matter system and optical system.In this thesis, we study the nonlinear dynamical behaviors of spinor BECs confined in a one-dimensional optical lattice. The fundamental background knowl-edge is reviewed in the first two chapters. In the first chapter, we mainly review the basic theories of BEC. In the second chapter, we introduce the optical lattice and the basic characters of dipole-dipole interactions, etc.In the third chapter, we study the nonlinear magnon excitation of this spinor chain in the optical lattice. We first analyze the characters of spin-wave excitation in the solid crystal system. Based on this we analyze the dynamical characters of the magnons generated by the static magnetic dipole-dipole interaction (MDDI) and the external laser induced dipole-dipole interaction (LDDI) in the optical lattice system. Specially, we choose a blue-detuned optical lattice and define an effective temperature for this system. We make a comparison between the genera-tion process of magnons and that of photons in optical vibration cavity. We show that by suitably choosing system parameters, we can reproduce the dynamical Casimir effect at finite temperature in this magnon system. Furthermore, we can obtain ferromagnetic magnon squeezing state, which shows remarkable tunability. Our results provide useful references for the study of magnon squeezing state in solid ferromagnetic system. At last, we analyze the statistical properties of the excited magnons.In the fourth chapter, we study the nonlinear magnetic soliton excitation of the spinor chain in the optical lattice. Being different from short-range exchange interaction of electrons in solid crystal system, the MDDI and the LDDI have long-range characters. To find how these two types of dipole-dipole interactions affect the soliton excitations, we choose a blue-detuned optical lattice, and an external laser field is imposed into the system to induce the LDDI. When the external laser field is absent, we analyze how the long-range character of the dipole-dipole interaction affect the excitation of magnetic solitons and study the existence conditions of magnetic soliton solutions for three types of distance ap-proximations. We find that the magnitude and the sign of the effective mass of the magnetic solitons can be tuned by changing the system parameters. This means that we present another way to obtain soliton solutions when only MDDI exists in the system. In addition, we analyze the impact of anisotropy character of the dipole-dipole interactions on the magnetic soliton excitations in detail.Finally, we conclude and give an outlook for investigations in future.