简谐外势约束下二维玻色气体的玻色-爱因思斯坦凝聚（BEC）
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摘要
1925年，爱因斯坦曾经预言理想玻色气体在德布罗意波长大于粒子间的平均距离时会发生相变，这时将会有相当数量的粒子处于最低的能量状态，并出现量子简并，这就是著名的玻色-爱因斯坦凝聚(Bose-Einstein Condensation简称BEC)。在此以后的几十年时间里人类并未在实验上观察到BEC也未能对BEC的特性给出确切的回答，但科学家们为实现BEC的努力却从未停止。直到二十世纪八十年代，与实验上形成BEC相关的激光冷却和捕获原子技术趋于成熟后，才使科学家们的BEC实验研究得到了实质性进展。终于在1995年相继有三个小组在实验上实现了BEC，从而又极大地推动了BEC研究的新高潮。
当前研究的重点多集中在三维的情况，而实际上由于二维情况的特殊性使得二维体系有着三维体系所没有的特点。低维系统的独特性质已经在新材料新器件的研制中得到了广泛的应用，这激发了人们对二维体系研究的强烈兴趣。对二维系统理论上的研究也将有利于从实验上实现二维BEC。
本文首先概括性地谈论了最先实现BEC的三个实验以及导致实验上形成BEC的几个关键技术：气室中的磁光阱技术、原子束塞曼冷却技术以及用于检测BEC形成的吸收成像技术，接着对二维自由理想玻色系
MW 4D7rtotMchkfgCAsMdeftte
统的性质进行了理论计算，然后对简谐势阶中的二维理想玻色气体进行
了讨论。Mll的重点是通过数伯计算方法比较准确地得到了二维理想玻
色气体在有夕＋势情况下的一些特性，如：系统的粒子数对BEC特性的影
响；零点ggX＼］－ar－－xW的影响：热容与温度之间的关系以及激发态上的粒子
与温度之间的关系等等。通过研究发现，粒子数有限时，基态上粒子数
NO的宏观积累是在某个温度区域内和缓平滑上升的。在温度／下
tNtN／叫达到极大值。系统的热容表现为非单调变化并在某一温度几
下潍u极大值，颇／。几。鲤结果与热力学极限条件下的靓不
同。托用洲用数值计算的结果与其他近似方法的解析结果作比较，将会
对h维囚禁玻色气体的BEC特性u的了解。在接下来的章节里我
们主要调研了当前考虑外峻瞧祝下有相互作用玻色气体的影釉碾。二
维BEC81－mW上的研究将对实验上实现BEC有一定的指导意义。最后本
文对BEC的广阔应用前景作了展望。
Einstein had predicated that the ideal bose gas would occur phase change when the thermal deBroglie wave-length becomes larger than the average spacing between particles, then a macroscopic population gathers in the lowest energy quantum state and quantum degeneracy happens, which is the famous Bose-Einstein condensation (BEC). During the following several decades, scientists did not find the BEC in experiments and could not give a definite answer for the phenomena, but the endeavors for realizing the BEC in experiments have never been stopped. Until 1980', scientists got substantial developments in BEC after the techniques of laser cooling and atom-captured method related to realizing BEC in experiments became mature. Finally, three groups realized BEC in experiments successfully in 1995, which motivated greatly scientists to do researches in BEC.
The current researches are focused on three-dimensional system while the unique characteristics in two-dimensional system make 2D system have some particularities which 3D system has not The particularities of low-dimensional systems have been widely applied in manufacture of new materials and new devices, which agitates people's strong interests in 2D system. The theoretical
in
researches in 2D system will promote the realization of BEC in experiments.
In this article, we begin with 1he original Ihree experiments on realization of BEC and several critical techniques for these experiments: magnetic-optics in gas cell, atomic Zeeman cooling technique and adsorbing imagination technique for BEC detection. Then we go on theoretical calculations on the characteristics of two-dimensional ideal bose system, followed by the discussions of two-dimensional ideal bose gas in a simple harmonic trap. We focus on getting characteristics of two dimensional ideal bose in potential through numerical calculations, such as the effect of the number of particles in bose system on the characteristics of BEC; the effect of fugacity on zero energy; the relations between heat capacity and temperature, the relations between temperature and the number of particles in the first excited state. Comparing the results obtained through numerical calculations and that obtained through other approximate analytic calculations, we can get further understanding on the 2D bose gas trapped in a simple harmonic trap. In the following chapters, we mainly concentrate on investigating and studying the latest progresses in bose gas with interaction between particles. The researches of 2D BEC in some degree guide the BEC in experiments. At last, we look into the future of BEC applied into our society.
