原子激光传输的ABCD形式理论研究
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摘要
激光问世以来以其高强度、高单色性和高相干性在科学和技术中获得广泛应用，改变了整个光学的面貌。原子激射器自产生以来，也一直是科学研究的热点。这种在玻色-爱因斯坦凝聚(BEC)的基础上产生的新型激光器，具有和以往的光子激光类似的特征：单色性、相干性和方向性。原子物理学家也在期望着原子激光能像光子激光那样产生一个新的光学革命。但要想把原子激光付诸于应用，首先就要了解原子激光的特性和传输规律。原子激光作为一种新型的光源，目前还没有形成一套能够被大家普遍接受的传输理论。我们试图提供一种研究原子激光传输的一般方法，为进一步的实验和理论预测提供一个理论工具。
本文主要是利用含时量子系统传输的Schrdinger形式理论初步探讨原子激光的传输问题。本文在含时量子系统传播子的ABCD形式理论的基础上，引入某一方向的束宽、发散角、曲率半径和品质因子等光束传输参数来表征横向势能满足x和y分量相互独立的原子激光的传输。对于某一方向上品质因子守恒的情况，可以通过解析某一方向品质因子守恒的充要方程，找到原子激光的ABCD形式的传播子。如果考虑原子激光内部原子间相互作用的影响，就要引入描述原有品质因子和原子间相互作用综合作用效果的物理量——有效品质因子和有效复数曲率半径，利用Heisenberg图像得到传播子的有效ABCD形式。无论是否考虑原子激光内部原子间的相互作用，传播子表达式和ABCD变换矩阵都仅与所通过的势场有关，可以用以研究各种原子束的传输，而且计算极为简单。
第一章：本章回顾了原子激光器的理论和实验方面的进展；比较了原子激光和激光之间的差异，为进一步选择适当的方法来研究原子激光的传输奠定了基础；最后，综述了原子激光传输的研究进展并总结了本人的工作。
第二章：本章介绍了傍轴光束在实数折射率介质和复数折射率介质中传输的Schrdinger形式理论，同时指出由于原子激光的传输满足Schrdinger方程，与含时量子系统的传输方程类似，故利用含时量子系统的Schrdinger形式理论来研究原子激光的传输是可行的。
第三章：本章在含时量子系统的Schrdinger形式理论的基础上初步讨论原子激光——这种品质因子不守恒的含时量子系统的传输问题。本章讨论在忽略了原子间的相互作用情况下，横向可分原子激光的传输，这个问题实际上仍属于品
质因子守恒系统的范畴。文中还把所得到的结果与Le Coq等人的工作进行了比
较。
第四章:由于原子激光内部原子间相互作用的存在，原有的品质因子不再守
恒，需要引入描述原有品质因子和原子间相互作用综合作用效果有效品质因子来
讨论原子激光的传输。因此，本章在Schrsdinger形式理论的基础上，引入守恒
的有效品质因子来讨论有效品质因子守恒的的含时量子系统—原子激光(考虑
了原子激光内部原子间相互作用)传播子的有效 ABCO形式。
第五章:本章总结了前几章的内容，并在前文的基础上提出可以进一步深入
的工作。
The realization of atom lasers opens up the new intriguing perspectives in coherent atom optics. These novel atom sources are based on Bose-Einstein condensates (BEC) from which a coherent matter wave beam is extracted, which have coherence, monochrome and high direction. It is important to characterize the qualities and the propagation of these atom beams. The propagation of an atom laser, which is outcoupled from a Bose-Einstein condensate, is investigated within the framework of quantum mechanics. We discussed the propagation of the atom laser beam which had non-conservation quality factor based on the theory of ABCD formulation of propagation of a time-dependent quantum system. The equation for the quasi-continuous atom laser beam wave-function is given by Schrodinger equation, the beam width, divergence curvature radius and quality factor were introduced to represent the atom laser beam whose transverse potential could be separated. For the conservative quality factor system in some direction, we can ge
t the propagator of ABCD formulation. We considered the propagation of the atom laser beam during the interacting with BEC, the propagation with the trap on and the free propagation with the trap off which potential can be separated by x and y. Within the frame of the ABCD formulation of propagator of a time-dependent quantum system, the ABCD law is extended to the so-called effective ABCD system which effective beam quality factor is conservative; we discussed the system like the propagation of the atom laser beam. The effective beam quality factor and the effective parameter are introduced to obtain the effective ABCD formulation of propagator by means of Heisenberg picture. The atom-atom interaction in the atom laser beam makes the beam quality factor nonconservative, and the effective beam quality factor describes the beam quality factor and the atom-atom interaction together.
