Laser-manipulating Macroscopic Quantum States Of Bose-Einstein Condensates Held In Periodic Potentials

Bose-Einstein condensaton is a very fantastic quantum phenomenon,which has been an attractive subject in recent decades.Actually,it is a physical phenomenon referring to many aspects,such as atomic and molecular physics,quantum optics,statistical physics,and condensed matter physics.Shortly after the first experimental realization of Bose-Einstein condensates in 1995 in dilute gases of alkali metals,intense efforts have been devoted to the study of the new properties of Bose-Einstein condensates.Since the realization of Bose-Einstein condensates of alkali-metal atoms in an optical lattice,the studies of Bose-Einstein condensates in periodic potentials have been the subject of an explosion of research,both theoretically and experimentally.In the framework of mean-field theory,we study the Bose-Einstein condensates held in periodic potentials,which are governed by Gross-Pitaevskii equations.We investigate the regular or chaotic macroscopic quantum states of the systems by theoretical analysis and numerical method,and how to manipulate the macroscopic quantum states by using laser potentials.Some meaningful results are obtained after the investigations.This paper is organized as the foolowing six chapters:In the first chapter,we give a simple introduction of the background of Bose-Einstein condensates,and review the research history of it.We also introduce the mean-field theory of dilute gases of alkali metals,the research status and improvements about the regular states of Bose-Einstein condensate and chaos.In chapter 2,We investigate the boundary value problem(BVP)of a one- dimensional Gross-Pitaevskii equation with the spatially periodic Kronig-Penney potential(KPP)of period d,which governs a quasi-one-dimensional repulsive Bose-Einstein condensate.Under the zero and periodic boundary conditions,we show how to determine n exact stationary eigenstates{R_n}corresponding to different chemical potentials{μ_n}from the known solutions of the system.The n-th eigenstate R_n is the Jacobian elliptic function with period 2d/n for n=1,2,…,and with zero points containing the potential barrier positions.So R_n is differentiable at any spatial point and R_n² describes n complete wave-packets in each period of the KPP.It is revealed that one can use a laser pulse modeled by aδpotential at site x_i to manipulate the transitions from the states of{R_n}with zero point x≠x_i to the states of{R_n'}with zero point x=x_i.The results suggest an experimental scheme for applying Bose-Einstein condensate to test the BVR and to observe the macroscopic quantum transitions.In chapter 3,the spatial chaos of the Bose-Einstein condensate perturbed by a weak laser standing wave and a weak laserδpulse is studied.By using the perturbed chaotic solution we investigate the new type of Melnikov chaotic regions,which depend on an integration constant co determined by the boundary conditions.It is shown that when the|c₀|values are small,the chaotic region is corresponded to small values of laser wave vector k,and the chaotic region for the larger k values is related to the large|c₀|values.The result is confirmed numerically by finding the chaotic and regular orbits on the Poincar(?)section for the two different parameter regions.So for a fixed c₀ the adjustment of k from a small value to large value can transform the chaotic region into the regular one or on the contrary,that suggests a feasible method for eliminating or generating Melnikov chaos.In chapter 4,we investigate an attractive Bose-Einstein condensate perturbed by a weak traveling optical superlattice.It is demonstrated that under a stochastic initial set and in a given parameter region the solitonic chaos appears with a certain probability which is tightly related to the zero-point number of Melnikov function,and the latter depends on the controllable potential parameters.Effects of the lattice depths and wave vectors on the chaos probability are studied analytically and numerically,and different chaotic regions of parameter space are found. The results suggest a feasible method for strengthening or weakening chaos by modulating the potential parameters experimentally.In chapter 5,we study the spatiotemporal chaotic evolution a Bose-Einstein condensate in a moving optical lattice considering damping effects.Melnikov chaotic solution and chaotic region of parameter space are found by using the direct perturbation method.Due to the damping effects,there is a regular region with zero chaos probability for the system.In the chaotic region,the spatiotemporal evolution of the chaotic solution is analytically bounded but unpredictable between the superior and inferior limits.We demonstrate theoretically and numerically that the chaotic region reduces as the propagating velocity of the optical lattice increases.On the other hand,adding a second lattice,we find the chaotic region would widen as the intensity of it increases.In the double lattice case,the chaotic region is also strongly influenced by the velocity of the lattices.This is helpful for eliminating or generating Melnikov chaos.In the last chapter,we give a simlpe summary and discussion to the abovementioned works,and also discuss the prospects of applicions of the Bose-Einstein condensate systems.Our main works are involded in the sencod,third,fourth and fifth chapters.