Chaos And Soliton In Bose-Einstein Condensates

Bose-Einstein condensates have been an attractive subjects in recent decads. They not only offer the perfect macrosciopic quantum systems to investigate many foundmental problems in quantum mechanics but also have extensively application foregrounds such as in atom laser and quantum computation. In the framework of mean-field theory the Bose-Einstein condensates is govonered by the Gross-Pitaeviskii equation. Based on the Gross-Pitaeviskii equation we shall study the spatial chaos and time-space chaos in Bose-Einstein condensates which are loaded into a optical lattice, and the dynamical behaviors of matter wave solitons. Meanwhile, we will also discuss the problem of chaos control in the tight binding Bose-Einstein condensates by using the OGY feedback control method.Our paper is organized as the following six parts. In the first part we shall give a simple introduction to mean-field theory and the research status, improvements and applications of chaos and soliton in Bose-Einstein condensates. In the second part, The spatial structure of an enlongated Bose-Einstein condensate atomic cloud loaded into a weakly static optical lattices is investigated. Characters of Melnikov chaos in the system are revealed by using the direct perturbation method. The distribution of atomic number density is demonstrated and the superfluid properties is discussed, meanwhile the spatial chaos involved in these physical quantities is researched. We find for appropriate boundary condition of the number density the distrbution of the condesate is like to a sationary dark soliton. For this practical system we prove theoretically the exsistance of the finite chaotic attractor in the generalized phase space. On the other hand, the analytically unsolvability and numerically uncomputability are also revealed in the evolutions of number density and superfluid velocity with spatial coordinates, these properties prove the exsitance of spatial chaos. Through theoredical analyses, we obtain a weak chaotic region of the density distribution. This region provide a criterion to eliminate the physical invalid part, which is caused by the numerical instability, in system evolutions.In part three, we analyze the spatially chaotic distribution of the condensatesin a strongly static optical lattice. Applying the tight binding approximation, the Gross-Pitaeviskii equation can be turned to a discete nonlinear Schrodinger equation. Considering the case of stationary solution, we obtain a two-dimensional map about the number density and phase angle. Investigations to the map show that the distributions of the condensates on all sites are random and chaotic. In order to control the chaotic distributions, we superimpose a weak and highly focused laser beam on the optical lattice and use it as a control signal. In the scheme of the OGY feedback control method, we find for a appropriate laser beam (proper strength and position) the system can be drived to a desired target state such that we can obtain a regular structure of the Bose- Einstein condensates.Apart from the static lattice, we also studied the space-time evolution properties of the Bose- Einstein condensates held in a travelling optical lattice. In this charpter, we focuse on the features of spatiotemporal chaos of the Bloch-like solution of the system. When the damping effects are not considered, we simply described the procession from regular motion to chaos. Meanwhile, we also discoveried that the enhancement of the speed of the travelling optical lattice has a superessive role to the oneset of the spatiotemporal chaos. When the damp is in our considerations, we discussed the Melnikov chaos of the system analytically, and presented the chaotic region in parametrical space. Features of the transient chaos are studied through numerical method. The transition procession from transient chaos to stationary one has been numerically simulated. In this procession, we find that the final attrators of the transient chaos undergo a series of period-doubling bifurcations. Moreover, the baffling role, which is caused by propagation of lattice and damping, to the onset of chaos is also exsit for the damping case. Due to the facility and accuracy in the control of the travelling speed of the optical lattice, this kind of superessive effects to chaos provides us a experimentally possible way to control the choas in the Bose-Einstein condensate system.In the investigation to the matter wave soliton, the dynamical behaviors of the soliton-like in a time dependent harmonic trap with cigar type are investigated. In the case of quasi one-dimension, we obtain the single bright soliton solution, bisoliton soluton and even the multi-solitons ones. The influences of the geometrical prop-...