当前研究的重点多集中在三维的情况，而实际上由于二维情况的特殊性使得二维体系有着三维体系所没有的特点。低维系统的独特性质已经在新材料新器件的研制中得到了广泛的应用，这激发了人们对二维体系研究的强烈兴趣。对二维系统理论上的研究也将有利于从实验上实现二维BEC。
本文首先概括性地谈论了最先实现BEC的三个实验以及导致实验上形成BEC的几个关键技术：气室中的磁光阱技术、原子束塞曼冷却技术以及用于检测BEC形成的吸收成像技术，接着对二维自由理想玻色系
MW 4D7rtotMchkfgCAsMdeftte
统的性质进行了理论计算，然后对简谐势阶中的二维理想玻色气体进行
了讨论。Mll的重点是通过数伯计算方法比较准确地得到了二维理想玻
色气体在有夕＋势情况下的一些特性，如：系统的粒子数对BEC特性的影
响；零点ggX＼］－ar－－xW的影响：热容与温度之间的关系以及激发态上的粒子
与温度之间的关系等等。通过研究发现，粒子数有限时，基态上粒子数
NO的宏观积累是在某个温度区域内和缓平滑上升的。在温度／下
tNtN／叫达到极大值。系统的热容表现为非单调变化并在某一温度几
下潍u极大值，颇／。几。鲤结果与热力学极限条件下的靓不
同。托用洲用数值计算的结果与其他近似方法的解析结果作比较，将会
对h维囚禁玻色气体的BEC特性u的了解。在接下来的章节里我
们主要调研了当前考虑外峻瞧祝下有相互作用玻色气体的影釉碾。二
维BEC81－mW上的研究将对实验上实现BEC有一定的指导意义。最后本
文对BEC的广阔应用前景作了展望。
Einstein had predicated that the ideal bose gas would occur phase change when the thermal deBroglie wave-length becomes larger than the average spacing between particles, then a macroscopic population gathers in the lowest energy quantum state and quantum degeneracy happens, which is the famous Bose-Einstein condensation (BEC). During the following several decades, scientists did not find the BEC in experiments and could not give a definite answer for the phenomena, but the endeavors for realizing the BEC in experiments have never been stopped. Until 1980', scientists got substantial developments in BEC after the techniques of laser cooling and atom-captured method related to realizing BEC in experiments became mature. Finally, three groups realized BEC in experiments successfully in 1995, which motivated greatly scientists to do researches in BEC.
The current researches are focused on three-dimensional system while the unique characteristics in two-dimensional system make 2D system have some particularities which 3D system has not The particularities of low-dimensional systems have been widely applied in manufacture of new materials and new devices, which agitates people's strong interests in 2D system. The theoretical
in
researches in 2D system will promote the realization of BEC in experiments.
In this article, we begin with 1he original Ihree experiments on realization of BEC and several critical techniques for these experiments: magnetic-optics in gas cell, atomic Zeeman cooling technique and adsorbing imagination technique for BEC detection. Then we go on theoretical calculations on the characteristics of two-dimensional ideal bose system, followed by the discussions of two-dimensional ideal bose gas in a simple harmonic trap. We focus on getting characteristics of two dimensional ideal bose in potential through numerical calculations, such as the effect of the number of particles in bose system on the characteristics of BEC; the effect of fugacity on zero energy; the relations between heat capacity and temperature, the relations between temperature and the number of particles in the first excited state. Comparing the results obtained through numerical calculations and that obtained through other approximate analytic calculations, we can get further understanding on the 2D bose gas trapped in a simple harmonic trap. In the following chapters, we mainly concentrate on investigating and studying the latest progresses in bose gas with interaction between particles. The researches of 2D BEC in some degree guide the BEC in experiments. At last, we look into the future of BEC applied into our society.
引文
[1] Huang K., Statistical Mechanics, 2nd Edition(Wiley), New York,1987,p263
[2] 王谨，詹明生，高克林，原子的玻色爱斯坦凝聚，大学物理，1998，17，p6
[3] London F., On the Bose-Einsrein condensation, Phys. Rev., 1938, 54,10,p947
[4] Hecht C.E., The possible superfluid behavior of hydrogen atoms gases and liquids, Physica, 1959, 25, 11, p1159
[5] Hulin D., Mysyriwucz A., Guillaume C., Evidence of Bose-Einstein statistics in an excition gas, Phys. Rev. Lett., 1980,45, 24, p1970
[6] Lin J. L., Wolfe J. P., Bose,-Einstein condensation of paraexcitons in stressed Cu_2O, Phys. Rev. Lett, 1993,71,8, p1222
[7] Wieman C., Chu S., J. Opt. Soc. Am. B, 1989, 6, 11, (Special issue on laser cooling and trapping of atom)
[8] Anderson M. H., Ensher J. R, Matthews M. R, et al., Observation of Bose-Einstein condensation in a dilute vapor, Science, 1995, 269, p198
[9] Bradley C. C., Sackett C. A., Tollett J. J., et al., Evidence of Bose-Einstein condensation in atomic gas with attractive interactions. Phys. Rev. Lett., 1995, 75, p1687
[10] Davis K., Mewes B.. Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Leto, 1995,75, p3969
[2] 王谨，詹明生，高克林，原子的玻色爱斯坦凝聚，大学物理，1998，17，p6
[3] London F., On the Bose-Einsrein condensation, Phys. Rev., 1938, 54,10,p947
[4] Hecht C.E., The possible superfluid behavior of hydrogen atoms gases and liquids, Physica, 1959, 25, 11, p1159
[5] Hulin D., Mysyriwucz A., Guillaume C., Evidence of Bose-Einstein statistics in an excition gas, Phys. Rev. Lett., 1980,45, 24, p1970
[6] Lin J. L., Wolfe J. P., Bose,-Einstein condensation of paraexcitons in stressed Cu_2O, Phys. Rev. Lett, 1993,71,8, p1222
[7] Wieman C., Chu S., J. Opt. Soc. Am. B, 1989, 6, 11, (Special issue on laser cooling and trapping of atom)
[8] Anderson M. H., Ensher J. R, Matthews M. R, et al., Observation of Bose-Einstein condensation in a dilute vapor, Science, 1995, 269, p198
[9] Bradley C. C., Sackett C. A., Tollett J. J., et al., Evidence of Bose-Einstein condensation in atomic gas with attractive interactions. Phys. Rev. Lett., 1995, 75, p1687
[10] Davis K., Mewes B.. Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Leto, 1995,75, p3969