论地球重力场对玻色—爱因斯坦凝聚的影响
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摘要
1995年,物理学实验取得了一个重大突破,通过激光冷却、静磁阱与蒸发冷却等技术,在87Rb、7Li和23Na原子气中直接实现了玻色—爱因斯坦凝聚(BEC)。该凝聚被认为是二十世纪实验物理学上最重大的成就之一。玻色-爱因斯坦凝聚体具有宏观相干、隧穿和量子超流等奇特的量子特性,它的实验实现为物理学研究打开了一个崭新的领域。BEC的研究不仅对基础研究有重要意义,而且在原子激光、原子钟、原子芯片技术和纳米技术等领域均有美好的应用前景。因而,BEC及其相关问题的研究已成为当前国际物理学界的前沿热点之一。
外囚禁势场的形式、粒子的空间维度和有限粒子数效应(非热力学极限效应)的理论研究为实验上成功实现BEC奠定坚实的基础。本文选地球重力场作为外囚禁势场,在一维和三维情况下,利用量子态密度积分(热力学极限下)和分离量子态求和两种方法,分别研究它对BEC形成的影响,重点讨论有限粒子数效应。同时,研究了与一维谐振势场相关的BEC若干问题。
全文共由五章内容构成。在第一章中,本文首先综述了玻色-爱因斯坦凝聚及其研究进展,然后综述了BEC中的有限粒子数效应的研究进展。第二章研究了一维地球重力场对玻色-爱因斯坦凝聚的影响。理论结果显示:在热力学极限下和有限的温度范围内,该体系会出现BEC现象,其临界温度TC随粒子数N增大而升高,热容量和熵本身是连续的,该相变是三级相变。当粒子数有限时,体系的临界温度高于热力学极限下的临界温度;粒子数有限时的临界温度与热力学极限下的临界温度的比值TC / TC0与粒子的质量无关。该体系与我们已知的理想玻色子被囚禁在一维谐振势场的情况不同:一维谐振势场中,在热力学极限下,标准的半经典近似结果显示该体系没有BEC,但当粒子数有限时,BEC可由基态凝聚分数来定义。
第三章分析了与一维谐振势场相关的若干BEC问题。(1)、研究了在一维谐振势场中的玻色气体化学势。我们发现Mungan给出的化学势的封闭表达式只是一个高温下的近似结果。当温度低于某一特征温度值时,基态上有宏观的粒子占据数,此时该化学势的表达式就不适用了。我们给出了在整个温度范围内化学势的正确结果。探讨了在低温范围内,Mungan的近似结果μapp和精确结果μ的偏差。当温度从Tc继续下降时,绝对误差μ?μapp会减小,但相对误差μ?μapp/μ会迅速从0.7增加到1.0。(2)、侦探了在一维谐振势阱中的玻色气体热容量。我们看到,基态凝聚分数N 0 /N随温度T /T0变化图展示BEC有限粒子数效应的一般特性,但是单粒子无量纲的热容量C / Nk B随温度T /T0变化图则没有显示这一特性。这是因为C / Nk B随T /T0单调上升,最后趋近于1,曲线无最大值。(3)、讨论了一维δ函数修饰的谐振势场中玻色气体的热容量。结果发现δ函数修饰可使少粒子数玻色气体体系的热容量发生明显的改变。当粒子数较少时,单粒子无量纲的热容量C / Nk B随温度T /T0的变化曲线会出现最大值。
第四章研究了一维半空间谐振腔中理想玻色气体体系的BEC问题。结果表明,该体系在热力学极限下,不会发生玻色—爱因斯坦凝聚相变。但当温度T 第五章研究了三维地球重力场对玻色-爱因斯坦凝聚的影响。如同一维情况,我们从热力学极限和有限粒子数两方面对其进行了研究。结果显示:在热力学极限下,在有限的温度范围内,该体系会出现BEC现象,热容量不连续,但熵本身是连续的,该相变是二级相变。当粒子数有限时,其临界温度低于热力学极限下的临界温度,此结果类似于玻色气体被囚禁在三维谐振磁阱中的情况,这一结果与非相互作用的玻色子在一维的地球重力场中的情况相反。同时,我们还看到当粒子数有限时,体系的临界温度与热力学极限下的临界温度的比值TC / TC0与粒子的质量m和势阱宽度L有关。固定玻色子的质量m = 7amu( 7 Li ),分别研究了L = 1,5,10微米的情况。结果显示:当L = 5微米时,体系的有限粒子数效应相对较小。固定L = 5微米,分别研究了m = 7amu(7 Li ),23amu( 23Na )和39amu(39K )的情况。结果表明:当m = 39amu时,体系的少粒子效应越明显。最后,我们对本论文进行了一简单总结。
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.
外囚禁势场的形式、粒子的空间维度和有限粒子数效应(非热力学极限效应)的理论研究为实验上成功实现BEC奠定坚实的基础。本文选地球重力场作为外囚禁势场,在一维和三维情况下,利用量子态密度积分(热力学极限下)和分离量子态求和两种方法,分别研究它对BEC形成的影响,重点讨论有限粒子数效应。同时,研究了与一维谐振势场相关的BEC若干问题。
全文共由五章内容构成。在第一章中,本文首先综述了玻色-爱因斯坦凝聚及其研究进展,然后综述了BEC中的有限粒子数效应的研究进展。第二章研究了一维地球重力场对玻色-爱因斯坦凝聚的影响。理论结果显示:在热力学极限下和有限的温度范围内,该体系会出现BEC现象,其临界温度TC随粒子数N增大而升高,热容量和熵本身是连续的,该相变是三级相变。当粒子数有限时,体系的临界温度高于热力学极限下的临界温度;粒子数有限时的临界温度与热力学极限下的临界温度的比值TC / TC0与粒子的质量无关。该体系与我们已知的理想玻色子被囚禁在一维谐振势场的情况不同:一维谐振势场中,在热力学极限下,标准的半经典近似结果显示该体系没有BEC,但当粒子数有限时,BEC可由基态凝聚分数来定义。
第三章分析了与一维谐振势场相关的若干BEC问题。(1)、研究了在一维谐振势场中的玻色气体化学势。我们发现Mungan给出的化学势的封闭表达式只是一个高温下的近似结果。当温度低于某一特征温度值时,基态上有宏观的粒子占据数,此时该化学势的表达式就不适用了。我们给出了在整个温度范围内化学势的正确结果。探讨了在低温范围内,Mungan的近似结果μapp和精确结果μ的偏差。当温度从Tc继续下降时,绝对误差μ?μapp会减小,但相对误差μ?μapp/μ会迅速从0.7增加到1.0。(2)、侦探了在一维谐振势阱中的玻色气体热容量。我们看到,基态凝聚分数N 0 /N随温度T /T0变化图展示BEC有限粒子数效应的一般特性,但是单粒子无量纲的热容量C / Nk B随温度T /T0变化图则没有显示这一特性。这是因为C / Nk B随T /T0单调上升,最后趋近于1,曲线无最大值。(3)、讨论了一维δ函数修饰的谐振势场中玻色气体的热容量。结果发现δ函数修饰可使少粒子数玻色气体体系的热容量发生明显的改变。当粒子数较少时,单粒子无量纲的热容量C / Nk B随温度T /T0的变化曲线会出现最大值。
第四章研究了一维半空间谐振腔中理想玻色气体体系的BEC问题。结果表明,该体系在热力学极限下,不会发生玻色—爱因斯坦凝聚相变。但当温度T
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.
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