In chapter 1, a review of the atom lasers and the atom laser beams is given. The compare between the atom laser and the optical laser is also represented, which is the foundation of the propagation of an atom laser beam. At the end of this chapter, we list our works about the propagation of an atom laser beam.
In chapter 2, a Schrodinger formulation for paraxial light beam propagation of
the real and comprehensive refractive indexes is represented. The equation for the atom laser beam wave-function is also given by a Schrodinger equation, which is similar to the equation of the time-dependent system. Therefore, it is possible to investigate the propagation of an atom laser beam.
In chapter 3, we discussed the propagation of the atom laser beam, with the transverse independent of gravitate direction, which had non-conservation quality factor based on the theory of ABCD formulation of propagation of a time-dependent quantum system. Actually, the atom laser beam with the transverse independent of gravitate direction is included in the conservative system. We also compare our methods and results with Le Coq's.
In chapter 4, the interaction between atoms within the atom laser beam will be included in the potential energy, which leads to the non-conservation of the quality factor. An effective quality factor representing both the interaction between atoms within the atom laser and the quality factor is introduced to study the propagation of the atom laser beam. We derive the ABCD formulation of the propagation of an atom laser beam with arbitrary beam profile including the interaction between atoms within the atom laser by means of the theory of quantum mechanics.
In chapter 5, there is a summary of the methods in this thesis, together with the further work.
本文主要是利用含时量子系统传输的Schrdinger形式理论初步探讨原子激光的传输问题。本文在含时量子系统传播子的ABCD形式理论的基础上，引入某一方向的束宽、发散角、曲率半径和品质因子等光束传输参数来表征横向势能满足x和y分量相互独立的原子激光的传输。对于某一方向上品质因子守恒的情况，可以通过解析某一方向品质因子守恒的充要方程，找到原子激光的ABCD形式的传播子。如果考虑原子激光内部原子间相互作用的影响，就要引入描述原有品质因子和原子间相互作用综合作用效果的物理量——有效品质因子和有效复数曲率半径，利用Heisenberg图像得到传播子的有效ABCD形式。无论是否考虑原子激光内部原子间的相互作用，传播子表达式和ABCD变换矩阵都仅与所通过的势场有关，可以用以研究各种原子束的传输，而且计算极为简单。
第一章：本章回顾了原子激光器的理论和实验方面的进展；比较了原子激光和激光之间的差异，为进一步选择适当的方法来研究原子激光的传输奠定了基础；最后，综述了原子激光传输的研究进展并总结了本人的工作。
第二章：本章介绍了傍轴光束在实数折射率介质和复数折射率介质中传输的Schrdinger形式理论，同时指出由于原子激光的传输满足Schrdinger方程，与含时量子系统的传输方程类似，故利用含时量子系统的Schrdinger形式理论来研究原子激光的传输是可行的。
第三章：本章在含时量子系统的Schrdinger形式理论的基础上初步讨论原子激光——这种品质因子不守恒的含时量子系统的传输问题。本章讨论在忽略了原子间的相互作用情况下，横向可分原子激光的传输，这个问题实际上仍属于品
质因子守恒系统的范畴。文中还把所得到的结果与Le Coq等人的工作进行了比
较。
第四章:由于原子激光内部原子间相互作用的存在，原有的品质因子不再守
恒，需要引入描述原有品质因子和原子间相互作用综合作用效果有效品质因子来
讨论原子激光的传输。因此，本章在Schrsdinger形式理论的基础上，引入守恒
的有效品质因子来讨论有效品质因子守恒的的含时量子系统—原子激光(考虑
了原子激光内部原子间相互作用)传播子的有效 ABCO形式。
第五章:本章总结了前几章的内容，并在前文的基础上提出可以进一步深入
的工作。
The realization of atom lasers opens up the new intriguing perspectives in coherent atom optics. These novel atom sources are based on Bose-Einstein condensates (BEC) from which a coherent matter wave beam is extracted, which have coherence, monochrome and high direction. It is important to characterize the qualities and the propagation of these atom beams. The propagation of an atom laser, which is outcoupled from a Bose-Einstein condensate, is investigated within the framework of quantum mechanics. We discussed the propagation of the atom laser beam which had non-conservation quality factor based on the theory of ABCD formulation of propagation of a time-dependent quantum system. The equation for the quasi-continuous atom laser beam wave-function is given by Schrodinger equation, the beam width, divergence curvature radius and quality factor were introduced to represent the atom laser beam whose transverse potential could be separated. For the conservative quality factor system in some direction, we can ge
