Analytic Study To A Kind Of Bose-Einstein Condensates In Near Harmonic Potentials

In this thesis, we mainly perform analytic study on a kind of Bose-Einstein condensates (BECs) held in the one-dimensional (1D) near harmonic potentials. Such near harmonic potentials agree with normal harmonic potentials for the spatial range near the center of magnetic field, and tend to some constants for far range from the center. We investigate the 1D atomic gas BECs loaded in two different near harmonic potentials, namely BEC guided by magnetic waveguide and condensed atoms trapped in cigar-shaped confining potential, respectively. In the waveguide system, the considered near harmonic potential is a square-Sech potential that is more general than Rose-Morse potential(V₀Sech²Ïƒx). In cigar-shaped trap system, we introduce a new near harmonic confining potential model which is more consistent with realistic confining potential than the normal harmonic potential.Under the mean-field approximation, the dynamic behavior of the considering systems can be described by one-dimensional stationary nonlinear Schrodinger equation, namely Gross-Pitaevskii equation(GPE). By solving the GPE, we obtain the exact solutions of the two systems. In magnetic waveguide system, we get both the bound state and transmission state; and in the cigar-shaped trap system, only the transmission state is found.Based on the exact solutions, we discuss the stability of stationary state solutions, transport properties of transmission state and calculate the quantity of condensate atoms. Through the analysis of local minimum of per atom, we find that the excited-state BEC is perfectly stable provided the temperature is close to absolute zero. As accords with the previous result about BEC stability investigation. Because of the non-zero flow density J in transmission state, We study the transport properties about J. Exact flow density shows that external confining potential suppresses the transport flow, on the contrary, the internal atom-atom interaction improves the transport of condensate atoms. This result is completely different from the previous nonlinear transport study in other system. The more interesting thing in our systems is that the transmission state BEC is stable superfluid flow and condensate atoms can absolutely transmit through the considering obstacle, which corresponds to resonant transport. It is consistent with Lee's numerical result in the same system. According to the parameter values of the experiment, we investigate the quantity of condensate atoms in cigar-shaped trap. Our theoretical result accords with data from experiment well.