The Ground State Of Spin-orbit-coupled Bose-Einstein Condensates In Optical Lattice

In recent years,the physics of ultracold atomic gases have been attracted enormous attention.Especially,after the realization of the artificial spin-orbit coupling for ultracold atoms by using the interaction between light and atoms,many abundant and novel phenomena in the ground state and dynamics have been revealed,such as the plane wave phase,the stripe phase,the atomic spin hall effect,the spin-momentum locking and so on.Loading the spin-orbit coupled Bose-Einstein condensates into an optical lattice,a new ground state phase,i.e.,the endge-quasimomentum phase emerges.The spin-orbit coupled Bose-Einstein condensates in optical lattice provide an ideal platform for simulating the quantum phase transition of the strongly correlated system.However,the research of spin-orbit coupled Bose-Einstein condensates in optical lattice is preliminary.Particularly,the competition between the optical lattice and the spin-orbit coupling for determining the ground state phase transition and the effect of the atomic interactions on the energy band structure are still unclear.So based on the mean-field theory and the variational method,the ground state phase transition and the nonlinear energy spectrum structure in the system of the spin-orbit coupled Bose-Einstein condensates in optical lattice have been discussed in detail in this thesis.The dissertation is organized as follows.The relevant physical research background of the thesis is briefly introduced in the first chapter.The realization of spin-orbit coupling in the ultracold quantum gas,the research status of the ground state and energy spectrum structure of the spin-orbit coupled system are presented.The energy spectrum structure and dynamics of the Bose-Einstein condensates in optical lattice system are introduced.Then experiment principle and the ground state phase transition in spin-orbit coupled Bose-Einstein condensates in optical lattice are reveled.Finally,the main research content and structure of the dissertation are shown.In the second chapter,the phase transition of the Bose-Einstein condensates with spin-orbit coupling trapped in one-dimensional optical lattice is studied.Based on the variational and two-mode approximation,the critical condition for phase transition between polarized and unpolarized Bloch wave phase is obtained analytically,which explicitly reveals rich competitive relationship among the spin-orbit coupling,optical lattice and atomic interactions in determining the phase transition of the system.The variation of the energy minima and the phase transition point by the optical lattice depend on the strength of atomic interactions and the spin-orbit coupling.Our results provide a theoretical evidence for deep understanding the competition mechanism between the optical lattice and the spin-orbit coupling for the ground state phase transition of spin-orbit-coupled Bose-Einstein condensates in optical lattice.On the basis of the previous work,the nonlinear energy spectrum structure and the current density of the spin-orbit coupled Bose-Einstein condensates in optical lattice are detailedly investigated in the third chapter.When the parameters satisfy certain conditions,an interesting loop structure in the Brillouin zone edge will emerge.The Raman coupling and the optical lattice suppress the emergence of the loop structure,while the spin-orbit coupling and the atomic interactions promote the emerging of the loop structure in the Brillouin zone edge.Especially,the spin-orbit coupling can make the energy spectrum structure more complex.The nonlinear energy band structure is closely related to the current density of the system.The spin-orbit coupling causes a strong asymmetry of the current density and the separation of the current density distributions of different spin states in the momentum space near the boundary of the Brillouin zone.But the strength of the optical lattice and the Raman coupling can weaken the asymmetry.The appearance of loop structure breaks the Bloch oscillation,causes the Landau-Zener tunneling,and the separation of the current density distributions of different spin states in the momentum space means the emergence of the spin exchange dynamics.The fourth chapter summarizes the dissertation and outlooks for this field.