Influence Of Presence Of The Earth's Gravitational Field On The Bose-Einstein Condensation

In 1995, Bose-Einstein condensation (BEC) was realized in experiment for almost ideal gases of alkali atom 87Rb , 7Li and 23Na in a magnetic trap at the temperature of nano Kelven. The condensation was regarded as one of the most important physical achievements in the 20th century. Pure BEC forms a novel states of matter behaving like a giant atom of unusual properties such as macroscopic coherence, tunneling, superfluidity, etc. BEC research is not only fundamental but also practical, and therefore is currently a hot research topic in atom physics, laser, statistical mechanics even astronomy.Studies on the forms of the external potential trapping the Boson gas, the space dimension and the finite number effects (one of the non-thermodynamical limit effect), etc, have been experimentally examined on various BEC experiments. This dissertation chooses earth's gravitational field as the external potential for atomic gases, and investigates its influence on the formation of the BEC in a 1D case and a 3D case respectively, by theoretical means of both the thermodynamic limit—continuous limit of quantum states, and a sum over discrete quantum states, with emphasizing on the finite number effects. Moreover, we also study some theoretical problems relating to BEC in one-dimensional harmonic potential.The dissertation is organized as the following five parts. In the first part, we review the present advances of BEC study and finite number effects. The second part investigates influence of presence of the Earth's gravitational field on the Bose-Einstein condensation in a 1D case. The results show that microscopic bouncing balls display usual BEC in the thermodynamic limit, and it is a third-order phase transition. Furthermore, the results also reveal that the critical temperature with a finite number of particles is higher than that in the thermodynamic limit. The ratio of critical temperature TC for finite-particle system over the critical temperature TC 0 of system taken thermodynamic limit ( TC / TC0) is irrelevant to mass of particle. This system is different from Boson gas in the one-dimensional harmonic potential one for which the standard result indicates that there is no BEC.The third part analyzes theoretical problems relating to BEC in one-dimensional harmonic potential. Fisrt, by studying chemical potential for the Bose gases in a one-dimensional harmonic trap, we find out that a closed expression for the chemical potential, reported by Mungan, is approximate for higher temperature and not applicable for temperature lower than a characteristic value below which the ground state becomes occupied by a macroscopic number of particles. We address the correct behaviour of the chemical potential in the whole range of temperature. We also discuss the error between approximate value(μapp) and exact value (μ) at low temperature. As the temperature is further lowered from TC , the absolute errorμ?μapp decreases, whereas the relative errorμ?μapp/μincreases quickly from 0.7 to 1.0. Secondly, by studying the heat capacity for the Bose gases in a one-dimensional harmonic trap, we see that there is clearly an abrupt increase of population in the ground state, whereas the curve of C / Nk B for finite N has no maximum, and C / Nk B increases monotonically with temperature T /T0 . Thirdly, by studying the heat capacity of Bose gases in a harmonic trap decorated with Diracδfunctions, we see that modification of the shape of the trapping potential changes obviously the heat capacity with a few Bose particles,which the curve of C / Nk B for finite N has maximum.The fourth part investigates BEC in a positive one-half-dimensional harmonic potential. The standard result indicates that the BEC is not possible unless the number of particles is finite. However, for the boson gas trapped in a positive one-half-dimensional harmonic well, there is still a certain nonzero characteristic temperature T0 below which the ground state becomes occupied by a macroscopic number of particles. We also see that there is clearly an abrupt increase of population in the ground state, whereas the curve of C / Nk B for finite N has no maximum, and C / Nk B increases monotonically with temperature T /T0 .The fifth part discusses influence of presence of the Earth's gravitational field on the Bose-Einstein condensation in a 3D case. We study it in the thermodynamic limit and in the case of a finite number of particles, in the same way as the 1D earth's gravitational field. The results show that microscopic bouncing balls display Bose-Einstein condensation in the thermodynamic limit. The heat capacity is discontinous, whereas entropy is continuous at critical temperature. So it is a second-order phase transition. Results also show that with finite number of particles, the critical temperature is lower than that in this continuum limit, which is similar to the result for the bosonic gas confined in the 3D harmonic magnetic traps. In addition, we find out that the ratio TC / TC0 of two critical temperatures is relevant to mass of particle m and width of trap L , where TC and TC 0 are determined by finite-particle system and the thermodynamic limit respectively. For m = 7amu ( 7 Li ), we note that finite number effects for L = 5 microns is less significant than that for L = 1,10microns. For L = 5 microns, we note that finite number effects for m = 39amu( 39K ) is more explicit than that for m = 7amu( 7 Li ) and m = 23amu( 23Na ). In the last part of this dissertation, we give a summary to the above-mentioned works.