Chaotic And Regular Transport Of Spin-orbit Coupled Bose-Einstein Condensates In Optical Lattices

Spin-orbit coupling?SOC?plays an important role in many condensation phe-nomena,such as spin Hall effect,topological insulators,spintronic devices and so on.The Bose-Einstein condensates?BECs?system provide an ideal platform for simulating the related condensate properties in solid state system because of its experimental controllability.Since the realization of SOC in an⁸⁷Rb Bose-Einstein condensate?BEC?,more and more research focuses have been turned to this field,and many novel physical phenomena have been discovered,such as quantum tricrit-icality and phase transitions,Skyrmions,gray solitons,Dirac monopoles,vortices with or without rotation and so on.In this thesis,based on the mean-field theory of BEC and related research methods,we mainly discuss and demonstrate the chaotic spin-motion entanglement of the SO coupled BECs and the transmission problem of a matter wave in a nonlinear Kronig-Penney superlattice with negligi-ble SOC,and obtain some interesting conclusions and put forward some feasible control schemes.The thesis is divided into five chapters.In the first chapter,we give a brief introduction about the research history and experimental realization of the Bose-Einstein condensation,and BEC theory-the mean field theory.Meanwhile,the chaos of BECs and the effect of chaos on atom transport are briefly described.Moreover,the theoretical basis and experimental realization of SO coupled ultra-cold atoms are also introduced.Finally,we briefly introduce quantum entangle-ment and the connection between quantum entanglement and chaos.In the second chapter,we study the spatially chaoticity-dependent spin-motion entanglement of a SO coupled BEC with a source of ultracold atoms held in an optical superlattice.In the case of phase synchronization,we analytically demon-strate that a)the SOC leads to the generation of spin-motion entanglement;b)the area of high-chaoticity parameter region inversely relates to the SOC strength which renormalizes the chemical potential and c)the high-chaoticity is associated with the lower chemical potential and the larger ratio of the short-lattice depth to the longer-lattice depth.Then we numerically generate the Poincar?e sections to pinpoint that the chaos probability is enhanced with the decrease of the SOC strength and/or the spin-dependent current components.The existence of chaos is confirmed by computing the corresponding largest Lyapunov exponents.For an appropriate lattice depth ratio,the complete stop of one of?or both?the current components is related to the full chaoticity.The results mean that the weak SOC and/or the small current components can enhance the chaoticity.Based on the insensitivity of chaos probability to initial conditions,we propose a feasible scheme to manipulate the ensemble of chaotic spin-motion entangled states which may be useful in coherent atom optics with chaotic atom transport.In chapter three,it was experimentally demonstrated that for a periodically kicked cold atomic system the presence of classical chaos led to greater entan-glement generation between the electron and nuclear spins[S.Chaudhury et al,Nature,2009,461:768].Here we study the effects of chaos on the spin-motion entanglement for a two-frequency driven BEC with SOC in a single-well potential.By using the well-known Melnikov's chaos criterion,we directly obtain the chaotic regions in parameter space,which is consistent with that of Ref.[S.Rong et al,Chaos,2009,19:033129],and the only difference is that the accurate boundaries between chaotic and regular regions are not given here.We find that increasing SOC strength can reduce the area of the chaotic region in the?E₂<0 plane.Par-ticularly,it is observed that the presence of chaos can assist or suppress entangle-ment generation,depending on the initial phase differences between two motional states,which could be manipulated by using the known phase engineering method.The main effects of initial phase on entanglement generation are summarized as follows.On the one hand,for initial phase??0?=₂^?,there is more entanglement generation for initial states localized in chaotic sea of phase space than those in the regular island.On the other hand,when the initial phase is set appropriately to??0?=?,the converse effect called the chaos-suppressed entanglement generation,may occur.Such an interesting phase effect of wave function extends the impor-tant conclusions reported in recent years that chaos helps cold atoms to improve quantum entanglement.The results will be physically significant in designing the quantum information processing protocols and in understanding the many-body entanglement.In chapter four,We study the transmission of a matter wave based on BEC-s in a nonlinear Kronig-Penney optical superlattice.Here we apply an integral equation[W.Hai et al,Phys.Rev.A,2000,61:052105]to seek concise exact so-lution of a one-dimensional nonlinear KP model,which contains a simple nonlinear map connecting transmission coefficient with system parameters.Consequently,we propose a scheme to manipulate probability distribution and transmission by adjusting the system parameters.A new quantum coherence effect is evidenced by the strict expression of transmission coefficient,which results in the aperiod-ic probability distributions and different transmission coefficients including the approximate zero transmission and total transmission,and the multiple transmis-sions.The method based on the concise exact solution can be applied directly to investigate atomic transport in some nonlinear cold atomic systems?In chapter five,we give a simple summary of this work.We also make a brief research prospect on the chaotic transport of SO coupled BECs and the practical application of spin-motion entanglement generation.