Critical Temperature And Ground State Properties Of Low-Dimensional Bose-Einstein Condensation

Bose-Einstein Condensation (BEC) is one of the hotspot in the physics research recently. It reveals a new form of matter in the universe, a large number of dense atoms evolves coherently, bringing the quantum phenomenon from the microscopic world to the macroscopic world. Excitons BEC is the Bose-Einstein Condensation realized in the semi-conductor materials. The physists lay more and more emphasis on it due to it's speciality and wide prospect, such as the application on high-powered and low-consumption light-emitting devices, ultra-fast logic devices, and quantum computing. Critical temperature is the phase transition temperature at which the common bose gas transforms into the Bose-Einstein Condensation. This thesis calculates the critical temperature in the low dimensional spaces according to the first-excited-state energy of particles in the traped condensate being different from zero, investigates the effect of first-excited-state energy on the critical temperature as well as the relationship of the critical temperature and the ground-state faction versus the total number of particles in different dimensional spaces. The results we get are consistent with the experiment data. It is shown that the critical temperature in low-dimensional spaces increases more quickly than that in high-dimensional space with density of particle number increasing, and the critical temperature of system has close relation to the first-excited-state energy. Our theoretical method can explain the phenomena that even under the condition of absent special external potential, such as harmonic potential and power-law potential, the Bose-Einstein Condensation can be realized at several K for excitons gas trapped in GaAs quantum well. The wave function of BEC can be described by the Gross-Pitaevskii equation. In the last part of the thesis, we study the effect of total particle number, the interaction between particles, the frequency of harmonic potential and the power-law potential on the distribution of the particles in BEC and the ground state energy of the condensate by solving the G-P equation using the Fourier-Grid-Hamiltonian method numerically. The results we get show that with the intensity of power-law potential or the frequency of harmonic potential increases or the repulsive interaction between particles decreases, the particle density in the condensate center will increase and the radius of the condensate will decrease. The ground state energy of the BEC will rise with the increase of the total particle number, the repulsive interaction between particles, the frequency of harmonic potential and the intensity of power-law potential. Thomas-Fermi approximation grows more and more resembling with the numeric result when the particle number increases, which shows Thomas-Fermi limit is a good method in the condition of large particle number, if the particle number in the system is less, the numeric way should be use when study the BEC problem.