Localization In One-dimensional Quasi-periodical Optical Lattices

The realization of Bose-Einstein Condensation(BEC)marks the beginning of a new field of ultra cold atoms.The cold atoms trapped in the optical lattices can be used to simulate a variety of crystal structures,which offers new opportunities for studying the properties of traditional solid physics.Disorder and interaction of quantum systems are extremely important topics in condensed matter physics.According to Anderson's groundbreaking work,non-interacting particles with disorder exhibit Anderson localization.In three dimensions,a mobility edge which separates localized and extended states can exist.In one and two dimensions,any small random disorder is sufficient to localize all single-particle states.Recently,a kind of quasi-periodical potential between random disorder and periodic potential has attracted great attention,and the transition of extended and localized states has been detected in one-dimensional quasi-periodical systems.In this thesis,we firstly discuss the experimental realization of bichromatic optical lattices,and introduce some important order parameters,such as entropy,inverse participation ratio,and structure entropy.We analyze the characteristics of single-particle localization of some special quasi-periodical systems by numerical diagonalization,such as the one-dimension of Mathieu-like potential and slowly varying potential.On this basis,we applying the exact diagonalization method to study the localization phenomenon in the bichromatic optical lattices,which is formed by the superposition of two incommensurate lattices.We take one-dimensional Aubry-Andre model as an example to study the localization characteristics of the system.Two pairs of almost mobility edges are shown in one-dimensional Aubry-Andre model for the incommensurate parameter?(?)1 For the case of the amplitude of the modulation ?<2t,we define the energy edges between extended region and critical region as E1c± and the ones between critical region and localized region as E2c±.We study the varying of almost mobility edges with the parameters of the systems.The results show that the positions of Elc± are independent of the incommensurate parameter ?,Elc±=±|2t-?|.The positions of the almost mobility edges E2c± depend on both the incommensurate parameter and modulation strength,and when ??0,Elc± and E2c± coincide with each other.In the region ?>2t,the eigenstates in the band center and band edge exhibit different localized behaviors by numerical calculation of inverse participation ratio and Shannon entropy.On the other hand,the system with nearest neighbor interaction only shows ergodic phase and many-body localization transition is not found for small sized system.However,for the case of ?=((?)-1)/2,many-body localization transition remains in the interacting system.We believe that the phenomenon of ?(?)1 depends on the system size.