Gap Solitons In One Dimensional Periodic Optical Lattices

Ultracold atoms as the hot contents in physics has made great progress. Because it has the advantage of high purity and adjustability, the cold atomic system provides the ideal platform for researching new fields. The development of capturing and cooling contributes to realizing ultracold-atoms experiments in optical lattices. Moreover, loading ultracold atoms in optical lattices has become an alternative way to build the topological insulators. As one-dimensional(1D) system without additional symmetries, it is generally thought to lack topological nontrivial phase. In article called Edge states and topological phases in one-dimensional optical superlattices,it introduces properties of trapped fermions on 1D quasiperiodic optical lattices and shows that this systems display nontrivial properties, which share the same physics origins of topological phases of 2D quantum Hall effects on periodic lattices. What is the topological properties in 1D bose system? Does it exist boundary states? The paper is organized as following:Firstly, it introduces the development process of the ultracold atoms and its relevant technologies and the process of discovering solitons and its classification and gives the contents of this subject.Second, it introduces the energy spectrum of trapped fermions system with periodic boundary condition and open boundary condition. And the system with open boundary conditions has edge states. By contrast, the fermions system can be mapped to two dimensional quantum Hall effect.Finally, we study the properties of 1D bose gas system. In the condition of ignoring the interaction, we study boundary states, states in band and energy band of the system. Then we take the interaction into account, the system is nonlinear. So the system has edge gap solitons. We study the type of solitons, their chemical potential and their wave function under different parameters.