t the propagator of ABCD formulation. We considered the propagation of the atom laser beam during the interacting with BEC, the propagation with the trap on and the free propagation with the trap off which potential can be separated by x and y. Within the frame of the ABCD formulation of propagator of a time-dependent quantum system, the ABCD law is extended to the so-called effective ABCD system which effective beam quality factor is conservative; we discussed the system like the propagation of the atom laser beam. The effective beam quality factor and the effective parameter are introduced to obtain the effective ABCD formulation of propagator by means of Heisenberg picture. The atom-atom interaction in the atom laser beam makes the beam quality factor nonconservative, and the effective beam quality factor describes the beam quality factor and the atom-atom interaction together.
In chapter 1, a review of the atom lasers and the atom laser beams is given. The compare between the atom laser and the optical laser is also represented, which is the foundation of the propagation of an atom laser beam. At the end of this chapter, we list our works about the propagation of an atom laser beam.
In chapter 2, a Schrodinger formulation for paraxial light beam propagation of
the real and comprehensive refractive indexes is represented. The equation for the atom laser beam wave-function is also given by a Schrodinger equation, which is similar to the equation of the time-dependent system. Therefore, it is possible to investigate the propagation of an atom laser beam.
In chapter 3, we discussed the propagation of the atom laser beam, with the transverse independent of gravitate direction, which had non-conservation quality factor based on the theory of ABCD formulation of propagation of a time-dependent quantum system. Actually, the atom laser beam with the transverse independent of gravitate direction is included in the conservative system. We also compare our methods and results with Le Coq's.
In chapter 4, the interaction between atoms within the atom laser beam will be included in the potential energy, which leads to the non-conservation of the quality factor. An effective quality factor representing both the interaction between atoms within the atom laser and the quality factor is introduced to study the propagation of the atom laser beam. We derive the ABCD formulation of the propagation of an atom laser beam with arbitrary beam profile including the interaction between atoms within the atom laser by means of the theory of quantum mechanics.
In chapter 5, there is a summary of the methods in this thesis, together with the further work.
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邓锡铭，丁丽明，叶陈春．ABCD定律的推广[J]．中国激光 1990，17(4)：259-264．
郭弘，刘承宜，胡巍等．多阶强度非线性条件下的光束传输研究[J]．中国科学(A辑)，1999，29(3)：276-282．
何向阳，刘勃．原子激射器[J]．现代物理知识，2002，14(2)：25-26．
李师群，周义东，黄湖．原子激射器——相干原子束发生器[J]．物理1998，27(1)：11-17．
刘承宜 郭弘 胡巍 邓锡铭．光束传输的Schroedinger形式理论研究[J]．中国科学．A．2000，30(1)：54-62．
刘承宜，郭弘，胡巍，邓冬梅．复数折射率介质中光束传输的Schrodinger形式理论[J]．中国科学 A．2002b，32(4)：355．
刘承宜，郭弘，刘勇．稳态等离子体电子密度的X光激光干涉测量方法研究[J]．强激光与粒子束．1999，11(5)：601-604．
田野，刘承宦，郭旗，胡巍．用信息熵描述非线性光传输的研究[J]．光学学报．2002，22(7)：813．
王裕民．原子激光[J]．缴光与光电子进展，2000，416(8)：1-5．