Lorenz Chaos And Soliton In Bose-Einstein Condensates

The realization of Bose-Einstein condensation in a dilute atomic gas has attracted enormous attention in recent decades. It has provided physicists with a new fertile ground for exploring many aspects of this fascinating phenomenon including Josephson effect, chaos, soliton, vortices, dynamical instability, superfluidity and quantum phase transition. Their intrinsic nonlinearity and the interaction with externally applied fields make them a kind of classical chaotic system. In the framework of mean-field theory the Bose-Einstein condensates are governed by the Gross-Pitaeviskii equation.This paper consists of four parts. In the first chapter, we shall give a simple introduction to chaos and soliton in BEC. In the second chapter, we investigate the dynamics of a weakly open Bose-Einstein condensate (BEC) with attractive interaction in a magneto-optical double-well trap. A set of time-dependent equations describing the complex dynamics are derived by using a two-mode approximation. The stability of the stationary solution is analyzed and some stability regions on the parameter space are displayed. In the symmetric well case, the numerical calculations reveal that by adjusting the feeding from the non-equilibrium thermal cloud or the two-body dissipation rate through the mechanism like the Feshbach resonance, the system could transit among the periodic motions, chaotic self-trapping states of the lorenz model and the steady-states with the zero relative atomic population or with the macroscopic quantum self-trapping (MQST). In the asymmetric well case, we find the periodic orbit being a stable two-sided limited-cycle with MQST. The chaotic or periodic relative atomic population and total number of condensed atoms mean the chaotic or periodic atom-tunnelling and fluctuate of the condensate.In the third chapter, we have studied the dynamics of two-dimensional (2D) trapped and un-trapped Bose-Einstein Condensates (BECs) with a rapid periodic modulation of the scattering length via a Feshbach resonance technique, a→a₀+ a₁ sin(Ωt) with an attractive (negative) mean value and the large constants a₀, a₁ and 0. Applying a variation approximation (VA), the critical threshold for the collapse of the 2D trapped vortex BEC is predicted and the collapse is prevented by causing the scattering length oscillating rapidly. On the other hand, with analytical...